Advances in Heat Transfer, Volume 4

Advances in

HEAT TRANSFER Edited by James P. Hartnett

Thomas F. Irvine, Jr.

Department of Energy Engineering University of Illinois at Chicago Chicago, Illinois

State University of N e w York at Stony Brook Stony Brook, Long Island N e w York

Volume 4

@ 1967 ACADEMIC PRESS

New York

-

London

COPYRIGHT 0 1967, BY ACADEMIC PRESS INC.

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LIST OF CONTRIBUTORS E. R. G. ECKERT, Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota A. J. EDE, National Engineering Laboratory, East Kilbride, Scotland

C. FORBES DEWEY, JR., University of Colorado and Joint Institute for Laboratory Astrophysics, Boulder, Colorado

JOSEPH F. GROSS, Department of Geophysics and Astronomy, R A N D Corporation, Santa Monica, California E. PFENDER, Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota ROBERT SIEGEL, Lewis Research Center, National Aeronautics and Space Administration, Cleveland, Ohio ALICE M. STOLL, Aerospace Medical Research Department, U. S. Naval Air Development Center, Johnsville, Warminster, Pennsylvania

1 Present address: Department of Mechanical Engineering, University of Aston in Birmingham, England.

V

PREFACE In the preface to Volume 1, we noted that heat transfer research has grown at an amazing rate during the past decade, primarily due to problems associated with the growth of the atomic energy industry and the aerodynamics and astronautics efforts throughout the world. We also noted that while the results of these research efforts are normally published as individual articles in national and international journals, it is often difficult for the nonspecialist, or even the specialist, to make engineering use of these individual papers. It was our hope that review articles which start from widely understood principles and develop the topics in a logical fashion would be of value to the engineering and scientific communities. The interest aroused by the first three volumes of “Advances in Heat Transfer” seems to us to be an indication that this function is being fulfilled.

J. P. HARTNETT T . F. IRVINE, JR.

October, 1967

vii

Advances in Free Convection A. J. EDE* National Engineering Laboratory East Kilbride. Scotland

I. Introduction . . . . . . . . . 11. Laminar Flow . . . . . . . . A. The Classical Problem of the Flat Plate B. The Vertical Circular Cylinder . . . C. Nonuniform Surface Temperature . . D. Nonuniform Physical Properties . . E. Nonsteady Conditions . . . . . F. Very Low Grashof Numbers . . . G. Effect of Vibration . . . . . . 111. Turbulent Flow . . . . . . . . A. Instability of Laminar Flow . . . . B. Turbulence . . . . . . . . Symbols . . . . . . . . . . References . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . .

. . .

1 4 4 . 1 7 . 22

. .

. . . . .

. . . . . . . . . . . .

28 33 4 3 4 7 5 0 5 0 5 4 62 62

I. Introduction

T h e average treatise on heat transfer dismisses free convection in a single chapter. It is, nevertheless, potentially as large a subject as forced convection, and has developed rapidly within the last decade. The present review acknowledges this situation by confining its attention to one aspect only, that of free convection on a vertical surface. T h e fluid is assumed to be of infinite extent and devoid of any motion or temperature variations other than those associated with the free convection; the surface is assumed to be free of any obstruction which might disturb the flow. Although papers falling within this category are still appearing, it would seem that the vein has now been almost worked out, and that attention is being increasingly diverted

* Present address: Department of Mechanical Engineering, University of Aston in Birmingham, England. An excellent survey of a much wider field was prepared by Ostrach (111) in 1959. 1

2

A. J. EDE

elsewhere. T h e subject has therefore reached a convenient stage for a general review. T h e early development of the subject was characterized by the appearance of many papers dealing with experimental work ; theoretical papers were rare. I n recent years the position has been reversed, and most recent publications have been largely concerned with analytical work. T h e considerable ingenuity displayed by early mathematicians has largely given way to solution by computer. Progress is faster but the papers are not so attractive to read! Analytical work has been preponderantly concerned with laminar flow, no

' Y

FIG.1. Coordinate system.

doubt because turbulence is less tractable. Fortunately laminar flow is very important in free convection. Velocities are comparatively low, and the leading edge is not a serious source of disturbance. I n this review laminar flow in all its aspects will be taken first, and instability and turbulence will be examined later. T h e system of coordinates is displayed in Fig. 1. T represents the absolute temperature of the fluid, and takes the values T, at the surface and T , in the bulk fluid. 0 denotes the excess of the fluid temperature at a point over the bulk temperature, i.e., T - T , ;it can of course be negative, in which case the direction of the convective flow will be reversed. No further reference will be made to this possibility.

ADVANCES IN FREE CONVECTION

3

In much analytical work it is convenient to dedimensionalize the variables, and this can be done in many ways. In some papers a succession of transformations is carried out, and the variety of nomenclature then employed adds considerably to the difficulty of following the argument and of comparing one paper with another. A common device is to allot special symbols to the basic variables, calling them 3, a, 19or A’,U, 0 in order that x , y, 8, etc. can be reserved for the dimensionless variables in which the greater part of the analysis is to be conducted. Sometimes an author will leave his final conclusions in terms of these transformed variables, and it may be suspected that he was more interested in the elegance of his mathematics than in the usefulness of his results. It is impossible in a review paper to maintain a consistent set of symbols and yet cover all possibilities, and symbols will sometimes be defined where they appear without regard for the fact that they may already have been used with a different significance. The basic symbols x, y, u, zi, 8, etc. will however always refer to the straightforward dimensional variables. Application of dimensional analysis to free convection on a flat vertical surface reveals that the Nusselt number Nu, = h x / k is a function of the Grashof number Gr, =glg8,x3/v2 and the Prandtl number Pr = q / k . Here h is the local heat-transfer coefficient at the position x ; the average coefficient h for the whole surface is also of interest, and can be used to form a Nusselt number NuL = h L / k and a Grashof number GrL = glg8, L3/v2.The Rayleigh number Ra = G r - P r is also widely used. For the vertical circular cylinder, the only other shape which will arise in this review, generality can be preserved by introducing a further group D / L , and the diameter can be used to form alternative Nusselt and Grashof numbers NuD = h D / k and Gr, = glgdWD3/v2.These usages arose from the early work in which the surface temperature 8, was nearly always assumed to be uniform. When the surface heat flux qzu is assumed uniform, a modified Grashof number Gr,+ = glgq,,x4/v2can be employed. In later work with more complicated boundary conditions these groups are not quite so useful. All analytical work is based on the following set of equations, representing the conservation of mass, momentum, and energy in a fluid moving under the influence of a body force : ap+ at

a ax] (pu,) = 0

4

A. J. EDE

F a y -+hi axj axi

ax,

auj) - 32(aU,)~j

+r)-

axj

(5)

In all the cases discussed here the flow either is two-dimensional or has cylindrical symmetry, so that only two momentum equations have to be considered. If there are no sources of heat within the fluid, Q is omitted. If viscous dissipation can be neglected, the term prefaced by 7 in Eq. (5) is omitted. If the flow is steady terms in a/& are omitted. The boundary conditions specify that the temperature or heat flux is some function of x at y = 0, and is zero at infinity. The velocity components are zero aty = 0 and infinity. The spatial derivatives of temperature and velocity are zero at infinity. The boundary conditions for velocity differ from their forced convection counterpart, and have the effect that the velocity and temperature profiles are quite different in shape. Velocity changes from zero through a maximum to zero again, whereas temperature varies monotonically from maximum to zero. T h e peculiarity which distinguishes free convection from forced, as far as analytical work is concerned, is that the equations of momentum and energy are coupled ; they cannot be treated separately. The motion is, of course, directly caused by the transfer of heat, and has no independent existence. As a consequence the principle of superposition cannot be used to develop solutions for complicated situations from solutions for simple, idealized cases. 11. Laminar Flow

A. THECLASSICAL PROBLEM OF THE FLAT PLATE The flow is assumed to be steady ;the surface temperature is uniform and the physical properties of the fluid (apart from density) are unaffected by temperature. Dissipation is neglected and there are no heat sources within the fluid. Two principal methods of analysis have been developed: direct solution of the differential equations, with the aid of similarity transformations, and solution of the equations in an integral form, by assuming plausible expressions for the velocity and temperature profiles. Both procedures assume that the profiles are of essentially the same form for all values of x. T h e set of equations previously quoted, with the simplifications mentioned, were assembled by Oberbeck (1) in 1879. He aligned his axes so that one was vertical, giving F , = -g, F2 = F3 = 0, and put p = po(l -PO). I t

5

ADVANCES IN FREECONVECTION follows that ap/ax = --pop aO/ax, etc., and if p become

= -pogx

+p', the equations

au + av + aw = p u -ae + v -ae + w ax ay aZ ( ax ay axas>

~- -

(

av

av

ax

ay

ae

ae

po u - + v - + w -

(

poc u - + a - - + w -

ax

ay

-

avj--+-+;: $) az - q (;;: ae - k az )-

;

--

(-+-+7 ay ax aa 2x 20

a20

Oberbeck's attempts to obtain a solution as a power series in/3 do not include any cases falling within the scope of this survey. Two years later Lorenz (2) greatly simplified these equations by means of certain sweeping assumptions, and reduced them to the following two, which he solved in series : O=jSgPoO+T pocue = kL

(d2uldy2)

(d26/dy*)

(11) (12)

His solution leads to the following expression for the Nusselt number : Nu,

= 0.548(GrL.Pr)''4

(13)

a result which is astonishingly close to experimental data and more accurate solutions. The simplifications introduced, however, have the effect that the local heat transfer coefficient is not a function of x, which is contrary to experience.

1. The Dzfeerential Equation Method It is necessary to advance 50 years, passing over some not very successful theoretical work and the gradual emergence of dimensional analysis, to reach the next stride forward. This was achieved by Pohlhausen, who in collaboration with Schmidt and Beckmann (3)solved the Oberbeck equations in a less primitive form : azl

av

.&+ay=o

A. J. EDE

6

au (. + pcu-++( iz P

au

D a y ) = PgPe

i:)

=K-

azu

+ 77 ay2

a2e

ay

T h e zero suffix to p has now been dropped as its variation with temperature is accounted for by the introduction of /3. T o obtain these equations they applied Prandtl's boundary-layer approximations, assuming that the effects of the free convection were confined to a thin layer of fluid adjacent to the heated surface (as their experiments clearly showed). A more rigorous derivation of these simplified equations has since been given by Ostrach (4). I n view of the continuity equation (14) Pohlhausen introduced a stream function such that u = a+/ay, v = -a+/ax, and then sought a similarity transformation in terms of a new variable = Axmy". It is found, on subthe equations become stitution, that if m = -*, n = 1, A = (g/3p2/4q2)1/4,

+

c

f"'+ 3fs" - 2f '2 + g = 0 g" + 3Pr .fg'

+

=0

(18)

where f andg are the functions of 5 in which and 0 are expressed. It follows from this that the temperature and velocity distributions can be expressed in terms of y/xli4, and this accorded with the experimental measurements made by Schmidt and Beckmann. Pohlhausen attempted a solution of these equations in series, but encountered convergence difficulties and was therefore compelled to take boundary values of f n and g' from the velocity and temperature gradients obtained experimentally, and solve numerically for a single value of the Prandtl number, namely 0.733, appropriate to air. The heat-transfer coefficient resulting from this analysis is a function of x, and the results can be expressed as follows : or

Nu,

= 0.39(Gr,.Pr)114

NuL = 0.52(GrL.Pr)li4

After Saunders (5) and Schuh (6) had obtained further solutions by approximate methods which did not have to fall back on experimental data, Ostrach (4) virtually concluded the study of the Schmidt and Beckmann equations by obtaining exact solutions for eight values of Pr, from 0.01 to 1000, by means of a computer. The solution may be put in the form or

Nu,

=

(3A/4)(Gr,.Pr)li4

NuL = A(GrL-Pr)Ii4

ADVANCES IN FREECONVECTION

7

where A is a function of Pr. Further calculations of the same type have been made by Sugawara and Michiyoshi (7), Sparrow et al. (8),and Gebhart (9). Le Fevre (10) considered the extreme cases where Pr tends to zero or infinity. Starting with Eqs. (17) and (18), a change of variables leads to Pr(fl”’ +gl) +flfl’’ - +flz +gl g1”

=0

+ gl‘fl = 0

(23) (24)

which facilitates the task since for Pr -+ 0 Eq. (23) becomes 4f12

f1f1”-

+g1 = 0

(25)

and for Pr -+03 it becomes

+g1 = 0 (26) Solution by computer gives Eq. (22) with A = 0.670 327 for Pr -+ w and A = 0.800 544Pr’/4 for Pr + 0. fl”’

TABLE I COMPUTER SOLUTIONS OF THE CLASSICAL PROBLEM Pr

A

0 0.01 0.03 0.09 0.5 0.72 0.733 1.o 1.5 2.0 3.5 5.0 7.0 10 100 1000 10,000

0.800 564Pr1I4 0.240 279 0.308 0.377 0.496 0.516 492 0.517508 0.534705 0.555 059 0.568033 0.589916 0.601 463 0.611 035 0.619 0.653 349 0.665 0.668 574 0.670 327

co

0.849 126Pr11Z 0.080 592 0 0.136 0.219 0.442 0.504630 0.507 890 0.567 140 0.651 534 0.716483 0.855 821 0.953 956 1.05418 1.168 2.191 40 3.97 7.091 30 0.710 989Pr114

Table I lists the best values at present available for A in Eq. (22). The value of the derivativeg’ at 5 = 0 is also tabulated. This is the generalized form of or (32/2Prli4/4)A. the heat-transfer coefficient and is equal to N~,/(Gr,/4)l/~ It will be seen that, for Pr > 100, A is almost constant, in conformity with

8

A. J. EDE

the long-recognized fact that, for so-called “creeping motion,” where the inertial terms can be omitted from the momentum equations, dimensional analysis indicates that the solution must have the general form Nu =f(Gr-Pr), so that A must be constant in Eq. (22). It has also been argued that, for very low Prandtl numbers, viscosity should not appear in the general solution, which must therefore have the form Nu = j(Gr.Pr2). The solution for Pr + 0 is evidently in conformity with this conclusion. Le Fevre also proposed an empirical expression which fits the computer solutions very closely and facilitates interpolation to other values of the Prandtl number :

A 4 = Pr/{2.43478 + 4.884Pr1/’ + 4.952 83Pr) A simpler form of slightly reduced accuracy is A 4 = 2Pr/5( 1 + 2P1-l~’+ 2Pr) T h e various solutions are displayed in Fig. 2.

log,,Pr

FIG.2. Flat plate: comparison of theoretical solutions.

2. The Integral Equation Method With the work of Ostrach, the classical problem may be regarded as completely solved. I t is, however, of more than merely historical interest to examine alternative methods that have been proposed, and in particular the integral equation method, because they may be used for more complicated problems. Following the ideas of von Karman (11) and others, the usual boundary layer equations (15) and (16) are formally integrated with respect

ADVANCES IN FREECONVECTION

9

t o y across a boundary layer of thickness 6, assumed to be the same for both velocity and temperature. They become

These equations can also be obtained quite simply from first principles (12). In order to proceed further it is necessary to adopt approximate expressions for u and 0 as functions of y. The following simple polynomials give profiles which correspond quite well with experimental results : = u(y/6) (1 -y/ti)2, e = e,(i - y / q 2 (31) where U and 6 are as yet undetermined functions of x. On substituting these expressions into the integrals, Eqs. (29) and (30) reduce to

T h e variables can be separated by means of the substitutions U = p x m , 6 = qx", and the following solution is obtained :

U = 5.17v(Pr + 20/21)-'~2(g/3B,/v2)'~2 x1j2 6 = 3.93Pr-'I2(Pr + 20/21)"4(g~8,/v2)-'~4x ' / ~

whence, in the usual form, Nu, = 0.51P1-'/~(Pr + 20/21)-'/4 GrL'4 or NuL = 0.68Pr'i4(Pr + 20/21)-'/4(GrL.Pr)1/4

(34) (35) (36)

(37) For air, this gives NuL = 0.55Rat/4,which is in adequate agreement with the more exact solution. This method was first used in free convection by Squire (13). I t has the advantage of simplicity, and gives a solution containing the Prandtl number explicitly. It is of course only approximate, and various improvements have been proposed. Merk and Prins (14) used more complicated expressions for the velocity and temperature profiles. Their solution has the form of Eq. (22) with A = 1.887y, where (Pr + 1 . 3 4 1 ) ~ ~1.292 x 1OP2(Pr+ 0.1602)y4+ 2.199 x 10W = 0 (38) Sugawara and Michiyoshi (15) introduced three modifications. In their

A. J. EDE

10

first method they assumed third-degree polynomials for u and 0, and in order to establish the coefficients they went back to the differential equations, from which they obtained ~

By this means they were able to arrive at expressions for the profiles which involved only one unknown parameter, namely 6, and this was found by solving the energy equation in the integral form. In their second method they assumed that the vertical component of velocity, u, was a function of temperature only. This implies that isothermals are also lines of constant u, which, in view of the Pohlhausen analysis and experimental results, is a reasonable assumption. By means of this transformation the differential equations were put into a form which could be solved by assuming polynomial expressions for 0. T h e integral energy equation is then solved as usual for 6. In their third attempt, Sugawara and Michiyoshi considered the possibility that the two boundary layers for velocity (6) and temperature (6') might be of significantly different thickness. The expressions for u and 0 obtained by their second method were put in terms of the two different deltas, and substituted into the integral equations, which were then solved for 6 and x = S'lS on the assumption that the latter was independent of x. Fujii (16) took a somewhat similar line, but used profiles expressed in terms of exponentials :

where s is determined from the boundary conditions. The solution then proceeds as usual. The first form of temperature profile gives acceptable results only for Pr < 0.1, but the second is reasonably satisfactory for all Prandtl numbers. Brindley (17) has applied Meksyn's asymptotic expansion procedure to the similarity solution. Starting with Eqs. (17) and (18) he expressed f and g as infinite power series in f ; consideration of the boundary conditions leads to

f=

130

C r=2

02

g=l+

a,.?,

C a,.? r=l

These are put into the equations and, by equating to zero the sums of coefficients of corresponding powers of (, the values of the a's and a's are determined in terms of a 2 and a 4 .Writing F and G for the series expressions for f and g and substituting for f,f and g in Eqs. (17) and (18) gives I,

fiit+3Ff"=2F'2-G g" + 3Pr * Fg' = 0

(41) (42)

ADVANCES IN FREECONVECTION

11

which can be integrated to provide expressions for f and g in terms of exponential functions of the integral of F. These functions and their integrals are evaluated by a lengthy procedure, at the conclusion of which the boundary conditions at infinity are used to determine a2 and a 4 .Numerical results, based on the first three terms of the expansion, are obtained by computer for Prandtl numbers of 0.01,0.733, 7.0, 10, 100, and 1000. They agree quite well with the more exact solutions except at Pr = 0.01. As with the other methods which have been described in this section, this is not claimed to have any superiority over the Pohlhausen-Ostrach procedure, but it can be used in more difficult cases (combined forced and free convection is discussed in the paper). All of these solutions can be put into the form of Eq. (22) with A given as a function of Pr. Some of them are compared in Fig. 2. In considering the application of such analyses to fluids having Prandtl numbers differing considerably from unity, the assumptions made concerning boundary-layer thickness should be borne in mind. The boundary-layer approximations require that both boundary layers shall be thin ; some solutions assume that both are of the same thickness. When the Prandtl number is large the velocity boundary layer extends beyond the thermal boundary layer.

3. Comparison with Experimental Data A large number of experimental observations are available for testing these solutions of the classical problem. In making the comparison it should be realized that there are many difficulties in reproducing the idealized situation assumed in the theoretical treatment. Perfect precision requires that the only motion in the fluid shall be that resulting from the presence of the heated surface; in practice, convection currents caused by outside influences or other parts of the apparatus, and disturbances such as draughts, are extremely difficult to suppress entirely. While the heat put into the experimental body may easily be determined with precision, some is usually lost by processes other than by free convection from the surface under consideration. T h e rate of heat transfer by free convection in gases, particularly, is quite low, and comparatively large amounts of heat may be lost by conduction along supports, leads, and so on, and by radiation, and accurate estimation of these losses can be difficult. The “infinite volume of fluid at a uniform temperature” may in fact be a comparatively small bulk of fluid in a container, and it may be difficult to attain a really steady state or to be sure that the fluid temperatures measured are truly representative and entirely unaffected by the presence of the heated surface. A finite surface must have edges, and their shape and size is evaded in the theoretical treatment so far considered. It is difficult to insure that the flow is substantially two-dimensional.

A. J. EDE

12

Apart from such practical difficulties, the temperature distribution on the surface demands attention. The classical theory assumes that it is uniform. Some experimenters have attempted with varying success to achieve this condition, but most have simply produced a situation corresponding approximately to that of a uniform heat flux at the surface. Before considering the experimental data, therefore, it is desirable to examine the extent to which this factor is likely to affect the results. A useful guide can be obtained from a theoretical paper by Sparrow and Gregg (It?), who showed that a similarity solution can readily be obtained for the uniform heat flux case. T h e treatment is essentially the same as that for the uniform temperature ’~ problem, except that the similarity variable now has the form y / ~ linstead o f y / ~ ” Results ~. were obtained by computer for Prandtl numbers of 0.1, 1, 10, and 100, and mean Nusselt numbers were calculated on the basis of (a) an average temperature difference for the whole surface, and (b) the temperature difference halfway up the surface. In all cases they were found to be slightly higher than those for a uniform temperature ; the ratios are listed in Table 11. Since most experimenters have used method (b) in analyzing their data, it can be seen that this lack of correspondence between idealization and experiment is not likely to be a serious source of difficulty. TABLE I1

TEMPERATURE UNIFORMHEAT-FLUX~

COMPARISON BETWEEN U N I F O R M

AND

Ratio of Nusselt numbers

Pr

(a) Mean

(b) At L / 2

0.1 1 10 100

1.08 1.07 1.06 1.05

1.02 1.015 1.01 1 .oo

a The table gives the ratio of the Nusselt number for a uniform heat-flux to the Nusselt number for a uniform temperature.

The largest single group of data is concerned with the vertical flat plate in air. The available data are plotted in Fig. 3, and can be seen to form a smooth curve, concave upwards. The straight line represents Ostrach’s solution ; it is tangential to the curve, representing the data reasonably well for Rayleigh numbers between lo6 and lo8,but deviating from the curve at

ADVANCES I N FREE CONVECTION

13

either end. Even in the region of best agreement, almost all the experimental points lie above the line, and since the commonest source of experimental error-uncontrolled air currents-usually leads to increased heat transfer this is understandable. The divergence at high Rayleigh numbers may be due to the development of turbulence, and that at low Ra to a thickening of the boundary layer to such an extent that the boundary-layer approximation becomes invalid.

FIG.3. Flat plate: experimental data for air.

Data are also available for comparison with the theoretical velocity and temperature profiles. Figures 4a and 4b compare the Ostrach solution with the data of Schmidt and Beckmann. T h e agreement may be regarded as satisfactory, but a closer look will be taken later. Another important difference between theory and experiment is concerned with the physical property data required for calculating the various groups. In the theoretical work all properties other than density are assumed to have constant values, but no real experiment can be conducted under isothermal conditions and all properties vary to some extent with temperature. It is therefore necessary to decide at what “reference” temperature the physical properties should be evaluated. Most experimental work in gases is not sufficiently precise to warrant close attention to this point, but Eichhorn (29) has made careful measurements of velocity profiles near a heated plate

14

A. J. EDE

FIG.4. Flat plate: velocity and temperature profiles in air. [From Ostrach

(4.1

ADVANCES IN FREECONVECTION

15

38 cm high, using illuminated dust particles, and has compared them with the Ostrach solution, paying particular attention to the question of the reference temperature. He found that using the surface temperature or the bulk fluid temperature altered the position of the data points by about So$, and the precision of his data was quite sufficient for this to be appreciable. After a detailed discussion of the observations of Schmidt and Beckmann (3) and of Ostrach ( 4 ) on this question he concluded that the best reference temperature (expressed as the excess temperature) was 0.83 times the surface temperature. In a theoretical paper which will be discussed later Sparrow and Gregg (20) advocated a corresponding figure of 0.62, but they added that this was not greatly superior to 0.5. The latter corresponds to the simple arithmetic mean of the surface and bulk temperatures, and is often called the “film” temperature. Eichhorn’s results (19)indicate that the velocity in the outer regions of the boundary layer is a little higher than expected, particularly towards the lower part of the surface, and he suggests that possible explanations include stray aircurrents, a starting-length effect, and failure of the boundary-layer approximations at small distances from the leading edge. A useful description of the techniques available for measuring temperature and velocity distributions in free convection is given by Kraus (21). A smaller amount of data are available for fluids other than air, and can be used for examining the effect of the Prandtl number. The data for each fluid can be represented reasonably well by the same type of equations as Eq. (22). T h e data for all fluids can therefore be correlated by plotting A = NuL/Ra2l4against Pr, provided the Rayleigh numbers are limited to a range within which Eq. (22) is valid. Figure 5 illustrates the difficulty of obtaining data of sufficient precision to test the effect of the Prandtl number. Only the small quantity of data for liquid metals reveals a significant effect, and this is in line with the theoretical prediction. In view of the great preponderance of data for air, individual points have not been plotted for this fluid ; instead, the vertical line indicates the region within which almost all the points would lie, the circle indicates the point representing a leastsquares fit of the data, and the cross indicates a mean value obtained from the relevant data of Saunders (22), probably the most precise and comprehensive individual set of results. In calculating the dimensionless numbers plotted in Fig. 5 the physical properties have been taken at the arithmetic mean of the surface and bulk fluid temperatures. Other reference temperatures have been tried but in view of the scatter of the data this is not a very profitable exercise. The matter becomes very important however for liquids of large Prandtl number because of the marked effect of temperature on viscosity. In Fig. 5 a group of data obtained with oil is plotted with physical constants taken also at the

A. J. EDE

16

bulk and surface temperatures. Correlation is evidently poor when the bulk temperature is used, but there is little to choose between the mean and surface temperatures. Agreement with the Ostrach solution is slightly better when the mean temperature is used. This is not conclusive however since the velocity boundary layer for aviscous liquid, as already mentioned, may be thicker than is consonant with the boundary-layer approximation, so that close agreement may not be expected. Also shown in Fig. 5 is the curve corresponding to the interpolation equation (28). A few measurements have been made of velocity and temperature profiles in fluids other than air. Goldstein and Eckert (23) used a Zehnder-Mach

L Ede Elhacd v Sounders Mercury

2

I

0

I

2

3

%o R

FIG.5. Flat plate: experimental data, all fluids.

interferometer to make detailed observations of temperature profiles in water (Pr = 6.4). T h e boundary condition was that of a uniform heat flux, so the results were compared with the results of the Sparrow and Gregg analysis for the appropriate Prandtl number. A very satisfactory agreement was obtained (Fig. 6). Szewczyk (24)measured temperature distributions in water by means of a thermocouple probe, and also obtained excellent agreement with theory. Wilke et ul. (25) measured the rate of muss transfer from vertical plates, using electrolysis and rates of solution of organic solids. The Grashof numbers varied from lo4 to lo9, and the Schmidt numbers from 500 to 80,000. A general correlation of NuL = 0.66(Gr12.Sc)'i4 was obtained. This provides, in effect, data at a much higher Prandtl number than any obtained so far in heat transfer measurements, and the coefficient agrees satisfactorily with the theoretical value of 0.67 (Table I).

ADVANCES IN FREE CONVECTION

17

FIG.6 . Flat plate: temperature profiles in water. [From Goldstein and Eckert (23).]

B. THEVERTICAL CIRCULAR CYLINDER For sufficiently large values of D / L , the cylinder can be regarded as effectively the same as the flat plate. When D / L is small enough for the curvature to be significant analysis is more difficult and has not been brought to the same degree of completeness. An early attempt by Elenbaas (26) was based on the Langmuir (27) stationary-film hypothesis. By an ingenious train of argument, drawing on the consideration that any relation for the cylinder should reduce to that for a flat surface as D --f m, he deduced that NuD exp(-2/NuD)

= 0.6(D/L)i'4Rag4

(43) the numerical coefficient being determined empirically. Apart from unconventional procedures such as this the majority of attempts have been based on similarity methods, applied either to the differential equations or to their integrated form. Sparrow and Gregg (28) used the former method. The boundary-layer equations in cylindrical coordinates are

a(Yzl)- 0 -+-ax ay

(44)

A. J. EDE

18

The following substitutions are made :

f(5, C) = c3 x-1‘4 $4

g(5, C) = e/e,

where <,(,f,andg are dimensionless, a is the radius of the cylinder, and the constants cl, c2, and c3 involve only physical properties and lengths. T o solve the resulting equations,f andg are expanded as power series in 5, with coefficients which are functions of 5 ; these are inserted in the equations, the terms grouped according to powers of 5, and the groups set equal to zero. By this means a series of pairs of differential equations are obtained, the first being those resulting from the flat-plate analysis. Sparrow and Gregg solved the next two pairs by computer for Pr = 0.72 and 1.0. T h e results are given as graphs in which the ratios of the Nusselt numbers for a cylinder and a flat surface under the same conditions are plotted against 5 = 23/4(x/a)Gr;1/4. T h e integral equation method was employed by Le Fevre and Ede (29). The equations are

[Similar equations had been given previously by Merk and Prins (30),but they had assumed that 6 was small in comparison with the radius and had accordingly placed the r’s outside the integrals, thereby obtaining a solution identical with that for the flat surface.] For want of better information about velocity and temperature profiles, those used for the flat-plate analysis [Eq. (31)] were adopted, and differential equations in Uand 6 were obtained as usual. These variables were then expanded as power series in l/a, and the subsequent procedure is similar to that of Sparrow and Gregg, except that the pairs of equations can be solved much more easily. T h e first pair is the same as those obtained in the flat-plate analysis; solution of the second pair gives the following result as a first approximation for the cylinder : Nu

4

-3 -

7GrL.Pr2 [5(20 + 21Pr)(

\ll4 +

4(272 + 315Pr)L 35(64 + 63Pr)I)

(50)

In Table I11 the Sparrow and Gregg and the Le Fevre and Ede solutions are compared ; the figures given are the ratios of the mean Nusselt numbers

ADVANCES IN FREE CONVECTION

19

to their corresponding values for the flat plate, obtained from the Ostrach and Squire analyses, respectively. TABLE 111 MEANNUSSELT NUMBER FOR A CYLINDER OF HEIGHT L THAT FOR A FLATSURFACE OF THE SAME HEIGHT

RATIO BETWEEN AND

D

Pr

(GrL.Pr)'14

100 30 10 6

=

Pr = 1.0

0.72

S-G

LeF-E

S-G

LeF-E

1.02 1.06 1.17 1.27

1.01 1.03 1.10 1.16

1.02 1.05 1.16 1.26

1.01 1.03 1.09 1.15

With laminar boundary-layer flow the effects are assumed to be confined to a thin layer adjacent to the surface. It is reasonable to conclude that, for cylinders at fairly high Grashof numbers, effects due to the curvature of the surface may still be negligible even if the diameter is quite small. T h e criterion must presumably be based on the ratio between the boundary-layer thickness and the diameter, and it may be deduced from analyses such as that of Squire that the criterion can be put into the form Ray4.D/L > M , where M is suitably chosen. Since a sufficiently large value of Ra;f4*D/L may equally well be achieved by increasing RaL as by increasing D, it follows that a graph of NuI, against RaL, for cylinders of various D/L, should exhibit a scatter due to curvature effects when RaL is small but not when it is large. For a particular value of D/L, therefore, a sequence of results covering a wide range of Ra should agree with flat-plate data for large RaL but should begin to diverge from them a t some point as the Rayleigh number is dedecreased. T h e flat-plate data agree with an equation of the form NuL = A . Ra;/4 over a considerable range, but diverge for sufficiently low RaL , and this has been regarded as resultingfrom a thickening of the boundary layer to a degree sufficient to invalidate the approximations. Since a divergence of the same type seems likely to arise also as a consequence of D / L effects, it may be expected that with cylinders it may be difficult to distinguish between the two possible causes of divergence from the simple relation. This is illustrated by Fig. 7 , which presents a selection of experimental data for cylindrical and flat surfaces. The divergences from the Rail4 relation occurring with cylinders of small D / I , are of the same type as occur

20

A. J. EDE

with the flat plate, but are more marked and occur at higher values of the Rayleigh number. It can readily be shown that the general solution to the cylinder problem can be put in the following form :

Equation (50), for example, can be written as

Restricting attention to air, so that the effect of Pr can be neglected, it should be possible to correlate experimental data by plotting NuL/Ray4

log,, RO,

FIG.7. Cylinder: selected experimental data.

against Rap4.D/L. I t can easily be seen that this is almost the same as plotting NuDagainst RaD.D/L, a more direct method employed by Elenbaas (26) ; the latter however has the disadvantage that both groups tend to infinity as D becomes large, so that the flat surface cannot be shown on the graph as a limiting case. Experimental data from a number of authors are plotted in this way in Fig. 8, together with curves representing the solutions of Elenbaas, Sparrow and Gregg, and Le Fevre and Ede. T h e latter may formally be evaluated for any value of Raii4.D/L,though the approximations employed are not appropriate for small values of D/L. It is of interest to see that there is nevertheless a reasonable degree of accord with the experimental data over the whole range.

ADVANCES IN FREECONVECTION

21

Eigenson (32,32)has discussed the vertical cylinder problem, referring to experiments with cylinders of widely varying sizes but giving no detailed results. He presents equations which he claims to represent the data of himself, Carne (33),and Koch (34).He states that curvature has no detectable effect if GrD > lo6;for GrL = lo8(say) this would mean D/L = (GrD/GrL)”3 = 10-2’3,so that (GrL.Pr)1/4. D/L = 20, which does not conflict with Fig. 8. Eigenson’s recommended formula for cylinders having GrD < 0.14, i.e., for “wires,” is NuD = 0.45, whence NuL = 0.45L/D, and NuL/RaLi4= 0.45Rai’4.D/L. The line representing this relation is drawn in Fig. 8, and is not in serious disagreement with the other available data. Assuming that

FIG.8. Cylinder: experimental data for air.

GrL = lo8, the limit GrD< 0.14 corresponds approximately to RaZ4.D/L < 0.1. Sparrow and Gregg (28) also considered the point at which the mean Nusselt number for the cylinder diverges noticeably from that for the corresponding flat plate. Their criterion for a 5% difference can be expressed as RaL‘4*D/L33> for Pr = 1 (see Fig. 8). There is a special difficulty in carrying out experimental work on cylinders with small values of D/L, as the slightest movement in the bulk of the fluid is sufficient to deflect the rising column of heated fluid away from the upper part of the cylinder. The result is, of course, an increase in the measured heattransfer coefficient which may be very large. It will be apparent that the classical problem has been very thoroughly explored both theoretically and experimentally, though there are still a few areas where solutions are not as comprehensive, or experimental verification

A. J. EDE

22

as precise, as could be wished. Within the last decade, however, attention has shifted to a series of problems which differ in various ways from the idealizations of the classical case. Most recent work has been analytical. Many of the investigations have been limited, through mathematical difficulties, to the study of small effects, which are therefore difficult to verify experimentally. Some of the problems studied appear to have been chosen because they yielded to analysis rather than because they had any real practical importance. A considerable library of mathematical solutions has been accumulated, and it is difficult for a reviewer to avoid the style of a mere catalog. Both the differential and integral equation procedures have been extensively used. A very popular and successful technique has already been encountered ; the variables are expanded as series in terms of a parameter which is assumed to be small; these expressions are substituted into the equations, and terms of the same degree in the parameter are grouped and set equal to zero. The outcome is a series of sets of equations, the first of which will previously have been encountered in the solution of a simpler problem. By solving one or more of the other sets of equations (usually by computer) a solution valid for small departures from the simple case is obtained. This technique will be called the perturbation method, and details of its application will be omitted in the remainder of this review.

C. NONUNIFORM SURFACE TEMPERATURE Attention has thus far been restricted to situations where the surface temperature is uniform, except that a brief reference has been made to the very similar problem where the surface heat flux is uniform. Extension to more realistic situations is difficult because the principle of superposition cannot be used. 1. The Dtfleerential Equation Method A series of studies of the similarity method have shown that solutions of this type can be obtained for a considerable variety of surface boundary conditions. Finston (35)suggested that the Pohlhausen transformation, which can be written as

$h = c , x3’4f(y/x”4),

0 = c2g(y/x1/4)

(53)

could be generalized to + = C 3 ~ j ( y x q ) ,

e=C4xng(yx*)

(54)

ADVANCES IN FREECONVECTION

23

If this is to be successful in separating the variables, it appears that the constants are not independent, and that the transformation must have the form which corresponds to the situation where the surface temperature varies as x".The uniform temperature case is given by n = 0, and the uniform heat-flux case by n = +. Evidently an infinite number of situations can be solved in this way, in principle at least, though most will have little practical importance. Foote (36) followed this up by discussing the possibility of solving the general problem in series, and solutions for a number of particular cases were obtained by Niuman and Pohlhausen (37) with the aid of a differential analyzer. The generalized forms of Eqs. (17) and (18) are

f

''1

+ ( n + 3)ff" - 2(n + 1)f' +g = 0 g"+ Pr(n -t 3)fg' - 4Pr anf'g = 0

(56)

(57) These equations were also developed at about the same time by Sparrow and Gregg (38),who obtained solutions for a number of values of n between 3 and -0.8, for Pr = 0.7 and 1.0, using a computer. Sparrow and Gregg pointed out that the nature of the transformation does not permit the specification of conditions at x = 0 once the conditions at y = 0 and co have been fixed. T h e general solution for 0, cc x" shows that the heat flux will then vary ~ , conversely, if the heat flux varies as xnthe surface temperaas ~ ( ~ " - l ) /and, ture will vary as x(~"+')/'. Sparrow and Gregg discuss the relation between the average Nusselt number and the Grashof number. It depends of course upon the definition of the temperature difference. Numerical values are given for certain cases where the mean temperature difference (obtained by integration) is used, and where the temperature difference at x = L / 2 is used. Temperature and velocity profiles are given ;the latter are of similar shape, for all values of n, but the former change completely as n passes through zero ; when n is negative the temperature a short distance away from the surface is higher than at the surface. This results from the existence of higher surface temperatures lower down the plate. Correspondingly, the local heat transfer coefficient as ordinarily defined becomes negative. The analysis for n < 0.6 reveals that an infinite source of heat is needed to maintain the correct temperature at the leading edge, so the problem has no practical significance. They also obtained a similarity transformation for the situation where the surface temperature varies as exp(mx) ; this leads to the equations f +ff" - 2fQ + g = 0 (58) 'I'

6'' + Pr(fg'

-

4f'g) = 0

(59)

A. J. EDE

24

It is of interest to note that m does not appear in these equations; it reappears through the similarity variables when a particular case is evaluated. Solutions were obtained for Pr = 0.7 and 1.0. For Pr = 0.7, Nu, = mx'i4.(0.735/&).Grii4 (60) Yet another general examination of the circumstances under which similarity solutions can be obtained has been made by Yang (39).He showed how the various solutions obtained by earlier workers fitted into a general pattern, and that solutions are in principle possible for a surface temperature distribution of the form ( u bx") ; a heat flux distribution of the form ( a + bx") is also amenable to solution, and leads to the same equations if m = (5n - 1)/4. In the discussion of this paper the limitations of the similarity transformation, particularly in connection with conditions at the leading edge, were further examined. Thus one solution required a finite, nonzero boundary-layer thickness at the leading edge, with prescribed profiles, and it was suggested that the analysis might more nearly apply to a situation where free and forced convection were combined. In the same paper Yang showed that a similarity solution can be obtained for the vertical cylinder if the surface temperature is proportional to (u + x). The almost identical case where it is proportional to x had been solved earlier by Millsaps and Pohlhausen (40). Starting with the boundary-layer equations in cylindrical coordinates [Eqs. (44), (45), and (46)J the following transformation variables are introduced : a stream function +, r' = Y / U , X' = x/u, u' = ua/v, 71' = va/v, 0' = g/3a30/v2, = x'f(r'), 0' = ( x ' / a 4 )g(r') ;the equations then reduce to r3f + ~ 2 f f - r2f" - ~2 f' - r$f' + rf' + g = 0 (61)

+

J! , I

111

'I

r2g"+ (Pr-f - 7)rg'

+ (16 - 4Pr.f-

Pr-rf') = 0

t 62)

where the primes have been dropped from the dimensionless variables and now represent derivatives. A further reduction is achieved by putting r = exp(s), and the equations are solved by computer. Profiles and Nusselt numbers are given for Pr = 0.733, 1, 10, and 100. 2. The Integral Equation Method The various modifications of the integral equation method devised by Sugawara and Michiyoshi (15) have already been mentioned. They applied their first method to the situation where the surface temperature varies as

They also considered the case where 0,

=Q

+ bx"'

ADVANCES IN FREECONVECTION

25

Sparrow (41)struck a rather different note by assuming that the heat flux varied according to the relation q/qo= 1 f e(x/L)', where qo is the heat flux at x = 0. The usual expressions, Eq. (31), were taken for the profiles, and the integral equations converted to differential equations in the ordinary way. T h e velocity variable and the boundary-layer thickness were expressed as power series in el which was assumed small, and a perturbation procedure was followed. Solutions were computed for the first five of the resulting sets of equations which were obtained, the number chosen being based on a study of truncation errors. A similar approach was made to the corresponding problem where the wall temperature is given by

e,/ow,o= 1 rf: + I L ) ~ The results were applied to determining the distribution of the surface temperature when the heat flux is specified, and conversely, the results being presented graphically. The graphs cover ranges of Pr from 0.01 to 1000, q/qo from 0.5 to 1.5, S,/SW,, from 0.7 to 1.3, and Y from 0 to 3 . In general, the effects of Pr and of Y are not large, but q/qo and Bw/B,,o have a considerable effect. Tribus (42) showed that the integral equation method could also be generalized. T h e equations corresponding to Eqs. (32) and (33) are

where q is any function of x.Equation (64) can be integrated. Eliminating U and changing variables gives a simple first-order differential equation. A solution is given in terms of the integral of q ; for q cc xn it reduces to

A table of comparative figures shows that agreement with the exact solution is quite good except for n < 0, when the nature of the profiles assumed does not correspond to reality. It is also shown that if q 5c xn, 0, K x(4nf1)'5as in the exact solution. It has further been shown by Bobco (43)that this method may be applied to the particular heat flux distribution studied by Sparrow, and indeed in principle to any other distribution. The integral method is developed with

26

A. J. EDE

complete generality, and expressions for the velocity, boundary-layer thickness, surface temperature, and Nusselt number are obtained in terms of the indefinite integral I = j” Pq-’l3dx, where Q = JE qdx and c = (35Pr + 16)/12. The problem is solved if I can be evaluated, but this is possible only for certain forms of heat flux distribution. An approximate value can be calculated, however, by integrating by parts and estimating the residual by numerical or graphical methods. The procedure is found to give a solution agreeing to within about loo/, with that of Sparrow. Fujii’s (16) modification of the integral equation method has been mentioned previously. He used it to examine the situation where the surface temperature is proportional to x” and (1 + ax). The solutions were used to determine the point on an isothermal surface at which the heat-transfer coefficient would equal the mean coefficient on the nonisothermal surface.

3 . Experimental Work The only published experimental work which could be used to check these analyses appears to be that of Sugawara et al. (44); they attempted to reproduce the condition 8, = a bxm, for m = 0.13, and claimed satisfactory agreement with their own theoretical results. Schetz and Eichhorn (45) carried out an experimental study of the quite different but very practical situation where the temperature distribution includes an abrupt change from one temperature to another at some position on the surface. They obtained temperature profiles in air with an interferometer, and velocity profiles in water by means of flow visualization with tellurium dye. T h e tests in air were made with a surface 50 cm high; the upper and lower halves were maintained at different temperatures. A single test with a uniform temperature over the whole surface gave temperature profiles agreeing well with the Ostrach solution. The ratio of the two values of B,, was then varied from 0.8 to 4.8, and the temperature profiles were used to determine local heat transfer distributions. Figure 9 shows some of these for the upper half of the surface. In all these tests, both halves of the surface were hotter than the surrounding air, so that the convective flow was in an upward direction over the whole surface. Some tests were made with the upper half cooled to below the ambient temperature, resulting in opposing flows from the two halves. Flow visualization showed that the resulting motion was unsteady and three-dimensional.

+

4. Miscellaneous

A few rather specialized papers may be considered at this point. Lemlich and Vardi (46) explored the consequences of a body force which varied in

ADVANCES IN FREE CONVECTION

27

intensity with x. The case discussed is that which would arise in a spinning satellite, with the axis of rotation passing through the leading edge of the plate; the value of "g" will then be proportional to x. They studied the uniform temperature problem by means of the integral equation method, and found that the local heat-transfer coefficient was independent of x ; since heat flux equals heat-transfer coeficient times temperature difference,

x/xo FIG. 9. Flat plate: nonuniform temperature. (The total height of the surface is 2x0; the lower half is maintained at O1 and the upper half at 02.) [From Schetz and Eichhorn (45).]

the problem is equally that of the uniform heat flux. T h e local Nusselt number is given by

A. J. EDE

28

No mention has been made so far of radiative heat transfer. In theoretical work it is ignored ; in experimental work, especially with gases, it is of significant magnitude and must be estimated, so that the measured heat flux can be corrected to give the true convective component. Cess (47)has pointed out that radiation can affect the temperature distribution when the heat flux is specified, and can thereby affect the convective heat transfer. He uses the differential equations with the following boundary condition at the surface : k(aB/+),=o=-q+ U E ( T , ~Tm4) (68) Here u is Stefan's constant and E is the emissivity of the surface. I t is assumed that the surroundings and the ambient fluid are all at the same (absolute) temperature T,. This is a modified form of the uniform heat flux problem, and new variables are introduced : f

= c1y/x'/5,

5 = acTm3x'/5/Kq,

c1 =

(g&/5kv*)'/5

(69)

Solutions are assumed to have the form

Ifi = f o ( O

+ l f i ( 4 ) + 5"2(0 +

(70) for the stream function, and a similar expression for the temperature; a perturbation procedure is followed, and it is found that, for Pr = 1, *

+

Nu,/GrLi4 = 0.456 - 0.1805 * The corresponding equation for Pr = 0.72 is estimated to be

(71)

NU,/GT:/~= 0.407 - 0.1615 + *

(72) This solution is valid only for small values of 5. An approximate solution for larger values of 5 is obtained by a similar method, except that a simpler, linearized radiation component is used in the boundary condition. It is found that Nu,/GrAi4 = 0.357 + 0.0075-5'4 (73) Experimental data are presented which are in general agreement with these predictions, though the effect is small and the scatter of the results is considerable. A number of papers have reported work on free convection in an electrically conducting fluid in the presence of a magnetic field. This topic was reviewed recently by Romig (48) and will not be discussed further here.

D. NONUNIFORM PHYSICAL PROPERTIES In all the theoretical work so far considered it has been assumed that the physical properties, other than density, do not vary with temperature. In

ADVANCES IN FREE CONVECTION

29

general this is a reasonable assumption, but situations may be encountered where it is not ; for example, when the viscosity of the fluid is high, or when the temperature difference is unusually large. 1. Large Temperature Dtference Hara (49) considered the effect of a large temperature difference in air. H e put ( T , - T,)/T, = E , and assumed the following relations for the variation of the physical properties with temperature: p/pm= C-I, r)/r), = k/k, = C", c = const, 5 = TIT,, m = 0.76. He used the differential equations with a similarity transformation, obtaining the following equations :

where

f"'+Pf"+ Q=O g" + Rg' = 0

(74) (75)

R = 3Pr *f 15" -t m('/< These were solved by successive approximation. After formal integration, two functions were selected arbitrarily for f and g, substituted in the equations, and first approximations obtained ; the process was then repeated until sufficient accuracy had been attained. Solutions are given for 5 = 2 and 4 ;the resulting Nusselt numbers are approximately equal to Nuo( 1

+ 0.055~)

where Nuo is the value for E = 0, i.e., the constant physical property solution, with the bulk temperature used as reference temperature. Sparrow and Gregg (20) considered a number of hypothetical fluids, using the differential equations and a similarity transformation. T h e resulting equations are

These differ from Eqs. (17) and (18) in that 77, k, etc., are functions of temperature ; y w , k,, etc., are the values at the surface temperature. I t is interesting to note that, for a gas which has the properties p =pIRT, pr) = const,

A. J. EDE

30

pk = const, c = const, the equations reduce to the ordinary constantproperty form and the usual solution can be used. Numerical solutions are obtained for fluids whose properties conform to the relations listed in the accompanying tabulation. T h e results are compared ~~

p

k

7 c

Pr

~

A

B

C

D

P/RT T3” x T3’4 Const Const

P/RT T2’3 % T2I3 Const Const

PIRT z l/p z l/p Const Const

P/RT x T3’*/(T A , ) x T3I2/(T + A,) Const Const

Gas

%

%

E P/RT c T3I2/(T+A , ) cc T3”/(T+ Az) a+bT Variable

+

with constant-property solutions for various values of TWIT,,and the best reference value T, is found for using the constant-property solution as an approximation. ItisconcludedthatT,- T, -0.38(T,Tm),i.e., 8,=0.628,, gives the best results for gas A, and that the same temperature gives reasonably good results for the other gases as well, the deviation being of the order of lo(,or less. It is also found that the conventional film or arithmetic mean reference temperature gives results that are not much less accurate, and probably good enough for most practical purposes. Similar calculations are given for a fluid with physical properties corresponding to those of liquid mercury, and the best reference temperature is found to be T, = T, - 0.3(1;, - Tm),i.e., 8, = 0.78,. Use of the film temperature would again be sufficiently accurate for many purposes.

2. Near-Critical Conditions The physical properties of a fluid are particularly sensitive to temperature near the critical point, and it is to be expected that unusual results will be obtained if experiments are carried out under such conditions. A number of recent investigations have confirmed this supposition. Fritsch and Grosh (50,51) made a study of the effects produced in water. For their analytical work they used the differential equations with the similarity transformation for the uniform temperature case, except that the effect of pressure on density was included ; the resulting equations are

-f’g’2(A2

- 2@,,)

-

g”

PO,‘,f’g” = 0

(78)

=0

(79)

+ 3Pr .fg‘

ADVANCES IN FREECONVECTION

31

where y = (l/p)d2p/dT2. These were solved by computer. T h e viscosity and thermal conductivity were assumed to be constant, and were evaluated at the film temperature. Specific heat and density were constrained to vary with temperature according to the best experimental data available, tabulated values being stored in the computer. Solutions were obtained for 35 different cases, covering a range of pressures from 22.1 to 23.4 MN/m2 at bulk temperatures from 373 to 381"C with temperature differences from 0.4 to 11°K (the critical point is 22.1 MN/m2 and 3742°C). The bulk temperature was first varied through a range of values while the temperature difference was held constant; this was done for two pressures, and it was established that heat transfer reached a maximum when the bulk temperature corresponded to the point at which c and p attained their maxima. T h e remaining examples were solved only for a bulk temperature within 0.14"K at this point. The results were found to conform approximately to the following equation : Nu, = 0.37Pr-li8( Tm/0w)1/9 Ra:I4 (80) The employment of such an equation, involving the choice of single values for the physical constants, is clearly bound to lead to difficulties in view of the rapid changes with temperature, and attention is paid in the paper to the selection of the best reference temperature. The above equation fails to reflect some of the detailed features of the results. Comparison with the Ostrach constant-property solution shows that the local heat-transfer coefficient calculated on the variable-property basis can be almost twice as great. The experimental work of Fritsch and Grosh covered the same conditions. The surface was a strip of platinum foil 1.2 cm high, heated by direct conduction of electricity. Temperatures were determined by thermocouple. The results show good qualitative agreement with theory, though a systematic deviation of about 203$ was found, the experimental data being the higher. The variation of heat flux against temperature difference at various pressures was very similar to that predicted theoretically. A rather different technique was adopted by Simon and Eckert (52) for experiments in carbon dioxide. Very low heat fluxes were used, giving temperature differences of the order of 0.005"K, and temperatures were determined by means of a Zehnder-Mach interferometer. Heat-transfer coefficients and estimates of the thermal conductivity of the fluid under various conditions were deduced. In general, tests were made at fixed heat fluxes over a range of pressures. When plotted against density, both the heat-transfer coeficient and the thermal conductivity exhibited marked peaks as the critical density was approached; the height of the peak

32

A. J. EDE

increased with increasing heat flux. It appears from these results that the thermal conductivity of COz increases approximately linearly with heat flux for densities close to the critical value. These experiments were really more concerned with the more subtle problem of heat transport in a near-critical fluid than with free convection in the ordinary sense. The results, particularly the form of the interference fringes, were employed in a discussion of the possibility that unusual convective patterns or cluster formation in the fluid might be the cause of the anomalous behavior of the thermal conductivity. It is clear that an analysis such as that of Fritsch and Grosh, which assumes a fixed value for thermal conductivity, can hardly be expected to account completely for experimental results of this type.

3. Viscous Dissipation The motion produced by free convection is comparatively feeble, and does not normally produce a significant amount of heat by viscous dissipation ; the corresponding term is customarily omitted from the energy equation. In theory, the effect might be significant at very large Grashof numbers, but the flow would then normally be turbulent. Gebhart (53)has nevertheless examined the effects which might be expected in laminar flow, and has considered in what circumstances the analysis might have practical significance. ) ~ must be The differential equations now include the term q( & ~ / a ywhich added to the right-hand side of Eq. (16). A perturbation analysis is performed in terms of a dissipation number E =gPx/c, which is assumed to be small. It will be seen that a new dimensionless group is involved ; this has an unfamiliar appearance because, for the only time in this review, the conversion of mechanical energy into heat is contemplated. The first of the resulting series of pairs of equations are the usual equations for zero-dissipation ; solutions are obtained for the second pair, by computer, for the uniform temperature and uniform heat flux cases, for PrandtI numbers of 0.01, 0.72, lo2, and lo4 ( lo2 only for uniform heat flux). The effect of dissipation is to inhibit heat transfer when heat flows from surface to fluid, and vice versa. I t is zero at the leading edge and increases along the surface. ‘The ratio of the mean coefficients with and without dissipation is (1 & A*gt’3L/c),where A is as follows: 0.72 102 104 Pr: 0.01 A: 0.074 0.25 0.38 0.41 Since the Grashof number is proportional to the first power ofg and to the cube of L, the product gL in the dissipation number is proportional to

ADVANCES IN FREE CONVECTION

33

for a fixed Grashof number. It is therefore possible to conceive of a situation where the force field is so intense that dissipation might be significant even at Grashof numbers low enough to permit the flow to be laminar. The only way in which such a situation could readily be obtained is when centrifugal forces are involved, and the “free vertical surface” might then be regarded as an impossible abstraction. The analysis might however be applied to closed cavities of considerable width. Possible effects of an external pressure gradient are also discussed in the paper. 4. Miscellaneous Merk (54) employed a modification of the integral equation method to examine the thawing of a vertical surface of ice as a result of immersion in warmer water. There are two abnormal features of this situation: the melting ice produces mass transfer, and the anomalous behavior of the density of water in the region of 4°C causes a severe distortion of the flow pattern. Experimental evidence supporting this analysis was provided by Ede (55). Acrivos (56) and Na and Hausen (57) have discussed the situation when the fluid is non-Newtonian, so that the viscosity varies with the rate of shear. Similarity solutions exist for certain special cases, but not for uniform surface temperature.

E. NONSTEADY CONDITIONS 1. Negligible Thermal Capacity An early analysis of nonsteady free convection was made by Illingworth (58). He considered the effect of a sudden change in the temperature of a large surface which had previously been in equilibrium with the surrounding fluid. The flow was assumed to be of the “parallel” type, so that velocity profiles are invariant with respect to x. A solution was obtained in terms of exponentials and error functions, giving the velocity as a function of position and time. This type of solution is, for a surface of finite extent, only valid for a short period after the transient has been started. A similar analysis was carried out by Sugawara and Michiyoshi (59), who concluded that in air the “parallel flow” regime lasted for less than a second under typical conditions. A substantial advance was achieved by Siege1 (60). He started with the integral equations, modified to include time-dependent terms :

A. J. EDE

34

Introduction of the conventional profiles leads to

(which reduce to the usual form if the time derivatives are omitted). T h e remainder of the analysis is based on the method of characteristics. T h e behavior at any position is considered as passing through three stages. I n the first, heat transfer is by pure conduction : the velocity and temperature distributions are independent of x and convective heat transfer in the ordinary sense has not started. This is the “parallel flow’) regime previously mentioned. In the third regime, the steady convective flow is fully established and the profiles are of the normal type and independent of time. T h e second, intermediate, stage is one of transition, in which conditions are influenced by the leading edge but have not yet become steady. T h e boundaries between these three stages are set by particular characteristic lines. T h e solution affords equations for these lines, so that the time can be found for each stage to be reached for any value of x ; the profiles and heat transfer conditions are given for the first and third stages, but no information is obtained about the transitional stage. By repeating the analysis with a slightly different velocity profile it is established that the results are not very sensitive to the form adopted. T h e following is a summary of the results obtained with the usual form of profile : the conductive regime ends at a time t = 1.80(1.5

+ Pr)112(glg0,J-1i2x1i2

(85)

and the steady regime is attained at a time

+

t = 5.24(0.952 Pr)lI2 (glgOzL.)-1’2x1/2

(86) T h e steady-state conditions are of course the same as those obtained in earlier work. In the conductive regime the temperature profiles are the same as those obtained by the ordinary theory of conduction in a stationary medium. T h e velocity profile is given by

L-

-

2s

2gp& t - 1.5 + Pr

(1 - s)2

where s = (~py~/12kt)’~~ ; the corresponding heat-transfer coefficient is

h = (pCk/nt)”2 (88) The heat-transfer coefficient for the conductive regime is a function of time ; it decreases rapidly from its initial value of infinity towards the steady-state

ADVANCES IN FREE CONVECTION

35

value, but at the time corresponding to the termination of this regime it has fallen to a slightly lower level than that of the steady state. It is to be presumed that at some stage in the intermediate regime it must increase again. This corresponds to an “overshoot” in surface temperature, but it is not clear from the analysis whether this is a real phenomenon or merely a consequence of the approximations used. This point will be considered again later. T h e same method was also applied to the uniform heat flux case, with similar results. Sparrow and Gregg (61) considered the situation where the temperature (uniform) of a surface is caused to vary slightly about a mean level which is higher than the ambient temperature. They used the differential equations with the time-dependent terms retained, but with the usual boundary-layer approximations, and introduced a modified stream function +1, so that 24 = a+,/ay,

ZI = -(a+,/ax

+ ayjat)

After the usual similarity transformation, a perturbation analysis was carried out in terms of the amplitude of the temperature variation; this produced the usual steady-state equations together with further sets, of which the first pair was solved by computer for Pr = 0.72. T h e ratio between the instantaneous rate of heat transfer at x and the corresponding steady state was found to be

and for the over-all heat transfer on a plate of height L the ratio is

If an accuracy of 5;:) in the estimation of heat transfer is sufficient, the steady-state solution is adequate provided that

T h e same analysis was developed further by Chung and Anderson (62), using a slightly different perturbation expansion. They obtained two more terms in the expansion, enabling the solution to be used for larger variations in temperature. They also investigated the effect of fluctuations in the force field. Their results are presented graphically. Nanda and Sharma (63) considered the effect of a sinusoidal variation in the (uniform) surface temperature. By writing the temperature and the velocities as sums of steady and oscillatory components, the differential

A. J. EDE

36

equations are converted into two sets, the first being of the usual type and the second containing the oscillatory terms. Solutions are obtained for large and for small values of the frequency. For the latter the integral equation method is used with conventional profiles for the steady-state equations and fourth-degree profiles for the oscillatory equations. For the high-frequency case the Lighthill approximation is adopted, whereby all the nonlinear inertia terms are neglected and the equations reduced to a simple form which can readily be solved. T h e results are presented chiefly as graphs, but the following is the expression for the over-all heat transfer as a ratio to the steady-state value :

:( 1 36 625

q’/q= 1 + - 1 + ---A2 where few is the amplitude and tion, and

w

‘I2

f

cos wt

the frequency of the temperature oscilla-

132N2+ 216Pr + 953) A = --(15)’12 MN( .297N2 + 112(14 + 3Pr) ~

where M = u L 2GrL1/’/v and N = (Pr + 20/21)”2. This solution can be compared with that of Sparrow and Gregg. For Pr = 0.72, N = 1.29, A = -2.88M, and the ratio becomes 1 + (5/4) (1 + 0.4781c12)”2E cos wt

If the temperature oscillation is dl, sin wt, the rate of change is €wowcos w t , so that Sparrow and Gregg’s solution gives a value of 1

+ 0.918~w(L/g/36’,,)’~~~ cos wt = 1 + 0.918Mc cos wt

for the same ratio. Putting, as a typical set of conditions, w = 50 cps, = lo8, v = 1.5 x lop5m2/sec,it is found from the Nanda and Sharma solution that the ratio is 1 + 726 cos w t , and from the Sparrow and Gregg solution it is 1 + 7 6 cos ~ wt. Yang’s (39) general exploration of possible similarity solutions has already been referred to. He also discussed the nonsteady problem, and showed that similarity solutions were possible in principle for the following cases: surface temperature proportional to l/(x + at) and (x + u)/(l - Z I ~ ) ~ , which correspond, respectively, to heat flux proportional to l/(x + and (x + u ) / (1 - bt)ji2.He also considered the situation where x is large so that temperature and heat flux are functions of time only (the parallel flow case) and showed that solutions are possible if the temperature is proportional to (1 + at)P or the heat flux proportional to (1 at)“,where q = p - 3. Puttingp = 0 gives another solution for the stepwise change in temperature.

L = 0.5 m, Gr,

+

ADVANCES IN FREE CONVECTION

37

The stepwise change has also been studied by Hellums and Churchill (64, 65) who solved the nonsteady differential equations by finite difference methods on a computer. Figure 10 shows the variation of Nu/(Gr)l14 with the dimensionless time variable t(g/3ew/x)l/’ obtained by this means. A number of interesting points arise from this analysis. First, the temperature profiles and heat-transfer rates for small values of time agree well with those obtained by the pure-conduction analysis. The new solution diverges from the conduction solution at about t = 2.4(x/g/3BW)’/’; Siegel’s analysis indicated a coefficient of about 2.7. The steady state predicted by the new

0

3

2

I

Conduction alone

4

Dimensionless lime t ( g p ~ , / x ) ” 2

FIG. 10. Flat plate: stepwise change in temperature: theory. [From Hellums and Churchill (65).]

solution agrees to within 2% with the exact solution, and it appears that this stage is reached at about t = 3.5(x/g/30,)”’; Siegel’s value for the coefficient was about 7.1. Finally, it seems that the local heat-transfer coefficient passes through a minimum shortly before the steady state is attained, in accordance with Siegel’s results. This was the first analytical treatment to reveal the whole transient behavior. Schetz and Eichhorn (66) discussed the initial conduction regime in greater detail. If all the velocity derivatives in the x direction are set equal to zero, the differential equations reduce to

&/&

=

0,

SO

that

&/dy

=0

(92)

A. J. EDE

38

ae

pc-=kat

a2e ay2

(94)

The third of these is the ordinary linear, nonsteady heat conduction equation and can be solved by means of the Laplace transform. The result is substituted in the second equation, which can then be solved in the same way. Solutions are obtained for seven different forms of transient. For the stepwise change to a temperature of Ow, the temperature profile is 8, erfc { y / ( 4 ~ t ) l / ~ ) (here K = k / c p ) and the velocity profile is

unless Pr = 1, when it becomes

Other cases considered include a double stepwise change in temperature, a temperature varying as t1I2, a linear rise in temperature, a sinusoidal variation in temperature, a sinusoidal variation in g, and a stepwise change in uniform heat flux. The same problem has also been discussed by Menold and Yang (67) in a paper published almost at the same time. The same procedure is followed and several of the solutions just mentioned are obtained. Additional cases include a linear rise in uniform heat flux and a sinusoidal variation in heat flux. Goldstein and Briggs (68) made a fresh estimate of the time required before the effect of the leading edge reached a given position, thus terminating the conductive regime, for a stepwise change in heat flux. They assumed that the leading edge effect was propagated according to the velocity field obtained in the parallel-flow solution of Schetz and Eichhorn and others. The distance penetrated in time t is j-6 udt, and this can be evaluated and its maximum value determined with the help of the Laplace transform; it is found that the distance is x, = 0 . 0 5 7 9 ~ ~ q ( t 5 / k p c )The 1 i 2 . local heat-transfer coefficient at this value of x, for Pr = 1, is given by Nu/(Gr*)'/' = 0.502, which may be compared with the steady-state uniform heat flux value of 0.534. This again confirms the existence of a slight overshoot. Solutions are also given for the stepwise change in uniform surface temperature. It is found that the amount of overshoot decreases with Prandtl

ADVANCES IN FREECONVECTION

39

number and tends to be larger for a step in temperature than for a step in heat flux. According to this analysis, the time to the end of the conductive for air ; previous estimates by Siege1 and by regime is t = 2.97(~/g;SO~)”~ Hellums and Churchill gave values of 2.7 and 2.4, respectively, for the coefficient.

2. S@n$cant Thermal Capacity The theoretical work so far discussed may be regarded as somewhat unrealistic, since an arbitrary behavior of the surface temperature or heat flux is postulated, whereas in any practical arrangement it will be affected by the thermal capacity of the body whose surface is transferring heat to the fluid. A true stepwise change, for example, is an unattainable abstraction. Goldstein and Briggs (68),in the paper just mentioned, briefly discussed the effect of the thermal capacity of the body, but the most substantial contribution has been made by Gebhart in a series of papers. In the first of these (69) the basic equations are established. T h e body is assumed to be a circular cylinder, though in subsequent developments it is assumed to be flat ;its thermal diffusivity normal to the surface is assumed to be large so that the temperature is uniform throughout at any value of x. The first major approximation is that conditions are averaged over the full height L of the surface. Energy and momentum balances provide the following equations :

Lq” - c ” LL F at d x + k

=

(&I a8

y=o

dx = 0 (95)

where q” is the rate of generation of heat per unit area of surface, c” is the thermal capacity of the body per unit area, and uL and eL are, respectively, the velocity and temperature at the top of the surface. These are transformed by a series of substitutions. They cannot be solved directly because insufficient information is available for evaluating the integrals ; the profiles are not known.

A. J. EDE

40

T o avoid this difficulty it is assumed that the profiles do not vary significantly with either x or t. This is an extension of the similarity assumption, and its justification is considered at some length. It enables the equations to be simplified very considerably, but a number of integrals still remain; in effect they average the dependent variables over the height of the surface. Estimates of these integrals are obtained by considering the values they would have in the initial conductive stage and the final steady stage, and taking average values. The equations are thereby further simplified, and in their most compact form they are as follows :

*

=-

Y

d ( $-Y )a-

dT

TXIJ = 0

x d -SIJP- U=- ,,(XY) Y

-

(98) (99)

Here S, U , W, and cc depend only on the Prandtl number ; the thermal flux quantities q" and qm" are the instantaneous and asymptotic values. The remaining symbols represent generalized variables, as follows : T , time ; Q, thermal capacity ;$, temperature ; F, boundary-layer thickness ;X,velocity. Bars indicate that average values have been taken over the height L. In the first paper solutions are obtained by computer for the steady state and the stepwise change in heat input when the thermal capacity is assumed to be zero ;these are problems considered by earlier workers, and are primarily discussed in order to establish confidence in the method, having regard to the many assumptions made. The steady-state solution agrees to within 9(:" with the Sparrow and Gregg solution. The stepwise-change solution breaks new ground in that it provides information about the intermediate regime for arangeof Prandtl numbers from 0.01 to 1000. They are presented in tabular and graphical form in terms of the generalized variables listed above. For Pr < 0.5 a slight overshoot is detected. In the second paper (70) the same equations are solved for a stepwise change in heat input with a range of values of thermal capacity, i.e., of Q. For convenience of interpretation and comparison with experimental results they are expressed in terms of the thickness of a nichrome plate which would have the correct value of Q for a particular fluid. The plate is assumed to be 8.9 cm high, giving a Grashof number of lo', with convection taking place on both sides. Solutions are obtained by computer, using a successive approximation method, taking the temperature profile for onedimensional conduction as the starting point.

ADVANCES IN FREE CONVECTION

41

The following general conclusions may be drawn from the results. If Q is small, so that the stepwise change in heat input corresponds closely to a stepwise change in heat flux, the greater part of the temperature change takes place within the initial, conductive regime. If Q is large, the surface temperature responds so slowly that the first and second regimes are almost without significance ; the situation at any moment could be closely approximated by putting the instantaneous value of the surface temperature into the ordinary steady-state solution. This is called the "quasi-steady" condition. T h e range of Q within which the true transient regime has any importance is relatively small, extending only from 0.1 to 1.0. The results are presented as tables and graphs. T h e initial rise in the surface temperature is found to be proportional to 1/Q. The effect of the Prandtl number is almost entirely included in the generalized variables used. In the third paper Gebhart (71) discussed the effect of a linear rise in heat input ; in other respects the assumed conditions were the same and the same equations were used. The value of q"/q," was put equal to qT until qT = 1, after which a constant value of qT = 1 was assumed, thus representing a rough approximation to an asymptotic rise to the steady value. Calculations were carried out for Pr = 0.72, for Q = 0.5, 0.1, 0.01, and 0.001, and for various values of q. The results are presented graphically, particular attention being paid to the values of q and Q for which quasi-static values were produced even while the heat input was being varied. I t is shown that, for a stepwise increase in heat input, the quasi-static solution closely approximates to the true conditions only if Q > 1 .O ;for the linear rise in heat input, however, the quasi-static solution is adequate for much lower values of Q provided that q is less than about 0.2. Gebhart (72) has applied the same methods to the situation where a plate of significant thermal capacity which is transferring heat by steadystate free convection suddenly has its supply of heat shut off. In this case there is of course no initial purely conductive regime. Computer solutions are given for Q = 0.01, 0.1, and 1.0, all for Pr = 0.72, The quasi-steady solution is found to be adequate unless Q < 1.

3. Experimental Work Experimental work on unsteady free convection seems to be largely confined to two investigations. Goldstein and Ecltert (73) studied the effects of a stepwise change in heat flux, using a metal foil 10 cm wide, 16.5 cm high, and 0.025 cm thick in water, and observing the development of the temperature profiles with the aid of an interferometer. Very small heat fluxes were used, producing temperature differences of the order of 1S"K. Excellent

42

A. J. EDE

agreement with the steady-state predictions of the Sparrow and Gregg analysis confirmed the reliability of the technique. Because of the thinness of the foil the effect of thermal capacity should have been small, and the results should be comparable with the simpler theory in which it is ignored. The transient experiments confirmed the three-stage process predicted by Siege1and others. For a short time after switching on the heating current, the thermal boundary layer grew at the same rate over most of the surface,

min

I.o

4

FIG.11. Flat plate: stepwise change in heat flux: experimental. [From Goldstein and Eckert (73).]

so that its thickness was a function o f t but not of x.Later, the layer ceased to grow near the leading edge, and this cessation of further growth gradually spread upwards, giving rise to a regime in which 6 was a function both o f t and x.Finally a steady state was reached in which 6 was a function of x only. Figure 11 shows the excellent agreement between the local heat-transfer coefficients at the beginning and end of the transient, and the theoretically predicted values from refs. (18,60). The figure indicates that the coefficient passes through a very slight minimum. The existence of overshoot was confirmed by detailed examination of the interference fringes. The times required for the conclusion of the conductive regime were in good agreement with theory. The other major experimental investigation is that of Gebhart and Adams (74, who studied the stepwise change in uniform heat input to a series of

ADVANCES IN FREE CONVECTION

43

metal foils having significant thermal capacity. Surface temperatures were measured by an infrared detector of special design, capable of detecting about 1.5"K at 150°C. Temperature differences of the order of 100°K were used, the fluid being air. The foils were from 3.8 to 7.5 cm high.

FIG.12. Flat plate: stepwise change in heat input: significant thermal capacity. [From Gebhart and Adams (74).]

T h e results are compared with Gebhart's theoretical work in Fig. 12, using his generalized coordinates. Also shown are data obtained by Goldstein and Eckert and two other investigators (75, 76), all of whom used water. The data, which were obtained by various techniques and covered a range of thermal capacities represented by Q from 0.0014 to 30.8, are in good agreement with the theory. The data for air are all within 496 of the common curve for large Q, namely the quasi-static regime. T h e data for water corwspond to the conductive regime ;the plotted points are for the early part of the transient. T h e conclusion is drawn that under all conditions likely to arise in practice the quasi-static solution is an adequate approximation for free convection in gases. This is not the case for liquids, however, for which it is readily possible to obtain much lower values of Q.

I?. VERYLOWGRASHOF NUMBERS It has been shown that the thickness of the boundary layer is proportional to xCr;1/4. It may be concluded that, for sufficiently low valuesof the Grashof

44

A. J. EDE

number, the boundary-layer approximations will no longer be tenable. Since Gr, = gflOWx3/v2,a low value can arise from a number of causes, including a small heat flux, a low density, and a high viscosity. There is little experimental data available for these conditions, but some results due to Saunders (22) and Madden and Piret (77) which were obtained in gases at low pressures, may be considered. They have already been presented in Fig. 7. Saunders used very small surfaces, 0.3 cm high, at pressures down to 1 mm Hg; Madden and Piret used vertical wires about 40 cm long at pressures down to 0.05 mm Hg, where accommodation coefficient effects become important. The validity of the measurements of bulk temperature under such conditions is also a matter for consideration ; Saunders maintained that his bulk temperature measurements were valid because surfaces of different sizes gave results which were closely correlated on the Nu versus Gr.Pr graph. T h e progressive departure of the results from proportionality to (Gr.Pr)’I4 as the Grashof number falls has already been noted. It is reasonable to suppose that, for a sufficiently low Grashof number, all motion would cease and heat transfer would take place by conduction only. I t would then be proportional to the thermal conductivity and the Nusselt number would be constant. T h e Madden and Piret results appear to have approached more closely to this stage ; as already discussed, however, this may be due to the effect of curvature. T h e process of heat transfer at a very low Grashof number must be somewhat similar to that proposed by Langmuir (27) ;heat flows through a layer of fluid which is so stagnant as to render the predominant process that of pure conduction. Such considerations lead to equations involving 2/Nu or exp (2/Nu) (26,78). It has already been mentioned that, for “creeping” flow, dimensional analysis indicates that the Nusselt number is a function of the Rayleigh number only, so that a graph of Nu against Ra for a variety of fluids should exhibit no scatter due to Prandtl number effects. There are no suitable data for checking this point. When the boundary-layer approximations cannot be used the differential equations are much more difficultto solve and comparatively little analytical work has been accomplished. Yang and Jerger (79) have obtained a firstorder perturbation solution of the following equations :

-au+ - =avo ax

ay

ADVANCES IN FREE CONVECTION

(

;: ;)-

pc u - + v -

-

k

45

-+--) ; (Z

The perturbation parameter used was Gr:l4- ; the first set of equations were the same as those of the boundary-layer approximations, together with = 0 ; the next set are as follows :

The solution of these equations, even by computer, presents many difficulties, and reference must be made to the original paper for a full discussion. I t is necessary to obtain a solution to an ancillary potential-flow problem, outside the boundary layer, in order to provide acceptable boundary conditions. Results are given for Pr = 0.72 and 10. There are unfortunately no sufficiently accurate experimental data at appropriate Grashof numbers with which this courageous solution may be properly compared. In the paper, the velocity and temperature profiles predicted for a Grashof number of lo7 are compared with the profiles obtained experimentally by Schmidt and Beckmann (3),and a somewhat better fit is claimed than that given by the Ostrach solution. .When plotted in the usual dimensionless form (Fig. 13), the new solution gives a different velocity profile for each value of x/L,and these appear to represent the experimental data better than does the single line of the simpler theory (Fig. 4). The corresponding temperature profiles differ only very slightly from one another and from the Ostrach profile, and all agree with the experimental data. The matter is slightly confused by uncertainty about the choice of a reference temperature for the physical properties. When, however, heat-transfer coefficients are calculated on the basis of the new theory, the surprising conclusion emerges that, for small Rayleigh numbers, they are somewhat lower than those given by the simpler theory. In fact, of course, experimental data show an increasing divergence in the opposite direction as the Grashof number falls. It has been suggested by Gebhart that, in practice, a flow is induced from below the leading edge, which gives rise to increased heat transfer and is not taken into account in the theory. The possibility that the abnormal conditions near the leading edge might play a more important role than is generally realized has been examined

A. J. EDE

46

theoretically by Scherberg (80). He employs the usual boundary-layer equations in the integral form with third-degree polynomials for the profiles. Instead, however, of using simple expressions such as U = px" and 6 = qx", which automatically result in zero values at the leading edge, he assumes initial values of Uo and 6, at x = 0. For the special case where U0-7gPp Pr 7 20+UPr

(108)

602 -

the solution is of the conventional form except that x is replaced by (x + a), indicating that the whole boundary layer has been displaced upwards. The

0.28 0.24

- am

w r

0.16

0.12

am

--

Present solution Boundary -layer solution Experimental data

0.04 0 0

0.5

1.0

1.5

2.0

2.5

3.0

35

E FIG.13. Flat plate: velocity profiles in air: comparison of experimental data with Yang and Jerger's solution. [From Yang and Jerger (79).]

value of a depends of course upon the value assumed for Uo or a,,. For the general case where U o and are independent, a variety of numerical solutions is obtained and presented as graphs ; they can all be approximated by the conventional solution, with x replaced by (x a), provided that x is not too small. Figure 14 shows calculated isotherms and streamlines for a surface in which the temperature excess is uniform for x > x,, but declines to zero by a sigmoid path between x and xo . Scherberg's analysis takes account of the longitudinal conduction of heat within the boundary layer. This process is normally ignored, but could be significant near the leading edge where the flow is very slow.

+

ADVANCES IN FREECONVECTION

4O

47

1.2

\

4

0.4

' 0

0.4

0.8

1.2 1.6 2.0 p = y / 6 , . , n = 3 Pr=0.7

2.4

28

FIG.14. Flow structure near leading edge. [From Scherberg (80).]

G. EFFECT OF VIBRATION T h e effect of vibration upon convection has been extensively studied in recent years, but comparatively little attention has been paid to free convection on vertical surfaces, and the present limitation to this configuration precludes a proper discussion of this interesting subject. The vibration can be produced either by the motion of the heated surface or by the injection of acoustic vibration into the fluid by some external agency. A group of papers relating to the former process has been published by Clark and his co-workers. Schoenhals and Clark (81) considered a vertical flat plate made to vibrate transversely with velocity aQsinQt. The coordinate system is assumed to be attached to the surface and to move with it. The effect of the vibration is represented by an additional term in the momentum equation, derived as follows. The momentum equation in the y direction, usually ignored, is now

because of the velocity un of the wall; hence apjay is pgo cos Qt, where go = aQ2, the maximum acceleration of the surface. Integrating with respect t o y to a position outside the boundary layer, with x constant, gives

A. J. EDE

48

Differentiation with respect to x gives the pressure gradient parallel to the surface :

ax

ax

Now the pressure gradient in the x direction outside the boundary layer is = pm(1 - Be), so that

-prng, and p

aP

-=

ax

-pa q -

whence the following set of equations is assembled :

-aU+ - =avo ax

ay

A perturbation analysis, based on the parameter E = g/goGr:I4, is used to solve these equations. The first of the resulting sets of equations are of the usual Pohlhausen type ;solutions for the second set are obtained by approximate methods for low and for high frequencies. For the latter it is assumed that the convective terms are negligible in comparison with the rate of change of velocity or temperature. The validity of this assumption depends also upon the distance from the leading edge, so the group y = fi(4x/g/3eW)'/' is adopted as a criterion. By this means the equations are greatly simplified and analytical solutions can be obtained. For small values of y , the full equations are solved by a series method based on the solution for y = 0. T h e results are presented graphically for large and small y , with an arbitrary section added to link the curves together. They indicate that the effect of the vibration tends to be greatest near the leading edge and to decay as x increases. For a given value of the vibratory acceleration the disturbance increases as the frequency is reduced. Comparable experiments, very briefly described, apparently revealed no significant effect dpon heat transfer. Blankenship and Clark (82)extended the analysis to include the secondorder perturbation. Using essentially the same equations, they expanded the dimensionless velocity and temperature functions in the form

ADVANCES IN FREE CONVECTION

49

where E = aLQ/vGrt/*. In the usual way three sets of equations are obtained. The first are of the ordinary form and, in order to simplify subsequent calculations, their exact solutions are replaced by simple analytical expressions which represent them approximately. The first-order equations are then solved for high frequencies, and the range of frequency for which the solution is valid is examined. T h e second-order equations are also solved ;the analysis is too lengthy to be summarized usefully, and the original paper must be consulted for the details ; they include order-of-magnitude studies. Expressions are given for the velocity, shear-stress, normal temperature gradient, and local Nusselt number ; the latter has the form

where A is the value for a stationary surface. Graphs are provided for finding B and C for Pr = 0.72 and 10, for a range of values of frequency. It will be seen that the first-order term makes no contribution to the net effect of the vibration upon heat transfer. The second-order term however produces a very slight reduction in the Nusselt number. In contrast, the wall shearstress is slightly increased. The same workers (83)carried out an experimental study, using a 25-cmsquare plate with a 15-cm-square heated area located centrally; it was mounted vertically in air and driven at 50 cps by a vibration exciter. Guard heaters and screens of resin-impregnated cloth served to suppress edge losses and stray air currents. The results showed the expected slight decline in Nusselt number as the amplitude was increased, until a fall of about 2O/, had been produced ; with further increase in amplitude the Nusselt number rose sharply to as much as SOo$ above the normal value. Smoke studies revealed that this was due to the development of turbulence in the boundary layer. Blankenship and Clark also published a further paper (84) in which they repeated their analysis, but this time reached the conclusion that very little effect was to be expected as a result of the vibration. They stated that this was in accordance with the experimental results obtained earlier by Schoenhals. The relation between these papers is not altogether clear (and the reader should beware of misprints). As part of the same program, the effect of longitudinal oscillations has been studied by Eshghy et al. (85).A perturbation analysis in terms of the amplitude is used, and solutions are obtained for high and low frequencies for the first-order equations, and for high frequencies only for the secondorder equations. It is concluded that the Nusselt number will be reduced by the movement, but only to a very small extent. Experiments with a vertical cylinder confirmed this prediction, but once again if the amplitude was

A. J. EDE

50

increased sufficiently the nature of the flow changed and the Nusselt number increased considerably. The other process, where sound waves are injected into the fluid, has been studied experimentally by June and Baker (86). They used a plate 90 cm high, in air, at Grashof numbers between lo6 and lolo, and set up a sound field by means of a siren. Precautions were taken to insure that the plate itself was not set into vibration and that no air currents reached it. The frequency of the sound field was varied from 200 to 1000 cps, and the sound pressure level from 130 to 163 dB with respect to 20 pN/m2. It was found that the sound field significantly affected heat transfer ; the extent increased with increasing intensity and with decreasing frequency. There was a small but consistent effect of temperature difference, the effect of the sound being slightly greater at lower temperature differences. The greatest increase obtained was 220%, at 163 dB and 200 cps. Plots of Nu against Ra showed no sign of incipient turbulence. A reasonable correlation was achieved by plotting the ratio of the Nusselt numbers in the disturbed and undisturbed states against the ratio of the amplitude a of the sound field to the “viscous wave thickness” a0, a term frequently used in this field; it is defined as tio= (v/52)l12, where SZ is the frequency. The resulting correlation may be written as NuL+/NuL = 1

+ 0.037(d?2/v)’i2

in any consistent units.

(118)

111. Turbulent Flow

Theoretical work on laminar free convection is based on the assumption that motion is confined to a thin layer ; it should, therefore, become increasingly reliable as the Grashof number increases. In fact experimental results show a growing divergence from the Gr1I4 relation as Ra increases above about lo9 (Fig. 3). It has been established that this is due to turbulence, which develops first at the top of the surface and gradually extends downwards as the Grashof number is increased. The existence of turbulence seems first to have been noticed by Griffiths and Davis (87), who inferred it from observations of temperature and velocity profiles and the variation of the local heat-transfer coefficient with x.

-4. INSTABILITY OF LAMINAR FLOW The criterion for the onset of turbulence in forced convection is the Reynolds number ; it is generally assumed that the Grashof number or the Rayleigh number performs a similar function in free convection. Experimental evidence is not inconsistent with this, though it can hardly be said

ADVANCES IN FREE CONVECTION

51

to have been systematically tested. It is generally assumed that the state of the flow at x depends only upon Gr, , and is unaffected by the existence of further heated surface above this position. T h e critical value of Gr, has not been established unambiguously by experiment. It has in fact been found that turbulence develops gradually over a considerable range of x under given conditions, and that its onset is readily affected by disturbances in the surrounding fluid. Among early attempts to determine the critical Grashof number may be noted that of Hermann (88) who found that it varied for air between 0.5 x lo9 and 1.0 x lo9, and of Saunders (5, 22) who found that turbulence was fully established at Ra, = 1.5 x 1O'O in air, and began at about Ra, = 2 x lo9 in water. Eigenson (32) suggested a transition zone extending from Ra, = 7 x lo8 to 1.2 x 1O'O. Eckert and Jackson (89) said that under normal conditions the flow would be turbulent if Ra, > 1O'O. Eckert and Soehngen (90) gave the first detailed description of the development of turbulence ; with increasing height at a given value of GrL the flow in the boundary layer begins to develop a wave-like motion ;the waves roll up, then break up, and finally the motion becomes completely irregular. These observations were based on a study of interference fringes, and the superb photographs have been widely reproduced. A variety of experimental techniques have been brought to bear on the problem of incipient turbulence. Fujii (91) used suspended aluminum particles and a schlieren method for examining the flow in water (Pr = 3.5) and ethylene glycol (Pr = 28 to 65). He observed that a regular series of ascending vortices developed, with new vortices being formed continually at a position corresponding to Ra, = 4 x lo9. With increasing height the vortices curled over and broke like waves, and the regular vortex-street gave way to irregular turbulence at about Ra,= 1O'O. Before this regime was reached the vortex-street was quite regular, and the wavelength and velocity could readily be measured. Fujii also measured local heat-transfer coefficients, and found that the region in which the regular vortex-street existed gave values differing very little from those obtained with ordinary laminar flow formulas. At the point of transition to full turbulence, however, the local coefficients increased rapidly, by a factor of about 1.6. Holman and co-workers (92, 93) used an interferometer, but injected instability by placing a wire in the boundary layer near the lower edge and heating it with a pulsed electric current. They also made some interesting observations on the suppression of turbulence by placing various objects near the surface of the plate. Szewczyk (94) studied the development of turbulence on a 150-cm-high plate in water, using thermocouples to determine temperature profiles and a dye-injection technique for observing the flow. T h e latter afforded a more

A. J. EDE

52

detailed insight into the process than had been previously obtained. With increasing x, the dye streaks were at first smooth and continuous, but presently a faint two-dimensional pattern began to appear, with waves of slowly increasing amplitude having a wavelength of about 5 cm. The streaks then broke up and became “twisted,” and the waves rolled up and turned into vortices, which gradually became three-dimensional. Vortices were seen to develop on both sides of the position of maximum velocity in the boundary layer, i.e., nearer and farther away from the surface. The outer vortices were much stronger and broke down into turbulence when the flow nearer the surface was still fairly regular. It seems therefore that any theory of instability should concentrate on the flow farther away from the plate than the region of maximum velocity. T h e critical value of the Grashof number in these tests was in the region of 1.4x lo9. Tritton (95) used a quartz fiber anemometer to explore the flow over a vertical plate in air. He systematically traversed the boundary layer at a series of positions and found that the fiber behaved in three ways :(1) mainly stationary, with a steady deflection but with occasional feeble movements probably due to disturbances in the room; (2) sometimes stationary, but with sharply contrasted bursts of vibration ; (3) vibrating vigorously all the time. I t is suggested that the quartz fiber technique would fail to detect the wave-like motion observed by others, and that the intermittent bursts of vibration represented the breakdown of vortices into turbulence. However, this regime started at a Grashof number at about 9 x lo6, which is much lower than any figure given by other workers. The region of full and continuous turbulence was reached at about Gr, = 1.5 x lo9. Theoretical work on the instability of laminar free convection leads to a modified form of the Orr-Sommerfeld equation. The following outline of the derivation is based on that given by Szewczyk (94). The basic flow is described by u(y),v = 0, p ( y ) ,and 8(y).Superimposed is a two-dimensional disturbance S(x, y , t ) , etc., so that the resulting motion is given by u + c, 5,p + j,and 8 The full differential equations are set down and a perturbation stream function introduced : $(x, y , t ) = +(y)exp [G(x - t t ) ] , where I$ is a complex function representing the amplitude of the disturbance, 6 = 2r/X is the wavenumber of the disturbance, E = I?, + iEi, where 2, is the propagation velocity of the disturbance wave in the x direction and 2, is the amplification factor. The wave is amplified or suppressed according as t i is positive or negative. Then

+ e.

u“ = r$’(y) exp [&(x - Et)]

and

d

=

-i6$(y) exp [&(x - Et)]

and similarly e(x,y, t ) may be expressed as [ ( y )exp[i&(x- Et)]. The following substitutions are introduced to render the equations dimensionless : 5 =y/6,6 = x2/2/Gr:I4, f ’= ux/2vGr:/*, 13 = 8/8,, = r$x/2~6Gr;/’,E = fie,,

+

ADVANCES IN FREE CONVECTION

53

c = Ex/2vGr:I2, and a = cis. On substitution, and neglecting the nonlinear terms with respect to 4 and 5, the following equations are obtained :

(f‘- c)($”

- a’$)

-f”’d = -

i aG

P W5 (4““- 2a24”f a44)- i__

aFr

I

(119)

45where G = (64G1-,.)’~~ is a modified Grashof number analogous to the Reynolds number in forced flow, and Fr = 4v2Gr,/gSx2 is the Froude number. Equation (119) is the equivalent of the Orr-Sommerfeld equation with an additional term representing the body force, and Eq. (120) is the counterpart for the temperature fluctuation. A variety of methods have been used for solving these equations; the analyses are lengthy and will not be detailed here. Most workers have assumed that the body force term in Eq. (119) can be neglected; the second equation is then ignored, and the problem reduces to the solution of the ordinary form of the Orr-Sommerfeld equation on the basis of a velocity profile obtained from standard free-convection theory. TABLE IV CRITICAL GRASHOF NUMBER FOR ONSET OF INSTABILITY

0.733 1 1.5 2 3.5 5 7

1.93 x 3.08 x 6.55 x 1.21 x 5.15 x 1.39 x 3.90 x

lo6 106

lo6 107 107 108 108

The equations were apparently first obtained by Plapp (96),who attempted a solution based on a polynomial expression for the velocity profile. Szewczyk (94)used an asymptotic method of solution in terms of expansions about the two “critical” positions-i.e., those positions where the velocity approaches the phase velocity of the assumed perturbation. Kurtz and Crandall (97) and Nachtsheim (98) used numerical, finite difference methods; the former adopted a matrix approach and obtained a solution for Pr = 0.733 ; the latter used a step-by-step, forward integration procedure and obtained solutions for P r = 0.733 and 6.7. Both used computers.

54

A. J. EDE

Nachtsheim also considered the more difficult problem in which the body force term is not neglected and account is taken of Eq. (120) ; this leads to a sixth-order equation. This procedure takes account of the possibility that a perturbation of temperature may, by producing a buoyancy perturbation, introduce instability. It appears from the analysis that this possibility is of practical significance, i.e., leads to a lower critical Grashof number, for sufficiently small values of the wave number c(. That such a possibility is likely is disputed by Sparrow et al. (99),who improved upon the analysis of Kurtz and Crandall and obtained solutions for seven values of the Prandtl number, using the first equation only. T h e critical Grashof numbers obtained are given in Table IV.

B. TURBULENCE Early workers proposed a number of generalizations concerning turbulent free convection, based largely on dimensional considerations. Jakob and Linke (100)asserted that, when turbulence was fully established over all but a negligibly small part of a surface, the local heat-transfer coefficient should be independent of x ; it follows that the general equation Nu, = f (Gr, , Pr) must have the form Nu, = GrlI3.f(Pr), which does not involve x. This argument, which was also suggested as a possibility by Nusselt (101), is not entirely convincing, though it is true that the experiments of Griffiths and Davis (87) afforded heat-transfer coefficients which were virtually independent of x towards the top of their plates, and Heilmann (102) made a similar observation. Another suggestion concerns the effect of the Prandtl number. With fully developed turbulence the effect of viscosity should be small ; this indicates that the general equation should have the form Nu, =f(Gr;Pr*), which does not involve viscosity. A graph of the usual Nu-Gr-Pr form should thus exhibit a greater scatter due to Prandtl number effects than would be expected for laminar conditions. Frank-Kamenickij (203) goes further, and suggests that since thermal conductivity should also be negligible the equation must take the form NU,^ = A(Gr,y.Pr2)''2.

1. Theoretical Work T h e first serious attempt at an analysis of turbulent free convection was made by Colburn and Hougen (104). They assumed that the boundary layer included a laminar sublayer, within which both velocity and temperature varied linearly with y ; at the outer boundary of this sublayer, i.e., at y = al, the variation of velocity withy was zero, and its magnitude was such that the Reynolds number formed from it and a1 corresponded to the critical

ADVANCES IN FREE CONVECTION

55

value found in comparable experiments on forced convection. A simple analysis of the momentum and energy balances for a layer of fluid bounded by the planes y = y and leads to the solution Nu,

= (6Re,)-'l3

GrLi3

(121)

where Re, is the critical Reynolds number. The local heat-transfer coefficient is thus independent of x. Since the flow at the bottom of a vertical surface is always laminar, the equation cannot be used for the whole surface unless the Grashof number is so large that the laminar region can be neglected. On this assumption a mean Nusselt number can be found ; Colburn and Hougen assumed that Re, = 130, so that NuL = 0.10S(GrL)1/3

(122)

A more direct solution was obtained by Eckert and Jackson (89),using the integral equation method. The first requirement is the adoption of suitable expressions for the profiles. By examining the experimental data of Griffiths and Davis (87) and comparing them with data for forced convection profiles they chose the following expressions :

e = ew{i- ( y / q q

u = u(Y/q1/7,

(123)

These were inserted into Eqs. (29) and (30). The terms q(du/dy),=o and k(dd/dy),,o in those equations, representing the shear-stress T, and the heat transfer rate qw, cannot however be calculated from these profiles because the laminar sublayer is not taken into account. Accordingly Eckert and Jackson replaced them by the following expressions, taken from forced convection work : 7,

= o.o225pu2( UIS/v)-1/4

qw = 0.0225 UpcB,Pr-2/3(U 6 ~ / q ) - * / ~

(124) (125)

T h e following equations resulted : 0.0523

d

( U 26) = 0.125g15dw6- 0.0225 U2(U8/v)-'I4

d 0.0366 dx (U6)= 0.0225 UPrP2l3(U ~ / V ) - ' / ~

(126) (127)

The substitutions U = px" and 6 = qx" were found to be capable of separating the variables, whence

+

Ux/v = 1.185Gr:I2( 1 0.494Pr2/3)-'/2

6/x = 0.565Gr~'i'0Pr-*~'5(1 + 0.494Pr2/3)'/'0

(128) (129)

A. J. EDE

56 It follows that

Nu,

+ 0.494Pr213)-2/5

= 0.0295G1-2~’ Pr7/15(l

(130) If the flow can be assumed to be turbulent over most of the surface a mean Nusselt number can be obtained by integration : or, for air,

NuL = 0.0246G1-;/~Pr7/15(1 + 0.494Pr2/3)-2’5

(131)

NuL = 0.021l(GrLPr)2i5

(132) Bayley (105) extended this analysis to fluids of low Prandtl number, the basic difference being the rejection of the above expression for qw as inapplicable to such fluids. The main steps in a rather complicated argument may be summarized as follows. The boundary layer, of total thickness 6, consists of a laminar sublayer of thickness S1 and a turbulent outer layer. Reasoning based on forced convection work leads to the expression 1 - 61 6 1 - - +0.186--NU,^ x x 1 + (E/v)Pr

(133)

where E is the eddy viscosity and v the kinematic viscosity. T h e integral equations are simplified, by a series of assumptions, to the following :

%( U 2 6 )= B6 - CU2

d -(US) dx

Ak 1

= -CP

6

(134) (135)

By taking new variables x = US and Y = U26,and dividing one equation by the other, there follows A Y B _k d_ _X _ 4_ c y cp d X = Y which is solved in series for Y in terms of X ; the adoption of a simpler expression which represents the solution approximately leads to Nu, = GrA/4(501-15/32Gr--7/32pr--1/16+ 0.97Z-‘14Pr-1/2) (137) where I = J: Ody/60L,;for a temperature profile of the y1I7form, I = 0.125, and finally, for small Prandtl numbers and Grashof numbers between 1O’O and lo1’, NuL = 0.08Gr94 (138) assuming as usual that turbulence extends over the greater part of the surface. Le Fevre (106) suggested that a solution for very large Grashof numbers could be obtained from certain results obtained in forced convection work.

ADVANCES IN FREE CONVECTION

57

T h e basic link is the assumption that the Reynolds number corresponds to the square root of the Grashof number. In laminar flow, for example, Nu cc Re'/' for forced convection, and Nu K Gr'/4 for free convection. T h e von Karman form of the Reynolds analogy may be expressed as follows : ReaPr 2 = - + ($)'"5(~r - 1 + I n NU Cf

---

6

(139)

and the logarithmic formula for the friction factor Cfis

Cf'' log (CfRe) = 0.242

(140)

Assuming that the Prandtl number is of the order of unity, and that the Grashof number is large, it can be shown that Pr. Grii2 NUL

P r - l + I n - - 5Pr 6

+

or, approximately,

'1 +

2.58 In 2NuL]Z ___Pr

(141)

Pr.GrL12

NuL =

3Pr Two more solutions of the same type have been obtained by Fujii (207). T h e first, which is intended to represent the region where vortices are developing in the boundary layer, assumes temperature and velocity profiles very similar to those used by Eckert and Jackson : u=

U ( ~ / S )(1I / ~y / 6 ) 2 ,

e = e,{i

-

(y/ql/7}

(143)

Instead however of adopting expressions for 7, and qrufrom forced convection work, he assumes that the profiles are linear up to a distance 6, from the surface, such that the corresponding Reynolds number is 160, and finds that 6' = 0.02036. T h e resulting solution is as follows: or, for air

Nu,

=

1.46(1 + 3.47/Pr)-'14 Ra,t14 Nu,

= 0.94Raii4

(14.4) (145)

This represents an increase over the laminar flow relation, Eq. (20), and Fujii points out that he did obtain such an increase experimentally ;however, the increase was stated to correspond with the start of full turbulence, whereas the analysis just outlined is supposed to correspond to the vortexstreet region. F u j i also studied fully developed turbulence, following the procedure of Eckert and Jackson rather closely, and obtaining a solution of very similar form.

A. J. EDE

58

Siege1 (208)has obtained a solution for the uniform heat flux case, giving

Lemlich and Vardi (46)extended this to the very specialized situation where g is proportional to x. They found that

2. Experimental W o r k

A rather limited amount of experimental data is available for turbulent flow. Figure 15 presents the data for flat surfaces, in all fluids. It also includes certain results obtained with large cylinders; they are included here, rather

I 1.6‘

A

I.

I

I

9

I I

10

I I

II

I

I

12

i

loqORQc

FIG.15. Flat plate and large cylinder: experimental data for turbulent flow.

than in a later graph dealing specifically with cylinders, because all the other data are for air and because the effect of D / L for large cylinders is small. In selecting the data for this graph a lower limit of lo8 has been set for the Kayleigh number in order to provide continuity with the laminar data. It

ADVANCES IN FREE CONVECTION

59

may confidently be assumed that points for Rayleigh numbers exceeding 10'O refer to predominantly turbulent flow. An empirical formula due to Eigenson (32), referring to flat surfaces in gases, is NuL = 0.148Grg3 - 127.6

(148)

This is represented by a dotted line in the graph. Also shown are lines representing the laminar data and the various theoretical results for turbulence, evaluated with a Prandtl number appropriate to air. There is a great deal of scatter, and it is impossible to choose with any confidence between the different formulas. The general trend is for the slope to increase as the Rayleigh number increases, and it seems quite likely that for higher values of Ra than are at present available the slope would correspond more to the index of Eckert and Jackson than to the 3 slope of Colburn and Hougen, and might eventually approach the of Le Fevre and Frank-Kamenickij. T h e scatter precludes any serious assessment of the effect of the Prandtl number. Fishenden and Saunders (109) have suggested that the very scanty data can be correlated by means of Gr .Pr", where n lies between 1.3 and 1.5. For this range of Grashof number the hypothetical regime of negligible viscosity, which requires that n shall equal 2, has not yet been reached ; this conclusion is in conformity with the slope of the data. Velocity and temperature profiles were measured in turbulent flow by Griffiths and Davis (87) and may be compared with the theoretical predictions of Eckert and Jackson (89). The general shape of the latter was of course based on the experimental work, so that the chief point of interest is in the values of U and 6 resulting from the theoretical work. T h e comparison is made in Fig. 16; the agreement is not very good, possibly due to the imperfections of the comparatively early experimental work. It must be remembered that the analysis assumes that the whole of the surface is covered with a turbulent boundary layer, whereas in the experiments it is likely that the flow was laminar over the lower third. Nevertheless, over the upper half of the surface the measured velocities varied hardly at all with x, whereas according to the theory U is proportional to x ' ' ~ . Data for vertical cylinders in air are shown in Fig. 17. They are plotted on the NuL versus RaL basis, a distinction being made between results relating to different ranges of D/L. Despite the considerable amount of scatter it is clear that the effect of reducing D / L is to increase NuL at a given value of RaL. The data do not, however, warrant a more precise statement, nor is there sufficient information to enable any conclusions to be drawn concerning the effect of D / L on the values of the Rayleigh or Grashof numbers at which turbulence starts.

+

4

A. J. EDE

60 100

00

.

60

E

3

40

20

0

100

00

c

60

E 40

20

0

FIG.16. Flat plate: velocity profiles for turbulent flow.

Eigenson’s empirical formula is NuL = BGrti3 (for gases), where B is a tabulated function of Gr, . Curves for D / L = 0.001 and 0.01 are indicated in the graph. Eigenson states that, for cylinders of small diameter, the transitional zone is small, and that for a cylinder of 0.24 cm diameter he found no transitional zone at all ;the critical point corresponded to Ra, = 1.3 x lo’. For “wires,” with GrD > 1, he states that the flow will be turbulent; for

ADVANCES IN FREE CONVECTION

61

GrD < 0.14 it will be laminar. He gives no information about the values of D / L in his experiments, but refers to tests on wires of diameters 0.03, 0.05,

FIG.17. Cylinder: turbulent flow.

0.1 cm, and larger. Experimentally determined local heat-transfer coefficients have been given by F u j i (91) for Rayleigh numbers up to 10".

ACKNOWLEDGMENTS In the preparation of this review, I have had to restrict my attention to those papers, mostly in English, which were readily obtainable. A complete bibliography would, without question, include many excellent papers, reports, and theses which I have not consulted. I offer my apologies to the authors. Nevertheless, I hope that the outline of the subject at least has been adequately portrayed. T h e figures and text relating to the older work have been partly based upon the treatment in an earlier review (110). I am indebted to the Director, National Engineering Laboratory, United Kingdom, for permission to use this material. The same review contains details of all relevant experimental work published prior to 1956.

62

A. J. EDE SYMBOLS radius of cylinder specific heat (constant pressure) diameter of cylinder transformed velocity function transformed temperature function acceleration (usually gravitational) local heat-transfer coefficient thermal conductivity height pressure heat flux radial space coordinate time absolute temperature velocity components velocity parameter spacial coordinates Grashof number, g/3Bwx3/u* (similarly Gr,, Gr,) modified Grashof number, gBqwx4/v2 Nusselt number, hx/k Prandtl number, cv/k Rayleigh number, Gr;Pr

Reynolds number, U6/u coefficient of expansion boundary-layer thickness viscosity temperature excess, T - Tm kinematic viscosity similarity variable density shear stress stream function SUBSCRIPTS c critical value 0 value at x = 0 r reference value w value at wall (y = 0) co value at infinity (bulk temperature) The above symbols have been used consistently. Other symbols have been defined in the text.

REFERENCES 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20.

A. Oberbeck, Ann. Phys. Chem. 7, 271 (1879). L. Lorenz, Ann. Phys. Chem. 13, 582 (1881). E. Schmidt and W. Beckmann, Tech. Mech. Thermodyn. 1, 341, 391 (1930). S. Ostrach, N A C A T N 2635 (1952); NACA Rept. 1111 (1953). 0. A. Saunders, Proc. Roy. SOC.A172, 55 (1939). H. Schuh, Aerodynamische Versuchsanstalt Repts. & Trans. 1007 (1948). S. Sugawara and I. Michiyoshi, Mem. Fac. Eng. Kyoto Univ. 13, 149 (1951). E. M. Sparrow, F. K. Tsou, and E. F. Kurtz, Phys. Fluids 8, 1559 (1965). B. Gebhart, J . Fluid Mech. 14, 225 (1962). E. J. Le Fevre, Proc. 9th Intern. Congr. Appl. Mech., Brussels 4, 168 (1956). T. von Karman, 2. Angew. Math. Mech. 1, 233 (1921). E. R. G. Eckert, “Introduction to the Transfer of Heat and Mass.” McGraw-Hill, New York, 1950. H. B. Squire, in “Modern Developments in Fluid Dynamics.” Oxford Univ. Press, London and New York, 1938. H. J. Merk and J. A. Prins, Appl. Sci. Res. A4, 1 1 , 195,207 (1954). S. Sugawara and I. Michiyoshi, Trans. , I S M E 17, 109 (1951). T. Fujii, Bull. J S M E 2, 365 (1959). J. Brindley, Intern. .I. Heat Mass Transfer 6 , 1035 (1963). E. M. Sparrow and J. L. Gregg, Trans. A S M E 78,435 (1956). R. Eichhorn, Intern. J . Heat Mass Transfer 5 , 915 (1962). E. M. Sparrow and J. L. Gregg, Trans. A S M E 80, 879 (1958).

ADVANCES IN FREE CONVECTION

63

21. W. Kraus, “Messung des Temperatur- und Geschwindigkeit Feldes bei Freier Konvektion.” Braun, Karlsruhe, 1955. 22. 0. A. Saunders, Proc. Roy. SOC. A157, 278 (1936). 23. R. J. Goldstein and E. R. G. Eckert, Intern. J . Heat Mass Transfer 1, 208 (1960). 24. A. A. Szewczyk, Intern. J . Heat Mass Transfer 5, 903 (1962). 25. C. R. Wilke, C. W. Tobias, and M. Eisenberg, Chem. Eng. Progr. 10,663 (1953). 26. W. Elenbaas, Physica, ’s Grav. 9, 1, 285, 665, 865 (1942). 27. I. Langrnuir, Phys. Rev. 34,401 (1912). 28. E. M. Sparrow and J. L. Gregg, Trans. A S M E 78, 1823 (1956). 29. E. J. Le Fevreand A. J. Ede, Proc. 9th Intern.Congr. Appl. Mech., Brussels4,175 (1956). 30. H. J. Merk and J. A. Prim, Appl. Sci. Res. A4, 11, 195, 207 (1954). 31. L. S. Eigenson, Zh. Tekh. Fiz. 1,228 (1931). 32. L. S. Eigenson, Compt. Rend. Acad. Sci. U R S S 26,440 (1940). 33. J. B. Carne, Phil. Mag. 24,634 (1937). 34. W. B. Koch, Beih. Gesundh. Ing. 22, l(1927). 35. M. Finston, Z. Angew. Math. Phys. 7, 527 (1956). 36. J. R. Foote, Z. Angew. Math. Phys. 9, 64 (1958). 37. F. Niuman and K. Pohlhausen, Z. Angew. Math. Phys. 9,67 (1958). 38. E. M. Sparrow and J. L. Gregg, Trans. A S M E 80, 379 (1958). 39. K. T. Yang, J . Appl. Mech. 27,230 (1960). 40. K. Millsaps and K. Pohlhausen, J . Aeronaut. Sci. 23, 381 (1956). 41. E. M. Sparrow, N A C A T N 3508 (1955). 42. M. Tribus, Trans. ASMESO, 1180 (1958). 43. R. P. Bobco, J . Aerospace Sci. 26, 846 (1959). 44. S. Sugawara, I. Michiyoshi, M. Kudo, and R. Matsumoto, Trans. J S M E 19, 9 (1953). 45. J. A. Schetz and R. Eichhorn, J . Fluid Mech. 18, 167 (1964). 46. R. Lemlich and J. Vardi, J . Heat Transfer 86, 562 (1964). 47. R. D. Cess, Advan. Heat Transfer 1, 38 (1964). 48. M. R. Romig, Advan. Heat Transfer 1, 290 (1964). 49. T. Hara, Bull. J S M E 1,251 (1958). 50. C. A. Fritsch and R. J. Grosh, Intern. Develop. in Heat Transfer 5, 1010. Am. SOC. Mech. Engrs., Boulder, Colorado, 1961 51. C. A. Fritsch and R. J. Grosh, J . Heat Transfer 85, 289 (1963). 52. H. A. Simon and E. R. G. Eckert, Intern. J . Heat iVIass Transfer 6 , 681 (1963). 53. B. Gebhart, J . Fluid Mech. 14, 225 (1962). 54. H. J. Merk, Appl. Sci. Res. A4, 435 (1954). 55. A. J. Ede, Appl. Sci. Res. A5, 458 (1955). 56. A. Acrivos, J . Am. Inst. Chem. Eng. 6, 584 (1960). 57. T. Y. Na and A. G. Hausen, Intern. J . Heat Mass Transfer 9, 261 (1966). 58. C. R. Illingworth, Proc. Cambridge Phil. SOC.46, 603 (1950). 59. S. Sugawara and I. Michiyoshi, Proc. 1st Joint Congr. Appl. Mech. p. 501 (1951). 60. K. Siegel, Trans. A S M E 80, 347 (1958). 61. E. M. Sparrow and J. L. Gregg, J . Heat Transfer 82, 258 (1960). 62. P. M. Chung and A. D. Anderson, J . Heat Transfer 83, 473 (1961). 63. R. J. Nanda and V . P. Sharma, J . Fluid Mech. 15,419 (1963). 64. J. D. Hellums and S. W. Churchill, Intern. Develop. in Heat Transfer 5, 985. Am. SOC.Mech. Engrs., Boulder, Colorado, 1961. 65. J. D. Hellums and S. W. Churchill, J . Am. Inst. Chem. Eng. 8, 690 (1962). 66. J, A. Schetz and R. Eichhorn, J . Heat Transfer 84, 334 (1962). 67. E. R. Menold and K. T. Yang, J . Appl. Mech. 29,124 (1962).

64 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111.

A. J. EDE R. J. Goldstein and D. G. Briggs, J . Heat Transfer 86, 490 (1964). B. Gebhart, J . Heat Transfer 83, 61 (1961). B. Gebhart, J . Heat Transfer 85, 10 (1963). B. Gebhart. Intern. J . Heat Mass Transfer 6 , 951 (1963). B. Gebhart, Intern. J . Heat Mass Transfer 7 , 479 (1964). R. J. Goldstein and E. R. G. Eckert, Intern. J , Heat Mass Transfer 1, 208 (1960). B. Gebhart and D. E. Adams, J . Heat Transfer 85, 25 (1963). J. H. Martin, cited in ref. 74. H. Lurk and H. A. Johnson, J . Heat Transfer 84, 217 (1962). A. J. Madden and E. L. Piret, Proc. Gen. Discussion Heat Transfer, p. 328. Inst. Mech. Engrs., London, 1951. E. R. G. Eckert, “Introduction to the Transfer of Heat and Mass,” p. 165. McGrawHill, New York, 1950. K. T. Yang and E. W. Jerger, J . Heat Transfer 86, 107 (1964). M. G. Scherberg, Intern. J . Heat Mass Transfer 5, 1001 (1962). R. J. Schoenhals and J. A. Clark, J . Heat Transfer 84, 225 (1962). V. D. Blankenship and J. A. Clark, J . Heat Transfer 86, 149 (1964). V. D. Blankenship and J. A. Clark, J . Heat Transfer 86, 159 (1964). V. D. Blankenship and J. A. Clark, J . Appl. Mech. E31, 383 (1964). S. Eshghy, V. S. Arpaci, and J. A. Clark, J . Appl. Mech. E32, 183 (1965). R. R. June and M. J. Baker, J . Heat Transfer 85, 279 (1963). E. Griffiths and A. H. Davis, DSIR Food Invest. Board Spec. Rept. 9 (1922, 1931). R. Hermann, 2. Physik. 33, 425 (1932); 2. Angew. Math. Mech. 13, 433 (1933). E. R. G. Eckert and T. Jackson, N A C A T N 2207 (1950). E. R. G. Eckert and E. Soehngen, Proc. Gen. Discussion Heat Transfer, p. 321. Inst. Mech. Engrs. London, 1951. T. Fujii, Bull. J S M E 2 , 551, 555 (1959). J. P. Holman, H. E. Gartrell, and E. E. Soehngen, J . Heat Transfer 82, 263 (1960). J. P. Holman, K. E. Stout, and E. E. Soehngen, J . Aerospace Sci. 27, 463 (1960). A. A, Szewczyk, Intern. J . Heat Mass Transfer 5, 903 (1962). D. J. Tritton, J . Fluid Mech. 16,417 (1963). J. E. Plapp, J . Aerospace Sci. 24, 318 (1957). E. F. Kurtz and J. H. Crandall, J . Math. and Phys. 41, 264 (1962). P. R. Nachtsheim, N A S A Tech. Note T N D-2089 (1963). E. M. Sparrow, F. K. Tsou, and E. F. Kurtz, Phys. Fluids, 8, 1559 (1965). M. Jakob and W. Linke, Forsch. Arb. Ing. Wes. 4,75 (1933). W. Nusselt, Gesundh. Ingr. 38, 477, 490 (1915). R. H. Heilmann, Trans. A S M E 5 1 , 287 (1929). D. A. Frank-Kamenickij, Compt. Rend. Acad. Sci. S S S R 17, 9 (1937). A. P. Colburn and 0. A. Hougen, Ind. Eng. Chem. 22, 522 (1930). F. J. Bayley, Proc. Inst. Mech. Engrs. (London) 169, 20, 361 (1955). E. J. I,e Fevre, Private communication (1950). T. Fujii, Bull. J S M E 2 , 559 (1959). R. Siege], Gen. Elec. Co. Tech. Inform. Ser. R54GL89 (1954) [cited Ref. (46)]. M. Fishenden and 0. A. Saunders, “An Introduction to Heat Transfer,” p. 94. Oxford Univ. Press, London and New York, 1950. A. J. Ede, Natl. Eng. Lab. (U.K.) Rept. Heat 141 (1956). S. Ostrach, in “High Speed Aerodynamics and Jet Propulsion,” Volume IT’, Theory of Laminar Flows (F. K. Moore, ed.), Section F. Princeton Univ. Press, Princeton, New Jersey, 1964.

Heat Transfer in Biotechnology ALICE M. STOLL Aerospace Medical Research Department U.S. Naval Air Development Center Johnsville, Warminster, Pennsylvania I. Introduction . . . . . . . 11. Natural Environments . . . . A. Terrestrial. . . . . . . B. Terraqueous . . . . . . C. Extraterrestrial . . . . . 111. Artificial Environments . . . . A. NormalIndoors . . . . . B. Underwater Vessels . . . . . . . . . C. Spacevehicles IV. The Role of the Skin in Heat Transfer A. Functions and Characteristics . B. Thermal Sensation . . . . C. Injury and Protection . . . V. Concluding Remarks . . . . . References . . . . . . .

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65 66 . 66 . 90 . 97 . 103 . 103 . 109 .111 . 115 . 115 . 121 . 123 . 137 . 137

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I. Introduction

This chapter is the first in this series of volumes to be concerned with biotechnology. It seems proper therefore to begin with a definition of the term, biotechnology, to delineate its scope, and to point out that portion of the field to be discussed herein. A. Biotechnology may be defined as that area of knowledge which concerns scientific techniques appropriate to studies in the field of biology. Biology, of course, includes all life : human, animal, plant, insect, bacterial, and viral ; in all environments : terrestrial, terraqueous, and extraterrestrial, both planetary and interplanetary, if indeed life exists there. Clearly, a tremendous area is involved and throughout the whole of it, heat transfer is a basic process, for without it life is suspended or obliterated. 65

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B. It is obviously impossible in a work of this nature to cover all pertinent areas ; the present article is intended to provide general information in the field of heat transfer between the human organism and its various environments and particular information concerning the medium supporting the thermal interface between the organism and its surrounds, viz., the skin. As might be expected in interdisciplinary subjects, the units of measurement are sometimes English and other times metric. Since the reader who wishes to work in this field must be conversant with both systems, conversions to a single system have not been made, instead, the units of the original study have been maintained throughout. Indeed, such conversions often would result in awkward numbers and expressions because the original figures represent quantities rounded off for convenience of manipulation.

11. Natural Environments

A. TERRESTRIAL Until the past few decades man has been bound closely to the surface of the earth, rising into its enveloping atmosphere on the peaks of its mountains, a few thousand feet, and descending into its waters rarely more than a hundred feet. Thus, it was only natural that his “environment” was first characterized by local geological features, ground-level atmospheric conditions, and the effect of the great heat-giver, the sun. I t was, is, and no doubt will continue to be described principally by air temperature and humidity, wind velocity and insolation or its daytime corollary, cloudiness. From these data the average human learns to evaluate the environment sufficiently well to provide for conscious regulation of his heat loss by donning or doffing clothing or seeking shelter, as conditions may demand. However, man’s natural curiosity and his need to regulate artificial environments within shelters prodded him to analyze, quantitate, and evaluate the importance of each component of his environment and its effect on the thermal equilibrium of the human organism responding to it. Thus began the science of environmental physiology: the study of heat transfer between man and his surroundings. More recently, as man’s environment has begun to extend into space and down to previously unattainable depths, that phase of the study dealing with his customary earth-surface habitat has become known as bioclimatology while the more general term, environmental physiology, continues to apply to all environments whether terrestrial or extraterrestrial, natural or artificial. The principles governing heat transfer are common to all situations however, and variations in the environment may be accounted for by appro-

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priate variations in the terms of the heat-transfer expressions. While this concept is simple, it is frequently difficult, if not impossible, to apply because of complexity and variability of some environments. Nevertheless, useful approximations usually can be achieved even in the most complicated situations and the resolution of such expressions constitutes an important aspect of bioclimatological study. The bioclimate is defined as the average course or condition of weather with respect to life, so that one may speak of the bioclimate as being tolerable or intolerable, beneficial or deleterious, depending upon the effect of the climate on living organisms. Bioclimatology is specifically the study of the bioclimate and its effects, and bioclimatological heat exchange is the basic process by which the effects are produced. The measurement of this process requires quantitation of the influence of individual climatic components in producing heat loss or gain in living creatures. A complete description of the environment includes evaluations not only of air temperature, air velocity, and humidity, but also of total radiation. Of these measurements, that of radiation is the most difficult since it is a composite of radiations of different wavelengths and of different effectiveness with respect to the receptive object. In fact, until the development of appropriate instruments and methods within the past 15 years, direct measurements of climatic radiation components in terms of standard physical units had not been accomplished in the outdoor environment. Prior to this time, instruments which gave very useful indices of relative effects including radiation had been used (1-5) ; methods of combining effects of various climatic components had been worked out in carefully controlled laboratory chambers (6-9) ; and various radiation exchange balances were computed from measurements of insolation, temperatures of the earth, and pertinent atmospheric data (10, 11).Such indirect procedures were limited in usefulness, however, because they failed to provide thermal radiation data in standard physical units directly applicable to the living being exposed in the given surroundings. Results obtained with one instrument could not be compared directly with those obtained with another. Individual elements of the total heat exchange complex could not be extracted to determine their importance in the complex. These shortcomings were the principal reason for the initiation in the 1950's of a new approach in instrumentation and methods of measurement of bioclimatological heat exchange which has not been superseded since. The radiation exchange between man and his outdoor environment can be visualized in Fig. 1 which depicts the most important sources of radiation as : (1) the high-temperature radiation direct from the sun, Ro(99"/" at X less than 3 p ) ; (2) the scattered light from the sun, y R o ; (3) the reflected light from the sun, aRo; (4)the low-temperature radiation exchange between the

68

ALICEM. STOLL

R

=

Total radiotion effective on man

Ro

+

Direct. solar radiation

yRo Scattered solar radiation

t

aRo

H,

2

Reflected ’ Radiation solar to or from radiotian terrestrial objects

Hs Radiation to or from sky exclusive of sun

FIG.1. Radiation exchange between man and the outdoor terrestrial environment.

man and the sky, H,; objects of the terrestrial environment, H,. The radiant heat lost or gained by the man is the algebraic sum and may be expressed as

+

R = Ro + yR0 aRo ? H, f H, (1) where R is the total radiation effective on the man, Ro the direct solar radiation (9976 at h less than 3p), crRothe reflected solar radiation, y R o the scattered solar radiation, H , the low-temperature radiation to or from the sky, and H , the low-temperature radiation to or from terrestrial objects. To obtain the total radiation it is necessary to measure separately the intensity of these radiations at the place occupied by the man and then to combine their effects after correcting for reflecting power of the man’s

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skin and clothing. T o these quantities then may be added the more readily obtained air temperature, velocity, and humidity. The instruments developed were : the panradiometer for separation and measurement of solar and low-temperature radiation exchange and convection effects (12); the thermoradiometer for the measurement of surface and environmental temperatures (13);and the thermistor radiometer (14) which supplanted the thermoradiometer by reducing the area viewed in measuring such temperatures and extending the range of ambient operation down to the level of the lowest naturally occurring bioclimatic temperatures. The panradiometer and the thermistor radiometer provide all the data

--

Radiometer asremoly

Aluminum case / Aluminum cone

SIor

Side

BOC k

Front

Cone aperture positioning Thermistor

U

Mounting plate

FIG.2. Radiometer assembly.

required for an exhaustive analysis of the radiation and convection characteristics of the environment. With only these two instruments, it is possible to measure direct solar radiation, reflected and scattered solar radiation, the average radiant temperature of the total surroundings, the radiant temperature of the sky and of the ground separately, the air temperature, and the wind velocity. Detailed descriptions of the instruments and their method of use have been published (12,13,14).Briefly, the thermistor radiometer, the simpler of the two, utilizes four thermistors mounted as shown in Fig. 2 and arranged in a bridge circuit. Two thermistors are exposed to the radiation and two are hidden from the radiation by means of the collecting cone. The output of the bridge is amplified and fed to a meter which may be calibrated to suit the user. In Fig. 3, calibrations are shown for ranges up to f 135°C at

70

ALICEM.

STOLL

ambients from +20 to -50°C. The discrepancy between the slopes for the same d T above and below zero is due to the smaller energy exchange which occurs for a given temperature difference at lower temperatures in accordance with the Stefan-Boltzmann law. Thus, the energy exchange between radiators at -50°C and 0°C is twice as large as that for an equal d T between radiators at -50°C and -100°C. Over shorter ranges and at higher temperatures, this discrepancy is somewhat less and the slopes of the calibration curves for temperature above and below ambient are more alike. Since the introduction of this instrument commercially a number of similar devices have appeared. While the original instrument is used for the AT 601C

CB- 50% 'RG 5OT CB- 5°C CB-27'C RG 509:

I

,/RG

CB-509: 135'C

20

0

- 20, I

-40 -60

CB- 50°C RG 1359:

CB- 5O'C RG 509:

a o c

20 oc

C B - 59: C B -27% RG 5OT

FIG.3. Calibration curves for the thermistor radiometer: CB = temperature of reference blackbody; RG = temperature difference above ambient to produce full-scale deflection.

measurement of skin and surface temperatures as well as that of environmental temperatures, the later instruments are usually more limited in range. In any case, operation is simple : the meter is zeroed with the radiometer directed toward a blackbody standard which is maintained at ambient temperature ; the radiometer is then pointed at the surface to be measured and the meter deflection is read and added to the ambient temperature to obtain the radiant temperature of the observed surface. Alternatively an internal standard is maintained at some fixed temperature and the observed temperature is read directly on a scale calibrated on an assumption of an emissivity of 1.O. As the emissivity of the observed surface approaches unity, the radiant temperature so measured approaches the true temperature of the surface. For the human skin, having an emissivity of 0.99 (15)in the infrared, there is no better measure of true temperature than a radiometric one. In measurements of the radiant temperature of the ground and terrestrial

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objects, the emissivity in the infrared also is close to 1.0 so that, by suitable corrections for reflected and scattered solar radiation, the temperature of the surroundings may be determined from such simple measurements. These corrections are easily made by repeating each measurement with a glass filter in front of the receivers. The glass absorbs infrared and transmits visible radiation. Thus, the measurement taken through the filter represents the contribution of scattered and reflected solar radiation and may be subtracted from the reading without the filter to yield the radiant temperature of the region surveyed. The method for surveying the total surrounds (13) consists of temperature measurements at ten points, five in the upper hemisphere, N, S, E, W, and zenith, and five in the lower hemisphere, N, S, E, W, and nadir. Measurements are made with and without the glass filter, the instrument being held or shielded so that direct radiation from the sun into the radiometer is always avoided. From these measurements, the average temperature of the sky and of the ground may be computed as separate entities. Further computations yield the scattered solar radiation. and the reflected solar radiation as separate entities. At night, or during complete heavy overcasts, when direct solar radiation is zero, this instrument alone can provide all the radiation and temperature data required to determine the radiant heat exchange between man and his environment. When solar radiation is present in appreciable amounts, the panradiometer provides all necessary data to determine radiant heat exchange and, in addition, the effect of convective heat loss by virtue of its measurements of air temperature and velocity. The operation of this instrument is based upon the fact that most of the solar radiation reaching the earth lies at wavelengths in the visible and near infrared while the radiation from man to his surroundings lies at wavelengths in the far infrared. Thus, it is possible by selection of heat receivers of appropriate emissivities to separate the highfrom the low-temperature radiations. Figure 4 shows the basic heat receivers of the panradiometer. These are hollow chromium-plated silver spheres, 64 mm in diameter and of identical construction. Each is provided with a thermocouple for measuring its temperature and an internal heating coil for adjusting its temperature. One sphere is painted dull black, one white, and two are highly polished, as shown in Fig. 5 . The emissivity in the infrared and the visible wavelengths is determined for each sphere. Only three spheres are required for a radiation measurement ; the second polished sphere is used as a hot-sphere anemometer. The instrument is provided with the following : selector switches for comparing the temperature of any sphere with that of another, and adjustments for supplying heat to each of the spheres ; meters for measuring the heat supplied ; a null point galvanometer to indicate when all spheres are at the same temperature and a second galvanometer to indicate wind velocity changes during a measurement.

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ALICEM. STOLL

When these spheres are placed in a clear sunny outdoor environment the black sphere of emissivity close to 1.0 in both the visible and the infrared range comes to the highest temperature ; the polished sphere, being almost totally reflecting in the visible and only slightly less so in the infrared, becomes only slightly warmer than the ambient air ; and the white sphere, I

H

L

C B S’

FIG.4. Basic heat receivers of the panradiometer mounted on cross-arm support: CSS = chromium-plated silver sphere; C S T = chromium-plated steel tube; CBS = chromium-plated brass support; LA = Lucite adaptor; PH = polished housing; T = thermocouple; L = leads; T P = terminal posts; I = insulation; CA = cross-arm; H = heater.

which reflects much of the visible radiation and emits strongly in the lowtemperature range, becomes slightly cooler than air temperature due to heat losses by radiation to a cool sky. By construction, conductive and convective heat losses are equal for all the spheres when they are at the same temperature. The temperature of the warmest sphere is measured and heat is added to the cooler spheres to bring them to the same temperature. This heat is measured and serves to separate the radiation absorbed from the environ-

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ment from the radiation lost to the environment. For example, at. equilibrium, heat gained by the black sphere from the sun equals heat lost by the sphere to the environment through conduction along the support, convection losses to the cooler ambient air, and radiation losses to the cooler areas of the environment. Thus, as shown in the following equation, the total radiation from the sun, R, times the emissivity of the sphere in the visible, is equal to

FIG.5. Panradiometer spheres. Emissivities in the infrared and in the visible ranges, respectively: white 0.89, 0.16; black 0.96, 0.96; polished 0.04,0.28.

the Stefan-Boltzmann low temperature radiation exchange plus convection and conduction heat losses. For the black sphere :

RE^" = So ~ b j e ~Tb4 i ( - Ts4)+ C -ID (4 where R is the total radiation from the sun incident upon the sphere (direct and reflected and scattered solar radiation), Ebv the emissivity of the black

ALICEM. STOLL

74

sphere in the visible and near infrared (0.4-3p), Sothe Stefan-Boltzmann radiation constant, Ebi the emissivity of the black sphere in the infrared region beyond 3p, eSi the emissivity of the surroundings in the infrared region beyond 3p, Tb the temperature of black sphere, T,the radiant temperature of surroundings, C the convection heat losses from sphere to ambient air, and D the conduction heat losses along steel tubing support. Similar equations may be written for the white and the polished spheres by adding the term for heat input ( I ) to the total radiation. For the white sphere : Iw Rewv= So eWieSi(Tw4- Ts4) C D (3)

+

+ +

For the polished sphere :

1,

+ Repv= So

~ , i (Tp4 -

+ +D

TS4) C

(4)

It is a simple matter then to solve the equations simultaneously to find the total radiation, R, and the average temperature of the surroundings, T, . Thus,

As the relationships between the emissivities of the spheres are constants, Eqs. (5) and (6) can be written more simply as

R = (Iw- Ip K ) / ( a- bK)

(7)

T, = 4Tb4-t[(Iw - R,)/Sob] - Ewj)/(Ebi- Epi) = K , (Ebv - EWv) = U , and

(8) where (Epi (Ebv - E,j) = 6. Thus, R in the sunlight may be calculated from the heat balance between any two spheres. Also, by shading the spheres with polished disks and measuring R, the total radiation in the shade, the direct solar radiation may be determined. If the white and the polished spheres are balanced against the black sphere in the sunlight, the total radiation is obtained as follows:

+ R0(y+cr)-(€wi--pi)(Tb4-

(9) where all terms are as before and Ro is divided by 4 to account for the projected area of the spherical surface exposed to direct sunlight and to yield all radiation values in terms of kcal/m2 hr for practical purposes, In the shade, and correcting R’ by 10% to account for the solid angle of the sky cut off by the shading disks :

R=(R0/4)

TJ4)So

1.1R’= Ro(y + a) - (ewi - epi)(Ti4- Ts4)So

(10)

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75

when Eqs. (9) and (lo) are subtracted and solved for R,:

&/4= R - 1.1R'

+ (ewi - e,,J

(Tb4- TL4)So

(11) All the quantities on the right of the equation are known. Also if Tbis equal to Tb' within f 1°C as is usually the case, then

Ro=4(R- 1.lR') (12) On clear, cloudless days it is possible to determine the total reflectivity of the terrestrial environment. Under these conditions R,, the scattered radiation is small in comparison to the reflected radiation and may be neglected. Then LY

= R/Ro

and when LY has been determined for a given environment then y may be determined for any subsequent condition of the sky as

Y = (R/Ro)- a Wind velocity is measured by supplying a fixed quantity of heat to one of the spheres and comparing its temperature with that of the unheated polished sphere (16). The difference in temperature between the spheres is related to wind velocity by the factor K i n the relationship derived from Nusselt's and Reynold's equations : V = (H/KAT)* (13) where V is the wind velocity, H the heat supplied, AT the difference in temperature, and K a constant. K is obtained on appropriate calibration of the instrument in a wind tunnel (1). Before applying the panradiometer to field work, thorough checks were carried out by comparisons with direct measurements of the various quantities involved, i.e., total radiation in sun and in shade, the radiant temperature of the surroundings, and solar radiation (12). One comparison was that of the wall temperature as ascertained by the panradiometer in a temperature-controlled walk-in chamber with the temperature of the walls as indicated by thermocouples. It was found that in still air, the measurements are accurate to +O.l"C which was the limiting accuracy of the calibration of the thermocouples. With turbulence produced by an electric fan directed at the spheres, the scatter was increased to kO.3"C. In a comparison of measurements of the outdoor surroundings in sun and shade as made with the panradiometer and by direct survey with the thermoradiometer or the thermistor radiometer described earlier, the average difference was about f 1.8"C. Finally, the accuracy of the panradiometer measurements was checked by comparison with direct readings of solar radiation. For this purpose, an automatically tracking radiometer was set up and directed at the sun (17). This instrument measured continuously the intensity of the direct solar radiation as it is received at the earth's surface. It consisted of a thermopile

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ALICEM. STOLL

mounted on a movable base provided with electronic control by which it was automatically oriented so that the thermopile was normal to the path of the radiation at all times. The thermopile was composed of eight copperconstantan thermocouples connected in series. The warm junctions were soldered to blackened silver foil which received the radiation while the cold junctions were protected from the radiation by the radiometer housing. A polished aluminum collecting cone was affixed in front of the thermopile. In front of the cone a tube 9 inches long containing blackened baffles was set up to eliminate all but the direct radiation (Fig. 6). The radiometer was mounted on a circular brass plate which could be turned by means of gears

FIG.6 . Radiometer with shields and photocells of the positioning system: B = baffles; S = shields; SL = slits; VP = vertical positioner photocell (one of pair labeled); HP = horizontal positioner photocell (one of pair labeled); T = terminal posts; H = holder for radiometer and baffle tube; R = radiometer.

to pivot the radiometer in the horizontal plane. Another set of gears provided for movement in the vertical plane. Two pairs of photoelectric tubes, one in the horizontal plane and one in the vertical plane, were mounted behind narrow slits close to the sides at the base of the radiometer. The circuitry was arranged so that when the solar radiation falling on both pairs of tubes was the same, no emf was generated; when the radiation was unequal, the emf generated was amplified and fed into the gear control circuit to move the radiometer in the plane of the phototubes until the radiation was again equal (Fig. 7). This system kept the radiometer properly aligned to measure direct solar radiation continuously as the sun moved across the sky. The emf from the radiometer was recorded throughout the entire experimental period.

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77

The radiometer was calibrated against a U.S. Bureau of Standards lamp and against a muffle furnace. A factor of 0.327 k 0.001kcal/m2/hr/pv was obtained. Typical of the data recorded with this instrument are those shown in Figs. 8, 9, and 10. Section A of Fig. 8 represents a clear day; section B, radiation through an overcast of variable depth; section C, the effect of rapidly moving clouds in an otherwise clear sky. It is seen from section B that the overcast reduced the radiation intensity to about 60% of that received on the clear day (section A). The effect of discrete clouds (section C) c1

FIG.7. Semiblock diagram of radiometer and controls: VP = vertical positioner photocell (one of pair labeled); HP = horizontal positioner photocell (one of pair labeled); R = radiometer; S = support; HG = horizontal gear; VG = vertical gear; M = motor.

was to vary the incident radiation between these two levels. The extent to which the clouds reduce the radiation depends, of course, upon the depth of the cloud formation. The clouds present at the time of this recording were of medium depth. In Fig. 9, a record obtained on a typical “clear” day in this locale (New York City), the effect of a few rapidly moving heavy clouds may be seen. Here the direct solar radiation is momentarily reduced almost to zero. This is a common observation in this area where a few clouds are usually present for short periods even on the clearest days of the year. Figure 10 represents a typical overcast sky. The occasional dips in the curve may be associated with relatively thick sections in the overcast.

78

ALICEM. STOLL

c

d

."u .-

E

a

a,

E

L.

E c

'D

c

.*.

a2

,

i

I

i

,

L

d

.._ -.

HEATTRANSFER IN BIOTECHNOLOGY

r

79

80

ALICEM. STOLL

HEATTRANSFER IN BIOTECHNOLOGY

81

Comparisons of such readings with those obtained with the panradiometer at the same time are shown in Fig. 11. It is seen that the maximum deviation of the mean is +lo% and the average deviation is +4%. Such agreement is considered to be good for this environment where clear days are the exception and the solar radiation at the earth’s surface fluctuates fairly widely and rapidly. Nevertheless, violent wind and cloud changes render it difficult, and sometimes impossible, to obtain a reading because of the difficulties of rapid manual adjustment of heat balance inputs. It has been suggested to the engineers (18)that a task worthy of their undertaking would be Direct vs indirect measurements of solar radiation

Direct by solar radiometer

FIG.11. Comparison of measurements of direct solar radiation by the panradiometer and the solar radiometer.

the development of a servo-operated panradiometer, for with automatic operation remarkable results could be achieved quickly. The essential requirements in converting to automatic operation are :

(1) automatic selection of the warmest sphere and continuous recording

of its temperature ;

(2) automatic control of the input of heat to the two spheres of lower temperature to bring them to the temperature of the sphere of highest temperature ;

82

ALICEM. STOLL

(3) recording of the current through the heaters at the moment of thermal balance ; (4)a duplicate system for simultaneous measurements with the spheres shaded from direct sunlight ; and (5) continuous recordings of reference temperature, air temperature, and of wind velocity. Another desirable feature would be the addition of a humidity-measuring device to record continuously and thus provide a complete and detailed description of the environment by use of this single instrument. Of these requirements, the heat input is the most exacting. This implies control of the input of the heat to the two spheres of lower temperature by means of the temperature difference between these and the sphere of highest temperature. This balance must be accomplished fairly rapidly and without affecting the temperature of the sphere of highest temperature since the latter must depend entirely upon the radiation exchange with the environment. The rest of the requirements are relatively simple, involving for the most part only continuous recording of emf, a constant heat input to the hot sphere, and the addition of a humidity-sensing device, of which there are several types available. It is quite possible that thermistors would be more suitable than thermocouples for temperature detection and incorporation into servo-mechanisms by virtue of their electrical characteristics and adaptability to control circuits. Perhaps the automatic panradiometer would embody little of the present instrument except the principles on which its operation is based. I n any event, it has been amply shown by use of the present instrument that the system is scientifically sound and thoroughly reliable under operable conditions. Despite the difficulties involved in normal operation, it has been used successfully in the arctic, in temperate zones, and in the desert and has provided a wealth of information previously attainable in physical units only by means of the simultaneous operation of at least five separate instruments. With the instruments described, systematic measurements were made in New York City in all seasons, in Nome and Fairbanks, Alaska in the summer, and Fairbanks in the winter (29).A group led by Norman Sissenwine of the Office of the Quartermaster General collected data in Death Valley during the hottest part of one summer (20). Isolated readings have been made in other localities from time to time, but such observations are too sparse to permit systematic analysis for these regions. Climatic physical data are related to biology by casting them in a form whereby the physiological knowledge obtained in controlled laboratory environmental chambers becomes applicable to the naturally occurring

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climatic situation. One physiologically important quantity so used, is the radiant heat load (21, 22) :

solar contribution

low-temperature exchange

+

where Hm is the radiant heat load, Ro the direct solar radiation, ( a y ) Ro the scattered + reflected solar radiation, E,, the emissivity of man in the visible ( O S ) , the emissivity of man in the infrared (l.O), Sothe StefanBoltzmann radiation constant, T, the surface temperature of the man, and T, the radiant temperature of the surroundings. This quantity depends upon the total solar radiation absorbed by the man, minus the heat he is able to lose by radiation to his surroundings. It is computed from the measured values of total solar radiation which includes reflected and scattered, as well as the direct radiation, and the radiant temperature of the surroundings averaging that of the sky, ground, terrestrial objects, and any other sinks or sources of low-temperature radiation in the vicinity. The surface temperature of the man and the emissivity of his surface may be measured but can be reliably assumed for various conditions. A second parameter of importance is the operative temperature, a concept developed at the Pierce Laboratory of Hygiene (23):

where Tois the operative temperature, Kr the radiation “constant,” T, the radiant wall temperature, Kc the convection “constant.” and T, the air temperature. This system combines the effect of radiation and convection weighted properly to represent their respective physical influences on the body in accordance with Newton’s cooling law. It constitutes a temperature scale which serves to indicate, with respect to the nude or lightly clothed body, the physiological thermoregulatory processes brought into action by the net effect of radiation, air temperature, and air movement. Thus, at operative temperatures below 29”C, the body is cooling. Between 29 and 3 1“C, vasomotor regulation maintains heat balance while, above 31”C, thermal balance is maintained by evaporative regulation. This scale provides an important bridge between the physical features of the outdoor climate and their physiological consequence. Table I presents data to illustrate this correlation of physical features and physiological response: on the left appear the data on season and region; in the center, the physical description, the physiologically significant values, and the response of the body; and, on the right, the action required to maintain thermal balance. It may be pointed out, for instance, that in the New

00

P

TABLE I BIOCLIMATOLOGICAL DATAFROM THREE LOCALITIES Characteristics

Physical

Season Summer Summer Summer Winter Winter day Winter night

Locale New York City Fairbanks Death Valley New York City Fairbanks Fairbanks

RO (a+y)Ro (kcal/m2 hr) (kcal/m2 hr)

T, ("C)

T, ("C)

Physiological

Wind vel. Hm (mph) (kcallmz hr)

To ("C)

Action to Thermal maintain thermal effect balance

92

26.4

31.3

7.5

+ 80

31.0

Neutral

None

794

103

20.3

9.2

11.0

-1

23.7

Cooling

588 506

147 21

46.1 3.8

47.4 -4.0

3.1 5.0

+192 - 73

53.0 6.8

Heating Cooling

236 0

53 0

-31.5 -30.0

1.2 3.4

-154 -259

Slight increase metab. or insul. Shelter Increase metab. and insul. Shelter Shelter

582

-37.2 -42.0

-26.0 -33.2

Cooling Cooling

2

F

c4n 0

r

l-

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York City summer, with a moderately high solar radiation intensity and moderately high temperature of air and surroundings, the radiant heat load is also moderate; with an ordinary wind speed, the operative temperature occurs at the upper limit of the zone of vasomotor regulation and no additional action is required to maintain thermal balance. However, in Fairbanks, with a higher solar radiation, but cool air and surroundings, the radiant heat load is almost zero, and with an unusually high wind speed for this location, the operative temperature falls in the near portion of the cooling zone requiring slightly increased metabolic output or additional clothing to maintain thermal equilibrium. In the desert summer, with a solar intensity about equal to that of New York City which lies at about the same latitude, the air and surroundings are hot, the heat load very high, and the operative temperature so high that shelter is indicated. The winter analyses are equally revealing. All radiant heat loads are negative and the operative temperatures fall far out in the cooling range, particularly in the Arctic night. Figure 12 presents data selected from actual observations to point up specific effects. Thus, times were chosen when the solar radiation intensity was the same in all three regions so that the solar radiant contribution was equal throughout. Similarly, the wind velocity was the same and convective heat loss differences depended solely on temperature differences, not on velocity differences. Quite striking effects are seen. The effective sky temperature in the Arctic is below zero, while it is well above zero and approximately equal in New York City and Death Valley. T h e air temperature is greatly different in the three environments while the ground temperature reflects the influence of the solar radiation by being above air temperature, and the influence of the relatively low sky temperatures by being not greatly above air temperature, except in the hot desert. The effect of the temperature of the sky and ground in providing sinks and sources of low-temperature radiation is reflected in the radiant heat load. Remembering that the solar contribution is the same in all three environments, it is seen that the radiant heat load progresses from slightly negative in the Arctic desert to positive in the city and greatly positive in Death Valley. Thus, the cool surroundings of the Arctic region provide a radiation sink while the hot surroundings of the temperate zone desert act as an additional source of radiation. With respect to the man, the net result of all the factors shown may be expressed in terms of the operative temperature. It is seen that the man in Fairbanks has a light radiant heat load to contend with and is exposed to an operative temperature of 2 2 T , somewhat below the neutral zone of thermal balance. Cooling proceeds by radiation and convection and is easily controlled by light clothing. This is a very pleasant environment. The man in New York City has a fairly appreciable radiant heat load to carry but still finds himself, by virtue of some losses to the surroundings, in the region of the upper limit

00

cn Solar radiation

Wind velocity 5mph

Tsky

-9OC

Toir

+19oc

+21oc Fairbanks,Alaska

\

im;

+53oc Valley,Calif. Death

--?zy5

Radiant heat load Operative temp. Scale-Operative temp. Thermal z o n e - - - - - - - - Thermoregulo tory process

+ 30 OC New York City, N.Y.

Shelter

---_I

~pl~~~l-t71

kg cal/m2hr

I

--Cooling+22oc

I

I

I

I

Increased metabolism

kg cal/m2hr I

I

I

t1 I , , , , , ,

+29% +31% motion

I

,

. . , , , ,

--+ 51 OC-

Heoting

Evaporation

b

SheIte r

FIG.12. Selected data from three localities to show interaction of bioclimatological components.

HEATTRANSFER IN BIOTECHNOLOGY

87

of the neutral thermal zone where vasomotor changes alone suffice to maintain thermal balance. The man in Death Valley has the greatest burden for his solar load is increased by radiation from the surroundings whose temperature is extremely high and indicates that shelter is required to protect him against the body heating which evaporation alone is insufficient to prevent. The possibilities suggested here are quite obvious, for once the extremes and the mean values for the physical parameters have been established by systematic measurement, it is a relatively simple matter to draw up charts to indicate the amount of insulation or heat, or clothing, or whatever may be required to maintain life and comfort in any given environment. T h e importance of such an approach is readily apparent in military logistics where it might well prevent tragedies brought about by the right clothing in the wrong place. There are additional implications of importance to the fabric construction and design field, an area of recently renewed interest from a biotechnological viewpoint. In a current work for instance, there appears a fine analysis of the impact of climatic factors on the efficacy of clothing. Each component of the climate, air temperature, wind, radiation, and humidity, is evaluated with respect to its influence on the heat transfer between the clothed man and his environment. Special consideration is given to vapor pressure gradients, absorbency of clothing, and the effect of sunlight on damp, penetrating cold. The introduction of equations for heat load incorporating the insulation and moisture permeability properties of clothing provides a bridge for relating the heat exchange data developed for the nude body to that pertaining to the clothed body. Woodcock (24) provides for this transition in the equation

I A H = A T , + Si, AP, (16) where I is the insulation ("C/m*/hr/kcal), AH the heat loss (kcal), AT, the change in skin temp ("C), S the physical constant (2"C/mm Hg), AP, the change in vapor pressure of skin surface (mm Hg), i, the moisture permeability index (dimensionless) equivalent to k / K , and where k is the convection coefficient and K is the dry heat transfer coefficient. This equation is derived from more basic expressions representing the heat transfer due to convection and evaporation combined with a similar equation applicable to the total heat loss :

1

k (T,- T,) + S - ( P , - Pa) K

where T, is the air temperature ("C), Pa the vapor pressure of the air (mm Hg), K the coefficient of total dry heat transfer, and k the convection coefficient.

ALICEM. STOLL Also, I , the insulation value, is defined (25)as the reciprocal of K , and k/K and i,,, are defined as above. The basic equation for heat transfer from man’s skin through his clothing to the environment then becomes

T h e value of H is a minimum when Ps= Puand the man is not sweating ; it is a maximum when the skin is wet with sweat and P, is the saturated vapor pressure of sweat at skin temperature T, . There are further refinements to be applied depending upon the air velocity and whether or not the clothing is windproof. The influence of air velocity may be appreciated on application of the concept of operative temperature mentioned earlier (23).An illustration of this influence is seen in Fig. 13. Here is shown the change in operative temperature with air velocity calculated by Eq. (11) for two levels of air temperature T, and Tu‘. T, and T,‘ are the respective radiant wall temperatures in the sunshine and T, and T,‘ are the radiant wall temperatures in the shade; To and To’ of course are the operative temperatures. Air velocity is indicated on the abscissa in miles per hour, and the temperatures in degrees centigrade and degrees Fahrenheit on the ordinate. It is seen that when the wind velocity is less than 4.5 mph the operative temperature approaches the effective radiant temperature, T, , in the sun, or T, in the shade. As wind velocity exceeds 4.5 mph, the operative temperature rapidly becomes asymptotic with air temperature. The importance of this observation is emphasized in the lower curve where a comfortable level of temperature at 1-2 mph wind velocity is rapidly reduced to a very uncomfortable level at 9 mph in surroundings of low temperatures. In the shade the effect is reversed. The effects are thoroughly familiar as indicated by the common observations that cold days are not uncomfortable “until the wind blows” and, conversely, warm days are less uncomfortable when there is a “cooling breeze.” It is significant that the various parameters dealt with in the foregoing considerations are all in physical units so that actual heat loss or gain may be computed for virtually every situation in the normal outdoor environment and any clothing assembly of known characteristics. The measurements are then independent of any psychophysical or psychophysiological scales. While the latter have many important applications, as the military particularly are aware, they are greatly influenced by individual variables such as acclimatization, morale, motivation, and the like which are not amenable to generalizations. Thus do the biotechnological aspects of heat transfer in the terrestrial outdoor environment encompass considerations in a variety of disciplines. In the physiological field these include metabolic heat production, skin

HEATTRANSFER IN BIOTECHNOLOGY

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temperature and emissivity, vapor pressure of sweat, individual size, surface area, and, in some instances, pigmentation and hairiness. In the climatological area complete data must include air temperature and velocity, humidity, or, more specifically, vapor pressure, solar radiation rate, and radiant temperatures of the sky and of the ground and important terrestrial objects in the surrounds. In the textile field, for the clothed individual, such data are *F

201

63

W

a

49 47

a

a

w

n

I W

t W

-

> I-

*ac

25.

W

L 0

13

-I5

I

1

0

4.5

I

9 A I R

I

I

13.5 18 V E L O C I T Y

I

2 2.5

1

2 7 MPH

FIG.13. Operative temperatures in sun and shade.

the emissivity, insulation value, and air and vapor permeability of the clothing. Finally, and importantly, in the field of engineering the principles and mathematics concerned in heat transfer by radiation, convection, conduction, and evaporation are indispensable to analyses of thermal exchange between the man and his surroundings.

90

ALICEM. STOLL

B. TERRAQUEOUS About 75% of the planet earth is underwater. In the depths of its seas vast resources of foodstuffs and minerals await exploitation. Serious efforts to capitalize on these potentials have begun within just the past few years : their success depends upon man’s development of a capability for working for extended periods of time totally immersed in water, alternated with substantially normal routine existence for longer periods at depth in submerged shelters. Current research in this area concerns the determination of the metabolic cost of diving (26), long-term underwater swimming (27), the heat loss involved (26-29), and the physiological consequences of the underwater regimen (26,27). The analysis of heat-transfer processes occurring in these activities is fundamental to the development of protective gear and adequate shelters. It has been customary to express the total heat exchange between man and his environment as a balance of rates :

where M is the metabolic heat production, Wthe work output, E the evaporative heat loss, R the radiation heat gain (+) or loss (-), C the convection (or conduction) heat gain (+) or loss (-), and S the heat storage (+) or deficit (-). The physiologist uses units of kcal/m2 hr and the engineer, Btu/hr. The terms M , W, and E also deserve a word of clarification. Metabolic heat (IM) is generated within the body at a rate which is minimal during sleep and maximal during strenuous exercise. The maximum may be as much as 50 times the minimal rate for short bursts of a few seconds at a time, or about 20 times the minimal rate for longer periods. Since the mechanical efficiency of the body is low, most of the energy generated during exercise, in addition to the basal metabolic energy, must be dissipated as heat if the body temperature is to remain constant. In Eq. (19), (W) is the amount of this energy actually used in performing the work. Evaporative heat loss ( E )occurs from the lungs via the expired air and from the skin by vaporization of moisture as “insensible” heat loss at rest under comfortable conditions. During exercise or under hot environmental conditions active sweating occurs and “sensible” heat loss proceeds by vaporization of sweat. Should conditions be such that vaporization does not occur, as in sweat which runs off in streams or droplets, then only the heat carried off as a function of the thermal capacity of water is lost, a much less effective mechanism than evaporative loss. When Eq. (19) is satisfied, the metabolic heat production is exactly balanced by environmental and work heat loss; no heat is stored and the

HEATTRANSFER IN BIOTECHNOLOGY

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temperature of the body remains constant. I n the normal terrestrial surface conditions conduction heat losses are usually negligible by virtue of the low thermal capacity of air and the small percentage of the body surface ordinarily in contact with solid objects in the environment. The aqueous environment alters this situation by introducing a conduction, or, more accurately perhaps, a fluid convection term which is of great importance in the totally submerged condition. Since no seas are warmer than the normal temperature of the skin, this term always represents heat lost from the body and therefore carries a negative sign. The respiratory gas supply carried by a swimmer is at approximately the temperature of the water, consequently, the evaporative heat loss from the lungs is increased by the amount required to warm the inspired gas to the temperature of the body and to raise the water vapor content to saturation. However, given the ventilation rate and the required temperatures, this heat loss may be calculated as the sum of these two (29). On the other hand, heat loss by radiation becomes negligible. Equation (19) then becomes

hi- W - E - C = + S = O

(20)

for the balance to be maintained. The term C has been treated both as conduction cooling (29) and as convective cooling (28). Beckman (29) estimates from measurements on subjects immersed to the neck that heat loss from the lungs accounts for about 15% of the total heat loss and the bulk of the loss may be attributed to conduction according to the equation

where Hd is the heat loss via conduction (kcal/hr), Tcthe core temperature of the body ("C), T, the water temperature ("C), A the immersed surface area of body (m2),I , the thermal insulation of skin and subcutaneous tissue (Clo), I , the thermal insulation of the clothing (Clo), and 5.555 the conversion constant for converting the total insulative value into Clo. The Clo unit (25) is one of convenience which may be defined as about equal to the insulation provided by ordinary business clothing, or precisely 0.18"C/m2hr kcal which permits a transfer of 5.555 kcal/m2hrat a temperature difference of 1"C. While it is customary today to use Clo units for the insulation value of clothing, it is less customary to use them for skin and tissue. Therefore, it is usually necessary to convert from thermal conductivity in kcal/m2hr "C or its reciprocal, insulation in "C/kcal m2 hr, to Clo, 5.56 times the conductivity or 0.18 of the insulation value. Alternatively,

92

ALICEM. STOLL

the body insulation may be retained in cgs units and the Clounits converted to cgs units in which case Eq. (21) becomes simply

in which all units are in the cgs system. Nevins et al. (28)treat the problem of heat loss from the totally immersed body as one of convection, and the loss due to convection, C, as where

C = UA,(T, - Tw)

and A , is the convective area, ho the water-side film coefficient, x the thickness of the swimmer's suit, k the thermal conductivity of the suit, Cithe conductance of the interface between suit and skin, T, the skin temperature, and Twthe water temperature. It is seen that there is no difference between Eqs. (22) and (23) if U equals the reciprocal of the sum of the insulation of the body and its clothing. Inspection reveals that I, of the conduction equation is equivalent to (ho+ Ci) of the conduction equation and I , is equivalent to x / k . Further examination of the convection equation leads to the conclusion that the water-side film resistance, l / h o , as determined for a cylinder is negligible in comparison with the resistance of the suit and its contact resistance, and may be omitted. Therefore, I, = Ci and Eq. (22) is identical with Eq. (23) for the conditions which prevail in swimming at depths. The insulation of the body is the reciprocal of its conductance and is directly proportional to the thickness of the subcutaneous fat (26), Fig. 14, and inversely proportional to the specific gravity of the body (30),Fig. 15. It may be calculated by means of the equation Tb - Ts I, = ___ (24) C where I, is the insulation ("C/kcal/m2 hr), Tb the rectal temperature ("C), T, the skin temperature ("C), and C the heat loss through the skin (kcal/m2 hr) . Carlson et al. (30)in their use of Eq. (24) employed for C a value of 76% of the total heat production, attributing the remaining 24% to evaporative heat loss. The latter is too high by perhaps Sodl (29)since evaporation from the skin is eliminated in the totally immersed state. However, the variation introduced in Is by a loo/:, difference in C is well within the experimental variations due to individual differences.

HEATTRANSFER IN BIOTECHNOLOGY

93

I , is customarily given in Clo units. It is measured usually by the guarded ring thermal conductivity procedure which utilizes a flat slab of material. While the torso of the human body is sufficient in diameter to permit the application of I , values so derived without great error, the rest of the body C/hr/rnz/kcal 0.25

t

FIG.14. Direct proportionality of conductance and subcutaneous fat (26). 1.080

1.0701.060%

t

y

=

B

9

0

- 11

T

*AT\

-+

1.050-

'\-

1.040 -

0.

1.000

I

k

\

-27

r',

1.010 I

-19

c f

\\

1.020 -

,o

-23

\

1.030 -

-15

\'

S

-32

- 36

I

more nearly resembles a collection of cylinders (arms and legs, fingers and toes) and a sphere (the head). The variation in insulation due to geometric shape differences is well illustrated by van Dilla et al. (34, Fig. 16. Therefore, the insulation value of full-coverage clothing such as that needed in underwater work is best assessed by use of a manikin suitably instrumented for the purpose similar to the one described by Belding (32).Beckman et al.

94

ALICEM. STOLL

(33)present data obtained in this way on the &inch foamed neoprene underwater swimmers wet suit widely used by SCUBA divers. Flat surface measurements had indicated an insulation value of 1Clolkinch thickness, but the manikin measurements yielded a value of 0.86 in air, 0.62 in still water, and 0.60 in stirred water, considerably lower than the value obtained by the standard method. Another variable introduced by the water environment is the increase of pressure with depth which results in compression of the foamed neoprene and reduction in its insulation value (29).

Thickness of fabric in inches

FIG.16. Variation in insulation due to geometric shape differences [redrawn from van Dilla et al. (31)].

Returning now to Eq. (ZO), the total heat balance in the aqueous environment, it is seen that the evaporation loss and the conduction-convection heat loss through the skin must be balanced by an increase in metabolism if heat storage is not to become negative. Beckman’s review of available data (29)indicates that even marathon and channel swimmers (34) can maintain a metabolic heat output ( M - W )of approximately 275-350 kcal/m2 hr for not more than 10 or 12 hours, while trained frogmen are expected to be able to maintain a rate of 200 kcal/m2 hr. In one detailed experiment (27) the caloric consumption (111+ W ) of a trained underwater long-distance swimmer, traveling at 1.0-1.2 mph in water at 27-29°C and at a depth of 15 ft, was calculated from the air used. It was determined to be 132-140

HEATTRANSFER IN BIOTECHNOLOGY

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kcal/m2 hr (264-280 kcal/hr) maintained remarkably constant over a period of 4.5 hr following an initial rate of 205 kcal/m2 hr during the first 40 min of the swim. While the conditions are not exactly comparable, these data agree well with those observed on the ama, the diving women of Korea (26). After 40 min, working in water at 22-26"C at a depth of 15 ft, the caloric consumption, averaged for three divers, was 177 kcal/m2 hr, just about the same as that of the long-distance swimmer for a like period at the start of his swim. The lower energy level maintained after the initial period probably reflects the swimming rate found by experience to be the most efficient for this individual. Although it may be pure coincidence, it is interesting to note that in respirometer studies of the swimming energetics of salmon (35) the most efficient swimming speed for a typical adult sockeye salmon is 1.1 mph. For the fish at least, this is the velocity at which the ratio of speed attained to energy spent is optimal. I n any event, the metabolic rates of the underwater swimmer and the ama are all much below those cited earlier which were observed in marathon and channel swimmers. The latter may be considered to represent maximum values attainable by very few individuals whereas those of the ama are probably typical of the capability of a considerably larger segment of the population. For instance, along the southern coast and islands of South Korea alone 30,000 women are employed in diving daily, summer and winter (36). Many thousands more are similarly employed in Japan and other coastal areas. Presumably with adequate training a great portion of the world population could perform as well. However, this work is conducted only intermittently and interspersed with rewarming periods, for the energy produced is not sufficient to prevent a gradual fall in body temperature. Nevins et al. (28) assumed an output of approximately 600 Btu/hr (approx. 150 kcal/hr) for ( M - W ) and calculated, as a function of suit resistance, the minimum water temperature for thermal balance using the equation

where C, is the skin conductance (Btu/hr ft2 O F = 1.82), Tb the body temperature, approximately 98.6"F, A , the 19.5 ft2, T, the water temperature (OF), k the thermal conductivity of suit (Btu/hr ft2 O F = 0.087), and x the thickness of suit (&inch nylon-rubber). These data are shown in Table 11. (Equivalent values of suit resistance in Clo and metric units and water temperature in "C have been added to the original data.) It may be noted that the data commensurate with the SCUBA suit insulation as measured in air or derived from flat surface measurements (33),i.e., 0.86 Clo manikin in air, 1 Clo flat surface, fall between lines 3 and

96

ALICEM. STOLL TABLE I1

SUITRESISTANCE (Nevins et al. (28)) Suit resistance O F

ft* hr/Btu

0 0.1 0.2 0.3 0.4

Water temperature

"C m2 hr/kcal

Clo

"F

"C

0 0.067 0.134 0.216 0.269

0 0.4 0.7 1.2 1.5

81.6 78.6 75.4 72.4 69.3

27.55 25.89 24.11 22.44 20.73

4 of Table 11. From these calculations then, the minimum water temperature for thermal balance would be about 23°C or 73"F, an impractically high figure for dives of any appreciable depth. Temperatures of 10°C (50°F) are more realistic. Extrapolation of the data in Table I1 to this level reveals that a suit resistance of 1.04"F ft2 hr/Btu or 3.9 Clo would be required for thermal balance. Translated into suit thickness, this value represents approximately 1 in, again an impractical figure, for a suit of this thickness would be so bulky that motion would be seriously restricted and the assumed heat production would no longer apply. Furthermore, heat loss from the hands and feet is the limiting factor under such conditions, as shown by Beckman et al. (33).These investigators found that, although such a garment weighed 40 lb and severely restricted the motion of the subject, it did protect the deep body temperature adequately, but failed to prevent the hands and feet from cooling to pain temperatures. They concluded that in an 8-hr period heat loss from the extremities would be such as to limit safe exposure to water at temperatures below 12"C, the temperature at which tissue damage occurs (31).As did one of the group (Goldman) in independent studies in the arctic (37), these investigators came to the conviction that any extended exposure necessitates a distributed heat supply for maintenance of thermal balance. Consequently, present efforts are directed toward provision of such a system. Various measures are under consideration : e.g., battery-powered resistance-wire heated suits; electrochemical cells, sea-water battery powered, thermostatically controlled systems in suits, gloves, and boots ; catalytic fuel cells and radioisotopic power generators. The potential advantages, disadvantages, and cost of each type are discussed by Beckman et al. (33)with respect to application in protecting aircrewmen downed in open water and awaiting rescue. However, many of the features discussed apply equally well in the submerged condition or, indeed, any cold environ-

HEATTRANSFER IN BIOTECHNOLOGY

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ment. At the present time all such systems except the resistance-wire heated suit are in a research status and the state of the art is rudimentary.

C. EXTRATERRESTRIAL In space, heat exchange between man and his environment is profoundly affected by his orientation with respect to the sun. This circumstance has the effect of broadening tremendously the amplitude of fluctuation of the radiative heat load, positively in full sun and negatively in complete darkness. In full sun that portion of the body upon which the solar radiation falls receives 2 cal/cm2 min (1200 kcal/m2 hr or 442 Btu/ft2 hr), the value of the solar constant. In darkness in deep space a body loses heat at a rate commensurate with an infinite sink at -273°C or absolute zero. At the present time man’s ventures into space have not carried him beyond orbits about the earth. In these the above extremes are modified by the proximity of the earth which provides some radiant long-wave energy at all times and some reflected solar energy during part of each orbit. Furthermore, the practical necessity of wearing a protective space suit restores the evaporative, convective, and conductive heat loss terms to the metabolic balance equation, for without the suit these modes would be nonexistent. As it is, these modes are confined to the transfer between man and the cocoon in which he floats which in turn controls the loss of heat from its occupant. Great masses of data concerning methods of maintaining such heat balance are available in current literature. Two systems have proven successful for extravehicular activity (EVA) for short periods of time at least : the Russian cosmonauts for 10 min and the American astronauts for 20 min. Apparently, the EVA time was limited not by heat transfer problems but by limitation of the oxygen supply. A secondary limitation, in the type of suit used thus far, is the cooling system which must remove all the heat produced within the insulated suit by both the man and his equipment. It utilizes a coolant which eventually is ejected into space and is so expended. The rate at which it must be expended may curtail the mission for the amount of coolant carried aloft is necessarily limited. For this reason, much effort in the field of bioastronautics is directed toward devising suits of flexible materials of such physical, thermal, and optical characteristics as to provide passive control of heat balance under all environmental conditions anticipated in earth orbits. As early as 1960 it was determined that passive suit-temperature control could be achieved by proper choice of materials and surface spectral properties (38).For this purpose, the primary requirement was a low ratio of absorption of incident radiation (u)to emissivity ( E ) , such that u/e approximates 0.1. In more recent work (39) E was based on the same physical

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assumptions, viz., conditions pertaining to orbiting at 300 nautical miles, including orientations which produce the greatest heat gain and heat loss. The latter studies considered internal heat generation as the limiting factor in passive temperature control. These limits were calculated to be about 1800 Btu/hr for a noon orbit and 1550 Btu/hr for a twilight orbit. Below these levels passive temperature control appears feasible with materials currently available. Since internal heat generation including 500 Btu/hr imposed by subsystems was calculated to reach as much as 2500 Btu/hr during work, Richardson (39)considers rejection of this heat to be the primary problem in this analysis. He proposes a solar parasol as a means of shielding the worker and providing an additional 500 Btu/hr allowance. Actually there is disagreement as to the metabolic rates that will result from work performed while wearing pressurized suits in low-traction environments. In free space it is likely that there will be a reduction as compared with values at one g due to weightlessness and the relatively light work to be done (40),while on the lunar surface it appears that an increase of perhaps 30% may occur (41). Burriss et al. (42) consider the adoption of entirely passive thermal control methods unlikely because of the complexity of man’s environmental requirements and the wide variations in metabolic output and environmental thermal characteristics. However, their treatment includes extravehicular activity in orbits about the moon and Mars and on their surfaces, a much greater range than assumed by Richardson. None of the EVA suit studies have been refined to the point of considering hands and feet as particular entities, or finger and toe temperatures as limiting factors in space missions as they are in terrestrial and terraqueous missions. There are good practical reasons for this neglect : the current missions are short and the greatest problem anticipated is getting rid of the heat generated internally and derived from subsystems inasmuch as all extravehicular activity will be performed in sunlight with the astronaut returning to the spacecraft during periods of complete darkness. Eventually, however, it is to be expected that situations analogous to those facing the aquanauts will occur and the flow of heat out of the body will need to be impeded while the metabolic heat will be supplemented from an external supply. Perhaps by the time this need arises many of the technological problems will have been solved for the aquanaut and the solutions may aid the astronaut. In this regard it is interesting to compare the bioclimatological aspects of their respective environments with those of the earth’s surface (approximate values) (Table 111). It is immediately obvious from this compilation that, from the standpoint of calculations of heat transfer, the space environment is the simplest of the three. It may be represented graphically as in Fig. 17 showing the astronaut as a sphere orbiting the earth and subjected to direct solar radiation on one

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TABLE I11 BIOCLIMATOLOGICAL ASPECTS OF TERRESTRIAL, TERRAQUEOUS, AND EXTRATERRESTRIAL ENVIRONMENTS Underwater

Surface

Space

Ambient temp. range ("C) -0.5 to $32 -65 to +SO - 273 Ambient pressure 760 (760/10 m 620-790 0 depth) (mm Hg) Moisture content (gm/m3) 10,000 0 0.004-92.30 Solar radiation, max. 0 below 100 p 1060 1200 (kcallm2 hr) Solar radiation, reflected 0 0-3 5 5 In earth orbit 0-480 and scattered (reflected) (kcal/m* hr) Long wave radiation from 0 350 180 earth, max. (kcal/m* hr) 1 compensated to 1 0 Gravity (g force) nearly 0 Ventilatory gases 4% 0 2 16% N2 80% 20% O2/79% 100% 01 He at 62.5 m (200 ft) N2 Metabolic rate (walking) 176428 166-287 166 f 30% (kcal/m2 hr) Heat loss modes: Evaporation Evaporation Conduction man to gear Conduction Conduction Convection Convection Convection Evaporation gear to environs Conduction [{Radiation {Radiation

+

9

9

side and to reflected solar radiation and long-wave emitted radiation from the earth on the opposite side. Since there is no atmosphere there is no scattered solar radiation, no wind, and therefore no air temperature and velocity to affect the heat balance. Radiation from nearby objects need be considered only when the astronaut is assumed to be in the proximity of his vehicle or Noon orbit earth-sun line

Lepsnd: --Solar

energy

Albedo -Earthshine

FIG.17. Radiation exchange in earth orbit.

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protected by a parasol. Thus Eq. (14) for the radiPnt heat load on man on the ground applies with deletion of y R o ,the scattered solar radiation, and suitable adjustments for view factors, while Eq. (1 1) becomes entirely unnecessary. Similarly, the expression for heat balance in space is simpler than that of the aquanaut by virtue of the constancy of the environment as compared with the multiple variations in temperature, pressure, and breathing mixtures necessitated by changes in depth. Of the equations for the heat balance of an astronaut in an earth orbit proposed by various investigators (38,39, 41, and others) the simplest perhaps is that presented by Richardson (39):

where u is the Stefan-Boltzmann constant, To the temperature of outer surface of astronaut, Q the heat generated within space suit, A. the area of outer surface of astronaut, Eo the emissivity of outer surface of astronaut, a0 the absorptivity of outer surface of astronaut, qs the solar radiant flux, Fs-.o the radiant exchange view factor (s - 0), qa the earth-reflected solar radiation (albedo), FaPothe radiant exchange view factor ( u - 0 ) , qe the earth thermal radiation (earthshine), and Fep0the radiant exchange view factor ( e - 0). This basic equation is the reduced form of that expressing heat balance on the astronaut using a parasol and may be manipulated to apply to whatever orbital conditions may be of interest. A more difficult part of the problem is that which deals with the transfer of heat from the astronaut to the suit when his metabolic output exceeds the limits of passive control. Richardson considers varying conductance of the suit by alternately evacuating and filling with helium. Laboratory tests of this system indicated that a wide degree of control of the heat-transfer rate could be realized by this means. Burriss et al. (43)consider liquid-loop conduction cooling to supplement passive temperature control in comparison with radiative exchange between the man and his suit and find conduction far more effective. However, they point out that this method by-passes the thermoregulatory mechanisms of the body and therefore introduces the need for active control of the cooling rate by feedback from some physiological variable such as skin or deep body temperature, sweat rate or metabolic rate, or possibly manual control based on comfort sensations. Figure 18 represents the model used by Burris et al. in their thermal analyses. The suit configuration and ventilating gas distribution is typical of current suits such as those used in the Gemini missions. T h e gas distribution values shown are those used by these investigators in their calculations and are not intended to be specific for any given suit. The effectiveness

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Venti loting gas outlet

Vent i Io t i n g gas in

Vent ilot ing

FIG.18. Cooling system for space suit.

of heat transfer due to ventilation for the case where body temperature is constant was analyzed as

E=

To - Ti T, - Ti

~

=

[1

- exp

-hA

where E is the effectiveness, Tothe gas outlet temperature, Tithe gas inlet temperature, h the heat-transfer coefficient, A the area, rri the mass flow, and C,, the specific heat.

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T h e heat-transfer coefficient, h, was derived from the dimensional representation of laminar gas flow parallel to a flat plate : Nu = 0.664Pr0.33Re0.’ (28) where Nu is the Nusselt number = hL/k, Pr the Prandtl number = Cpp/k, Re the Reynolds number = 60VpL/p,L the length of gas flow path, k the thermal conductivity, p the viscosity, p the density, and V the velocity. An analogous derivation for mass transfer effectiveness assuming water vapor partial pressure at the evaporating surface was given as

where Pois the vapor pressure at the outlet, Pi the vapor pressure at the inlet, P, the vapor pressure at the surface, and hd the mass transfer coefficient. The mass transfer coefficient was determined from the expression for laminar flow over a plane surface : h dL/D = 0.664

(30)

where D is the diffusion constant for water vapor in air and Sc the Schmidt number = p/Dp = pRT/PD. For conduction cooling by liquid transport, the effectiveness was given as

where T, is a constant temperature of coolant tube wall and -hn-dL/mC, the integral of the heat balance equation for heat-transfer fluid flowing through an element of tube length dL :

hn( T, - T )dL = lizCp dT

(32) where T is the temperature of the fluid. Since only part of the coolant tubes will be in contact with the skin, only that portion of the total surface area will contribute to this heat loss and appropriate corrections for this function must be applied in any specific case. Solutions of the foregoing equations for the configuration in Fig. 18 led to the following conclusions : Cooling by ventilation would remove a maximum of about 1300 Btu/hr mostly by evaporation of sweat at the optimal ventilation rate of 15 ft3/min at 3.7 psia. At high metabolic rates, body temperatures would be undesirably high for comfort and well-being. Liquid-loop cooling by conduction of heat from body to liquid appears capable of removing over 2000 Btu/hr under conditions where no sweating is produced.

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Liquid-loop cooling by radiative transfer from body to suit and subsequent transport by the liquid removes less than 400 Btu/hr. Finally, it was concluded that a combination of liquid-loop conduction or ventilating gas cooling systems and passive thermal control by radiative methods would represent the most advantageous arrangement. I t would reduce the evaporant required by 0.72 lb/hr and the fixed weight in equipment by 4-7 lb. Thus, at the present time independent thermal analyses have arrived at the same solution with respect to the efficacy of radiation loss from the suit surface as well as the probable need for a supplementary system to provide for disposal of the heat from a very high metabolic rate anticipated when further progress permits work to be performed. Laboratory studies have lent support to the predictions (44) and, just recently, the decision was made to incorporate the liquid-cooled concept in the Apollo Project moon explorer’s suit (45), probably on the basis of calculations similar to the above. It is estimated that in addition to the rejection of 2000 Btu/hr without perspiration an extra measure of safety will be provided by the ventilation loop. Should the liquid loop fail, the ventilation loop at 6 ft3/min is capable of absorbing 850 Btu/hr by perspiration cooling, which, with a tolerable heat storage of 450 Btu/hr by the man, will allow him a safe return to the spacecraft. Nevertheless, the search for practical materials with better a / €ratios will continue, for success with passive thermal control to the extent of rejection of 750 Btu/hr by this means in current suits has already been demonstrated. Along with the materials search, combinations of materials and construction design will be pressed to improve not only the heat transport features, but also the flexibility of the garment to reduce the metabolic production by reducing the work load imposed by the stiffness of the suit. At the present stage of development experimentation with a variety of prototypes is going on in laboratories with a view to further testing in space. 111. Artificial Environments

A. NORMAL INDOORS Artificial environments probably began with the utilization of fire to provide radiant heat and to warm the air in the open or in natural shelters such as caves or rocky alcoves. This simple arrangement provided the rudiments of thermal comfort and the beginnings of the indoor environment. Today, it is well within the technological capability to provide a high level of sophistication in artificial climates supplying not only heat input, but active removal as well, concurrently with control of humidity and air velocity designed for the optimal physiological thermal balance of the occupants of

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the area. While this balance depends upon thermoregulatory responses of metabolism, vasomotion and sudomotion and their delicate neural adjustments, comfort depends not only upon these physiological parameters, but also upon the more subtle ones of long-term acclimatization, individual preferences, and cultural customs. Therefore, the criteria for comfort may vary widely from person to person and place to place. For the practical purposes of temperate-zone inhabitants, however, the basic guidelines have been well established by studies carried out mainly in the 1930's. The principles concerned are not different from those applicable to natural environments but are confined to a much smaller range of variation with respect to both the environmental factors themselves and insulation due to clothing. The latter today may be considered to be constant at about 0.5 Clo (rather than 1.0 Clo characteristic of 30 or 40 years ago), while the normal indoor environment is confined within the limits of temperature between 66-86"F, relative humidity from 25-85%, and air velocity of the order of 20-30 ft/min. Early basic comfort studies in this country resulted in the concepts of operative temperature (23) mentioned earlier and of effective temperature (46). Both systems reduce a combination of environmental elements to a single value : The operative temperature (23) is entirely objective. It weights air temperature, radiant temperatures, and air velocity in such a way that the single resultant value satisfies Newton's law of cooling with a cooling constant appropriate for the nude human body exposed to air temperatures from 68-86"F, air movement from 7.5-30 ft/min, and wall temperatures equal to the air temperature. Humidity is not included since its effect is considered to be on the evaporative regulative process which is a physiological function of the body rather than a physical function of the environment. Recently, interest in the operative temperature concept has been renewed and provisions have been made for the incorporation of sources of high-intensity radiation in determinations of the mean radiant temperature (47).This adaptation is similar to the application of operative temperature in the natural environment discussed earlier (Section 11,A). The basic equation [Eq. (15)] remains unchanged, as does its usefulness in providing a single index of comfort under combined radiation and convection influences in both natural and artificial environments. In contrast to the operative temperature concept, that of effective temperature is based on subjective criteria. It was originated as long ago as 1923 (46)and has been improved upon over the years (48-50). The effective temperature originally was an empirical value derived from combinations of temperature, humidity, and air movement which produce the same sensation of warmth on entering the environment. These data were summarized in charts which relate wet- and dry-bulb temperature readings, air move-

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ment, relative humidity, and distribution of comfort responses in summer and in winter. Early experience in the use of these charts for predicting comfort during normal occupancy of rooms environmentally controlled in accordance with the charted values showed them to be somewhat inaccurate. Obviously this result followed from the criterion of “warmth on entering”

50

60

ro

DRY

~

euLe

80 TEMPERATURE .OF

90

100

FIG.19. Effective temperature chart.

which was used originally. During prolonged sojourns in the controlled surroundings thermal equilibrium is established and the sensation of “warmth on entering” did not assure comfort during occupancy. Additional studies were made in 1960 (49)in which thermal sensation judgments were made after three hours occupancy. From these data new lines were added to the comfort chart, Fig. 19 (51),which provided parameters of “slightly cool,”

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“comfortable,” “slightly warm,” and “warm” (solid lines) superimposed on the original (dashed lines). It is seen that relative humidity has a much reduced effect on sensations measured over a 3-hr period except in the warm areas where evaporative cooling becomes increasingly important. Current studies (50) conducted on a much larger population of subjects wearing

FIG.20. Revised effective temperature chart.

standardized clothing providing 0.52 Clo insulation agree fairly well with the 1960 data, but show a substantially linear effect of relative humidity (Fig. 20). Agreement is excellent at the upper end of the comfort curves while at the lower end there is a difference of about 2°F in mean radiant temperature for the same wet-bulb temperature. This difference may possibly be related to the difference in clothing as it affects evaporative cooling, as well as to the increased number of subjects.

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In any event these latest data are from the most meticulously controlled study yet conducted. Therefore, they may be expected to constitute the best available comfort data on environmental conditions for the normal room with several young occupants engaged in sedentary pursuits (e.g., reading, playing cards). Under the sponsorship of ASHRAE, studies designed to determine the effects of activity, of mean radiant temperature with both symmetrical and asymmetrical geometry, and of clothing are in progress and are expected to provide new charts or corrections for the current ones within the next several years (52). While more exact baseline comfort data are being sought, efforts also are being made to evaluate the performance of different types of heating systems in terms of their comfort-producing capabilities. A unique procedure for accomplishing this end (53)utilizes the concept that the ideal heating system should produce a completely uniform environment ;therefore, any variation from the standards chosen constitutes a deficiency in performance. Such variations are termed “thermal variability factors” (TVF) and may be determined through the variations of another parameter called “environmental temperature.” The latter quantity is evaluated from measurements of air temperatures and surface temperatures in representative volumetric subdivisions of the occupied portion of the space in question and related to any comfort index desired. This interchangeability of comfort standards is achieved by use of correlation factors derived from the weighting given to the environmental elements in setting up the original comfort index. Thus the environmental temperature” may be written (1

T, = (1 - g ) T, +gT,

+ s,

(33)

where T , is the environmental temperature, T, the air temperature, T, the mean radiant temperature, z, the velocity, and g the parameter defined as

and s is the parameter defined as

Values of g and s can be calculated for any comfort index desired and substituted in the equation for environmental temperature. Since the uniformity of the environmental temperature, rather than its absolute value, is the criterion of efficiency of a system, calculation of T, of many small areas within the larger conditioned space provides a measure of uniformity or comparison of the individual values.

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This method has the disadvantage of requiring computer calculations of relatively large numbers of measurements. However, computers are quite common now and the surface temperature measurements, at least, could be made very quickly if done radiometrically instead of by thermocouples. In general, the technique would appear to be particularly valuable in assessing comparative efficienciesof different types and placements of heating systems where a large number of constructions of similar design are contemplated. In fact, miniature models such as that described by the originators (53) could be made to provide laboratory assessments of any constructions for

(T. -T,)OF

FIG.21. Thermal variability factors related to several heating systems [redrawn from

Ref. ( 5 3 ) ] .

any climate before finalization of the choice of heating and cooling systems. In this regard, Fig. 21 [redrawn from Parczewski and Bevans (53)]is shown to illustrate the differentiation made possible by this concept. For each of five different systems the T V F was ascertained and plotted against the heating load, the difference between the arithmetic average of environmental temperatures in the heated enclosure, T,, and the outside temperature, To. It is seen that the T V F increases approximately linearly with the heating load in each instance and that in this particular model the placement of the baseboard under the window shows the lowest T V F throughout the range of measurement and therefore would be the best choice. Other models might be expected to yield different results. The general concept, however,

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has the distinct advantage of transcending specificity. Theoretically, it can be used with any comfort index for evaluation of heating and cooling features of any type of construction. Should practice fulfill the promise of this theory the system would be universally applicable to all environmentally controlled enclosures. B. UNDERWATER VESSELS Research on undersea or “inner space” shelters is at least as new as that of “outer space’’ platforms even though man has been traveling underwater far longer than he has in space. Underwater vehicles reached a high degree of sophistication with the success of nuclear-powered submarines, but it was not until free swimming at depth for significant periods of time became feasible that undersea shelters assumed any practical importance. While SCUBA and “hard-hat” divers were limited to short sojourns at depths in excess of 200 ft, underwater vessels were designed to be primarily observation posts rather than shelters. Prolonged “bottom times” became possible with the advent of the concept of “saturation diving.” This technique consists of saturating the blood and body tissues over a period of 24 hr with breathing gases at a pressure equivalent to that of the ocean water at the depth of the dive. Once saturation has been achieved man may live and work at the selected depth indefinitely. Lindbergh and Stenuit spent 48 hr at 432 ft ( 5 4 , four U.S. Navy men spent 11 days at 193 ft (55), 28 men spent various times, up to 30 days for Carpenter, at 205 f t with excursions to 300 ft (56),and two divers have remained 48 hr at a simulated depth of 650 ft (57). An actual dive to the 600-ft depth is planned and Cousteau speaks of “sending people swimming to work two thousand feet deep” (58). The latter at the present time seems wildly improbable, but then so did the 600-ft depth just a few years ago. In any event, at least 27 deep-sea vehicles and three underwater stations are already a reality (57, 59) and the future looks to mobile undersea stations that can pick up and move from one work site to another under their own power (57). Of the biological problems associated with such ventures, the most serious is, of course, respiratory, but the heat-transfer problems also are considerable. As mentioned in Section II,B, and in Sealab I (55) and Sealab I1 (56) reports, heat loss and resultant low body temperatures act as the limiting factor in free-swimming time in the underwater environment. Within the underwater vessels, however, temperatures on the high side of the comfort zone apparently are maintained without difficulty. Detailed analyses such as exist for normal room conditions are not yet available for undersea shelters although some theoretical work has been done on the effects of a helium environment as compared to air (28). In this work the heat-transfer

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equation was simplified by dropping the evaporative loss term on the basis that in open-bottom research vessels the RH will be 100yoand the evaporation loss will then be zero. Other simplifying assumptions were that the emissivity of the vessel walls is close to unity, that the temperature of the inside of these walls is within one or two degrees of the water temperature, and that the oceanaut’s bare skin temperature should be about 92°F when heat loss equals metabolic heat generated. For the surface temperature of the clothed oceanaut, 82°F was assumed. For radiative heat loss calculations the dimensions of the usual “average man” were used and the emissivity and shape factor were taken as unity. In calculating convective heat loss, the geometry of the human body was approximated by a vertical cylinder with a heat-transfer coefficient appropriate for natural convection. Combination of the equations expressing the foregoing relationships yielded the over-all heat balance equation :

M - W = 2.49 x 10-8(T,4- T:,J

+ 19.5 Cc(T, - T,) 1.25

(34)

where M - W is the metabolic heat to be dissipated, T, the surface temperature, T mn the mean radiant temperature, Cc the convection coefficient (air or helium), and T, the air temperature. From solutions of this equation for various depths and for atmospheres of air and of helium a series of charts was prepared to provide the dry-bulb temperature necessary to produce “comfort” for a given mean radiant temperature, metabolic activity, and mean skin or body surface temperature. However, comparison of these data with the temperatures actually observed in Sealab I and I1 indicates that these charts yield temperatures somewhat too high. For instance, for conditions approximating those of Sealab I, i.e., water at 69”F, depth 193 ft, and a helium atmosphere, an ambient temperature slightly over 90°F is indicated whereas dry- bulb temperatures between 83-86°F were reported as comfortable (55). Also, the latter temperatures compare rather well with those reported as “comfortably warm” in a study of thermal conditions in submarines operating in cold waters (60). I t is probable that the discrepancy is due to the assumption of zero evaporative heat loss, or lOOyo RH. In fact, the RH inside the research vessels ranged between 60-90%, averaging in the 70’s (55, 56). Thus, a considerable amount of heat could be lost by evaporation and this term should be retained in Eq. (28). Nevins et al. (28) themselves suggested Gagge’s equation (61) as suitable for this use although they did not use it :

E = (Wp)A(P,- RH x P,)

(35)

where Wis the fraction of body area completely wet with perspiration, p the coefficient of heat transfer by evaporation, containing the constants for

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vaporization, air motion, and direction (Btu/hr ft2 inches Hg), A the total body area (ft’), P,the saturated vapor pressure at skin temperature (inches Hg), RH the relative humidity, Pa the saturated vapor pressure at air temperature (inches Hg). This expression would enter Eq. (34) as a negative term and have the effect of reducing the dry-bulb temperature required for comfort at any given mean radiant temperature. A more general system proposed by Krantz (62) is appropriate here and elsewhere in environments which may be described as abnormal or special because of unusual levels of one or more thermal features. This set of calculations and charts permits a rapid estimate of the comfort of the environment from a calculation of the ratio of evaporative cooling in progress to the maximum possible through perspiration (61). This ratio is directly related to comfort (63)so that values below 10% indicate conditions too cold for comfort, 1 0 4 5 % are comfortable, 25-707/, are tolerable, 70-100% are unpleasant, and values over 100% indicate conditions unsafe for any extended period of time. A quick look at these charts for the conditions prevailing in the Sealab I (uncorrected for the difference due to a helium atmosphere) indicates that at 90°F it is possible to lose as much as 450 Btu/hr with a relative humidity of 70% ; also, an ambient temperature of 85°F results in an evaporative loss ratio of 16%, well within the comfort range, while an ambient of 90°F yields a ratio of 43.574, in the tolerable but warm region, in general agreement with the actual observations. Of course, precise values would require appropriate corrections for gas and pressure. When underwater shelters reach the envisioned sophistication of presentday buildings in the terrestrial setting, the comfort charts appropriate to the latter environment will become applicable on correction for the thermal characteristics of the ventilatory gases as they differ from air at sea level. The thermal-conditioning problems otherwise will be simpler than the surface environmental problems because of the greater constancy of the surrounding temperature. No great fluctuations, either diurnal or seasonal, will occur at depth and, in fact, this environment will be more constant than even that of space where the sun exerts such a profound, albeit predictable, effect. The greater challenge to be met in the heat-transfer aspects of the underwater environment then will be power generation and conversion in a new gaseous medium rather than unique biotechnological thermal phenomena.

C. SPACE VEHICLES Trips to other planets will take long travel times even at the incredible speeds of spacecraft. Therefore it becomes essential to provide a comfortable

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“shirt-sleeve” environment for the voyagers, and also for occupants of space stations and platforms which are now in the planning stage. Because of the relative simplicity of the space environment it would appear that the greatest deviations from other indoor heat exchange expressions will be due to the lack of natural convection and the enhancement of radiant temperature fluctuations under the influence of solar exposure. However, in the absence of convection, COz and water vapor would build up in a cloud surrounding the occupant. If for no other reason, then, forced convection becomes a necessity for dispersing these products of respiration. Actually the problem of maintaining physiological thermal equilibrium is secondary to the total heat-transfer problem in a manned spacecraft, for the heat generated by operating equipment is far greater than that of metabolic heat. Consequently, man must be worked into the over-all system as a subsystem having certain basic thermal requirements. These requirements must be met in the presence of a gas mixture of oxygen and an inert diluent differing in pressure and thermal characteristics from the terrestrial gas mixture. An early study (64) offers the usual equation for computing forced convection heat loss with constants suitable for air at 80°F and 14.7 psia, and for oxygen at 80°F and 5 psia:

Qc = 1.55v0.466(Tb - T,) (36) where Qc is the convective heat loss, v the air velocity, Tb the body surface temperature, and T , the air temperature (dry bulb). For computations of in air,

~ ~this ~. loss in oxygen, the convection constant recommended is 0 . 9 3 2 ~ O . In work it was concluded that the comfortable atmospheric temperature in the spacecraft would be somewhat lower than the equivalent on earth. Furthermore, it is pointed out that in a low-pressure oxygen atmosphere variations in “air” temperature have considerably less effect on the heat transfer from the human body than variations in mean radiant temperature. For example, at an air velocity of 10 ft/min a 1°F variation in the mean radiant temperature FIG.22. Environmental control system for the 30-day biosatellite flight which will carry a monkey as passenger; the ECS for the 21-day flight carrying rats will be quite similar. A capsule atmosphere of 8076 nitrogen, 207” oxygen is maintained. T h e heat exchanger at upper right keeps air temperature at 75 f 5”F, and, through control of its dew point, keeps relative humidity in the 40-7076 range. Lithium hydroxide absorbers keep the carbon dioxide level below 1yo.T h e cryogenic oxygen tank supplies both the fuel cell and the capsule, flow to the capsule being regulated by a partial pressure sensor. A total pressure sensor controls nitrogen flow. The by-product of the fuel cell furnishes the drinking water; excess water is stored in a waste tank or when necessary can be used to adjust the heat balance by allowing it to boil off into space. Primary thermal control is achieved with the two separate coolant loops at left.

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is equivalent to a 5°F variation in dry-bulb temperature and, at 20 ft/min, 1°F in mean radiant temperature is equivalent to 3.7”F dry bulb. However, the lOOyooxygen atmosphere of today’s spacecraft will be supplanted by a two-gas system, and the radiant temperature to be anticipated in a manned station is as yet undetermined. That the latter will be uneven follows from present designs which utilize the vehicle skin as a fin material in a fin-andtube system to provide a radiator for transporting heat for rejection to space (65).Pumps, fuel cells, cryogenic systems, and various electrical equipment also are distributed throughout the craft so that hot and cold spots must exist to a considerable extent. At this point in time, following the success of the Gemini 6 and 7 mission and anticipating the first Apollo flight scheduled for late 1969, the requirements for thermal control of the spacecraft occupant’s environment are largely restricted to those which will assure survival and unimpaired performance, with comfort per se as a secondary goal. For this purpose work is proceeding with minimal specifications, e.g., for the “shirt-sleeve” atmosphere present trends point to oxygen and a diluent gas at a total pressure of 380-500 mm Hg, the partial pressure of O2 being 150-165 mm Hg, and the diluent, probably, helium. The relative humidity, depending on temperature, will be between 30-60y0 at ambient temperatures from 6840°F. Air movement within the cabin will approximate 20-40 ft/min. Eventually, for long missions the environment will be subject to thermostatic control by the crew so that it may be adjusted for variations in radiant temperatures as vehicular attitude changes and variations in metabolic heat as activity level and number of occupants vary. Some thought has been devoted also to automatic controls monitored by continuous sensing of crew temperatures. These requirements are to be met concurrently with those of the other thermal and power considerations so that the entire system functions as an integrated heating and cooling plant. By way of illustration, Fig. 22 (66) presents a block diagram of the environmental control system for a 30-day biosatellite flight which will carry a monkey as a passenger. This capsule is designed to provide an earth-like atmosphere of 20% O 2 and SO?, N2 at 14.7 psi. In this regard it will differ significantly from the vehicles for human occupancy. For the latter it is planned that the Apollo vehicle systems will be built along the lines proven in the Gemini capsules even though more distant plans may feature radical departures from these. A description of the Gemini systems is available in the open literature (67). The principles and practices evolved in the development of manned spacecraft have been collected into a new discipline known as “life support” served by “life support systems”. These systems concern not only thermal protection and respiration, but also myriad others contributing to the maintenance of life, e.g., water reclamation, O2regeneration, nutrition, waste disposal and utiliza-

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tion ( 6 9 and energy conversions of micro- and macroorganisms. Any discussion in depth of the biotechnology of heat transfer in such systems, or even a listing of pertinent references, would require considerably more space than can be devoted to it here. Suffice it to say that these topics are in the very turbulent forefront of current research and are best followed on a day-today basis. Which of the many proposals of systems (69) and vehicles themselves (70) competing for implementation and trial today will prove to be true “advances” is yet to be seen.

IV. The Role of the Skin in Heat Transfer A. FUNCTIONS AND CHARACTERISTICS I n the present writing attention has been centered on the broad picture of heat exchange between the body as a whole with the total environment, and, the effect of protective garments or thermal barriers interposed between the body and its various surrounds. In this section the focus is on the skin, that boundary layer which encloses each individual and forms his interface with the environment. Indeed, it is the skin rather than the body as a whole, which, by virtue of its many sensory receptors, determines the physical characteristics of an environment which is interpreted as comfortable. In addition to this role as a detector, the skin functions thermally as a heat generator, absorber, transmitter, radiator, conductor, and vaporizer. It is not often appreciated as a separate organ of the body, as vital to life as the heart, yet if 4004 of the skin is destroyed, the life of the entire individual is severely jeopardized, while death is inevitable on destruction of the total skin area. If, however, the individual survives extensive dermal trauma, the skin is remarkably regenerative and recovers, though frequently requiring the assistance of grafts. The complexity of this organ may be illustrated to some extent in Fig. 23. It is seen that the skin is a composite of morphologically distinct tissues. Within these layers of tissue are situated many discrete structures, including hair follicles, sweat glands, sensory receptors, capillary blood vessels, and an extensive network of fine nerves which contact these structures and relay signals to and from the central nervous systems. The whole skin is comprised of two main portions-the epidermis and the dermis. The epidermis is subdivided into four layers, the stratum corneum, stratum lucidum, stratum granulosum, and stratum germinativum. One or two of these layers (viz., stratum lucidum and stratum granulosum) may be absent in some areas of the body. The dermis is subdivided into two layers, subepithelial or papillary layer and the reticular layer, both of which are composed of connective

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tissue. All layers except the stratum corneum are living, growing tissue and therefore produce metabolic heat. Because of the variations in composition and properties of the skin in different regions of the body, its physical dimensions are difficult to state in precise terms. They also are highly resistant to direct measurement in the intact individual. Therefore, the physical dimensions of the whole skin shown in Table IV (71-74) are for the most part calculated values. From these values it is readily appreciated, however, that, by weight as well as by surface area, skin is one of the largest organs of the body. It comprises about 694 of

Stratum corneum Duct of sweat gland Stratum lucidum Stratum granulosum

Blood vessel

Stratum germinativum Duct of sweat gland Papillary layer of derrna

‘Reticular layer of derma

FIG.23. Section of human skin showing all layers (adapted from photomicrograph by Maximow, “Textbook of Histology” (Maximow and Bloom, eds.). Saunders, Philadelphia, Pennsylvania, 1948).

the total body weight and exceeds the weight of the liver, the brain, the heart, and the kidneys put together. For its area, its volume is small and consists mostly of water. The specific gravity of skin excised from several sites on the body was measured directly (71) and found to vary little with age, sex, or color. The thickness of the whole skin varies widely over the surface of the body. It may be more than 5 mm on the back and only 0.5 mm on the eyelids. T h e usual thickness is 1-2 mm. Of greater interest in the present consideration are the thermal and optical properties which are even more difficult to measure and therefore are known only approximately. Some of these are shown in Table V. Although these quantities are frequently referred to as “constants” it is obvious that

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they must be far more variable than constant since the medium itself is not only heterogeneous but of varying composition and thickness from place to place on the body surface. Furthermore, because of the presence of thermal and other sensory detectors in the skin, its thermal properties are influenced TABLE IV OF APPROXIMATE VALUES

THE

PHYSICAL DIMENSIONS OF SKIN

(70 kg, 170 cm or 154 lb, 5 ft. 7 in.) Dimension Weight Surface area Volume Water content Specific gravity Thickness

Value 4 kg 1.8 mz 3.6 liters 70-75 "/u 1.1 0.5-5 mm

8.8 lb 20 s q ft 3.7 qt 0.02-2 in.

TABLE V

THERMAL PROPERTIES AND OPTICAL PROPERTIES OF SKIN Property

Value

240 kcal/day 9-30 kcal/m* hr "C (1.5 i-0.3) x 10-3 cal/cm sec "C at 23-25°C ambient 7x cm?/sec (surface layer 0.26 mm thick) Diffusivity (K/pc) Thermal inertia ( k p c ) 90-400 x 10-5 calz/cm4 sec 'C' 0.8 cal/gm Heat capacity (c) Emissivity (infrared) 0.99 Reflectance (wavelength dependent) Maximum 0.6 to 1.1 p Minima < 0.3 and > 1.2 p Transmittance (wavelength Maxima 1.2, 1.7, 2.2, 6, 11 p dependent) Minima 0.5, 1.4, 1.9, 3, 7, 12 p Heat production Conductance Thermal conductivity (k)

by an efficient feedback system operating through the cutaneous blood-flow system, as well as through a voluntary or cortical control whereby the entire organism is activated to remove itself from hazardous environments. For example, local cooling causes a shutdown of blood flow to the skin to a minimum estimated at about 0.015 cm3/min/cm2(73). Local heating brings about local vasodilation, flooding the heated skin with blood at over

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100 times the minimal rate, resulting in measured thermal conductivities as much as four times the value determined at normal room temperature. In addition, prolonged local heating brings about sweating, not only of the heated area, but also of nonheated areas of the skin (75), indicating that remote as well as local effects may be produced by signals relayed from the skin detectors to the central nerve centers. More intense heating may stimulate pain receptors, resulting in reflex withdrawal of the limb or perhaps of the entire body. Changes in skin temperature are accompanied by variations in heat production, as well as heat loss, in the skin. The secretion of sweat, for instance, requires energy ; therefore, the metabolism of the skin must be increased during such activity. T h e amount of heat produced by the skin itself is unknown although it is possible to make some approximation from its oxygen consumption and related data. In vitro measurements of oxygen consumption indicate that the metabolic rate of this tissue is perhaps one fourth that of the cerebral cortex and about one fifth that of the liver (76). T h e latter is one of the most metabolically active tissues of the body which, in the dog, at least, accounts for approximately one third of the total oxygen consumption of the whole body (77). Since in man the weight of the skin is about twice that of the liver, and the water content about equal (4,a very rough estimate of the contribution of the skin to the total heat production may be calculated as (0,consumption) x 2 (weight) x 33% (heat production) = 1204, of the total metabolic heat production. On this basis, the estimated contribution of the skin is about 240 kg cal/day for the “average man,” a not insignificant amount. T h e thermal conductivity of the whole skin is so much under the influence of blood flow that it proved desirable to deal with it in two parts: (a) the specific conductivity referring to the bloodless tissue, and (b) that caused by the flow of blood to the skin surface. Used in this sense thermal conductivity becomes obscured in the term conductance, Table V (72, 78) referring to the heat flux per unit gradient decrease in temperature through the skin and the underlying tissues of the body. It becomes clear that to specify a firm value for thermal conductivity of skin in the usual sense it would be necessary to separate the skin into homogeneous layers and measure each separately. Since this technique cannot be employed in the intact body, approximations have been made by other methods involving the measurement of the product of conductivity, heat capacity, and density as one entity, or by determining average diffusivity with respect to depth, and deriving the conductivity from this. The value noted in Table V is one specified for intact living skin (79). It was obtained by comparison of the product (kpc) of skin with that of a known substance (glass) and applies at normal room temperature (23-25°C). Other deter-

+

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minations (80, 81) have shown values of kpc varying from 90-400 cal2/cm4 sec deg Cz, depending upon the physiological and physical conditions prevailing. Heat capacity has been measured in excised pigskin (82) and, because of the histological similarities between this and human skin, it has been assumed to be applicable to the latter as well. Since this value is fairly close to unity it has relatively little effect on the computations of thermal conductivity. However, as determinations of the latter become more dependable, direct determinations of heat capacity in human skin should be made. The importance of emissivity, reflectance, and transmittance of the skin has long been recognized. T h e fact that the skin emits heat very nearly like a blackbody, Table V (83),has permitted the development of various radiometric devices for the measurement of skin temperature, and mapping of the entire body surface temperature by thermography. The latter technique is being used more and more widely for the detection of a variety of pathological conditions which formerly would have required surgery or other drastic procedures (84). Reflectance and transmittance' are of particular importance in determining the extent and depth of heating of the skin exposed to energy of various wavelengths. Sources of such energy range from natural emitters such as the sun and the natural environment to artificial sources such as atomic flash, infrared and tungsten lamps, carbon arcs, furnaces, open fires, and myriad others. The determination of the effect on the skin of each of these types of energy constitutes a complicated study in itself since the optical properties vary with the degree of pigmentation of the skin, thickness of the layer, surface condition, and so on. One effect in common, however, is that prolonged or intense exposure results in death of the skin in whole or in part. As pointed out earlier in this chapter (Section II,B), the thickness of subcutaneous fat is of great importance in maintaining body temperature during immersion in cold water. Although this fat is not part of the skin proper, it exerts a great influence on the conductance of the body surface. Recent studies on autopsy subjects (85) have shown that caliper measurements of skinfold thickness can be correlated with the actual thickness of fat, but that different coefficients must be applied for different sites of measurement, age, and sex. In general, in this study, the thickness derived from measurements of subcutaneous fat in the triceps areas was representative of the mean thickness in nine representative sites of the body surface although the actual thickness of this layer varies from virtually zero in some areas to 1.5-2.0 cm in others, the thigh being fattest. Under the age of eleven years no variations I The maxima and minima noted in Table V are rough average values obtained from a variety of sources too numerous to list. Many of thesc are to be found in issues of the Joitrrral

of Applied Physiology.

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with sex were found, but in older subjects the female had thicker subcutaneous layers and thinner skin than did the male. These observations are of importance in estimates of physical insulation such as those of Rennie et al. (86),Lee et al. (87), and Buskirk et al. (88)in which skinfold measurements were used for this purpose. The latter study suggested also a possible active role of the body fat mass in the metabolic response to cold. Vapor pressure influences the diffusion of fluids from the body surface and therefore affects insensible heat loss. Recent attempts have been made to measure this pressure and assess its effectiveness as a diffusion barrier (89)in normal skin in comparison with burned skin. In the latter, the barrier is destroyed so that the vapor pressure over the burned skin is the same as that of water at the same temperature and is 15-25 times greater than that over normal skin. This observation gives some idea of the rapidity with which the burn victim becomes dehydrated. Of course, the entire burn syndrome is a complicated vicious circle of which the accelerated evaporative loss is but one important feature and altered heat transfer serves to enhance the body fluid losses induced by destruction of the skin. In the normal skin, insensible perspiration may account for as much as 25q/;, of the total heat loss, while active sweating may increase the evaporative loss to 10006 under extreme conditions. Sweat production itself varies widely among individuals and with acclimatization, race, and sex as shown recently in wide-ranging studies by Wyndham and his colleagues (90-92). In fact, the technological achievement of a rapidly responding man-size calorimeter (93) provided for measurements of total physiological heat transfer of the human adult in the transient state for the first time. In 1959, Benzinger’s interpretation of data gathered on this apparatus (94) triggered a controversy regarding the roles of sensory skin receptors and internal temperatures in thermoregulation which continues even today. Certainly it stimulated renewed effort and infused new life into a field of investigation which lay quite dormant at the time. Such efforts were directed particularly toward correlations of sweating and its onset with temperatures observed at various sites on the body. Controversy still ensues over the technical value and the physiological meaning of temperatures measured by various methods (95, 96). Thus, much is yet to be learned about the skin and its important functions in heat transfer. No mention has been made of minor variations which may be attributed to differences in pigmentation, hairiness, roughness, etc. It is obvious that visible radiation will be absorbed to a greater extent by pigmented than by nonpigmented skin, that the hirsute surface offers more insulation than the glabrous surface, and the rough skin is less reflective than the smooth. While it is possible that in rare instances such variations might seriously affect heat exchange, on the whole they are of little importance, especially in the clothed individual.

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B. THERMAL SENSATION The range of sensation due to thermal effects extends above and below the temperature levels of comfort or thermal neutrality into cold and pain on the one side and heat and pain on the other. Each limit is reached when the tissue is destroyed. For practical purposes these limits may be stated as instantaneous tissue temperatures of 0°C (or slightly below) where the cellular fluids crystallize, and 72°C where the full thickness of the skin is destroyed [(81); Section IV,C of this article]. Within the area of temperatures above the neutral zone it has been demonstrated that the sensation of warmth is evoked at very low heat inputs which produce a change of skin temperature at the rate of +0.001 to +0.002"C/sec (97). Similarly, a rate of change of -0.005 to -0.006"C/sec evoked the sensation of coolness. As tissue temperature is raised sensations of warmth give way to those of burning, then pain, intolerable pain, and finally numbness when sensation is lost on destruction of the skin. Destruction of the tissue by heat depends on both temperature and time (98),while the pain intensity depends on the rate of tissue damage (99, 100) which is related to the instantaneous tissue temperature. Analogous studies in the region below the neutral zone have not been done. It has been shown, however, that there is a marked change in the stimulus strength required to evoke a threshold sensation as the skin temperature changes from 32-34°C (101, 102) even though the data cited disagree as to a comparable effect in the warmth sensation. This disagreement significantly affects the arguments as to the mechanism of mediation of the sensations. A comparable change would support mediation by a single neural pathway while dissimilar effects would tend to support the existence of different pathways. Technically it is more difficult to perform definitive experiments with cold stimuli than it is with heat stimuli. Also, cold sensation and cold pain have been less of a problem area than that of warmth and burns. Therefore, it is unlikely that the lower end of the sensation scale will receive the intensive study accorded the upper end until such times as practical considerations force the issue. Perhaps problems associated with the terraqueous and extraterrestrial habitats of the future will provide this stimulus. In the meantime, much has been learned about the perception of burning pain and it is interesting to speculate on its application to cold pain. Hardy and his colleagues, working in this field for more than a decade, produced much of the present body of knowledge. It is set forth in detail in Hardy et al. (203) which describes the use of radiant energy as a warmth and pain stimulus, the elicitation of 21 just noticeable differences (JND) in sensation correlated with stimulus intensities and the fractionation of sensation into 10 levels of pain intensity comprising the "Dol" scale. If it were possible to

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THERMAL SENSATIONS AND

Sensation Numbness

I

--t

ASSOCIATED EFFECTS THROUGHOUT RANGEOF COMPATIBLE WITH TISSUE LIFE

Skin color White

emperaturc “F

“C

Mottled red and white

Bright red

Severe pain

68

-

64 -

56 52 -

Light red

Threshold pain

Protein coagulation

Injury Irreversible

72--162

60--140

Maximum pain

Process

TEMPERATURES

Thermal inactivation of tissue constituents

Possibly reversible

Reversible

48 44--111

Hot Flushed Warm

40 36 -

-

Neutral

Flesh

32 -

28--

Blanched

93 82

Normal metabolism

None

24 20 16-

- 64 .

12 -

aBright pink

Numbness

I

4-

0--

mite

- 50

-4--

32 25

Physicochemical inactivation of tissue constituents Protein coagulation

Reversible Possibly reversible Irreversible

provide a convenient, precisely controllable radiant “cold source” or heat sink, comparable studies could be pursued in the range below the neutral zone. However, a flux from skin to sink equal to that of source to skin used above could not be achieved even with the sink at absolute zero, therefore exactly equivalent techniques are not possible. Conduction methods are not suitable for precision work of this nature because of their interference with

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the surface itself and the uncertainty of the true temperature produced in the skin should contact with the stimulator be imperfect. As visualized in Table VI, where the temperatures at both ends of the scale are considered to be tissue temperatures reached instantaneously, death of the tissue is ascribed to protein coagulation, from heat at the upper end and from intracellular fluid crystallization at the lower (104).In between, tissue damage depends on both temperature and time and is more or less reversible. The processes postulated in each area, however, are not the same. In the upper region, a systematic progression of thermal coagulation can be demonstrated (98, 81) while in the lower region injury has been ascribed to extravasation of plasma and red blood cells into the tissues and obstruction of small vessels by impacted blood cells. In frost bite this process is believed to follow crystallization of extracellular fluid which draws water from within the cells causing protein and electrolyte concentration to the point where the protein coagulates (105). In practical situations it is a much slower process than injury from heat burns. Sudden freezing might produce intracellular crystals which would give rise to mechanical trauma. Since it has been postulated (99) and demonstrated (100) that pain can be related to the rate of noxious stimulation rather than to the amount of damage or to the tissue, however, it is entirely possible that cold pain could be systematically fractionated even though the relationship is not directly dependent upon thermal processes. Until such time as a satisfactory technique is developed whereby cold pain can be elicited fairly quickly and sharply this sensation must remain inextricably bound to effects which are primarily vascular. Indeed, some investigators suggested that all thermal sensations are bound to vascular activity within the cutaneous tissue (102),whereas it is generally agreed that pain is mediated by free nerve endings. Thus, the development of an appropriate experimental technique could contribute significantly to the elucidation of the mechanisms of perception of both pain and thermal sensations.

C. INJURY AND PROTECTION 1. Injury Injury from intense, short-term thermal exposures may be confined entirely to the skin and therefore constitutes an entirely different situation from that due to long-term heat gain leading to systemic effects such as heat prostration. By the same token, protection from burns requires a different approach from that appropriate to long-term environmental thermal protection. This type of protection is best considered in terms of escape time, i.e., that time sufficient for the victim to escape or to terminate the exposure

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before injury occurs. It is this limit which must be considered as the essential design criterion for protection from an anticipated hazard and it is this limit which is dictated by both the injury processes in the skin and the heat-transfer mechanisms operative in each situation. Serious misconceptions have crept into this field of research through adoption of rule-of-thumb terminology which has lost its identity as such and become accepted as fact. A glaring example of this process is the “critical thermal load.” This quantity is defined as the total energy delivered in any given exposure required to produce some given endpoint such as a blister. Mathematically it is the product of the flux and exposure time for a shaped pulse. Implicit in this treatment is the assumption that thermal injury is a function of dosage as in ionizing radiation, so that the process obeys the rncal/crn2

4000

r

Threshold blister

P D

b

2000 1 100

I

I

300 lrrodiance (rncol/cm2sec)

200

I

400

FIG.24. Product of irradiance and exposure time to produce threshold blisters plotted against irradiance alone.

“law of reciprocity,” i.e., that equal injury is produced by equal doses. On the contrary, a very large amount of energy delivered over a greatly extended time produces no injury at all while the same “dose” delivered instantaneously may totally destroy the skin. Conversely, measurements of doses which produce the same damage over even a narrow range of intensities of radiation show that the “law of reciprocity” fails, for the doses are not equal. Figure 24 illustrates this fact. In the figure are shown the energy doses required to produce a threshold blister (equivalent to the least amount of energy productive of complete transepidermal necrosis) in blackened human skin (81).In these experiments the irradiance of the source was set at levels from 100400 mcal/cm2 sec. T h e exposure time was determined at each level. The product of the irradiance and the time is plotted on the ordinate against the irradiance alone on the

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HEATTRANSFER IN BIOTECHNOLOGY

abscissa. If reciprocity existed over even this limited range of energy, this product would be a constant ; clearly, it is not. Why it is not becomes clearer on consideration of the processes productive of thermal injury touched upon in the preceding section. The fine details of the processes have not yet been solved (106),but the broad outlines were drawn by Henriques in 1947 (82, 98) and Moritz (107) and further developed since then (81,108).In general, damage may be related empirically to the temperature-time parameters by the expression appropriate to physical rate processes :

where L2 is the damage which is 1.0 at complete transepidermal necrosis, dQ/dt the rate of damage, P a constant determined from experimental data, A E a constant determined from experimental data, R the gas constant, and Tt the basal-layer temperature. These processes are illustrated in Table VTI and Figs. 25, 26 and 27. TABLE VII IRRADIANCE,TIME, AND TEMPERATURE FOR BLISTER PRODUCTION Radiation intensity

Exposure time

(mcal/cm2 sec)

100

150 300 400

33.8 20.8 7.8 5.6

Maximum skin temp.

Total time elevated T/PT

Ratio Exp. t / t t

("C)

(set)

(%I

52.9 54.0 56.7 59.1

34.1 21.8 9.8 6.9

99 95 80 81

TableVII shows theirradiance, exposure time, maximumskin temperature reached, and the total time of elevation of the temperature of the basal layer of the skin above pain threshold required to produce a threshold blister at four of the levels of irradiance noted in Fig. 24. These are the essential elements for injury : exposure to heat, and elevation of the skin temperature to an injurious level for a sufficient time to produce damage. Note that the exposure time occupies a greater proportion of the total time above pain threshold at the lower irradiances than at higher ones (last column). The skin temperature histories for exposures to the highest and the lowest irradiances used in these experiments are shown in Fig. 25. At the lower intensity the temperature rise is relatively slow and the skin temperature does not exceed 53"C, whereas at the higher intensity this temperature remains elevated for a considerable time after termination of the exposure.

IRRAOIANCE = 100 mc/cm* sec

58 56 Severe

-

TIME-SECONDS

FIG.25. Temperature-time histories during and following thermal irradiation at two levels of intensity.

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This cooling time is important, for damage occurs at all temperatures above 44°C whether the temperature is rising or falling. Thus, it is evident that injury must be assessed from the temperature-time history of the skin rather than from the dosage which entirely neglects injury during cooling. T h e rate at which injury proceeds varies logarithmically with the temperature as shown in Fig. 26 (81).It is seen that the rate of injury at 50°C is

1x10454

46

'

48

' &

2;

Temperature T

4 ;

5k

! B

FIG.26. Tissue damage rates versus temperatures.

100 times that occurring at 45°C. Above this level the rate of increase per degree centigrade is considerably less, possibly due to the cessation of some process for which 50°C is a critical temperature. Nevertheless, at the higher temperatures damage occurs quickly and extrapolation of this curve indicates that the skin is destroyed instantaneously at a temperature of 72°C. The actual proportion of injury inflicted during heating and cooling in the exposures discussed here is shown in Fig. 27. It is seen that, where equal damage was produced, indicated as SZ = 1, only lo%, of the damage occurred

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ALICEM. STOLL

during cooling at the lower irradiance and relatively lower temperature while more than a third of the total occurred during cooling in the higher irradiance exposure.

0.0840 -0.0560 0.028

0.0-,

0.056

, , ,

0.0

T h e essential factors with respect to thermal injury may be summarized :

(1) injury occurs at all basal tissue temperatures above 44°C regardless of exposure time ; (2) the rate of injury increases logarithmically with increase in teniperature level ; and (3) total damage is the sum of that inflicted during heating plus that sustained during cooling. 2. Protection ‘1’0 provide protection against thermal injury it is necessary to stipulate the characteristics of the causative agent, to analyze the resultant modes of heat flow, and to construct barriers designed to minimize the heat flow to the skin. An illustrative study is that of protection from flame contact. I n the knowledge that a significant number of pilots survive crash landings only to perish in fuel flames while disembarking from their aircraft, concerted efforts were made to provide adequate protection for this specific situation. Fundamental studies of heat transfer during flame contact revealed that

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several elements were highly significant in forming a barrier to heat transfer in flame contact : (1) a fire-resistant material, (2) a double layer of such material so that the outer layer constitutes a sacrificial layer which is a barrier to heat transfer until it is destroyed, and (3) an inner layer which also resists ignition. The practical operation of these elements was observed in experiments in which dummies clad in double-layer, fire-resistant polyamide coveralls AT *C

R

P

20

I\

15

FIG.28. Temperature rise within simulated skin versus width of air space between two layers of 3 oz/yd2 fabric.

I\

-

0

0

= 3 sec = 2 sec a = I sec

0

Fobric Perfwofed

Air Space (mm)

were drawn through fuel flames for a distance of 25-30 ft, commensurate with the situation presented by a fire surrounding an aircraft on a flight deck (109). In these experiments the material was not ignited ; the outer layer was destroyed, but the inner layer remained intact. Temperature measurements indicated that less than 2% of the body surface would be severely burned, and also that the time of 2.5-3 sec is sufficient for an uninjured man to reach safety by running through the flames. Laboratory studies of the heat-transfer mechanisms involved (110) showed that there is an optimal width of air space between the two fabric layers. When this optimum is exceeded the outer layer is destroyed and the inner layer then behaves like a single layer. In Fig. 28 it is seen that with flame contact the temperature rise measured in simulated skin (110) beneath the assembly decreases steadily with increase in air space until the optimum

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width, in this instance 4 mm, is reached. Thereafter, convection currents arising within the air space contribute to the heating, the outer layer perforates, and the inner layer then heats by convection, and the fabric conducts the heat to the skin. Heat exchange calculations show that at the optimal air space, heat is transferred to the inner layer entirely by radiation. Thus, the double layer is

Water cooled power connections Shutter Sample

t

I \ \ Flux redistributor

Radiation source chamber

element

FIG.29. Compound thermal-radiation imaging system and source.

effective in this situation by blocking convection, confining heat transfer to low-temperature radiation, and finally, on perforation of the outer layer, providing a low-conductivity cover, even though a very thin one. The problem of protection against thermal radiation is considerably more complex, for in this instance not only thermal properties but optical properties as well play significant roles (111). For the resolution of this problem, studies were made using a heated graphite element as the radiation source (112). This apparatus is shown in Fig. 29 and consists of agraphite element mounted under a bell jar and heated by direct current. 'The radiation from the element is concentrated by a clam shell mirror arrangement. T h e apparatus provides intensities ranging from 0-18 cal/cm2 sec over a ;-inch square area with an evenness of distribution of *l'Z).

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Figure 30 shows the brightness temperature and irradiance plotted against the current flowing through the element. The graphite element is essentially a gray body radiator ; therefore, the distribution of the energy with respect to wavelength follows that of a black body and may be plotted. Figure 31 shows this distribution for two levels of temperatures, 2600°K corresponding to 10 cal/cm2sec and 2300°K corresponding to 6 cal/cm2sec. This figure also presents the reflectances measured with a spectrophotometer and reflectance sphere on three fabrics of the same material in different Irradiance (cal/cm2 sec)

Brightness temp. ('C)

r

0

100

1

=Brightness temp. = lrrodiance

200

300

400

500

600-

Current ( A )

FIG.30. Brightness temperature and irradiance versus current through graphite element.

colors. I t is seen that the white material reflects significantly throughout and beyond the range of interest while the orange reflects throughout the range of the 2600" source and the green, only through the range of the 2300" source. As the temperature of the source increases and the peak radiation shifts to shorter wavelengths the differences in reflectance, and conversely in the total energy absorbed due to color differences, will increase greatly. Transmittance also plays an important role in heat transfer due to radiation for, unlike the situation with flame contact, the solid fabric is transparent to certain wavelengths whereas in flame contact, heating by transmission occurs only through open interstices. To assess the relative importance of the optical properties, four fabrics of the same material were used. These differed from one another only in color and woven construction. The fabrics in single and in multiple layers

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132

2300 O K

--

Reflectance (9-1 : Stlvery rhtte [O I 5 3 m m \ : Olive green (0 175 m m ) -----: Orange (0 192 rnm)

Wavelength (pl

FIG.31. Energy distribution and reflectance of selected fabrics.

72 -

68 64 -

lrrodionce =I0 col/crn2sec *'Natural (silver white) twill A=Green Twill o=Green ploin weove x = Oronge ploin weove

60 56 52 ~

-

48 44 ~

4036 32 28 24 20 . 16

-

12

-

84.

Fabric thickness(rnm1

FIG.32. Temperature rise in simulated skin beneath selected fabrics during radiant heating.

HEATTRANSFER IN BIOTECHNOLOGY

133

were applied to skin simulants and measurements were made of the temperature rise at depth within the simulants on exposure to an irradiance of 10 cal/cm2 sec. These data are shown in Fig. 32 plotted as temperature rise within the simulant. It is seen that the temperature rise is least under the green and the white twill fabric, while that under the green plain weave is considerably greater, and under the orange plain weave it is highest of all. These differences decrease steadily as the thickness increases, reflecting the shift in relative importance from transmission in the thinnest layers to absorption in the thickest layers. It is noteworthy that green and white twill fabric are equal as thermal barriers despite obvious differences in reflectance noted earlier. How this comes about is illustrated by the analysis of the data shown in Table VIII. T h e data refer to thicknesses of 0.3 mm and 0.6 mm for the four fabrics. Reflection and transmission are shown as percentages of the source energy integrated over the wavelength range of the source output. These figures refer specifically to the level of 10 cal/cm2 sec and multiplied by 10 yield the actual energy flux transmitted or reflected. At a thickness of 0.3 mm the white and the orange fabrics are similar in reflectance, while the two greens resemble each other, and reflect less than the lighter colors. The total temperature rises after 2 sec of irradiation are the same for the two twill fabrics and greater for the plain weaves. However, the white twill fabric permits more energy to be transmitted directly, and a much greater proportion of the total rise is due to this transmission. Similarly, the orange plain weave which transmits more than any of the others yields the greatest total rise as well as the highest proportion due to transmission. At the greater thickness, reflection is slightly increased and transmission drops to a low level as optical opacity is approached. Heating due to transmission decreases appreciably in all four instances while the heating due to conduction approaches a common value. The effectiveness of air spacing in providing a thermal barrier in building insulation and like applications has led naturally into applications of the same principles for clothing protection. Certainly air spacing is effective in flame contact. However, in high-intensity radiation where transmission occurs this efficacy is greatly reduced, as shown in Fig. 33 where the effect of air spacing is measured in the same fabric by the same technique except that heating is provided by the graphite source. It is seen that in radiation heating, the temperature rise decreases steadily with increase in thickness of the air space although the damage to the outside layer passes through a maximum at an air gap of 3-4 mm. In this arrangement, the heat is reflected from the second fabric layer and contributes to the temperature rise in the first layer. As the air space widens, less reflected heat reaches the outer layer from behind and the damage to this layer then diminishes, even though it is closer to the

TABLE VIII OPTIC.AL PROPERTIES OF

Fabric

FIRE-RESISTANT POLYAhlIDE FILAMENT FABRICS AND TEMPERATURE RISEI N BACKING SKIN 2 SECONDS' EXPOSURE TIME AT AN IRRADIANCE OF 10 cal/cm2 sec

Thickness (mm)

lo:;

Reflection (0;)

R-E

Transmission ( O 0 )

SIMULANT AT

jo:;:" T*E

Total measured

Due to transmission

Due to conduction

White twill Green twill Green plain Orange plain

0.3 0.3 0.3 0.3

77 64 63 76

16.5 12.0 16.2 20.6

37.2 37.2 42.8 47.8

32.8 23.7 32.0 40.8

4.0 13.5 10.8 7.0

Il'hite twill Green twill Green plain Orange plain

0.6 0.6 0.6 0.6

80 69 69 78

4.0 3.5 3.2 5.7

18.7 18.7 21.8 23.8

7.8 7.1 6.8 11.3

10.9 11.6 15.0 12.5

? E

scn 4 0

r r

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source when the air space is greatest. The significance of this comparison with respect to thermal injury may be summarized as follows : Calorie for calorie available at the source, radiation heating is less injurious than flame contact by virtue of the large proportion of the heat which can be reflected at the surface even by colored fabrics. However, analysis of the dissipation of that portion of the energy which is not reflected, and therefore is effective in heating the clothed skin system, reveals that the techniques which proved strikingly successful in protection from flame contact are relatively poor in radiation protection due to differences in the heat-transfer

Heating by Flame contact 0 Thermal radiation Outer perforated

, , Dornage slight

FIG.33. Comparison of effect of air spacing in flame contact and radiation heating.

Outer layer nperforated

-

0

Q

I

2

3

4

5

6

Air spoce(rnrn)

mechanisms. Air spacing is less effective because the directly transmitted energy is attenuated only as a function of the distance from the source which, in any practical clothing system, cannot be significantly increased. Thus the familiar fish-net spacer, so effective in providing comfort in naturally occurring hot environments, is of little value in protection from high-intensity radiation. Rather, it appears that for maximal protection, where total reflection is impractical, the interfabric space should be filled with a material that will stop direct transmission and confine heat transfer to the mechanisms of conduction and low-temperature radiation exchange. 'There remains to be considered pure conduction through a single, solid layer in contact with the skin, and mixed modes of heating. Obviously, only straightforward insulation can protect against conduction heating. For this use, the equations of Griffith and Horton (113) have been shown to apply precisely to conduction through a two-layer system (110). They are repeated

136

ALICEM.STOLL

here for convenience and because Griffith and Horton (113) contains a typographical error which is not easily detected but significantly affects the calculations. Temperature rise at depth in layer 2 :

-[x - a{l

-

( D 2 / D 1 ) 1(2. /2

i

+ l))]

x 1 - erf x - a{l - (D2/D1) 2(D2t )1/2 where subscript 1 refers to top layer 1, subscript 2 refers to base layer 2, and U is the temperature rise, H the heat flux perpendicular to surface, X the total thickness from surface to point of temperature rise measurement, a the thickness of layer 1, D the thermal diffusivity = k / S , k the thermal conductivity, S the volume specific heat (density x specific heat), (k232)”’ + (A1 y = ( k 2 S 2 ) 1 /2 ( k l S1)u2, X = ( k 20;”-k lDi/2)-1,and t the time. ~~

w2

Temperature rise at surface :

Temperature rise at interface :

Temperature rise at surface of semi-infinite homogeneous solid :

Uo = 2H{(t/S1k

l ~ ) }= ” ~ ( 2 H / k , ){(Dl t/7r)}1’2

(41) Mixed modes of heating, for example, the combined convection and radiation which occurs in fire storms, constitute special cases in which the relative weight of each type must be specified in order to determine the most

HEATTRANSFER IN BIOTECHNOLOGY

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effective means of protection. Such specifications are not usually possible and therefore no ideal solution exists. Combinations of reflection and insulation as incorporated in modern fire-fighter ’s equipment are “best available” solutions which are constantly under study. V. Concluding Remarks Much pertinent material is conspicuous by its absence. Notable in this category is any mention of the entire field of bioenergetics, i.e., energy transformations within living organisms, starting at the molecular level. Similarly neglected is bionics, the transport of heat through ionic action, as well as the older fields of thermoregulation, thermogenesis, nutrition, and many others which conceivably could be included within even the limited scope delineated at the outset. It is expected that subsequent writers will undertake discussions in these and additional areas. It is hoped that the present work will serve to introduce the field of biotechnology to some and to up-date some aspects of it for others. Because the biological systems and their terminology may be an entirely new area of interest to some, a bibliography of basic references has been appended. This listing is by no means comprehensive or exhaustive, but rather representative of several different approaches to the study of biology and human physiology which may be helpful in orienting the newcomer to the world of animate heat exchangers. ACKNOWLEDGMENT The kind permission to use data and figures granted by the many authors cited in the text is very gratefully acknowledged.

REFERENCES 1. T. Hill, Kata-thermometer in studies of body heat and efficiency. Med. Res. Council, Spec. Rept. Ser. 73,48 (1923). 2 . A. F. Dufton, The Eupathoscope, Dept. Gt. Brit. Sci. 2nd. Res. Bldg. Res., Tech. Paper No. 13 (1932). 3 . H. M . Vernon, J . 2nd. Hyg. Toxicol. 14, 328-338 (1932). 4. C. 0. Mackey and L. T. Wright, Heating, Piping, Air Conditioning 18,107-1 11 (1946). 5. C.-E. A. Winslow and L. Greenberg, Heating, Piping, Air Conditioning. 7, 41-43 (1935). 6. J. D. Hardy, E. F. DuBois, and G. F. Soderstrom, J . Nutr. 15,461 (1938). 7 . C.-E. A. Winslow, in “Temperature: Its Measurement and Control in Science and Industry,” p. 509, Reinhold, New York, 1941. 8. J. D. Hardy and E. F. DuBois, in “Temperature: Its Measurement and Control in Science and Industry,” p. 537. Reinhold, New York, 1941.

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9. C.-E. A. Winslow, A. P. Gagge, and L. P. Herrington, Am. J . Physiol. 127, 505-518 (1939). 10. W. M. Elsasser, HarvardMet. Stud. 6 (1942). 1 1 . B. Haurwitz, “Dynamic Meterology, Radiation.” McGraw-Hill, New York, 1951. 12. C. H. Richards, A. M. Stoll, and J. D. Hardy, Rev. Sci. Instr. 22,925 (1951). 13. A. M. Stoll and J. D. Hardy, J . Appl. Physiol. 5, 117 (1952). 14. A. M. Stoll, Rev. Sci. Instr. 25, 184 (1954). 15. J. D. Hardy, J . Clin. Invest. 13, 593 (1934). 16. V. Guillemin, Jr., Heated sphere anemometer. Engineering Division, Aero. Med. Lab., Rept. TSEAA-695-75. Wright-Patterson Air Force Base, Ohio, 1947. 17. A. M. Stoll, C. H. Richards, and J. D. Hardy, “The Solar Radiometer”. Unpublished. 18. A. M. Stoll, J. D. Hardy, and C. H. Richards, The servo-operated panradiometer, an engineering need in physiological heat transfer studies. Semiann. Meeting, Heat Transfer Div. Am. SOC.Mech. Engrs., San Francisco, 1957. 19. A. M. Stoll and J. D. Hardy, Trans. Am. Geophys. Union 36, No. 2, 213, (1955). 20. N. Sissenwine, R. Anateg, and P. Meigs, Environ. Protect. Rept. 178, Office of Quartermaster Gen., Dept. Army, 5, 58 (1951). 21. A. M. Stoll, Federation Proc. 10, Pt. I, 133 (1951). 22. J. D. Hardy and A. M. Stoll, J . Appl. Physiol. 7, 200-211 (1954). 23. A. P. Gagge, in “Temperature: Its Measurement and Control in Science and Industry,” p. 544. Reinhold, New York, 1941. 24. A. H. Woodcock, in ”Medical Climatology” (S. Licht, ed.), Chapter 21, p. 557. Waverly, Baltimore, Maryland, 1964. 25. A. P. Gagge, A. C. Burton, and H. C. Bazett, Science 94,428 (1941). 26. D. H. Kang, P. K. Kim, B. S. Kang, S. H. Song, and S. K. Hong, J . Appl. Physiol. 20, 46-50 (1965). 27. H. Hunt, E. Reeves, and E. L. Beckman, An experiment in maintaining homeostasis in a long distance underwater swimmer. Rept. No. 2, MR005. 13-4001.06. Naval Med. Res. Inst., Bethesda, Maryland, 1964. 28. R. G. Nevins, F. W. Holm, and G. H. Advani, Heat-loss analysis for deep-diving oceanauts. A S M E Meeting, Tech. Sessions Heat Transfer Div., 1965, Paper No. 65-WA/HT-25. 29. E. L. Beckman, Proc. 2nd Symp. Underwater Physiol. Publ. No. 1181, pp. 247-266. Natl. Acad. Sci., NRC, Washington, D.C. 1963. 30. L. D. Carlson, A. C. L. Hsieh, F. Fullington, and R. W. Elsner, J . Aviation Med. 29, 145-152 (1958). 31. M. van Dilla, R. Day, and P. A. Siple, in “Physiology of Heat Regulation and the Science of Clothing” (L. H. Newburgh, ed.), pp. 374-388. Saunders, Philadelphia, Pennsylvania, 1949. 32. H. S. Belding, in “Physiology of Heat Regulation and the Science of Clothing” (L. H. Newburgh, ed.), p. 361. Saunders, Philadelphia, Pennsylvania, 1949. 33. E. L. Beckman, E. Reeves, and R. F. Goldman, Current concepts of practices applicable to the control of body heat loss in aircrew subjected to water immersion. 36th Ann. Sci. Meeting Aerospace Med. Assoc. NezoYork City, 1965. Aerospace Med. Assoc., to he published. 34. I,. G. C. E. Pugh, 0. G. Edholm, R. H. Fox, H. S. Wolff, G. R. Hervey, W. H. Hammond, J. M. Tanner, and R. H. Whitehouse, Clin. Sci. 19, 257-273 (1960). 35. J. R. Brett, Sci. Am. 213, 80-85 (1965). 36. S. K. Hong, H. Rahn, D. H. Kang, S. H. Song, and B. S. Kang, .J. Appl. Physiol. 18, 457-465 (1963).

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37. R. F. Goldman, Science in Alaska. Proc. 15th Alaska Sci. Conj., Fairhnks, Alaska, March 1965, pp. 401-419. 38. T. F. Irvine and K. R. Cramer, Thermal analysis of space suits in orbit. ’ “.4DD Tech. Note 60-145, May 1960. 39. D. L. Richardson, Study and development of materials and techniques for passive thermal control of flexible extravehicular space garments. Air Force Sy-stems Command, AMRL-TR-65-156. Wright-Patterson Air Force Base, Ohio, September 196.5. 40. C. C. Adams, in “Medical and Biological Problems in Space Flight” (G. H. Bourne, ed.), p. 239. Academic Press, New York, 1963. 41. W. L. Burriss, S. H. Lin, and P. J. Berenson, Study of the thermal processes for man-in-space, Sect. 2-19. N A S A CR-216. NASA, Washington, D.C., April 1965. 42. W. L. Burriss, S. H. Lin, and P. J. Berenson, Study of the thermal processes for man-in-space, Sect. 6. N A S A CR-216. NASA, Washington, D.C., April 1965. 43. W. L. Burriss, S. H. Lin, and P. J. Berenson, Study of the thermal processes for man-in-space, Sect. 3. N A S A CR-216. NASA, Washington, D.C., April 1965. 44. W. C. Kincaide, Mech. Eng. 87, No. 11, 49-53 (1965). 45. J. H. Veghte, Aerospace M e d . 36, 964-967 (1965). 46. F. C. Houghten and C. P. J. Yaglou, A m . SOC. Heat. and Ventil. Engrs. 29, 361 (1923). 47. A. P. Gagge, G. M. Rapp, and J. D. Hardy, A S H R A E ( A m . SOC.Heating, Rejrig. Air-cond. Engrs.) J . 6, 67-71, (1964). 48. C. P. Yaglou, A S H V E ( A m . SOC.Heating Ventilating Engrs.) Trans. 53, 307 (1947). 49. W. Koch, B. H. Jennings, and C. M. Humphreys, A S H R A ( A m . Soc. Heating, Rejrig. Air-cond. Engrs.) Trans. 66, 264 (1960). 50. R. G. Nevins, F. H. Rohles, W. Springer, and A. M. Feyerherm, Criteria for thermal comfort. A S H R A E Semi-Ann. Meeting Houston, Texas, Jan. 24-27, 1966. 51. “ASHRAE Guide and Data Book, Physiological Principles,” Chapter 7. Fundamentals and Equipment for 1965 and 1966. A m . SOC.Heating, Rrjrigerating and A i r Conditioning Engrs., New York, 1965. 52. R. G. Nevins and F. H. Rohles, Jr., A S H R A E ( A m . SOC.Heating, Rejrig. Air-cond. Engrs.) J. 7, No. 5, 89 (1965). 53. K. I. Parczewski aad R. S. Bevans, A S H R A E ( A m . SOC.Heating, Rejrig. Air-cond. Engrs.)J. 7, NO. 6, 80-86 (1965). 54. R. Stenuit, N a t l . Geographic 127, No. 4, 534 (1964). 55. H. A. O’Neal, G. Bond, R. Lanphear, and T. Odum, Project SEALAB I, An experimental eleven-day saturation dive at 193 feet. ONR R e p . ACR-108, June 14, 1965. 56. Booda, L. L., Under Sea Technol. 6, 3 4 3 8 (1965). 57. Booda, L. L., Under Sea Technol. 6, No. 10, 21 (1965). 58. J. Y. Cousteau, Under Sea Technol. 7, No. 1, 25 (1966). 59. “Undersea Vehicles for Oceanography.” ICO Pamphlet No. 18, p. 18, October 1965. Interagency Comm. on Oceanography of the Fed. Council for Sci. and Technol. 60. J. L. Kinsey, Observations on the habitability of submarines in northern waters. Med. Res. Lab. R e p . No. 197, Vol. XI, No. 14, New London, Connecticut, 1952. 61. A. P. Gagge, A m . ,I.Physiol. 120, 277 (1937). 62. P. Krantz, A S H R A E ( A m . SOC.Heating, Rejrig. Air-cond. E t i g r s . ) ~ /8,68-77 . (1964). 63. A. P. Gagge, L. P. Herrington, and C.-E. A. Winslow, A m . .I.H y g . 26, 111 (1937). 64. J. E. Janssen, Thermal comfort in space vehicles. ASME Paper No. 59A-207, presented Nov-Dec 1959. 65. C. D. King, Environmental control. Space/Aeronatct. 44, No. 2, 122-126, R and D Tech. Handbook, 1965-1966. 66. I. Stambler, Space/Aeronaict. 44,No. 1, 46-54 (1965).

140

ALICEM. STOLL

67. W. J. Blatz, R. F. Pannett, E. L. Salgers, and G. J. Weber, Astronaut. Aeron. 2, 30-40 (1964). 68. NASA-NAS nutrition in space and related waste problems. NASA-SP-70 1964. 69. R. G. Neswald, SpacelAeronaut. 44,No. 2, 70-78 (1965). 70. J. B. Campbell, ed., Aerospace in Perspective (Spec. issue). SpacelAeronaut. 45, No. 1, 1966. 71. M. Feider and C. M. Bunche, A . M . A . Arch. Dermatol. Syphilol. 69, 563-569 (1954). 72. A. M. Stoll, .I. Heat Transfer 82, No. 3, 239-242 (1960). 73. J. D. Hardy and G. F. Soderstrom, J . Nutr. 16, 493 (1938). 74. P. Webb, ed., Properties of skin, in “Bioastronautics Data Book,” p. 122. NASA SP-3006. NASA, Washington, D.C., 1964. 75. R. Seckendorff and W. Randall, Federation Proc. 18, Pt. 1, 140 (1959). 76. W. S. Spector, ed., “Handbook of Biological Data,” pp. 77, 260. WADC TR-56-273. Wright-Patterson Air Force Base, Ohio, 1956. 77. S. E. Levy and A. Blalock, Am. J . Physiol. 118, 368 (1937). 78. C.-E. A. Winslow, L. P. Herrington, and A. P. Gagge, A m . J . Physiol. 120,l (1937). 79. A. J. H. Vendrik and J. J. Vos, J . Appl. Physiol. 11, 211-215 (1957). 80. M. Lipkin and J. D. Hardy, J . Appl. Physiol. 7, 212 (1954). 81. A. M. Stoll and L. C. Greene, J . Appl. Physiol. 14, 373-382 (1959). 82. F. C. Henriques, Jr. and A. R. Moritz, A m . J . Pathol. 23, 531-549 (1947). 83. J. D. Hardy, A m . J . Physiol. 127, 454-462 (1939). 84. Thermography and its clinical applications. Ann. N . Y. Acad. Sci. 121, Article 1, 1-304 (1964). 85. M. M. Lee and C. K. Ng, Human Biol. 37,91-103 (1965). 86. D. W. Rennie, B. G. Covino, B. J. Howell, S. H. Song, B. S. Kang, and S. K. Hong, J . Appl. Physiol. 17, 961-966 (1962). 87. D. Y. Lee, S. K. Hong, and P. H. Lee, J . Appl. Physiol. 20 No. 1, 51-55 (1965). 88. E. R. Buskirk, R. H. Thompson, and G. D. Whedon, in “Temperature: Its Measurement and Control in Science and Industry,” Vol. 3, pp. 429-442. Reinhold, New York, 1963. 89. J. S. Wilson and J. A. Moncrief, Ann. Surg. 162, 130-134 (1965). 90. C. H. Wyndham, J. F. Morrison, and C. G. Williams, J . Appl. Physiol. 20, No. 3, 357-364 (1965). 91. C. H. Wyndham, R. K. McPherson, and A. Munro, .I. Appl. Physiol. 19, No. 6, 1055-1058 (1964). 92. C. H. Wyndham, B. Metz, and A. Munro, J . Appl. Physiol. 19, No. 6, 1051-1054 (1964). 93. T. H. Benzinger and C. Kitzinger, in “Temperature: Its Measurement and Control in Science and Industry,” Vol. 3, pp. 87-109. Reinhold, New York, 1963. 94. T. H. Benzinger, Proc. Natl. Acad. Sci. 45, 645 (1959). 95. C. H. Wyndham, J . Appl. Physiol. 20, No. 1, 31-36 (1965). 96. R. D. McCook, R. D. Wurster, and W. C. Randall, .I. Appl. Physiol. 20, No. 3, 371-378 (1965). 97. E. Hendler, J. D. Hardy, and D. Murgatroyd, in “Temperature: Its Measurement and Control in Science and Industry,” Vol. 3, pp. 211-237. Reinhold, New York, 1963. 98. F. C. Henriques, Jr., Arch. Pathol. 43,489-502 (1947). 99. J. D. Hardy, ,I. Appl. Physiol. 5, 725 (1953). 100. A. M. Stoll, Ann. N . Y . Acad. Sci. 121, Article 1, 53 (1964). 101. F. G. Ebaugh, Jr. and R. Thauer, J . Appl. Physiol. 3, 173 (1950).

HEATTRANSFER IN BIOTECHNOLOGY

141

102. D. R. Kenshalo and J. P. Nafe, in “Temperature: Its Measurement and Control in Science and Industry,” Vol. 3, pp. 231-243. Reinhold, New York, 1963. 103. J. D. Hardy, H. G. Wolff, and H. Goodell, “Pain Sensations and Reactions.” Williams & Wilkins, Baltimore, Maryland, 1952. 104. M. B. Kreider, in “Medical Climatology” (S. Licht, ed.), pp. 453-454. Waverly Press, Baltimore, Maryland, 1964. 105. H. T. Meryman, Physiol. Rev. 27, 233 (1957). 106. F. B. Hershey, in “Research in Burns” (C. P. Artz, ed.), pp. 298-304. Davis, Philadelphia, Pennsylvania, 1962. 107. A. R. Moritz and F. C. Henriques, Jr., A m . J . Pathol. 23, 695 (1947). 108. A. M. Stoll, J. D. Hardy, and L. C. Greene, J . Appl. Physiol. 15,489 (1960). 109. A. M. Stoll, Aerospace Med. 7 , 846-850 (1962). 110. A. M. Stoll, M. A. Chianta, and L. R. Munroe, J . Heat Transfer 86,449-456 (1964). 1 1 1 . A. M. Stoll and M. A. Chianta, Federation Proc. 23,281 (1965). 112. D. L. Richardson, “A Thermal Radiation Heat Source and Imaging System for Biomedical Research.” Arthur D. Little, Inc., Contract No. N62269-1388, U.S. Naval Air Develop. Center, Johnsville, Warminster, Pennsylvania, March 1962, Rept. NADC-MR-6503, Dec. 1965. 1 1 3 . M. V. Griffith and G. K. Horton, Proc. Phys. Soc. (London) 58,481487 (1946).

BIOLOGICAL BIBLIOGRAPHY P. L. Altman and D. S. Dittmer, “Biology Data Book.” Fed. of Am. SOC.for Exptl. Biol., Washington, D.C., 1964. “Handbook of Physiology.” Sect. 4: Adaptation to the Environment Sectioned., D. B. Dill. Am. Physiol. SOC.,Washington, D.C., 1964. Arthur C. Guyton, “Textbook of Medical Physiology.” Saunders, Philadelphia, Pennsylvania, 1961. A. L. Lehninger, “Bioenergetics-The Molecular Basis of Biological Energy Transformations.” Benjamin, New York, 1965. Sidney Licht, “Medical Climatology”, Waverly, Baltimore, Maryland, 1964. William Montagna, “The Structure and Function of Skin.” Academic Press, New York, 1962. L. H. Newburgh, “Physiology of Heat Regulation and the Science of Clothing.” Saunders Philadelphia, Pennsylvania, 1949.

Effects of Reduced Gravity on Heat Transfer ROBERT SIEGEL Lewis Research Center National Aeronautics and Space Administration Cleveland. Ohio I . Importance of Studies at Reduced Gravity . . . . . . A . Space Applications . . . . . . . . . . . . B. Gravity as an Independent Parameter . . . . . . C . Elimination of Free Convection . . . . . . . . D . Present Objective . . . . . . . . . . . I1 . Experimental Production of Reduced Gravity . . . . . A . Drop Tower . . . . . . . . . . . . . B. Airplane Trajectory . . . . . . . . . . . C . Rockets and Satellites . . . . . . . . . . . D . Magnetic Forces . . . . . . . . . . . . 111. Free Convection . . . . . . . . . . . . . A . Fluid Flow in Reduced Gravity . . . . . . . . B . Heat Transfer . . . . . . . . . . . . . C . Transient Development Times for Boundary Layer . . IV . Pool Boiling . . . . . . . . . . . . . . . A . Nucleate-Pool-Boiling Heat Transfer . . . . . . . B . Critical Heat Flux for Pool Boiling . . . . . . . C . Transition Region for Pool Boiling . . . . . . . D . Minimum Heat Flux between Transition Boiling and Film Boiling . . . . . . . . . . . . . . . E . Film-Boiling Heat Transfer . . . . . . . . . F. Dynamics of Vapor Bubbles in Saturated Nucleate Boiling . G . Bubble Dynamics in Subcooled Pool Boiling . . . . H . Vapor Patterns for Film Boiling in a Saturated Liquid . . . . . . . . . . . . V . Forced Convection Boiling A . Reduced Gravity Effect on Two-Phase Flow . . . . B . Two-Phase Heat Transfer . . . . . . . . . . C . Designs Involving Substitute Body Forces . . . . . VI . Condensation without Forced Flow . . . . . . . . A . Laminar Film Condensation on a Vertical Surface . . . B . Laminar-to-Turbulent Transition and Turbulent Flow . C . Transient Time to Establish Laminar Condensate Film .

143

144 144 144 145 145 146 146 150 150 150 151 152 154 156 158 158 166 171 171 173 174 194 198 200 200 201 205 206 207 208 209

144

ROBERT SIEGEL VII. Forced Flow Condensation . . . . . . A. Flow Behavior in Low Gravity . . . . B. Pressure Drop . . . . . . . . . C . Vapor-Liquid Interface . . . . . . D. Noncondensable Gas . . . . . . . VIII. Combustion . . . . . . . . . . . A. Candle Flame . . . . . . . . . B. Fuel Droplets . . . . . . . . . C. Solid Fuels . . . . . . . . . . IX. Summary and Areas for Further Investigation . Nomenclature . . . . . . . . . . References . . . . . . . . . . .

. . .

.209

. . . . 210

. . . . . . . .

. . . . . .

.

.

.

. . . . . .

. . .

. . .

. . .

211 213 . 214 . 215 . 215 . 215 . 217 . 218 , 2 2 1 , 2 2 2

I. Importance of Studies at Reduced Gravity

T h e study of heat-transfer processes in reduced and zero gravity is of interest for reasons of both practical and basic research importance.

A. SPACE APPLICATIONS One of the fascinating aspects of flight in space is the weightless condition experienced by the material within a vehicle that derives its acceleration solely from the force of the local gravitational field. In this instance, the inertial force will exactly oppose the gravitational force, and the contained material will experience a zero-gravity condition relative to the vehicle. If not tied down, the material will “free float” within the vehicle. This would be the case in an orbiting satellite, space capsule, space station, or any freely coasting device in space. In space, a vehicle that has a small spin or is undergoing a small acceleration will have a gravity field equal to a fraction of that on earth. This would also be true for a device on the surface of the moon. As a consequence, the design of systems for space applications has made it necessary to consider low gravity effects with regard to factors such as fluid orientation, influence on the human body, and the present subject of heat transfer.

B. GRAVITY AS AN INDEPENDENT PARAMETER T h e gravity field is one of the independent parameters in many theoretical solutions and experimental correlations. When experiments are performed throughout a range of gravity fields, a means is provided for further evaluating the applicability of assumptions and theoretical equations employed in an analysis. Experimental correlations based on observations at earth

REDUCED GRAVITY ON HEATTRANSFER

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gravity can be tested to see whether they present the correct gravity dependence. Low gravity tests also provide a means by which gravity-dependent phenomena can be isolated from those that are independent of gravity, and this can further our basic understanding. For example, when studying bubble dynamics in nucleate pool boiling, the buoyancy force can be diminished or removed by low gravity testing. T h e remaining dynamic, drag, and surface tension forces can then be studied more clearly.

C. ELIMINATION OF FREE CONVECTION Low gravity tests can be used to remove unwanted secondary effects resulting from free convection. For example, if laminar-flow heat-transfer tests could be performed at zero gravity in an orbiting laboratory, larger temperature differences and lower flow rates than on earth could be utilized without free convection becoming of major influence. This could lead to additional substantiations of the laminar flow solutions and the momentum and energy equations from which they are derived. One of the difficulties in measuring the thermal conductivity of gases in a parallel-plate-type apparatus is the heat transferred between the plates by free convection. This unwanted component of heat flow could be eliminated by an orbital-type test. The use of very low gravity can provide a convenient change of scale by eliminating free convection. For example, the study of individual fuel droplet combustion in a very fine spray is made difficult by the small size of the drops. The drops are so small that negligible free convection develops around them when they are burned at earth gravity. For convenience, larger drops can be studied provided that the free convection which would develop around the drops in an earth environment is eliminated by utilizing low gravity tests.

D. PRESENT OBJECTIVE Although the gravity parameter appeared in many theoretical and experimental heat-transfer correlations throughout the development of the science of heat transfer, little had been done to experimentally verify the dependence at reduced gravities until the middle 1950’s. Previous to that time, some limited research had been done at high gravities-specifically, with reference to heat transfer in rotating machinery. It was the design of space devices, however, that sparked the interest in low gravity conditions and led to a rapid expansion of low gravity experimentation. This paper will review and summarize low gravity information up to about November 1966. Only heat transfer will be considered. Aspects of low

ROBERT SIEGEL

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gravity hydrodynamics or fluid orientation will be mentioned only when they are especially pertinent to the heat-transfer behavior. (A bibliography and brief review of hydrodynamic aspects is given by Habip (I).) The heattransfer subjects discussed are free convection, pool and forced flow boiling, condensation in stationary vapor and in forced flow, and combustion. Experimental and theoretical results will be compared when possible. When experimental results are not available, some of the theoretical correlations will be interpreted with regard to reductions in gravity. 11. Experimental Production of Reduced Gravity

T o experimentally study reduced and very low gravity behavior, a facility is needed that will provide a specified gravity reduction with reasonable convenience. Some of these facilities have been reviewed by Unterberg and Congelliere ( l a ) , and some pertinent characteristics will be considered here.

A. DROPTOWER Short periods of reduced gravity can be obtained with earth-bound equipment. A mass m that has a downward acceleration a can be regarded as having an upward inertial force ma. This force is in opposition to the earth gravity force; hence, when moving in the frame of reference of the accelerating mass, the effective acceleration is g = g, - a. Reduced gravity fields with magnitudes between 1 and Og, can be obtained by letting an experiment accelerate downward with an acceleration in the range from 0 to lg,. A body in perfectly free fall with no resisting force provides a zerogravity environment in the reference frame of the falling body. Therefore, to obtain reduced gravity by this means requires a structure of some type where the experiment can be hoisted and dropped under controlled conditions. A typical example of a free-fall drop tower is given in Fig. 1. The experiment could also be dropped from a natural elevation. For conditions approaching true zero gravity, air drag must be compensated for or eliminated. The drop tower could be enclosed so that it can be evacuated. Since this may not be feasible in most instances, the relative effect of air drag for a given falling package can be reduced by increasing the package weight. This, however, complicates stopping the package at the end of the fall. Air drag can be essentially eliminated by having the experiment freely falling within a larger falling container (2, 3) (Fig. 2). The outer container is acted upon by the air drag and falls slightly more slowly than the inner container. The experimental package could also be pulled downward mechanically to com-

REDUCED GRAVITY ON HEATTRANSFER

w

h

o

p

147

&

FIG.1. Schematic of 85-ft drop tower, NASA Lewis Research Center (2).

pensate for air drag (4).T o obtain a range of reduced gravities, the drop tower can be counterweighted as shown in Fig. 3 ( 5 ) or a braking rocket could be attached to the package. The disadvantage of using drop towers is that they provide a relatively short test time. For a zero-gravity test, the test time t as a function of drop tower height s is given by s = *g,P

A 1-sec test requires a height of 16.1 ft, a 3-sec test 144.9 ft, and a 5-sec test

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402.5 ft. A drop tower facility at the NASA Lewis Research Center consists of a 550-ft shaft (in the ground) that will provide a test time of -5 sec. The shaft can be evacuated to eliminate air drag. The test package can also be projected upward from the bottom of the shaft and thereby double the test time to 10 sec. Longer test times magnify extremely the difficulty in bringing the package to a stop without destruction at the end of the fall. Since this review is concerned with heat transfer, space will not be taken to dwell on the engineering difficulties involved in the operation of drop towers. Rather the purpose here is to emphasize that drop tower test times are limited by

-..

Music-wi re support

Wire-release mechanism 1

LBase rounded to reduce a i r drag (a) Before test drop, experiment at top of drag shield.

shield

II II II

~ L d Ld

Ld

lDeceleration spikes

(b) During test drop, experiment moves downward within drag shield.

(c) Just before drag shield is decelerated.

Frc. 2. Schematic showing position of experiment package within drag shield (a) before, (b) during, and (c) at end of test drop (2).

practical considerations to a range of several seconds in duration. This is a short time when considering thermal equilibrium requirements for most heat-transfer experiments. When interpreting drop tower experimental results, the fact that transient conditions may be involved should be kept in mind. Fortunately, some processes appear to have reasonably short transient times. For example, in boiling studies the bubble generation is very rapid and the bubbles appear to adjust quickly to the change from normal to reduced gravity. However, even for boiling there is some question as to whether the thermal layer of superheated fluid adjacent to the heated surface can adjust rapidly enough to the lower gravity condition. The

REDUCEDGRAVITY ON HEATTRANSFER

149

FIG.3 . Counterweighted drop tower used for boiling experiment (5). Drop height, 12.5 ft.

150

ROBERTSIEGEL

thermal-layer thickness is probably partially governed by free convection, which has a slower time response than the bubble generation.

B. AIRPLANE TRAJECTORY T o obtain longer test times at zero or fractional gravities an airplane can be used. For zero gravity the airplane flies in a Keplerian ballistic arc so that the experiment free floats within the cabin. By this means times of 20 to 30 sec can be obtained. The gravity level is low, but there are large initial disturbances incurred from the variation in acceleration as the airplane first dives and then turns upward into the zero-gravity flight path. For a large experimental package it is impractical to have it free float. With the package tied down it is possible to control the low gravity to within k O.O2g, with an experienced pilot (6, 6a). This gravity level is an adequate approximation to Oge for the many cases where other forces are present in sufficient magnitude so that a small gravity force has negligible influence. When the experiment is fastened to the airplane, fractional gravities can also be obtained, but again there is the difficulty of the pilot to control the desired gravity within a close tolerance. C. ROCKETS AND SATELLITES Longer durations of very close to zero gravity can be obtained by having the experiment on a rocket flight, in a satellite, or in a manned space capsule (7, 8). Such tests, however, become quite costly. Nevertheless, these tests may be the only way to finally determine the true long-term performance in some devices. 'The experiments must be designed to withstand the high launch acceleration, and the space vehicle has to be stabilized against tumbling or spinning. A planned orbital boiling test is described by Kirkpatrick (9).T h e feasibility of several orbital heat transfer and fluid mechanics experiments is examined by Nein and Arnett (9a).

D. MAGNETIC FORCES A technique that provides a range of gravity fields and can operate for long durations is the use of a magnetic field with a fluid that has magnetic properties. T h e fluid is placed in a magnetic gradient such as in the core of a solenoidal magnet, and the magnetic force can be adjusted to counteract all or part of the gravitational body force. Some difficulties are encountered in obtaining magnetic fields that provide their force in a perfectly unidirectional manner because of end effects in a finite-sized magnet. Also, the liquid and vapor phases of the fluid, as would be present in a boiling or condensation experiment, are affected differently by the magnetic field, and hence there is not a perfectly uniform reduced gravity simulation thoughout the two-phase mixture.

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151

Another means of utilizing magnetic forces is an electromagnetic technique discussed by Kirk (10, 11).This technique arises from the electromagnetic principle that an electric current density will interact with a component of a magnetic field transverse to it and yield a body force normal to the current-magnetic field plane. The experiment could consist of a container of liquid metal with electrodes on two sides so that a current is passed horizontally through the fluid. The container is then placed between the poles of a magnet so that the magnetic field lines are horizontal and normal to the current flux. If the electric current is in the proper direction, a force will be produced in opposition to the earth gravity field and hence can be used to obtain reduced gravity conditions. This method has a number of disadvantages such as the production of joule heating in the fluid, and the fluid motion tending to distort the magnetic flux lines. The disadvantages are discussed in detail by Kirk (11).One serious difficulty that would be encountered with this method in a two-phase system such as pool boiling is that the presence of vapor voids distorts the electric field. The resulting electromagnetic body force is then nonuniform. Also, the vapor is not influenced by the electromagnetic force in the same manner as the liquid so there is not a uniform reduced gravity simulation throughout the twophase mixture. 111. Free Convection

Gravity or a similar body force field appears in the analytical predictions and experimental correlations for free convection as a natural consequence of the buoyancy force being the defined driving potential for the flow. The important dimensionless group in free convection is the Rayleigh number Ra, which is equal to the product of the Grashof G r and Prandtl Pr numbers. The Rayleigh number contains a characteristic dimension L of the system that can be the height of a vertical plate, diameter of a heated wire, or thickness of the convective boundary layer. For a surface at a specified temperature T, , the Rayleigh number is given by

where throughout the present discussion the body force will be limited to that resulting from the gravitational field g. For a surface dissipating a specified heat flux q, a modified Rayleigh number is utilized where in the dimensional analysis T, - Tb is replaced by gL/k to give

152

ROBERTSIEGEL

I n free convection the magnitude of the Rayleigh number plays an important role in determining the threshold of convective motion, the limit where boundary-layer theory applies, and the transition from laminar to turbulent flow: it is also directly related to the Nusselt number. T h e gravitational field appears to the first power in Ra and Ra*; hence, when considering a gravity reduction it may be thought of as a proportionate reduction in Rayleigh number.

FLOWIN REDUCED GRAVITY A. FLUID 1. Threshold of Convective Motion Since a gravity reduction can result in low Rayleigh numbers, consideration must be given to the lower limit of free convective motion. T h e onset of free convection is a stability problem :extensive discussions of it are given by Ostrach (12) and Stuart (13), and a brief summary is given by Grober et al. (14). T h e stability is sensitive to the orientation of the heated layer. For a fluid layer confined between two horizontal plates and heated from below, Jeffreys (15)and Low (16) computed a critical Ra of 1706 based on the thickness of the layer. Above this value flow will start. This was confirmed by Schmidt and Saunders (17) and Malkus (Zd), among others, for a layer of water. Asa numerical illustration, if water at an average temperature of 100°F is contained between two horizontal plates 0.1 ft apart and the lower plate is 100°F warmer than the upper, a reduction of gravity field to 3.2 x 10-sge would cause the fluid motion to cease. If the upper surface of the layer is not bounded by a solid surface but is merely a free surface, the critical Ra is reduced to 1101 [Low (16) obtained 11081 according to the theory of Pellew and Southwell (19). For a vertical fluid layer the situation is not as clear as the horizontal case, and the onset of convection depends on the geometry enclosed between the vertical surfaces. A rectangular enclosure having two isothermal vertical walls at different temperatures, of height H , spaced L apart, and enclosed at the top and bottom by insulated plates has been studied by Eckert and Carlson (20). They found for air that the regime of pure conduction ended at RaL = 500H/L, which agreed fairly well with the analysis of Batchelor (21). A similar experiment by Emery and Chu (22)for water and oil (Pr = 30,000) and HIL = 10 and 20 indicated that the regime with conduction only is encountered for RaL < lo3.As a result of the direct dependence on Kayleigh number, the onset of the convective regime will be delayed in direct proportion to the magnitude of g as the gravity field is reduced. 2. Boundary-Layer 7’hrory When the Rayleigh number is sufficiently large so that fluid motion is present, a boundary layer may be established on the surface, depending on

REDUCED GRAVITY ON HEAT TRANSFER

153

the surface orientation. Consider a vertical surface at constant temperature with laminar free convection. The boundary-layer thickness at a distance x from the leading edge is given by Eckert and Drake (23) from an integral theory as -6 _- 3.93(0.952 + Pr)114 1 (3) X Pr1/4 (Ep With all quantities held constant except g, Eq. (3) indicates that the boundary-layer thickness depends on g-1/4so that the thickness becomes very large as gravity approaches zero. For the usual boundary-layer analysis, the convective layer along the wall is assumed thin relative to the characteristic dimensions of the system. Hence if the gravity field is small, care should be taken to examine the Rayleigh number to determine whether boundarylayer-type correlations can be utilized. This will be discussed later relative to the expressions for heat transfer. T o give a numerical example, if Ra, is lo4 and Pr = 1, Eq. (3) yields S/x = 0.465 and it is doubtful that the thinboundary-layer assumptions could apply. Of course even at earth gravity a low Ra, region is encountered near the leading edge of the plate, but generally it occupies a short length and thus is not very significant except for a body of small size. In low gravity this “leading-edge region” becomes proportionately larger. 3. Boundary-Layer Transition Transition from laminar to turbulent flow depends on the Rayleigh number and for a vertical plate at uniform temperature occurs in the vicinity of Ra,= lo9. For a plate with uniform heat flux the Ra,* at transition is about 10” (24).There is no reason to believe that the basic mechanisms of fluid stability and transition should be influenced by gravity, and consequently these transition Rayleigh numbers should apply in the reduced gravity range. As a consequence of thegx3 in Ra, , as gravity is reduced for a plate at uniform temperature the physical location of transition is shifted to a larger x in relation to gp1l3.

4. Hayleigh Numbers Encountered in Low-Gravity Applications As discussed by Chin et al. (24), it is of interest to explore the range of Rayleigh numbers that would be encountered in reduced gravity applications to provide some practical judgment for the flow regimes to be expected. This will be done for three fluids: air, water, and liquid hydrogen. Free convection heat transfer is often of importance in space applications with reference to heating of fuel inside storage tanks. Here the wall heat flux is specified as imposed by solar radiation. For this reason the modified

ROBERTSIECEL

154

Rayleigh number Ra," will be considered. Figure 4 shows Ra," as a function ofg/g, for a length dimension x of 1ft and for a wall flux q of 1Btu/(hr)(sq. ft). T h e liquid-hydrogen properties were taken at saturation conditions for atmospheric pressure and are as follows: cp = 2.3 Btu/(lb)("R), p = 4.43 lb/ cu. ft, k = O.O684Btu/(hr)(ft)("R),p = 320 x lb/(ft)(hr),v = 2.01 x 1OW sq. ft/sec, and p = 0.0158/"R. Figure 4 shows that for a length of 1 ft the

i: x

Im x

Liquid hydrogen at saturation

/

Fraction of Earth gravity, g/ge

FIG.4. Magnitude of modified Rayleigh numher as function of reduced gravity. Heat transfer per unit time and area, q , 1 Rtu/(hr)(sq. ft); length, s,1 ft; all fluids at atmospheric pressure.

modified Kayleigh number can still be fairly high for liquid hydrogen even when considering gravity fields as low as 1O-'ge. Larger surfaces would give much larger Ra" because of the fourth-power dependence on length. €3. HEATTRANSFER Since the local Nusselt number depends on the local Rayleigh number and

Ra,ydepends on the productgx3(or similarly hY* depends ongx'), the effect

REDUCED GRAVITY ON HEAT TRANSFER

155

of an increased or decreased gravity field on the Nusselt number can be simulated at earth gravity by a change in scale of the apparatus. This was considered for example by Schmidt (25) where the high gravity field g, in a rotating machine was simulated by scaling up the experimental equipment by a factor of (g,/ge)’’3when using an isothermal surface at earth gravity, Information concerning the effect of low gravity on free convection Nusselt numbers can thus be obtained by looking at an apparatus of reduced size. For example to simulate a reduced gravity of 1OP6ge, the apparatus can be scaled down by a factor of 100 to yield a geometrically similar boundary-layer formation in earth gravity. This follows from Eq. (3) where, for a given Prandtl number, S/x depends only on Ra,. Attention is then directed to experiments that have been performed with test sections having small dimensions such as fine wires or short vertical plates. For short vertical plates in air, data and a curve for natural convection are given by McAdams (26). T h e test results indicate that for lo4 < RaL < lo9,where the flow is laminar, the average Nusselt number over the surface varies as RaL’4. This one-quarter variation is the dependency predicted theoretically by utilizing the boundary-layer assumptions. For RaL < lo4,the NUL decreases less rapidly than Ra94, which indicates that the boundary-layer assumptions no longer apply and that the boundary-layer is becoming too thick compared with the characteristic length of the heated surface. The same behavior is observed for free convection from horizontal cylinders (26, p. 176) where below a Rayleigh number of lo4 based on the cylinder diameter the experimental data deviates from the boundary-layer prediction. Hence the limit can be assigned: if at low gravity fields RaL falls below lo4, boundary-layer theory should no longer be applied for computing the Nusselt number. T o examine the ranges of values where boundary-layer theory will not apply, consider a constant-temperature surface of characteristic dimension L and let [g$Pr/u2]L3(T, - Tb)(g/g,) = lo4. The quantity in square brackets has a specific value for any particular fluid under consideration. For a characteristic length L and a temperature difference T, - T b , the fractional gravity field can be determined below which boundary-layer theory no longer applies. Typical results are shown in Fig. 5 . For water, if T, - Tb= 100”F, and a surface of 1 ft characteristic size is considered, then gig, has to be below almost lo-’ for boundary-layer theory not to apply. If, however, the surface dimension is much smaller, say 0.01 ft, then for gig, below about 0.1 boundary-layer theory cannot be utilized. When RaI, is above lo4 it would be assumed, in the absence of any extensive low gravity information to the contrary, that the conventional boundary-layer free convection analyses and correlations would apply as given in texts such as McAdams (26) and Jakob (27). There are differences in interpretation, however. For example, in turbulent free convection from

ROBERT SIEGEL

156

a vertical plate at constant temperature, Jakob (27, p. 530) provides the expression valid for los < RaL < l o i 2 : -

NU,

= &L/k= 0.129(RaL)'i3

(4)

Since RaL contains the factor L3, Eq. (4)shows that the heat-transfer coefficient & is independent of L. Hence, if RaL is varied within the range of applicability by varying L, the & value is unaffected. However, if RaL is varied by changingg, then h will vary a ~ g ' / ~ .

When RaL < lo4 and boundary-layer theory does not apply, the data correlations can be used as given by McAdams (26, pp. 173, 176) or Senftleben (28). An analysis is also available by Mahony (29).

C. TRANSIENT DEVELOPMENT TIMES FOR BOUNDARY LAYER In reduced gravity, the thermal boundary layers are of greater thickness than in earth gravity, and consequently a longer time is required for them to develop after a transient change in thermal conditions. Transient times have been discussed relative to cryogenic storage tanks for space applications

REDUCED GRAVITY ON HEATTRANSFER

157

by Schwartz and Adelberg (30,32).The development of the free convective pattern in very low gravity can be fairly slow. During the transient process the externally applied thermal boundary conditions may continue to change, or the conditions within the fluid may be altered, for example, by an outflow from the tank. Hence, in some practical applications a steady condition may never be achieved, so it is necessary to deal with the transient process. Unfortunately, free convection circulation can depend on the geometry of the heated surfaces and it is therefore difficult to formulate general statements. Some insight can be obtained, however, by considering a simplified case such as the vertical flat plate. Transient free convection from a vertical plate in laminar flow was analyzed by Siege1 (32).Experimental results by Goldstein and Eckert (33) demonstrated that the analysis was reasonable. These results were for a plate that had negligible heat capacity. When appreciable capacity within the surface is involved, the transient times are longer, so the results given here provide a lower limit for the transient times. In discussing these results it is assumed that the Rayleigh numbers at reduced gravity are still large enough so that the thin-boundary-layer assumptions apply. Consider a position at height x on a vertical plate of total length L.The plate and the surrounding fluid are initially isothermal. Then the plate has a step in either temperature or heat flux suddenly imposed on it. The analysis shows that up to a certain time the position x dissipates heat only by pure conduction into the fluid. The end of the conduction transient occurs at time t, or at dimensionless time T, . Then there is an adjustment where the thermal layer established by only conduction changes into the steady-state free convection layer. The adjustment is completed and steady state is achieved at the end of time t, or T, . Hence, for T < T, ,the transient conduction solutions for stationary media can be utilized to obtain the temperature distribution in the fluid and the associated heat transfer. For T > T,, the steady-state convection equations are employed. The expressions for the transient conduction and steady-state times are given as follows : For a step in wall temperature : T~ =

or

+[l.SO(l.S + Pr)'12+ 2.48(0.6 + Pr)'i2](RaLPr)-'/2(x/L)1/2 ( 5 4

+ 2.48(0.6 + Pr)'/'] [gp(Tzc Tb)]-1/2~1/2 (5b) = +[5.24(0.952 + Pr)'I2 + 7.10(0.377 + Pr)112] (RaLPr)-112(x/L)112(6a)

t, = $[130(1.5 + Pr)'12 T~

-

or t, = )[5.24(0.952

+ Pr)'I2 + 7.10(0.377 + Pr)112][g/3(T'c- Tb)]-112x1/2 (6b)

158

ROBERTSIEGEL

For a step in wall heat flux: T, =

1.97(1 + Pr)2’5( RaL” Pr)-*i5 ( x / L ) ~ ’ ~

4.78(0.8 + Pr)2i5(RaLXPr)-’” ( x / L ) ~ ”

(7)

(8) Thus for a surface at constant temperature the transient times vary asg-‘I2, while for uniform wall heat flux the variation is as g-*l5. As a numerical example, the time is calculated to establish the free convection pattern in air at 70°F for the first foot of a plate that has been suddenly raised from 70 to 270°F. T h e Rayleigh number at earth gravity, computed from Eq. ( l ) , is about 2 x lo8 so that the laminar analysis can be utilized. Then from Eq. (6b) the steady-state time t, in earth gravity is about 2 sec. If the gravityfield is reduced to 10-4g,, the Rayleigh number is 2 x lo4 so that boundary-layer theory still applies. The steady-state time becomes 200 sec, or more than 3 min, for the boundary layer to become established. Further numerical examples are given by Schwartz and Adelberg (30, 34, who also discuss the transient development of a turbulent layer. T~ =

IV. Po01 Boiling Until about 1956 there had been practically no consideration given to a systematic study of the effect of gravity field on pool boiling. Pool boiling provides very high heat-transfer coefficients and hence can have useful applications in the high-performance heat-transfer devices employed in space applications. There are several factors of importance that must be studied in reduced gravity. It is necessary to know how the nucleate-boiling heat-transfer coefficient is influenced as gravity is reduced. T h e upper limit of the nucleate boiling flux (critical heat flux) is certainly very important for cooling surfaces where there is an imposed heating that must be dissipated. T h e upper limit is also important in the boiloff of cryogens from storage tanks where film boiling may be desirable as it will reduce the heat leakage. T h e behavior of the vapor formed during boiling is also of interest ; for example, in a liquid-cooled and moderated nuclear system, the distribution of voids in the moderator is important in determining the control of the reactor. In this section various factors in pool boiling will be treated individually with both theoretical and experimental results provided. .4. XCCLEATE-POOL-BOILING HEATTRANSFER

1. 7‘h~oql, for Kidpate Boiling ‘1‘0 obtain a possible indication of the gravity effects to he expected in nucleate pool boiling, two semitheoretical correlations can be considered as

REDUCED GRAVITY ON HEATTRANSFER

159

proposed by Rohsenow (34)and by Forster and Zuber [see Westwater (35), p. 19)]. T h e correlation of Rohsenow takes the velocity of a vapor bubble at the instant of breakoff from the surface as being the most meaningful velocity in the heat-transfer mechanism. Since this velocity depends at least in part on the buoyancy force of the bubble, the heat-transfer coefficient would be expected to be a function of gravity. T h e final result of the correlation, however, explicitly contains gravity to only a small power :

In the derivation of Forster and Zuber (35),it is postulated that the radial growth velocity of the bubbles while they are still close to the surface is the significant velocity governing the turbulent motion induced by the bubble action. As a result, this heat-transfer correlation appears to be independent of the gravity field. T h e gravity dependence indicated by these and other similar correlations must be viewed with considerable caution. The correlations were derived by utilizing physical models and bubble behavior characteristics obtained from boiling at earth gravity. Hence there could be other gravity dependencies implicitly contained in the derivations that have not been adequately accounted for.

2. Experimental Results In Table I are listed the experiments where nucleate boiling has been studied in reduced gravity. Several different fluids have been utilized such as water, liquid hydrogen, and ethyl alcohol. A variety of test section geometries were used: horizontal flat plates and wires, vertical wires, and spheres. These test sections were made from various materials such as platinum, lead, and copper. Thus, the experiments represent a diverse selection of geometry, surface material, and fluid combinations. Most of the experiments were performed by using drop towers of moderate height and hence were limited to only 1 or 2 sec at low gravity. Three experiments were conducted in airplanes that provided up to 17 sec of low gravity time. One experiment utilized a magnetic field to reduce the effective gravity on the fluid and provide results under steady-state conditions. The findings of these experiments will now be considered so that a conclusion can be made as to the behavior of nucleate boiling in reduced gravity. Our interest will be directed toward the variation of wall heat flux Q/A as a function of T,, - T,,, , which is a common way of displaying nucleate boiling

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160

data. In some preliminary nucleate boiling experiments in saturated water (37), it was found that for a fixed QIA no shift in T, - Tsatwas noted when the boiler was placed in near zero gravity for 0.75 sec. The instrumentation TABLE I REDUCED GRAVITY NUCLEATE POOLBOILINGEXPERIMENTS Authxe

Siegel and Usiskin

Ref.

Reduced gravity facilitv

(36) Drop tower

Usiskin and (37) Drop tower Siegel 9 ft high; counterweighied

Sherley

(38) Drop tower

Merte and Clark

(39) Drop tower 32 ft high;

Liquid Distilled water

Liquid condition Saturated

Cidelter Siege1 a d Keshoek

(5)

Test section material

Beaker

GimS

Horlzontal and Nichrome vertical ribbons: 0.006 in thick, 0.125 in. wide, 0.75 in long Horizontal wire: Platinum 0.0455-in. dlam, 2.5 in long Horizontal ribNickel ban: 0.010 in. thick, 0.2 in. wide, 2. 5 in. long Horizontal thin Lead

Distilled water

Saturated

Liquid hydrogen

Saturated

Liquid nitrogen

Saturated

Sphere: 1- and K-in. dlam

Copper

Distilled water

Saturated

Horizontal wire: 0.020-in. diam

Platinum

Airplane counterweighted ( 4 0 ) Drop tower 55 fl high

Test section immetrv

Drop tower Distiiled Saturated 12.5 It high water ethyl aleoh&. 6043 aqueoussucrose

film: 2-sq. in. area

Platinum Horizontal and vertical wires: 0.0197-indiam, 1.5 in. Long

Means of heating

Gravity range,

Test duration, see

e.

Hot plate -0 underneath Alternatingcurrent electricity

0.7

-0- 1 Directcurrent electrictty

0.75 for

zero gravity

Direct-0 1 current electricity 0 15 0.01 1 1.4 for .Heat storage of sphere zero test section gravIiy

-

-

Direct-0.01 Current electricity 0.014 Mrectcurrent electricity

1.85

0.9

MiUtiO"

Liquid nitrogen

Saturated

Sphere: K - and I-in. diam

Copper

Water

Saturated, becoming subcooled during test

Probably AlternatingStainlese current steel electricity

Schwvtz ( 4 3 ) Airplane and Mannes

Distilled water

Saturated

Rexand Knight

(44)

Ballistic

Propane (C,H. I

Vertical U-tube immersion heater: I- 1/16 in. Long, 5/16 in diam Horizontal ribbon: 2.15 in long.O.25 in. wlde.0.005 in thick Spherical tank: 25.4-em dlim

Papell and Faber

(45)

Magnet

Colloid of magnetic iron oxide in normal henane

Steel Saturated (pressure gradually inereasingj Horiaontal rtbChrome1 Saturated b n : 1/16 in. wide, 1 i n Long

Lewis e l 01. ( 4 1 ) Drop tower 31 It high; counterweighted Hedgepeth ( 4 2 ) Airplane and Z u a

missile

-

Heat storage C0.002of sphere 1 test section

NickelDirectchromium current iron alloy electricity

0

0.03 - 1

-

zero gravity

9 - 11

Direct(4.5x rvrrent 10-1 electricity Alternatingcurrent electricity

1.4 for

0-1

8

- 10

220

Steady state

of Usiskin and Siegel (37)was not able to detect a temperature shift of less than 6"F, but the indication was that gravity had little effect as long as nucleate boiling was sustained. Several more recent experiments summarized

REDUCED GRAVITY ON HEATTRANSFER

161

in the following paragraphs have utilized temperature measuring equipment of greater sensitivity. The data of Sherley (38) are shown in Fig. 6 where Q/A is plotted as a function of T, - T,,, for boiling of saturated liquid hydrogen. The data points include drop tower tests of 1-sec duration and airplane tests lasting 15 sec. There was a statistical scatter for both the 1 and Og, groups of data

FIG.6. Effect of gravity reduction on nucleate boiling curve for liquid hydrogen [Sherley (3811.

so a least-squares statistical line was fitted through each set. This revealed a very small shift in the data as a consequence of the gravity reduction. For zero gravity the T, - T,,, values were less than 0S"F smaller than at earth gravity. The data of Merte and Clark (39) for another cryogenic fluid, liquid nitrogen, displays a similar behavior. As shown in Fig. 7, the data in the nucleate region for lg, and 0.01-0.03ge are scattered within 1 or 2°F of each other. Forg/g, = 0.20 [(39),Fig. 101the data reveal a shift of about 1.5"F values than given by the near 0 and lg, results. toward larger T, - T,,,

ROBERTSIEGEL

162

For water, the data of Clodfelter (40) for a horizontal wire are given in Table 11. For a gravity reduction from 1 to O.O1ge the wall temperature decreased approximately 4"F, which indicates a higher heat-transfer coefficient ( h = (Q/A)/(Tro - TSat)) in the reduced gravity range. Table 111 from Siege1 and Keshock (5)includes data similar to those of Clodfelter (40) Fraction of Earth

inch 1/2

94, 0.01-0.03

1

0.01-0.03

1

0.20

1

0.33 0. 60

1

105

8 6

10"

Sphere

2

1 4

6

1/2

1

1

1

2 4 6 8 10' Temperature difference, ,T - TSat, "F

8 10'

2

4

6 8 lo3

FIG.7. Nucleate and transition boiling at earth and near zero gravity and film boiling at earth and fractional gravities for liquid nitrogen [Mertr and Clark (39)].

for a horizontal wire test section in water, and in this case T, - T,,, also decreased 2-4°F when gravity was reduced to 0.014ge. For a vertical test wire orientation, however, the temperature shift was in the opposite direction although of about the same magnitude as the horizontal case. The data of Schwartz and Mannes (43)for water were taken in an airplane providing a low gravity duration of 8 to 10 sec. The results are shown in

163

REDUCEDGRAVITY ON HEATTRANSFER TABLE I1 SHIFTI N SURFACE TEMPERATURE AS A RESULTOF GRAVITY REDUCTION FOR NUCLEATE BOILING OF SATURATED WATER& Heat flux Q/A Btu/(hr)(sq. ft) Fraction of earth gravity gig, = 1

1.28 x 4.36 x 5.22 x 6.09 x 6.38 x 6.87 x ~

Shift in surface temperature

Fraction of earth gravity g/ge= 0.01

1.27 x 4.33 x 5.20 x 6.09 x 6.35 x 6.82 x

104 104 104 104 104 104

T,,.(g/g, = 1) - T,,.(g/g, = 0.01) "F

3.5 4.4 4.3 3.0 4.6 4.1

104 104 104 104 104 104

~~

" From Clodfelter (40) TABLE 111 HEAT-TRANSFER DATA FOR NUCLEATE BOILING FROM AN ELECTRICALLY HEATED \TIRE" Horizontal wire Earth gravity, 1 ge Heat flux, Temp. Q/A difference,

Vertical wlre

Reduced gravity, 0.014 ge Heat flux, Temp. 4/A, difference,

Earth gravity, 1 4e Heat flux, Temp. Q/A difference, - B ~ n a t , ~ t u / ( h r ) ~w ;Teat, F (ss.ft) F

Reduced gravity, 0.014

Heat flux,

Q/A WU/(~)

gp

Temp. difference,

r,. sat,

~lu/b)

t ;Tmat,

Distilled water

30,300 50,100 90,700

15.3 16.1 18. 5

30,200 50,000 90,600

11.3 14.5 16. 5

29,100 48,700 62,800 69,400

11.3 14.2 15.9 11.3

29,100 48,900 62,900 69,700

13.8 11.6 19. 5 21.8

Ethyl alcohol

28,700 48,300 62,500

32.2 33.1 33.1

28,700 48,200 62,500

30.2 31.3 31.9

28,300 47,800

27. 1 30.1

28,300 47,800

28. 2 31.2

37.3

30,000 49,900 64,700 92,100

37.3 44.9 50.3 58.4

31,000 51,200 66,200 99,800

64.3 64. 1 79.0 87.1

31,000 51,400 66,400 94,000

65.8 61. 1 83.9 94. 5

(ss. ft)

60% by

weight aqueoussucrose solution a

30,000

49,900 64,700 92,100

F

44.9 50. 3 58.4

R U / ~ )

(scr.

ft)

~w

(6s.

It)

F

From Siege1 and Keshock ( 5 ) .

Fig. 8 and again reveal an insensitivity of the nucleate-boiling curve to the gravity reduction. Table I11 includes data for some additional fluids: ethyl alcohol and 60",, by weight aqueous-sucrose solution. For horizontal test wires, the gravity reduction caused a reduction in Tzc- T,,, of 2°F for alcohol, but had no effect for the sucrose solution. For vertical wires, T,, - T,,, increased somewhat when gravity was reduced-the increase for sucrose solution

ROBERTSIEGEL

164

being larger than for alcohol. However, the sucrose solution also required a substantially larger T, - T,,, for a given Q/A as compared with the alcohol. Figure 9 shows the data of Papell and Faber (45) who used a magnetic field to produce low gravity in normal heptane containing a colloidal suspension of magnetic particles. When this technique was used, boiling could be observed under steady-state conditions rather than for the short times available with drop towers. T h e results show a decrease in T, - T,,, of up to about 5°F when the gravity was changed from earth to near zero gravity.

loo

2

4 6 8 10' 2 4 Temperature difference, T, - lsat, "F

6 8 10'

FIG.8. Saturated nucleate pool boiling of water at 1 atm in earth and low gravity [Schwartz and Mannes ( 4 3 ) ] .

T h e data of Hedgepeth and Zara (42) revealed some interesting transient effects in airplane tests of up to 17 sec. When the experimental package was fixed to the frame of the airplane, the reduced gravity fields were i. 0.03ge. This was sufficient to keep the water mixed throughout the test and the fluid pressure and temperature remained close to equilibrium at saturation conditions. The test section temperature rose about 2°F during the low gravity period. This is consistent with the data for avertical surface by Siege1and Keshock (5)(see Table 111).When the experiment was free floating in the airplane, however, so that the gravity field was reduced to about O.O1ge,the bulk temperature remained constant but the pressure increased, thereby causing

REDUCED GRAVITY ON HEATTRANSFER

165

the bulk fluid to become subcooled. The increase in pressure was thought to be caused by a stratification of a warm layer of fluid adjacent to the vapor space in the boiler. During this transient process T, - T,,, decreased about 2.5"F.

FIG.9. Gravitational effects on nucleate pool-boiling heat transfer in magnetic iron oxide-normal heptane colloid. Saturation temperature, 205°F [Papell and Faber ( 4 5 ) ] .

An experiment using a ballistic missile is given by Rex and Knight (44) for boiling propane in a heated closed spherical tank. As in the free floating tests of Hedgepeth and Zara (42),the tank pressure increased during the flight so that the nucleate boiling was undergoing a slow transient. The

166

ROBERTSIEGEL

pressure rise may have had a tendency to suppress bubble formation. The temperature difference between the tank wall and the liquid was about 4.5"F at the very low gravity field during the test so it appears that nucleate boiling was maintained. There were no rapid excursions in tank wall temperature that would be indicative of a film-boiling condition. Rex and Knight (44) conclude that for the temperature differences that existed during the very low gravity test the wall heat fluxes dissipated were about one-third the values for lg,, and that the heat transfer in nucleate boiling is therefore adversely influenced by a very low gravity environment. Since the test period was long compared with the results from drop tower and airplane tests, this is an important finding. However, this writer would not consider the findings conclusive without further long-term tests. A close examination of the work of Rex and Knight (44) raised some questions. The T, - T,,, values were a few centigrade degrees in magnitude and were obtained from telemetered data of temperature values that were an order-of-magnitude larger. Thus a small telemetry error could lead to correspondingly much different values in T, - T,,,. Another more worrisome factor was that the lg, data used for comparison with the low gravity values were not taken with the same boiler under the same conditions of the heating surface as the low gravity tests. Rather, the lg, data were taken from Cichelli and Bonilla (46). The curve of Q/Aas a function of T, - T,,, is generally sensitive to the heated surface conditions. Hence there is some question as to whether the differences measured were a result of gravity effects or were caused by using two entirely different boilers to obtain the low gravity and lg, data. Also, the lg, data shown in Rex and Knight (44)[as taken from Cichelli and Bonilla (46)] are very sketchy in the small T, - T,,, range of the low gravity tests, so they do not provide as good a set of lg, reference data as would be desirable. T h e need for additional long-duration tests at very low gravity levels is evident. They would reveal whether the insensitivity of nucleate boiling to gravity, as found in short-duration tests, would be substantiated.

H. CRITICAL HEATFLUX FOR POOLBOILING The experimental results discussed in the previous section have shown within the frame of their limited scope that nucleate pool boiling is practically insensitive to gravity reductions. This is subject to the restriction that, when gravity is reduced, nucleate boiling is sustained and transition to another form of boiling does not occur. The heat flux imposed must not be high enough so that it exceeds the peak nucleate boiling flux for the low gravity condition being considered. This brings us to the consideration of how the peak nucleate boiling flux depends on gravity. This is a very important factor when nucleate pool boiling is to be applied in low gravity situations.

REDUCEDGRAVITY ON HEAT TRANSFER

167

1. Theory for Critical Heat Flux The theories and correlations of Kutateladze (47),Zuber (48),Noyes (49), and others have indicated that the critical heat flux depends on g1I4.For example, the relation derived for pool boiling from a horizontal surface takes the form (48)

If it is assumed that the fluid properties are not influenced by gravity reductions (they might be influenced by a change in hydrostatic pressure head when gravity becomes small), the ratio can be written as

This indicates that as the gravity field approaches zero it is no longer possible to sustain nucleate boiling as the peak nucleate boiling flux will approach zero. Experimental results will be examined to judge whether the dependency in Eq. (11) is valid. I t should be pointed out that Eq. (11) may be an oversimplification, as the theory is based mainly on a horizontal infinite flat-plate model. As discussed by Lienhard and Watanabe (50),there may exist an additional geometric effect so that Qc should be written as

where the functional dependence f would depend on the geometry involved. This could lead to a more complex gravity dependence than in Eq. (11). 2, Experimental Critical Heat Flux Behavior For the reduced gravity range, boiling experiments that provide information on the upper limit of nucleate boiling are summarized in Table IV. Experimental data are given in Figs. 10-12. The early experiments by Steinle (51) and Siege1 and Usiskin (36) indicated that for conditions close to zero gravity the critical heat flux would be substantially decreased from the values at earth gravity. When looking at the data points in Figs. 10 and 11 it is seen that there is a vertical spread in the data at each gravity for each series of tests. When a comparison is made with theory, the lowest data points should be particularly considered. The test durations are short and a period of time is required for the vapor pattern to be formed at the critical flux in order to obtain a transition from nucleate to film boiling. T o provide the necessary

ROBERTSIEGEL

168

vapor buildup in a test of short duration, it would be expected that the measured critical heat flux might often be greater than the critical flux that would be measured in a test of long duration. Data for horizontal surfaces and horizontal wires obtained using drop towers and airplanes are shown in Fig. 10. The measurements of Sherley TABLE IV CRITICAL HEATFLUXEXPERIMENTS IN POOLBOILING Authors Steinle

Ref.

ReduCsd gravity facility

( 5 1 ) Drop tower 9 ft high

tower Usiskin and (36. BOD 37) 9 i high; Siege1 counterweighted Merte and (39) Drop tower 32 ft high; Clark counterweighted (38) Drop tower Sherley

Liquid

Freon 114

Distilled water

Liquid condition Atmospheric pressure, probably saturated Saturated, atmospheric pressure

Test

kleansol heating

Gravity, Test range, duration,

-

see

R-

Horizontal Platinum Direct0 0.75 wire: 0.0015current in diam, electrlcity 1.281 in Long -0 - 1 0.75 for Horizontal Platinum Directwire: 0.0453current zero in. diam, electricity gravity 2.5 in long Heat storage 0.01 - 1.4 for Sphere: 1- and Copper % -in. diam of sphere 1 zero test sectlo" gravity

-

Saturated, atmospheric pressure

Liquid hydrogen

Saturaied, atmospheric pressure Saturated

Horizontal thin film: 2 sp. in. area

Diatllled water

Saturated, atmospheric pressure

Horizontalwire: Platinum Dlrect0.01 0.020-in. diam; current horizontal ribelectricity bons: %and % In. wide Horizontal wire: <0.01 0,020-in. d a m

Horizontalwire: Platinum Direct-0.015- 0.9 sec 0.020-in diam, current 1 for zero electricity gravity 1.5 in long Yertical wire: 0.020-in diam, 1.5 and 9.0 in

Airplane ( 4 0 ) Drop tower 55 ft high

Siege1 and Hoarell

(52) Drop tower

Saturated, atmospheric pressure

Lewtsef 01. ( 4 1 ) Drop tower 31 I t high; counterweighted Papell and ( 4 5 ) Magnet Faber

Saturated, 1,3,5-atmospheres of pressure Saturated, Colloid of mngnetic tron atmospheric oxide in narpressure mal beptane -Atmaspheric Liquid pressure oxygen

Airplane

Didilled 12.5 ft high; water ethyl counteralcohdl,608 weighted by weight aqueouseucrose solution

(53) Magnet

Test

Liquid nitrogen

Clodfalter

Lyon e f al.

gr:yR :pal

Liquid nitrogen

Lead

Directcurrent electricity

-

0

1

0

15 1.85

3.5 - 7

long

where: 1-, %-, Copper and )&-in diam

Heat dorage <0.002- 1.4 for of sphere 1 zero test section gravity

Horizontal rib- Chromei Alternatingban: 1/16 in. current electricity wide, 1 in Long Horizontal flat surface: 0.75in. diam

--

0 1

steady state

Platinum Axial heat -0.03- 1 Steady conduction state throwhcopper cylinder

(38)showed that close to Og, the critical flux had decreased to the range of 0.5 to 0.7 times the value at lg,. The tests of Clodfelter (40) which were at fields less than 0.01geyielded critical heat fluxes as low as 0.15 times the lg, value. The tests by Sherley (38) and Clodfelter (40) were confined to low gravities between 0 and O.Olg,. The precise values of the gravities were unknown ; for the airplane tests extremely close to zero gravity would be expected but large initial disturbances in the fluid may be present. The data

REDUCEDGRAVITY ON HEATTRANSFER

169

of Usiskin and Siegel (37)and Siegel and Howell (52)extend over a range of gravities and hence can be compared with the theoretical one-quarter power gravity dependence. As shown in Fig. 10 the data for water from Usiskin and Siegel (37) are located a little above the one-quarter-power line while those from Siegel and Howell (52) are somewhat below the line in the intermediate gravity range. The data €or ethyl alcohol follow the one-quarter power relation quite well. Hence for the range O.Olg, < g < lg, the onequarter power dependence appears to be a reasonable engineering approximation. For gravities lower than O.Olg, insufficient information is available to make a definitive conclusion. Although the critical heat flux was reduced considerably when zero gravity was approached, it still cannot be ascertained

c

/-

A 0.020-Inch-diam wire A 114-Inch ribbon

-$Zt

"1

10-2

1 2

Sherley(38)

0

Clodfelter (airplane) (40) Siegel and Howell (52) Water Ethyl alcohol

0 h

of zero gravity, g/ge = 0

1

V

, I Clodfelter (drop tower) 140)

A 118-Inch ribbon

2

l0-d'

0 Usiskin and Siegel 137)

I

l

l

I l l

1

4 6 8 10-1 2 Fraction of Earth gravity, g1ge

1

I

4

I

I 6

I l l -

8 1 8

FIG.10. Reduced gravity critical heat flux data for horizontal surfaces and horizontal wires obtained with drop towers and airplanes.

whether the critical flux is really zero at zero gravity. Tests of longer duration with carefully measured values of the very low gravity fields are still needed to extend the lower end of the logarithmic plot in Fig. 10. Figure 11 shows data for vertical wires and spheres obtained in drop tower tests of up to 1.4-sec duration. The data for liquid nitrogen (39) follow the one-quarter power line quite well, and so do the lower data points for water and sucrose solution given by Siegel and Howell (52). The data for ethyl alcohol, however, show a definite deviation above the theory as gravity becomes less than about 0. lge. The data in Fig. 12 were obtained by a different experimental means than those in the previous two figures. Boiling fluids were used that are influenced by a magnetic field. Hence when the fluid is placed in a uniform magnetic gradient it is possible to counteract all or part of the gravitational body force. This method has the advantage, as discussed earlier, that steady-state tests

ROBERTSIEGEL

170

Reference

10-2

4

2

6

8 10-1

2

Fraction of Earth gravity, 919,

4

818

6

FIG. 11. Reduced gravity critical heat flux data for vertical wires and spheres obtained with drop towers.

can be made. As shown in Fig. 12, the data are farther above the one-quarter power line than most of those obtained with drop tower or airplane tests. It was found by Lyon et al. (53) that if the magnetic force were increased sufficiently to provide a negative gravity (presumably adequate to lift the loo

A"

L

0

0

0

Reference

0 Papell and Faber (45)

2 Limit of zero gravity, 919,

10-1

,/-

2

=

0

0

Lyon et al. (53)

4 6 8 10-1 2 Fraction of Earth gravity, 919,

4

6

8 1

P

FIG. 12. Reduced gravity critical heat flux data for horizontal surfaces obtained by using magnetic body forces.

liquid from the surface), then when gravity was about - 0 . 0 3 ~the ~ critical heat flux decreased to very close to zero. Lewis et al. (41), provide a few data points giving the critical heat flux variation with gravity at pressures of 3 and 5 atm. T h e information is limited so that it would be speculative to try to make a definite conclusion

REDUCEDGRAVITY ON HEATTRANSFER

171

from it. Further experimentation is required on the gravitational effects at elevated pressures.

REGIONFOR POOLBOILING C. TRANSITION The experiments by Merte and Clark (39)and Lewis et al. (41)have yielded information in the transition region between the nucleate and film regimes for liquid nitrogen boiling from a sphere. A typical set of their data is shown in Fig. 7. The few points at near-zero gravity that fall in the transition region agree within a reasonable scatter with the transitional data at earth gravity. Based on this limited information it appears that the & / Adependence on T, - T,,, in the transition region is insensitive to gravity reductions, and in this respect is similar to the nucleate portion of the boiling curve.

D. MINIMUM HEATFLUX BETWEEN TRANSITION BOILINGA N D FILM BOILING The boiling curve of &/A as a function of T, - T,,, passes through a minimum Q/A in going from the transition region to the film-boiling region. A relation for this minimum derived by Berenson (54) is

which predicts a variation of ( & / A ) m i n as gravity to the one-quarter power. Equation (13) was derived for a horizontal flat surface. Data is provided for the minimum flux by Merte and Clark (39)and Lewis et al. (41)for boiling of saturated liquid nitrogen at atmospheric pressure from a 1-in.-diam sphere. Numerical values are listed in Table V and are plotted in Fig. 13. If the fact is considered that the experimental data is for a sphere, while the theory is for a horizontal plate, the agreement is remarkably good and the one-quarter dependence on gravity appears to be substantiated. For some test section geometries the one-quarter power dependence may not be valid. This wasdiscussed by Lienhard and Watanabe(5O)and Lienhard (55). It was found that for many geometric configurations other than a flat plate the ( [and perhaps also the critical heat flux as pointed out in Eq. (IZ)] can be obtained by multiplying the flat-plate expression by a geometric scale factor f(R’) : where

ROBERTSIEGEL

172

TABLE V MINIMUM HEATFLUX BETWEEN TRANSITION AND FILM BOILINGREGIME^

Fraction of earth gravity

Predicted from Eq. (13)

Experimental

1.o 0.6 0.33 0.2

2100 1850 1590 1400

1700-2100 1550 1300-1400 1300

0.03 0.01

875

0.003 0.001

491 374

gig,

a

__

Minimum heat flux (Q/A),,, Btu/(hr)(sq. ft)

666

870-1100 180-530

Lewis et al. (41)

2

4

2 4 6 810-l Fraction of Earth gravity, 919,

6 810"

2

4

6 810°

FIG.13. Minimum film boiling heat flux for a 1-inch-diam sphere in saturated liqL id nitrogen at atmospheric pressure [Lewis et al. (41)].

It is noted that R' contains a gravity factor in addition to the characteristic length L of the heated surface. Then instead of the simple type of relation as in Eq. (11) there results

REDUCED GRAVITY ON HEATTRANSFER

173

In (50)the factor f(R') for horizontal cylinders was derived as

f ( R ' ) = (2 + R'-2)'/2[R'2'3/(R'2 + 0.5)]3/4

(16) Equation (16)is valid for R' less than about 2 and the characteristic dimension L in R is the cylinder radius. Thus it is found that the gravity dependence in Eq. (15) can be quite complex. Evidently for the l-inch sphere discussed in the previous paragraph the geometric factor is not significant so that the flat-plate expression applied. E. FILM-BOILING HEATTRANSFER

1. Theoretical Relations T h e heat transfer through the vapor film in film boiling depends on whether the film is laminar or turbulent. There have been many analyses of film boiling including various surface orientations and types of boundarylayer assumptions. No attempt will be made to review them here. The present discussion is only intended to deal with the gravity dependence, and a few analytical expressions will serve to indicate this. For a laminar vapor film on a horizontal cylinder of diameter L, the filmboiling heat-transfer coefficient given by Bromley (56) is

where

Equation (17)indicates that as gravity is reduced the heat-transfer coefficient decreases in proportion to the one-quarter power of the gravity field. When the vapor film becomes turbulent as discussed by Hsu and Westwater (57) (and in the discussion at the end of their paper by Bankoff), the exponent increases and a two-fifths to one-half power variation is more applicable. An exponent of was proposed for a sphere by Frederking and Clark (58) who gave the relation analogous to Bromley's correlation

+

Since this is an average heat-transfer coefficient, it contains contributions from both the laminar and turbulent regimes.

2. Experimental Results Figure 7 shows some typical film-boiling data as given by Merte and Clark (39)and Lewis et al. (41). At a fixed T,, - T,,, there is a decrease in

174 174

ROBERTSIEGEL SIEGEL ROBERT

Q/A as gravity is reduced so that the heat-transfer coefficient decreases with gravity as expected. Figure 14 shows a correlation of the data as compared with the laminar and turbulent theories. T h e data, which extend over a range of reduced gravities from 0.17 to lg,, follow the one-third power variation quite well. Later in this review, in the section on bubble dynamics, some photographs will be shown of film boiling from horizontal and vertical wires. T h e pictures illustrate how important surface tension is in determining the configuration of the vapor film. This is especially true as gravity becomes very low as then Qld 2-

,213 '4

'ii

Frederking et al. (58)correlation:

GL-0. 14(Ra')1'3-,

103 8= 6: 4-

-

2-

4D 6-

'LBromley (56)type correlation for laminar flow:

2101

I " 1 1 1 1 1 1

I

fiI111111

KL 0. 62(Ra')114

I ' 1 1 1 1 1 1 1

1

' 1 1 i 1 1 1 1

'

1ll'Ll.L

FIG.14. Comparison of fractional gravity film boiling data for liquid nitrogen with correlations by Frederking and Clark (58) and Bromley (56). Gravity range, 0.17
surface tension begins to become the dominating force. At very low gravities the surface tension will tend to make the vapor film form into large vapor masses of a spherical shape, and the correlations derived on the basis of boundary-layer theory may no longer apply.

F. DYNAMICS OF VAPOR BUBBLESIN SATURATED NUCLEATE BOILING Another aspect of reduced gravity boilingthat isinfluenced by thechangein buoyancy is the dynamic action of the vapor bubbles being produced. Some of the nucleate-boiling heat-transfer theories are based on considerations of the actions of individual bubbles, and hence the characteristics of bubbles in reduced gravity must be studied. T h e quantities that will be

REDUCEDGRAVITY ON HEATTRANSFER

I75

discussed are bubble growth, the forces acting during growth, the size of bubbles at departure from the heated surface, and the rise of bubbles through the liquid. After the characteristics of single bubbles are presented, the behavior at high heat fluxes where many bubbles are present will be briefly considered. Siege1 and Keshock (59, 60) studied bubble dynamics in saturated pool boiling from a flat horizontal surface 2+ inches in diameter with a heated area in the center inch in diameter. This provided an area sufficiently large so that the symmetric growth of the bubbles would not be hampered. T h e surface had a very low roughness which limited the number of nucleation sites, and when a low heat flux was used, only a few sites were active. Single bubbles could then be studied without mutual interference effects. All of the bubbles discussed here were obtained in drop tower tests of approximately I-sec duration.

1. Nucleation Cycle and Coalescence of Successive Bubbles in Reduced Gravity As the gravity field is reduced, the detached bubbles begin to rise very slowly because of decreased buoyancy. This leads to a bubble coalescence mechanism that is much less frequently observed during earth gravity boiling. After a bubble departs and begins to move upward, if its rise velocity is small, the next bubble growing at the surface will collide with the rising bubble because of the rapid rate at which the diameter of the attached bubble increases during the early stages of growth. The succeeding bubbles formed at the nucleation site that contact the detached bubble and merge with it are thereby pulled from the surface before they can grow very large. This is illustrated by the sequence of photographs in Figs. 15(a) and 15(b). Several bubbles will rapidly feed into the larger bubble until it finally rises out of range. Then the next bubble will grow in an undisturbed manner. A bubble column in the low-gravity boiling regime is thus characterized by a distinctive cyclical behavior. An undisturbed bubble will grow to its final size and detach in a normal manner. Then several small bubbles will merge into it before the detached bubble can rise very far away from the surface. T h e large bubble thus serves as a temporary vapor reservoir near the surface and absorbs new bubbles while they are relatively very small. The bubble frequency at the surface is quite high when the small bubbles are merging into the larger one. This could greatly increase the turbulence induced near the surface and aid in promotinga high heat-transfer coefficient. Hence, this portion of the bubble cycle could play a significant role in reduced-gravity nucleate-pool-boiling heat transfer. At higher heat fluxes there will be many nucleation sites and a bubble will generally not have the opportunity to grow to completion without interference from adjacent

176

ROBERTSIECEL

FIG.lS(a). Comparison of bubbles growing in saturated water at atmospheric pressure for earth gravity and 6.1 04 of earth gravity. Time is measured from onset of growth for each - T,,,, 17°F bubble; heat flux, Q/A, 17,700 Btu/(hr)(sq. ft); temperature difference, T," [Siegel and Keshock (59)].

REDUCEDGRAVITY ON HEATTRANSFER

177

FIG.15(b). Continuation of the growth of the reduced gravity (0.061 g,) bubble in Fig. 15(a). It shows the merging of successive bubbles with the undisturbed bubble in Fig. 15(a).

bubbles. In this instance, the boiling process could be largely dependent on the small bubbles that form and rapidly merge with a vapor mass remaining near the heated surface. 2. Bubble-Growth Rates

When investigating the boiling process by examining the details of bubble dynamics, a factor of fundamental importance is the rate of growth of bubbles while they are attached to the surface.

178

ROBERTSIEGEL

a . Theoretical Relations. There have been many bubble-growth relations derived in the literature, and only a few of them will be discussed here to examine their gravity dependence. Fritz and Ende (61) considered bubble growth in an infinite uniformly superheated liquid. The heat conduction into the bubble was determined by having the temperature profile in the liquid adjacent to the bubble boundary equal to that for unsteady heat conduction in a slab. Their analysis resulted in the equation

Plesset and Zwick (62) included the influence of liquid inertia and accounted for the effect of the spherical shape of the boundary on the temperature profile, rather than using the temperature distribution in a plane slab as done by Fritz and Ende. This gave the same form as Eq. (19) except with an additional 2/3 multiplying the right side. Forster and Zuber (63)obtained 2 the right the same form as Eq. (19) except with an additional ~ r / multiplying side. Zuber (64) considered growth in a nonuniform temperature field and introduced a correction factor for sphericity to obtain

All of these expressions were derived for growth in an infinite medium away from solid surfaces. Surface tension, viscosity, and inertia were not considered to be important. Equation (19) indicates .a steady increase of diameter with the square root of time, while Eq. (20) predicts that a maximum diameter will be reached. The notable fact for our present purposes is that during bubble growth these expressions indicate that the functional dependency of bubble diameter on time is independent of gravity. 6. Experimental Results. Figures 16-18 show typical bubble-growth measurements for bubbles growing in liquids at saturation temperature and atmospheric pressure conditions. Figure 16 is for water, while Fig. 17 is for a 60°4 by weight aqueous-sucrose solution. The growth curves do not exhibit any definite trend with gravity, the differences between the curves being within the range of variations encountered between different bubbles in tests at a fixed gravity. This finding is in accord with the theory. The curves for water extend to much longer times as gravity is reduced because the bubble departure size in this fluid increases as gravity is reduced. This will be discussed in the next section. In Fig. 18, data for water are compared with the theoretical growth relations. The Fritz-Ende relation offers the best general agreement over the entire range of data. However, if the data are grouped into initial ( t < 0.02

REDUCEDGRAVITY ON HEATTRANSFER

179

Time, t, sec

FIG.16. Growth of typical single bubbles in saturated water at atmospheric pressure for seven different gravity fields. Heat flux, Q/A,10,900Btu/(hr)(sq.ft); temperature difference, T,,- T,,, , 11.1"F [Siegel and Keshock ( 5 9 ) ] .

,0002

,0004

,001

,002

,004

Time. 1.

sec

.01

.02

.c

FIG.17. Growth of typical single bubbles in 600: by weight aqueous-sucrose solution at saturation temperature and atmospheric pressure for five gravity fields. Heat flux, Q / A , 20,500 Btu/(hr)(sq.ft); temperaturedifference,T,$,- T,,, ,30.1°F [Keshockand Siegel(6O)l.

ROBERTSIEGEL

180

sec) and final ( t >0.02 sec) growth periods, the Fritz-Ende relation does not indicate the observed diameter variation during the final growth period. The data indicate D t3'* compared with t1I2 from theory. In the initial growth period, the time exponents were observed to range from 0.5 to 0.8. In the final growth period at low gravity, the bubbles in water become so large that they may extend out of the superheated thermal layer adjacent to the surface. In this instance the model used in deriving the Fritz-Ende relation N

FIG.18. Comparison of theoretical predictions with bubble growth data in reduced gravity for saturated water at atmospheric pressure. Heat flux, Q / A ,10,900 Btu/(hr)(sq. ft) ; temperature difference, T, - T,,,, 11.1"F[Siegel and Keshock ( 5 9 ) ] .

[(Eq. (19)]would not seem reasonable. In the final stage of growth, vaporization may actually occur from only around the cylindrical stem at the bubble base. If it is assumed that the heat transfer through the stem area is constant, since the base diameter is fairly constant as will be shown in a later figure, then

(E')

pUh --

=

const

or D t1I3.This is close to the three-eights power variation indicated by the data in Fig. 18. N

REDUCEDGRAVITY ON HEATTRANSFER

181

3 . Diameter of Bubbles at Departure a . Theoretical Relations. Several theoretical relations have been proposed for predicting the size of bubbles at departure from a horizontal surface, and two of these will now be reviewed so that the gravity dependence can be examined. T h e best known is the Fritz (65) equation

where the contact angle 8 is in degrees. A relation by Zuber (48)is

These relations indicate that the bubble departure diameter will increase with gravity reduction as g-"* or g-Il3, respectively, provided none of the

0 Water aq ueous-s ucrose

8

c

6 10-2

2

I

4

1

6

I

I

8 10-1

10

2

Fraction of Earth gravity, 919,

4

I

6

I

I

8 loo

FIG. 19. Effect of reduced gravity on diameters of bubbles in saturated liquids at instant of detachment from heated surface [Siegel and Keshock (59, 6 0 ) ] .

other factors in the equations, such as 8, depend on gravity. T h e Fritz relation [Eq. (22)] was derived by assuming that buoyancy alone is the force that overcomes the surface tension at the bubble base and pulls the bubble from the surface. Hence the negative one-half power function of gravity depends on the absence of appreciable dynamic forces to initiate bubble detachment. b. Experimental Results. Siegel and Keshock (59, 60) measured average bubble diameters in reduced gravity at departure from the heated surface and normalized the values with respect to the average departure size in earth gravity. The results are shown in Fig. 19 for distilled water and a 60y0by

182

ROBERTSIEGEL

weight aqueous-sucrose solution. T h e contact angle [B in Eq. (241 was found by Siegel and Keshock (59) to be independent of gravity, and hence Eq. (22) predicts D d ( g ) / D d ( l g e= ) (g/ge)-’lz. T h e data for water does have a negative one-half power slope for (gig,) < 0.1 ; hence, the large bubbles in this gravity range appear to be governed only by buoyancy and surface tension forces as this is the foundation for Eq. (22). For 0.1 < gige < 1, the slope for water in Fig. 19 tends more toward a negative one-third power variation, which indicates that additional forces are influencing bubble departure. For the 60n,, sucrose solution it is surprising that the departure diameter has hardly any variation with gravity. This indicates that buoyancy has practically no role in bubble departure for this fluid. T o examine why the bubble departure diameters have such different gravity dependencies in the two fluids, the forces acting on the bubbles were investigated.

4. Forces Acting on Bubbles during Bubble GiTowtA ‘The forces acting on bubbles during their growth in earth and reduced gravity have been computed by Keshock and Siegel (60) for saturated conditions and by Cochran et al. (66) for subcooled conditions. Since there are some differences in these two references in the way the forces were analyzed, the force expressions will now be reviewed. A typical bubble is shown in Fig. 20. Forces opposing bubble detachment will be taken as negative. T h e surface tension holding the bubble to the surface is given by Fs= -.rrUb 0 sin B (24) ‘The drag force was derived approximately by Keshock and Siegel (60) as

Fd = -(7~/16)bpLDdD/dt

(25)

where D is an equivalent spherical diameter for the bubble, and a value of

45 was dsed for b, which is a constant in the drag coefficient. A similar expression was used by Cochran et al. (66) with b = 48. T h e net buoyancy force is equal to the integral over the bubble surface of the vertical component of the liquid hydrostatic pressure force minus the downward-acting weight of the vapor in the bubble. For an unattached bubble this yields simply

When a bubble is attached to the surface, however, the buoyancy force must be modified to account for the fact that the base area does not have the

REDUCED GRAVITY ON HEATTRANSFER

183

ordinary liquid pressure acting underneath it. A correction to account for this was applied directly (60) to Eq. (26) to yield

T h e correction term does not contain gravity and will continue to aid bubble detachment as gravity is reduced. Equation (27) has the disadvantage that bubble contact angle 0 is needed, and for a growing bubble this cannot be measured with good accuracy. Also, the last term in Eq. (27) was derived by using some simplifying assumptions about the radii of curvature at the

FIG.20. Typical vapor bubble on heated surface.

bubble base. Since this term does not include gravity, there is some question as to whether it really should be included in the buoyancy force. Perhaps a better way of considering the buoyancy force is given by Cochran et al. (66). Here the buoyancy involves only the part of the bubble that has liquid pressure acting on both the upper and lower portions of the liquid-vapor interface. This constitutes only the unshaded volume in Fig. 20 and gives the net buoyancy as

The last term arises from the weight of the vapor in the bubble, and because of its gravity dependence is included in the buoyancy term. To account for

ROBERTSIEGEL

184

I

t

0

,*-

2

Time, sec (solidcurvesl

I

.02

I

.04

lime, sec (dashed curves)

I

.O 6

I

.08

FIG.21a. FIG.21. Variation of diameter, contact angle, and base diameter with time for bubble growth in saturated water. Heat transferred from solid surface to boiling liquid, 10,900 Btu/(hr)(sq. ft); temperature difference, T,, - T,,, , 11.1 "F. (a) Gravity fields, 1.O and 0.229ge. (b) Gravity fields, 0.061 and O.014ge. [Keshock and Siege1 (60).]

the pressure forces on the shaded volume V bin Fig. 20, the pressure force is utilized on the bubble interface area lying above the bubble base area. T h e pressure inside the bubble is higher than that outside so a net upward force is provided. Assuming the top of the shaded volume to be spherical with a radius of curvature R,, and p o and p i not to vary over this area, provides the pressure force

REDUCED GRAVITY ON HEATTRANSFER

185

.16 .12

.08 .04 0

.28 .24

.20

-32

.I6

.I2 .08

I

0

.d,

.Ib

.I

.2

20 2 5 Time, sec (solid curves) .115

J

.3

-.16

0.061 ,014

--O--

.04

0

- .24,

Fraction of Earth gravity,

I

.4

1

.5

Time, sec (dashed curves)

- .08

.& 1

.6

.is 1

.7

.40 O ~

1

.8

FIG.21b.

Since R, can be measured with good accuracy, the use of Eqs. (28) and (29) in preference to Eq. (27) has some advantage. Equation (29) emphasizes that part of the pressure force acting on the bubble, as given by F p , is independent of the gravity field. There are inertial forces developed during bubble growth primarily as a result of the liquid surrounding the bubble being placed into motion. The inertial force of the apparent liquid mass surrounding the bubble is given by Keshock and Siege1 (60) as

ROBERTSIEGEL

186

This equation represents an attempt to directly compute the liquid inertial force, but it involves a number of simplifying assumptions. Another approach, as given by Cochran et al. (66), utilizes the equation of motion stating that the sum of forces acting on the vapor in the bubble accelerates the center of mass of the vapor. This yields the dynamic force as

Fdy=

d (mv),,p,r - Fd - Fs - I i b - Fp dt bubble ~

Equation (31) provides the liquid dynamic force from the other bubble forces and hence is not as direct a calculation as is Eq. (30). Some typical .-2

0 8 -.

FIG.22. Variation of diameter, contact angle, and base diameter with time for bubble growth in saturated 60% aqueous-sucrose solution. Gravity fields, 1.0 and O.126ge;heat flux, Q / A ,20,500 Btu/(hr)(sq. ft); temperature difference, T,,,- T,,,,30.1”F [Keshock and Siege1 ( 6 0 ) ] .

Time. sec lrolid curvesl

0

.008

Time.

.016

.024

6

.032

rec ldashed curvesl

results will now be presented where the foregoing equations have been applied to observed bubble behavior in reduced gravity. T h e experimentally measured bubble dimensions, base diameter D,, contact angle 8, and equivalent spherical diameter D,are given for typical bubbles during growth in earth and reduced gravity for water in Fig. 21 and for sucrose solution in Fig. 22. T h e forces throughout bubble growth were computed from Eqs. (24), (25), (27), and (30), and they are shown in Figs. 23 and 24. T h e drag force was generally found to be negligible.

REDUCEDGRAVITY ON HEATTRANSFER

187

Fraction of

-.4

,

o

.ooe

,004

I

I

0

I

.02

.01

.012

.o16

I

I

I

I

I

I

.04 .05 sec (dashed curvesl

.03

Time,

.a

.024

.020

Time, SR (solid curves)

.O?

.06

(al 7 xlo-6

I

1

-I 0

I

0

I

I

.05

.I0

I

Time.

1 .I0

I

.20

I

.20

.I 5 SK

I .30

Time,

I

.30

I

.35

(solid curvesl

I .40

SK

I

.25 I SO

(dashed curves1

I

.60

I

.70

.

3 I

.80

(b)

FIG.23. Inertial, buoyancy, and surface-tension forces for bubbles growing in saturated water. Heat transferred from solid surface to boiling liquid, 10,900 Btu/(hr) (sq. ft); temperature difference, T,,, T,,, , 11.1"F. (a) Gravity fields, 1.O and 0.229ge; (b) gravity fields, 0.061 and O.014ge (60). ~

188

ROBERTSIEGEL

Figure 23(a) shows the forces for two typical bubbles growingin water, one at earth gravity and the other at a reduced gravity field of 0.229ge. T h e magnitudes of the forces are quite similar for the two bubbles, and the only effect of the gravity reduction is an increase in the total growth time. Because of the rapid initial growth of either bubble, the liquid inertial force reaches its maximum early in the growth period. By the time the inertial force reaches its maximum, however, the bubble base diameter has increased sufficiently to produce a surface-tension force that is somewhat largerthan the inertial force. Hence, the maximum liquid inertial force is insufficient to tear the bubble away from the surface. The inertial force then decreases while the buoyancy force continues to increase. T h e latter eventually surpasses the surface-tension force so that the bubble must detach. Since a finite time is required for the bubble base to form a neck and finally break loose, the bubble continues to grow, and at departure the buoyancy exceeds the surface-tension force. Figure 23(b) shows the forces on bubbles in water for the much lower gravity fields of 0.061 and 0.014ge. T h e total growth times are much longer than those of Fig. 23(a), and consequently the peak in the liquid inertial force occurs very early relative to the total growth period. As growth continues, the inertia decreases and the bubble base continues to spread so that the surface-tension force becomes quite large. T h e increase in buoyancy is slow because of the low gravity field, but as the bubble becomes quite large, the buoyancy comes into balance with the surface-tension force and departure occurs. Hence, for these bubbles in the very low gravity range, the departure is dependent on an equilibrium of buoyancy and surface-tension forces. It is reasonable that the departure diameter in this range should depend ong-112as was shown in Fig. 19, since this gravity dependence is predicted by a balance of surface-tension and buoyancy forces as in the Fritz equation, Eq. (22). Figure 24 shows the forces computed for two typical bubbles growing in 60°:, aqueous-sucrose solution, the solid lines being for a bubble in earth gravity while the dashed lines are for 0.126ge. The two sets of curves are quite similar to each other. One of the most important features of these curves is that the inertial forces are large compared with the surface-tension forces. This is a result of the larger growth rates characteristic of bubbles in sucrose solution. For these bubbles, the large inertial force soon overcomes the surface-tension force and hence initiates bubble detachment. This force unbalance occurs when the buoyancy force is still small. Consequently, the departure process is dominated by inertia. As gravity is further reduced below the values shown in Fig. 24, the buoyancy becomes smaller and is even less important in influencing departure. Hence, the departure of the rapidly growing bubbles observed in sucrose solution appears to be governed principally by inertial and surface-tension forces and should not exhibit a gravity dependence. This is in accord with the results in Fig. 19. Of course,

REDUCED GRAVITY ON HEAT TRANSFER

189

the removal of the detached bubbles from the vicinity of the heated surface is still gravity dependent. I t must not be inferred that all bubbles growing in water, for example, would be of the gravity-dependent type discussed here. If a particular 12TI0-5

3x10-6

--I -4

-3 -2 --I

-0

-2

-

-4

-

-6

-

-0

0

I

0

-

Fraction of Earth gravity.

- -I

1.0 0.126

- -2

----

- -3 .002

.004

.006

.000

DIO

.012

.014

Time, I& (solid curves)

I

.004

I

.000

I

.012

I

.016

1

.020

Time, rec (dashed curvesl

I

.024

I

.020

--4

.

6

I

.032

FIG.24. Inertial, buoyancy, surface-tension, and drag forces for bubbles growing in 60% aqueous-sucrose solution for 1.0 and O.126ge gravity fields. Heat flux, Q/A,20,500 Btu/(hr) (sq. ft); temperature difference, T,, - Tsa,, 30.1"F (60).

nucleation site emitted rapidly growing bubbles, then these would most likely exhibit a more significant inertial influence. It has been reported by Rohsenow (67) that for subcooled boiling, bubbles have sometimes been propelled away from the surface before condensing even for a horizontal surface facing downward. Usually, however, the bubbles grow and collapse while remaining either attached or very close to the surface. The differences in bubble behavior may result from the relative

190

ROBERTSIEGEL

magnitudes of the inertial, buoyancy, and surface-tension forces as discussed here. Some information on subcooled bubbles will be given a little later.

5. Rise of Detached Bubbles through Liquid Very limited experimental information in reduced gravity is available concerning the rise of boiling bubbles through the liquid after the bubbles detach from the heated surface. In Siegel and Keshock (59)boiling bubbles in saturated water were followed for approximately 1 inch of their rise away from the surface. At this height, the bubbles appeared to have reached a constant-rising velocity except for the lowest gravity tested, 0.014ge. An equation by Harmathy (68) for moderately distorted ellipsoidal bubbles predicts the rise velocity as (32) so that the steady velocity should decrease as gravity to the one-quarter power. The experimental data shown in Fig. 25 indicated that this power variation was correct.

Fraction of Earth gravity, g/g,

FIG.25. Effect of gravity field on velocity of freely rising bubbles for boiling water at saturation temperature and atmospheric pressure (59). Earth gravity, = 11.8 in./sec.

The drag coefficient for a rising bubble depends on the bubble Reynolds number. Harmathy (68) states that Eq. (32) is limited to a Reynolds number greater than 500 and hence will not apply for a low Reynolds number range where viscous forces are more significant. Some low Reynolds number information is given by Keshock and Siegel (60) for bubbles rising in 60” ,~by weight aqueous-sucrose solution, which is much more viscous than water. In this instance the duration of the drop tower experiments was insufficient

REDUCED GRAVITY ON HEAT TRANSFER

191

for the bubbles to achieve a steady-rise velocity. However, the motion of the bubbles accelerating for a short distance away from the surface demonstrated a considerable reduction in rise velocity resulting from decreasing the gravity field.

6. Some Effects of Higher Heat Fluxes T h e previous discussion of bubble dynamics has been limited to single bubbles that were observed at low heat fluxes. At low gravity, for higher fluxes the increased vapor formed coalesces into larger masses as it is not readily removed from the vicinity of the heated surface. This is illustrated in Figs. 26 and 27, which show nucleate boiling from horizontal and vertical wires at earth gravity and O.014ge.The heat flux is about 50,000 Btu/(hr)(sq. ft), which is still fairly low for boiling in earth gravity. However at O.014ge, the critical flux is only about 35% of that at lg, (using the relation Qc g1’4) so that the heat flux in Figs. 26 and 27 is relatively much closer to the critical heat flux when considering the low gravity. The photographs illustrate how after 0.5 sec in reduced gravity the average size of the vapor masses in the liquid has become much larger. Nucleate boiling is still present since the vapor masses have not grown around the wire. Schwartz and Mannes (43) studied the vapor formation for nucleate boiling in low gravity for heat fluxes from 8000 to 30,000 Btu/(hr)(sq. ft). T h e volumes of the bubbles were measured from photographs and the latent heat transported by the bubbles was then computed. This is given in Fig. 28 where the latent heat transport divided by the total heat transfer is given as a function of the total heat transfer. As gravity is reduced, the latent heat transport increases. I t is recalled from Fig. 8, however, which is also taken from Schwartz and Mannes (43),that the total nucleate-boiling heat flux was practically uninfluenced by a gravity reduction. This means that, although the latent heat transport increased as gravity was reduced, other means of heat transfer in the complex nucleate-boiling phenomenon must have been similarly decreased. Siege1 and Usiskin (36) and Clodfelter (40) show photographs for water at higher heat fluxes that are in the nucleate range at earth gravity but are above the critical heat flux in the reduced gravity tested. In this instance, the boiler in reduced gravity is quickly filled with large bubbles that envelop the heated wire or ribbon and cause a burnout of the electrically heated test section. This usually occurs at a local position along the test section, rather than the test section being completely blanketed with a vapor film. McGrew et al. (69) conducted an experiment to study bubble dynamics at zero gravity by a simulation of nucleate boiling. A porous rubber membrane was placed between a chamber of nitrogen gas and a pool of water, and N

FIG.26. Nucleate boiling for three fluids at saturation conditions from horizontal electrically heated wire (5). (a) Earth gravity. (b) After 0.50 sec in 0.014 earth gravity.

REDUCED GRAVITY ON HEATTRANSFER

193

Bu

x

ROBERTSIEGEL

194

the gas flowed through the membrane to produce streams of bubbles in the water. The apparatus was placed in a 100-ft drop tower yielding 23 sec of zero gravity. The gas flow was initiated as the free fall started. At first small bubbles grew at a number of positions on the membrane. Then neighboring bubbles coalesced into larger bubbles and eventually large bubbles formed that did not move away from the membrane. Thus, the surface tended to become covered with gas. The action was very similar to that of boiling bubbles in water and alcohol shown in Fig. 26. -Fraction

: 0

0

of Earth gravity, 919, 1

0.3

0.25 0.2 0.14

0.1

0.057 0.03-OM L

i2

-z

L

c ...

4-

-

a

B

d

c

F

2-

c L

5

c

1 0 4 r 8-

-

6-

-

4 0

.4 .6 .a Ratio of latent to total heat transfer

.2

1 I

FIG.28. Latent heat transport as function of total heat flux in lg, and low gravity fields [Schwartz and Mannes ( 4 3 ) ] .

G. BUBBLEDYNAMICS IN SUBCOOLED POOLBOILING The pool boiling discussion up to this point has been restricted to fluids

at or very near the saturation condition. There are a few reports on the influence of having the liquid subcooled, and these will now be discussed.

REDUCED GRAVITY ON HEATTRANSFER

195

The work of Cochran et al. (66) contains low gravity experimental data, while that of Rehm (70) interprets data at earth gravity to predict what would be expected when gravity is reduced. The tests by Cochran et al. (66) were carried out with distilled water at a pressure of 1 atm. A horizontal chrome1 strip 0.005 inch thick, 0.25 inch wide, and 0.50 inch long was heated with direct current. The strip was insulated on the back side so that boiling occurred only from the top surface. T h e boiler was mounted in an 85-ft drop tower equipped with a drag shield to reduce air friction. This yielded 2.3 sec at a gravity field less than 10-5ge. The boiler was also operated at earth gravity at the same heat flux, 28,900 Btu/(hr)(sq. ft), which was the value used throughout all the tests. T h e subcoolings tested were 5 , 10, 15, 25, and 40"F,and a few tests were also made close to the saturation temperature. For the low gravity test near saturation, a vapor mass formed above the surface and served as a collector for the vapor being generated at the surface. As time progressed, the vapor mass covered the heater and burnout seemed imminent. This behavior is similar to the results discussed in the previous section dealing with the saturated condition.

1. Bubble Growth Figure 29 shows typical bubble volumes as a function of time from the beginning of growth until the bubbles form a neck and break loose from the lWx10-8

Subcooling,

mc

##

0°

--* --*J.

"F

4

-------_______ --\

\

Fraction of Earth gravity, 919,

----14

Ti me, mil I iseconds

I

16

I

18

< 10-5 1

I

x)

I

22

I

FIG.29. Effect of gravity reduction on growth of bubbles for subcooled boiling of water at atmospheric pressure [Cochran et al. ( 6 6 ) ] .Heat transferred from solid surface to boiling liquid, 28,900 Btu/(hr)(sq. ft).

surface. At a large subcooling of about 38.5"F, the gravity reduction to less than 10-5g, had no appreciable influence on the bubble growth. At a smaller subcooling of about 5°F the gravity had little influence during the early

196

ROBERTSIEGEL

growth period. The significant influence was that bubble detachment was delayed and the maximum size of the bubble was increased somewhat. An average of several bubbles also showed that for small subcoolings the maximum size of the bubbles increased when gravity was reduced. This is consistent with the data for saturated water discussed previously.

2. Forces Acting on Bubbles Forces computed from Eqs. (24), (28), (29), and (31) are given in Fig. 30 for bubbles with large and small subcooling. The dynamic force in Fig. 30 should not be directly compared with the inertial force in Figs. 23 and 24 because of the different approaches used in computing the forces as discussed earlier in connection with (30) and (31). Figure 30(a) shows that for large subcooling there is little influence of the gravity reduction on the bubble forces. The buoyancy at lg, is already small because with large subcooling the bubbles condense before they can become very large. As a consequence, removing the buoyancy by going to zero gravity has little influence on the bubbles. Hence if subcoolings on the order of 30°F are maintained, it would be expected (based on the limited available data) that the nucleate-boiling heat transfer would be independent of gravity. Figure 30(b) shows the effect of gravity reduction for a smaller subcooling of about 5°F. At earth gravity the dynamic force is small; in fact, it becomes negative before departure and helps keep the bubble attached to the surface. The buoyancy in earth gravity becomes appreciable during the latter half of the bubble lifetime. At zero gravity the times involved are approximately doubled. The dynamic force becomes much larger than at earth gravity and must have a significant influence on bubble detachment. From Eqs. (29) and (24) the ratio of pressure to surface tension force is

For a bubble that is a truncated sphere, sin 8 = D,/2R, and FJ FS = - 1, which shows that for this geometry these forces would be equal in magnitude. A truncated spherical bubble shape would be expected for conditions of static equilibrium at zero gravity. The curves for pressure and surfacetension forces in Fig. 30 are similar in shape; their deviation from each other for zero gravity is evidently a result of the bubble distortion during growth. The findings by Cochran et al. (66) tend to agree with those of Kehm (70) who studied the bubble forces in subcooled boiling of water at earth gravity. Kehm found that as saturation was approached the removal force was due almost completely to gravity. This was not true for subcooled conditions, where the buoyancy force became of much less importance. For a horizontal

REDUCEDGRAVITY ON HEAT TRANSFER Fraction

I

1

I

2

1

3

I

(of

4

I

197

Sub-

I

5

6

Fraction Sub. of Earth CDolir OF gravity,

44 5.1

1

-

-1201 0

I

4

I

8

I

I

12 16 lime, millirexmdr

I

20

I

24

I

(b) FIG.30. Effect of gravity on bubble forces in water boiling at atmospheric pressure (66). Heat transferred from solid surface to boiling liquid, 28,900 Btu/(hr)(sq. ft). (a) Large subcooling. (b) Small subcooling.

plate facing upward, the ratio of the buoyant to the total removal force near bubble separation decreased from 0.8 at saturation to 0.15 at a subcooling of 30°F. Hence at zero gravity a subcooled condition is desirable when boiling is being used solely for cooling without the need for net vapor production.

198

ROBERTSIEGEL

FIG.31. Film boiling of saturated ethyl alcohol from electrically heated wire at earth and three reduced gravities (5). (a) Horizontal wire. (b) Vertical wire.

H. VAPORPATTERNS FOR FILMBOILING IN

A

SATURATED LIQUID

Some quantitative data giving the gravitational effect on film-boiling heat transfer were considered previously. Photographic results for filmboiling vapor patterns are given here for horizontal and vertical wires in earth and three reduced gravity fields. T h e fluid is ethyl alcohol and is at its saturation temperature at atmospheric pressure. The photographs in Figs. 31(a) and (b) indicate that as gravity is reduced there is a general increase in the size of the periodically spaced vapor masses along the wire. The vertical wire is especially interesting because the vapor does not rise in the form of a smooth boundary layer of increasing thickness with height. Rather, the circumferential component of the surface tension pulls the vapor into a series of regularly spaced vapor enlargements rising along the wire. When

REDUCED GRAVITY ON HEATTRANSFER

199

FIG.31b.

gravity becomes low, the influence of buoyancy and hence wire orientation is greatly reduced, and the vapor configuration around the vertical wire achieves an appearance similar to that for the horizontal wire. T h e principal difference is that for a vertical wire the entire vapor configuration moves axially along the wire. Additional information on the instability of a vapor layer surrounding a horizontal cylinder is given by Lienhard and Wong (71) and Siege1 and Keshock (5).

200

ROBERTSIEGEL V. Forced Convection Boiling

For low gravity applications in a system where there is a net vapor generation, a difficulty, arising as a result of the low buoyancy force, is the separation of the vapor from the liquid. This difficulty can be overcome by utilizing a forced-flow-type boiler in order to provide a substitute body force. If the pressure and drag forces exerted by the moving liquid and vapor are substantially larger than the buoyancy force, then the system performance should be independent of a gravity reduction. In this section two-phase flow boiling will be considered with reference to low gravity features. The limited experimental information on the effect of gravity on two-phase flow regimes and heat transfer will be reviewed and discussed.

A. REDUCED GRAVITY EFFECT ON TWO-PHASE FLOW T o obtain an indication of the importance of the body force in two-phase flow, the most simple case that can be considered is where no heat flow is present. An experiment of this type was carried out by Evans (72) who circulated a mixture of air and water through a $-in.-diam clear plastic tube 18inches long. The apparatus was flown in a zero-gravity airplane trajectory, which provided test durations of up to 15 sec. For some of the tests the apparatus was free-floating in the airplane, while for other tests it was tied to the aircraft frame and was subject to random accelerations of +0.02ge. The two different test conditions yielded the same results, so evidently the small gravitational body force of &0.02gewas negligible compared with the forces induced by the forced flow. Detailed descriptions of the flow patterns are given by Evans (72) as a function of percent air in the water and average velocity of the bubbles. T h e results reveal how the gravity field and tube orientation influence the distribution of bubbles over the tube cross section. For flow in a horizontal tube with approximately 30% by volume of air, the bubbles at earth gravity were mostly in the upper portion of the tube cross section with the smaller bubbles clustering near the wall. With a small volume of air the bubbles were quite small and were distributed a little more into the lower part of the tube cross section than for the higher air volume case. For either upward or downward vertical flow at earth gravity, the bubbles were distributed quite uniformly over the tube cross section except very close to the wall where the bubble population was somewhat diminished. In zero gravity the bubble distribution over the tube cross section was found to be very similar to the vertical case at earth gravity. This points out the possibility of utilizing a vertical-tube earth-bound experiment to simulate zero-gravity conditions. A second series of adiabatic visualization experiments were carried out by using twisted ribbons or coiled wires inserted into the tubes. These produced

REDUCED GRAVITY ON HEATTRANSFER

201

induced radial accelerations ranging up to 4ge. Swirling flow at earth gravity with coiled wire inserts in a vertical tube closely resembled swirling flow at Og,. T h e swirling increased the tendency of the air to coalesce in the central portion of the tube cross section. For swirling flow in a horizontal tube at lg,, the air masses still had a tendency to accumulate in the upper portion of the tube for the range of experimental conditions that were employed.

B. TWO-PHASE HEATTRANSFER What can be concluded from the adiabatic flow patterns concerning the two-phase flow behavior in low gravity with heat transfer ? Also, can tests at earth gravity be utilized to provide information applicable to the zero-gravity condition ? One difference fromadiabatic flow is that for the heat-transfer case there is bubble production at the wall. This is illustrated in a photographic investigation by Hsu and Graham (73) for a vertical tube at earth gravity. Hence the patterns given by Evans (72) have to be modified by postulating the generation of bubbles at the wall. Since in zero gravity the bubble generation would be symmetric around the tube periphery, it is reasonable that heat-transfer tests at earth gravity to simulate the zero-gravity case should be performed with a vertical tube orientation. Although a vertical-flow test at earth gravity will provide proper bubble symmetry over the flow cross section, there is still an effect of buoyancy along the tube length. If the flow is upward, the buoyancy force provided by the earth gravity field is aiding the bubbles in moving with the flow direction, Alternately, a test at earth gravity can be performed in downflow. The buoyancy force on the bubbles is then in opposition to the flow drag and there is an effective -lg, field with regard to the flow direction. In zero gravity the vapor bubbles leaving the wall should be dragged up to the speed of the liquid and then should move with the liquid. Hence in considering the simulation of zero gravity by earth-bound experiments, the interaction of the drag and buoyancy forces must be evaluated. Some estimates of the forces in forced convection boiling are discussed by Adelberg (74, 75). Ratios comparing drag, buoyancy, surface-tension, and inertial forces are formulated. There is so little heat-transfer data available on forced convection boiling in reduced gravity that at present generalized conclusions as to the relative importance of the various forces cannot be made. As an illustration consider one possible way to inquire into the gravity influence, at least for cases where there is not a high percentage of vapor in the liquid. This is by examining the terminal velocity u, for a bubble rising through a pool of liquid, as is given by expressions such as Eq. (32). For downflow in a vertical tube at earth gravity, the buoyancy

202

ROBERTSIEGEL

force can cause a bubble to move in opposition to the liquid with a velocity u , , ~relative to the liquid velocity. Thus if the liquid downflow velocity were u,,,, the bubble would remain at a fixed position along the tube length. If the liquid velocity were several times u,,,, the buoyancy effect should become negligible and the downflow case would perform the same as upflow. Some experimental work that approximately demonstrates the previous discussion is given by Simoneau and Simon (76). Boiling was studied from a heated wall in a vertical channel with liquid nitrogen in either upflow or downflow. The pressure was 35 psia and the bulk liquid was 6 to 10°F subcooled. For the range of bubble sizes encountered in nucleate boiling, the terminal rise velocity in a stationary liquid pool was computed to be in the range 0.53 < u , , ~< 0.75 ft/sec (76a). For a low flow velocity of 0.86 ft/sec, the downflow condition produced an appreciable vapor accumulation in the channel. At a higher velocity of 2.6 ft/sec, the downflow case had only slightly more vapor accumulation near the channel outlet than the upflow case. The behavior of the vapor patterns indicates that a flow velocity several times u, would be sufficient to cause buoyancy forces acting in the flow direction or opposite to it to have little influence on the vapor motion. Similar results were also found for film boiling. Some critical heat flux experiments to determine the influence of a positive or negative earth gravity relative to the flow direction have been carried out by Papell et al. (77). Liquid nitrogen was pumped in either upflow or downflow through a vertical heated tube 0.505 inch in diameter. Pressures ranged from 50 to 240 psia, the inlet velocity of the liquid from 0.5 to 11.0 ft/sec, and the inlet subcooling from 12 to 51°F. The critical heat flux was defined as the flux required to initiate an excursion of increasing temperature at any local position along the tube wall. Figure 32 shows a typical set of data for the critical heat flux as a function of liquid inlet velocity in upflow and downflow. In upflow the excursion of increasing wall temperature always started at the outlet end of the tube. Presumably this is caused by both the flow velocity and buoyancy force driving thevapor accumulation toward the upper end of the tube. The highest vapor accumulation would be reached at the tube outlet and tend to blanket the wall with vapor. When the inlet liquid velocity for upflow exceeds about 6.5 ft/sec for the particular conditions in Fig. 32, the upflow velocity begins to exert an effect on the critical flux. Below this velocity the influence of buoyancy must be appreciable. Turning now to the data for downflow, at very low velocities it was found that the critical flux was first reached at the inlet (upper end) of the tube. Evidently the flow velocity was insufficient to drive the bubbles downward, and consequently the vapor still accumulated near the top of the tube. For velocities higher than 2 to 3 ft/sec, the critical flux position shifted to the outlet (lower end) of the tube.

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Beyond a certainvelocity, about 6.5 ft/sec in the case of Fig. 32, the upflow and downflow data merge together so that there is no longer an influence of the buoyancy orientation relative to the flow direction. Since the reduced gravity range O,< gig, ,< 1lies between the lg, conditions tested, the merging of the curves indicates that above this inlet flow velocity the same performance should be obtained in reduced gravity as in the earth gravity tests. Results of the same nature as shown in Fig. 32 were obtained for a range

‘1

7 Upflow \

.20c u c

.-

-a -

Flow .16-

:‘

Inlet

Burnout independent of flow direction

xc

Z

c

‘Downflow

.08Inlet

m U .c ._ L

V Burnout at inlet

0 Burnout at outlet

V

@ 0

I

2

4 6 8 Liquid inlet velocity, ftlsec

10

FIG.32. Typical data for critical heat flux as function of inlet velocity for upflow and downflow of liquid nitrogen in O.SOS-in.-diamtube. Inlet pressure, 75 psia; inlet subcooling, 19°F [Papell et al. (77)].

of subcoolings and pressures. T h e velocities at which the effect of orientation vanishes are summarized in Fig. 33. T h e dotted lines are each for a different inlet pressure, and they show that for a given pressure the inlet velocity necessary to overcome the influence of gravity on the critical heat flux depends strongly on the inlet subcooling of the liquid nitrogen. When the pressure or subcooling is increased, the liquid velocities necessary to make gravity unimportant become smaller. This would be expected since an increase in pressure or subcooling should decrease the vapor volume thereby decreasing the influence of buoyancy on the critical heat flux.

ROBERTSIEGEL

204

Another illustration of the influence of gravity was found by Macbeth

(78).He gathered together data for boiling in forced axial flow through heated rod bundles at earth gravity. The majority of the data was for water at 1000 psia. The data for vertical axial upflow through the bundles were found to form a separate group from those for horizontal axial flow. T h e horizontal flow burnout heat fluxes were as much as SO:/, below those for the vertical upflow condition. Evidently in the horizontal case there was a stratification of vapor that would promote a burnout condition among the upper tubes in

1

Inlet static pressure, psia Constant inlet temperatur Constant inlet pressure

d

740

4

c

Buoyancy independent region

v1 m

I-

't 0

1

2

I

4

6

Liquid inlet velocity, ftlsec

a

1

FIG.33. Liquid inlet velocity at which critical heat flux becomes independent of upward or downward flow direction for liquid nitrogen in O.SOS-in.-diam tube (77).

the bundle. At very low gravity this stratification would not occur, and the critical heat flux would be expected to correspond more to the range of data for the vertical orientation. Some near-zero-gravity two-phase heat-transfer tests by Papell (79) revealed an instability effect. Distilled water at 50 psia and subcooled 175 to 198°F was pumped through an electrically heated stainless-steel tube that was 0.311 inch in diam. The flow rate was 0.20 lb/sec and the heat addition 1.40 Btu/(sec)(sq. in.). For higher flow rates the instability was not observed, and it also could not be induced at earth gravity. The apparatus was mounted in an airplane that flew a zero-gravity trajectory and produced

REDUCED GRAVITY ON HEATTRANSFER

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test times of 15 sec at 0 to 0.02ge. The instability was induced as follows: After the apparatus had been in zero gravity for 3 sec, the flow was stopped for about 1sec. During the flow interruption the wall temperatures increased about 200°F. When the flow was restarted, the wall temperatures decreased to a lower level than had existed previous to the interruption even though none of the test conditions had been changed. The increase in heat-transfer coefficient was about 16%. During the flow interruption there is an increase in the amount of vapor within the tube. It is postulated that the presence of more vapor changes an originally highly subcooled bubbly flow to a more violent slug-type flow which then persists for the remainder of the test. In earth gravity the buoyancy would help separate the vapor back out of the liquid and prevent the slug flow pattern from developing. T h e zero-gravity performance of a flow loop containing an evaporator and condenser was investigated by Feldmanis (79u,b). Water was pumped through a straight electrically heated evaporator tube [some photographic results with swirl devices in the tubes are also included in (79u)l and the resulting two-phase mixture passed into a tapered condenser. Zero-gravity airplane flights were used to free float the apparatus and provide 8 to 19 seconds of zerog, testing time. Water entered the 0.305-inch i.d. evaporator tube with a velocity of about 0.09 ft/sec and was only partially evaporated; the fluid left as a mixture of liquid and vapor slugs with a velocity of 72 ft/sec. Prior to entering the zero-gravity flight path, the electrical power was turned on and the boiling and condensing processes established. Two significant findings in zero gravity were found with regard to system pressures ; detailed heat transfer measurements were not taken. One finding was an increase in system pressure during the zero-gravity condition. The second was the damping of pressure oscillations that existed in the flow loop during earth gravity operation. After about five seconds in zero gravity the loop operation became quite stable and the pressure oscillations almost entirely disappeared. The reasons for this behavior are not yet clearly defined.

C. DESIGNSINVOLVING SUBSTITUTE BODYFORCES Instead of utilizing drag forces produced by forced flow in a straight tube, there are alternate means of providing a substitute body force to obtain proper performance at low gravities. An informative discussion of some of these ideas is given by Ginwala et ul. (80). One approach is to utilize centrifugal forces to provide an artificial gravity field. Twisted ribbons or coiled wires can be inserted into the boiler tubes to provide a swirling action. This was mentioned earlier in connection with the work of Evans (72) who observed swirling flow patterns for air-water mixtures at earth and zero gravities. Extensive work has been done with twisted inserts in tubes as a

206

ROBERTSIEGEL

means of improving boiling performance. An example is the work of Gambill and Greene (81)and Gambill et al. (82),where swirl was utilized to increase the burnout heat flux. Since the inserts can produce effective radial accelerations of several or more earth gravities, a satisfactory performance should result in reduced and zero gravity. Instead of using internal inserts to swirl the flow, the whole boiler can be rotated, although this would probably not be as convenient for a space application. A number of investigations utilizing centrifuges have demonstrated the nature of boiling in high effective gravity fields and the accompanying improvement in the critical heat flux. No attempt will be made to review the extensive work in this area; some references (83-88) are provided for the interested reader. Another approach that can be employed where net vapor generation is required for a power generation cycle is to use flash evaporation by employing an expansion engine or by having a sudden pressure drop occur on the heated liquid. This method of vapor production can eliminate the need of a body force to separate the vapor from the liquid. Instead of utilizing a body force in boiling, there is the possibility of controlling the liquid by surface forces that are independent of gravity. The liquid can be pumped to the heated surface by the use of capillary forces. A wick-type material of either fibrous or metal mesh would be placed adjacent to the surface to keep it in contact with the liquid. Some tests at earth gravity with wick materials have been carried out by Ginwala et al. (80) and Allingham and McEntire (89). In the latter, it was found that at low heat fluxes the wicking material covering the surface aided bubble nucleation and improved the heat transfer as compared with ordinary pool boiling. At higher heat fluxes, however, the boiling was hindered as the incoming liquid motion was interferred with by the escaping vapor. If the wicking is wrapped around the heater in such a way that an open path for vapor removal is provided, high critical heat fluxes can be obtained. Values up to 1.2 x lo6 Btu/(hr)(sq. ft) were attained for water by Costello and Frea (90). VI. Condensation without Forced Flow

The discussion of condensation will begin with situations where there is no forced-flow pumping pressure applied to move either the liquid or the vapor so that the fluid motion is produced only by the gravity field. As would be expected, for these conditions low heat transfer is obtained when the gravity field is substantially reduced. T o mitigate this difficulty, a forced flow can be utilized to move the liquid and vapor over the surface. In the following discussion some aspects will first be presented of condensation

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when only the gravity force is present. Then in the succeeding section some results will be given with forced flow, such as in a condenser for a space power plant.

A. LAMINAR FILMCONDENSATION ON A VERTICAL SURFACE When condensing is occurring on a stationary surface in a quiescent vapor, the process is gravity dependent as it is the gravitational force that produces the liquid flow along the surface. For example, consider the classic

FIG.34. Laminar condensate film on vertical flat plate.

case of laminar film condensation on a vertical flat plate as illustrated in Fig. 34. The thickness of the liquid film flowing down the plate is (91)

Equation (34) shows that the film thickness at a fixed x position on the surface increases with decreased gravity in proportion to g-ll4. Hence at very low gravities the liquid layer would become extremely thick. Since the heat transfer depends inversely on the film thickness, within the simplifying assumptions of the Nusselt condensation theory the local heat-transfer coefficient is given by (91)

208

ROBERTSIEGEL

The Nusselt theory and more refined boundary-layer theories of film condensation have generally utilized boundary-layer-type assumptions that require the liquid layer to be thin relative to the length along the surface. These assumptions can be violated for the thick layers produced in low gravity. When the boundary-layer assumptions are valid, Eq. (35) shows that h, decreases asg’I4leading to low heat transfer as gravity becomes very small. The integrated average heat-transfer coefficient over the length of the surface also has this gravity dependence.

B. LAMINAR-TO-TURBULENT TRANSITION AND TURBULENT FLOW For condensation on a vertical flat plate as in Fig. 34, the transition from a

laminar to a turbulent condensate film depends on the Reynolds number of the film which can be defined as

Reg = CSlv (36) For laminar flow the mean film velocity C depends on gl/’, while 6 depends on g-’I4. Thus the film Reynolds number in the laminar range depends on g1/4.As discussed by Grober et al. (14, p. 330) this Reynolds number can be written (after slight modification) as

The numerical value of the transition Reynolds number is somewhat indefinite, and for purposes of the present discussion an average value of 350 will be utilized. Then Eq. (37) can be rearranged to give at transition

As an illustration, if saturated steam at a 1-atm pressure is condensing on a wall at 62”F, then using average film properties yields (T,,, - T,)x(g/ge)li3z 820 ft O F Since (T,,, - T,) = 150°F in this illustration, ~ ( g / g , )=~5.5 / ~ ft. At earth gravity a laminar condensate film would be maintained for 5.5. ft of condensing surface. In a reduced gravity field of O.OOlg,, the length having laminar flow would be increased to 55 ft. When the boundary layer becomes turbulent, the heat-transfer coefficient is more strongly dependent on gravity than for the laminar case. For a turbulent condensate layer on a vertical flat plate of height L , the integrated average heat-transfer coefficient is given by Grober et al. (14, p. 342)

REDUCED GRAVITY ON HEAT TRANSFER

209

This equation indicates a dependence on gravity to the one-half power rather than the one-quarter power as in laminar flow. C. TRANSIENT TIME TO ESTABLISH LAMINAR CONDENSATE FILM Consider the time that would be required to fully establish the film condensation layer for a typical situation-laminar condensation on a vertical flat plate. A vertical plate is suspended in a large expanse of quiescent pure vapor at its saturation temperature Tsatas in Fig. 34. Then the plate temperature, which is initially also at T,,, , is instantaneously dropped to a lower temperature T , and condensation commences. It is assumed that film condensation occurs throughout the transient process. Sparrow and Siege1 (92) give the time for a steady-state condensation layer to be established at a position x along the plate :

As gravity is reduced, the transient time increases in proportion to g-’I2. As an example consider saturated water vapor at 1 atm, 212”F, condensing on a plate whose temperature is suddenly dropped to 100°F. For these conditions the time required to achieve steady state is about 0.63 sec at x = 6 inches, and 0.9 sec at x = 12 inches. If gravity is reduced to O.Olg,, the time at x = 12 inches increases to 9 sec, and for 10-6gethe steady-state time becomes 900 sec or 15 min. This of course assumes that the laminar boundary-layer theory still applies for the thicker condensation layers encountered is 0.22 at low gravities. Since the layer thickness at x = 12 inches for 1OW6ge inch as computed from Eq. (34), this seems to be a reasonable assumption for this example. VII. Forced Flow Condensation

An interest in forced flow condensation in reduced gravity stems from the design of electric generating plants for space use. A system that shows some promise is a Rankine cycle turbogenerator unit that utilizes a liquid metal as the working fluid. A forced flow condenser would be employed so that gravity would not be required to collect the condensate, and the system would be able to operate properly in the zero-gravity space environment. In addition to requiring heat-transfer information for the condenser design, two-phase pressure drop data is needed. Condensation occurs in the lowpressure portion of the Rankine power cycle. It is restricted to a low-pressure drop allowable from the turbine exit pressure to the pressure required at the suction side of the pump recirculating the condensate to the boiler.

ROBERT SIEGEL

210

A. FLOW BEHAVIORI N Low GRAVITY Some flow visualization tests have been carried out for mercury condensing in horizontal tubes. Cummings et al. (93)[a brief discussion is also given by Trusela and Clodfelter (94)]employed a 12-inch-long glass tube with a 0.196-inch i.d. An airplane flying a ballistic parabola was utilized to obtain 12 sec at zero gravity. Albers and Macosko (95)and Albers and Namkoong

-

Flow direction

FIG.35. Flow configurations at interface location for 1 and zero g, mercury nonwetting condensation (#-in. 0.d. glass tube.) (a) lg,; vapor mass flow rate, 0.031 Ib/sec. (b) Zero g,; vapor mass flow rate, 0.035 Ib/sec. [Albers and Narnkoong (96)].

(96) observed condensation of mercury in constant-diameter horizontal glass tubes with 0.27-, 0.40-, and 0.49-inch i.d. and a tube length of 84 inches. The vapor flow rate ranged from 0.03 to 0.05 lb/sec, and the tubes were convectively cooled, thus providing a uniform heat flux boundary condition. An airplane was used to provide test times of 10 to 15 sec. In all

REDUCED GRAVITY ON HEATTRANSFER

21 1

of these tests the mercury did not wet the tube wall. Motion pictures taken with high-speed cameras revealed the details of the condensation process. An illustration of the differences in the flow at 1 and Og, is shown in Fig. 35. At earth gravity, and particularly when the vapor velocities were low, the mercury drops would run diagonally down along the tube wall resulting in a liquid accumulation in the bottom portion of the tube cross section. At Og, there tended to be uniform distributions of the drops over the cross section of the vapor stream and around the wall circumference. In general, the drops were observed to form on the tube surface and travel along the wall; they increased in size as they coalesced with other drops along their path and in some cases were swept into the vapor stream. At high vapor inlet velocities of 250 ft/sec the entrained drops became very small and a mist flow resulted. As an aid in the analysis of dropwise condensation, a detailed photographic study was made by Sturas et al. (96a)of the geometry of individual mercury drops in 0, 1, 1.5, and 2ge environments.

B. PRESSURE DROP One of the important objectives in the condensation experiments was to find whether the differences in flow for the 0 and lg, conditions would influence the pressure drop in the condenser tube. Detailed pressure drop information was obtained by Albers and Macosko (95, 97) and Albers and Namkoong (96) for both constant-diameter and tapered stainless-steel tubes, and the findings will now be discussed. The constant-diameter tube had an internal diameter of 0.31 1 inch and was 87 inches long. The tapered tube was 84 inches long and had internal diameters of 0.40 inch at the inlet and 0.15 inch at the exit. The effect of gravity on the static pressure drop from the beginning of the condensing length to a local position along the tubepo -pi is given in Fig. 36. Near the entrance of the condenser there are high vapor velocities that produce large friction losses. This caused the value of p, - p, to increase within the first half of the condensing length. In the last half of the condensing length, the pressure rise derived from momentum recovery exceeded the frictional loss so thatp, -pi decreased. The important finding for the present discussion was that the change from the 1 to Oge environment had no discernible influence on the static pressure drop distribution throughout the tube length (see Fig. 36). The over-all static pressure drop from the start of the condensing length to the vapor-liquid interface at the end of condensation is given in Fig. 37 as a function of condensing length for a typical mass flow rate. Although the majority of the Og, data fall slightly above the lg, values, there is little difference for practical purposes between the 0 and lg, conditions. The values of

212

ROBERTSIEGEL

the over-all static pressure drop ranged from 0.2 to 2.2 psi for the straight tube, and from a 0.9-psi pressure rise to a 0.1-psi drop in the tapered tube for the range of flow rates and condensing lengths tested. From these measurements it appears that the change in distribution of mercury drops within the tube, caused by a reduction from earth to near Fraction ot Earth gravity,

.-m

V c c VI

0

12 24 36 48 60 Distance from condensing tube inlet I , inches

1

(b) FIG.36. Effect of gravity on local static pressure drop (96).Vapor mass flow rate, 0.038 Ib/sec. (a) Constant diameter tube (0.31-in. i.d.). (b) Tapered tube (0.40-in. inlet i.d. with taper ratio, 0.036 in./ft).

zero gravity, has no significant influence on the pressure drop during condensation. Hence there may be no need to resort to low gravity tests when pressure drop data for condensing mercury is needed for a space vehicle application. I t must be carefully noted that this finding is based on a range of flow and heating variables that is rather limited when considering the wide range of flow conditions that are possible in a two-phase forced-flow situation. For conditions considerably different from those discussed here

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there might be a significant effect of gravity reduction. Since the tests have all been for mercury, it would be desirable to conduct similar tests with fluids that wet the tube wall.

I 1.61

48

Fraction of Earth gravity,

gh,

0

0

52

I

56

I

60

I 64

Condensing length, {, inches

I 68

72

(b) FIG.37. Effect of gravity on over-all static pressure difference for entire condensing length (96).Vapor mass flow rate, 0.028 lb/sec. (a) Constant diameter tube (0.31-in. i d . ) . (b) Tapered tube (0.40-in. inlet i.d. with taper ratio, 0.036 in./ft).

C. VAPOR-LIQUID INTERFACE Another factor of importance in low gravity forced flow condensing of nonwetting mercury is whether the vapor-liquid interface that is established across the tube as shown in Fig. 35(b) will remain stable. This is discussed from a theoretical viewpoint by Lancet et al. (98). Consider two adjacent fluid layers having different densities. If a body force is oriented from the more-dense toward the less-dense fluid, the interface will tend to become unstable. T h e interfacial disturbances that have the greatest tendency to amplify have a wavelength (the critical wavelength) given by the Taylor instability theory

Equation (41) is interpreted with respect to the interface in a tube to imply that the interface will be stable (i.e., the liquid will not spill out from a vertical tube) if the condenser tube diameter is less than one-half the critical wavelength. A more detailed derivation applying the stability theory to a round tube geometry yielded the same functional form as Eq. (41) but with a lower constant of 1.84 instead of 2n. These relations indicate that the

214

ROBERTSIEGEL

maximum stable tube diameter increases as gravity is reduced according to

g-1/2*

As discussed by Lancet et al. (98) [see also Denington et al. (98a)], it has been found experimentally at earth gravity that the range of critical tube diameters for mercury is up to 0.168 inch, depending on the tube surface finish and adhesion between mercury and the surface. This is in agreement with the experimental results of Reitz (99), where the maximum tube diameter for stability at lg, was 0.15 inch. In a reduced gravity field of, for example, O.05ge, the critical diameter would increase by about 4.5 times so that it would be about 0.75 inch. For low gravity testing the tube used by Reitz (99)was 0.196 inch in diameter and was found to have a stable interface in zero gravity. The interface experienced small oscillations resulting from impacts of condensate drops, but did not exhibit any large-scale unstable motion (93).It should be remarked that there are some restrictions in the Taylor theory ;for example, the body force is normal to the interface between the two fluids. A horizontal tube [see Fig. 35(a)] in a finite gravity field can have a long interface (not normal to the gravity vector) where the condensate layer in the lower portion of the tube cross section gradually thickens along the tube length. T h e interface can be especially long if the liquid wets the tube wall very well. In the Taylor theory thereis no mass addition at the interface. In condensation, new condensate is continually being supplied at the interface and there is liquid flowing away from it. This may influence the stability, especially for high rates of condensation where the flows would be large.

D. NONCONDENSABLE GAS

An indication of the behavior of noncondensable gas in the forced flow condensation process was obtained by Lancet et al. (98). Condensation at earth gravity was observed in a below-critical-diameter tube that was tilted at a few different angles away from the horizontal. The vapor-liquid interface was held stable by surface tension. Noncondensables accumulated in the vapor region upstream of the interface. In the region of high noncondensable concentration, little condensation occurred. When the noncondensables occupied several diameters of the tube length, the mercury drops were slowed by viscous damping in the gas. The drops rapidly coalesced and eventually bridged the tube diameter before the liquid could touch the vapor-liquid interface. This entrained a slug of noncondensables and at the same time formed a new interface, thereby starting the process again. The same behavior was observed at zero gravity by Cummings et crl. (93). In zero gravity the flow was more evenly distributed over the tube cross section. This permitted the noncondensables to be more easily trapped by the liquid and carried away by the liquid flow.

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VIII. Combustion

Some combustion processes depend on gravity because free convection accounts for the circulation of some of the reactants into the flame zone. A. CANDLE FLAME A common example of gravity-dependent combustion is a candle flame. Here free convection carries most of the oxygen into the flame. I n zero gravity the question is whether diffusion alone would suffice to carry the combustion products in and out of the flame zone to sustain the reaction. Experiments on burning a candle in various gravity fields were carried out by Hall (100). A white paraffin candle and a camera were mounted in a box that could be free floated in an airplane during a zero-gravity trajectory. Test results were also taken during the 2ge turn entering into the zerogravity flight path. Thirty-five tests were made, each lasting from 25 to 28 sec. Two atmospheres were used, air at 14.7 psia and oxygen at 5 psia. During all the tests the candle was found to burn continuously. At 2ge the flame heightened and narrowed as compared with lg, but remained yellow in color. In zero gravity the flame became spherical and had a lightblue color. Thus, at least for the experimental durations considered, the combustion in a candle flame can be sustained in the absence of free convection.

B. FUELDROPLETS The burning of fuel droplets in reduced and zero gravity has been studied by Kumagai and Isoda (101-103). In combustion of a finely divided fuel spray the fuel droplets are so small that the influence of natural convection around each drop is negligible. Consequently, the flame surrounding an individual droplet is essentially spherical. One way to study droplet combustion would be to observe the actual very fine droplets, but this has the difficulty of dealing with the small size. An alternative procedure is to use droplets of larger size and eliminate the free convection, which would occur for them at earth gravity, by studying the combustion in a low gravity field. The latter of these alternatives will be considered here. The resulting experimental information can be compared with analyses that assume the combustion is spherically symmetric about the droplet. Kumagai (101) and Kumagai and Isoda (102) placed a chamber in a counterweighted drop tower. The chamber contained a vertical silica filament from which a fuel droplet was suspended. T h e droplet diameter range was from 1 to 1.5 mm. The droplet was ignited by an electric spark

ROBERTSIEGEL

216

at the beginning of the low gravity period. One photograph was taken during each test as the falling chamber passed a stationary camera 0.1 to 0.4 sec after ignition. Direct photographs were utilized to observe the flame boundary, and schlieren photographs were taken to reveal the hot-air zone. T h e fuels were ethyl alcohol and n-heptane. In earth gravity the flame had an oval shape and there was an upward flow of the hot air around the droplet.

.2

0

I

.1

t

.2

I

.3

I 1 1 I .4 .5 .6 .7 Fraction of Earth gravity, 919,

I

.8

I

.9

FIG.38. Evaporation constant and shape of luminous flame for burning droplets as function of gravity for two fuels [Kumagai and Isoda (IOZ)].

In zero gravity the combustion zone became spherically symmetric around the droplet. The dimensions of the flame and hot-air zones are given by Kumagai and Isoda (102)along with photographs at six different gravities. In Fig. 38 the outlines of typical flame zones are shown to illustrate the change toward a symmetric shape as gravity is reduced. The decrease of droplet size with time during combustion was expressed in terms of an evaporation coefficient K,, . This coefficient relates the rate of change of droplet mass to the droplet diameter. It was found experi-

REDUCED GRAVITY ON HEAT TRANSFER

217

mentally that there was a direct proportionality between these two quantities so that -dmf/dt = K,, 7rp (D/4) (42) Since the droplets are assumed spherical, Eq. (42) can be integrated to yield DO2 - D2 = K,, t (43) where Do is the initial size of the droplet. As shown in Fig. 38, the evaporation coefficient was found to decrease with gravity; for example, for n-heptane the value of K,, at zero gravity is about one-half that at earth gravity. A shortcoming of the experiments by Kumagai (101) and Kumagai and Isoda (102)was that the combustion was observed only at one instant during each test. T o observe the behavior of the combustion process as a function of time, another series of experiments was conducted (103). Here a movie camera was placed in free fall along with the combustion chamber and burning was observed in zero gravity for about a 1-sec duration. The fuels were n-heptane, ethyl alcohol, and benzene. Ignition was made just after the falling platform was released. The luminous flame boundary was completely spherical and concentric with the droplets. For benzene the luminosity of the flame obscured the view of the droplet so that droplet measurements could not be made. For n-heptane and ethyl alcohol, however, the burning rate was again found to obey Eq. (42). The K,, values at zero gravity for n-heptane and ethyl alcohol were 0.60 and 0.46 mm2/sec as compared with the corresponding higher values of 0.84 and 0.70 mm2/sec at earth gravity. These values are a little different from those in Fig. 38, but again reveal how the lack of free convection decreases the rate at which the droplets are consumed. The hot-air zone outside the flame expanded continuously with time during the zero-gravity period. The diameter of the luminous spherical flame increased at first, and then it decreased as the burning droplet became small.

C. SOLIDFUELS The combustion of several solid fuels in zero gravity was studied by Kimzey (104). An airplane was used to provide test durations up to 12 seconds. Some of the fuels tested were styrene, white foam rubber, paraffin, neoprene, and polyurethane. These were burned in pure oxygen, air, and oxygen-nitrogen mixtures at pressures from 5 to 14.7 psia. The fuel was ignited during weightlessness with an electrically heated nichrome wire, and motion pictures were taken with both color and infrared film. Ignition during weightlessness was found to be essentially unchanged from that in earth gravity, but the burning rate was considerably reduced in

218

ROBERTSIEGEL

zero gravity. For example, in 5-psia oxygen the burning rate of a tubular length of polyurethane was reduced from 0.6 to 0.08 inch/sec by a change from 1 to Og,. T h e reduction in burning rate is consistent with the decrease in evaporation coefficient discussed in the previous section. The flame behavior, however, for the several materials tested, was different from the continuous burning reported for a candle (100). Since one of the materials tested by Kimzey (104) was paraffin it is not clear why this difference was found. Soon after ignition the flame in (104) was found to reach a maximum size and brilliance; then, the flame receded and darkened. This was thought to be caused by an insufficient rate of diffusion of oxygen into the flame (the fact that the candle in (100) continued to burn indicates that the method of fuel supply, i.e., the use of a wick, may be significant). The flame darkening occurred within a few seconds after ignition. Often, the flame would appear to be extinguished as the zero-gravity period proceeded ;however, the flame would resume if gravity were restored even momentarily. The tests were of insufficient duration to determine whether the flame might eventually be completely extinguished in zero gravity. IX. Summary and Areas for Further Investigation

T o the harried reader who is endeavoring to cope with the explosive production of scientific literature is dedicated this capsule summary of some of the significant findings covered in this review. Within the summary, some areas for further research will also be brought to attention. Studies in reduced gravity are important (1) for specific applications in space devices, (2) to determine the validity of the gravity dependence in theoretical and experimental correlations, and (3) to remove unwanted effects of free convection in certain experiments. Experimental tests have been conducted in the following types of facilities (listed in order of providing increased durations of testing time) : (1) drop tower, (2) airplane flying ballistic trajectory, (3) rocket suborbital flight, (4) manned space capsule, (5) satellite (tests being planned), and (6) use of a magnetic field in conjunction with fluid having magnetic properties. In the area of free convection there is a lack of data at reduced gravities, so this provides a subject for future experimentation. A difficulty is that free convection boundary layers have a relatively slow time response so that tests cannot be conducted with convenient facilities such as drop towers or airplane flights. From analysis and correlations of lg, data some conclusions are the following : 1. Free convection depends on the Rayleigh number, which contains gravity to the first power. In laminar flow the free-convection heat-transfer

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coefficient should vary a ~ g ' /while ~ , for turbulent flow the exponent increases and tends toward g2lSas the Rayleigh number continues to increase. to g1/3 2. For laminar flow the boundary-layer thickness depends ong-'l4; hence, at very low gravities the thickness can become large enough so that thinboundary-layer theory no longer applies. 3. If a vertical plate initially at the same temperature as its surroundings has its temperature suddenly raised, the time required for steady state to be achieved for laminar flow depends on g-'/'. As a result, the transient times can become very long in low gravity. For pool boiling of a saturated liquid the following results have been found :

1. For nucleate boiling, both analysis and experiment indicate that the relation between temperature difference (T, - Tsst)and wall heat flux is insensitive to gravity. 2. T h e peak nucleate boiling (critical) heat flux was found experimentally to vary reasonably well as g1/4,which is in agreement with theory. 3. T h e relation between T, - T,,, and wall heat flux in the transition region between nucleate and film boiling appears from limited data to be insensitive to gravity reductions. 4. T h e minimum heat flux value where transition boiling changes to film boiling depends on g1/4. 5. In laminar film boiling the heat-transfer coefficientdepends ~ n g ' ' ~For . a turbulent vapor film the exponent increases and may be as large as 3 to $. The experimental evidence to substantiate these conclusions is mostly from tests of short duration. Additional experimental work is needed with longer testing times. This is especially true at very low gravities ;for example, it has not been conclusively demonstrated that the critical heat flux does go to zero as exactly zero gravity is approached. Longer test times would assure that the thermal layer near the surface has sufficient time to adjust to the reduced-gravity conditions. Photographic studies of reduced-gravity pool boiling have shown the following for saturated conditions :

1. Vapor accumulations tend to linger near the surface and collect new bubbles being formed thereby helping to remove them from the surface. 2. T h e dependency of bubble diameter on time during bubble growth is insensitive to gravity as shown by both theory and experiment. 3. The diameter of single bubbles at departure depends on g-'l2 for bubbles that grow slowly. For these bubbles, buoyancy is the force causing bubble departure.

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4. For rapidly growing bubbles, departure is governed by inertial forces and the departure size becomes insensitive to gravity. 5. There is a pressure force acting on the top portion of the bubble that lies above the area of the bubble base. This force aids bubble departure and does not depend on gravity (it is also present for subcooled conditions.) 6. The velocity at which detached bubbles rise through the liquid appears from limited data to decrease with gravity as predicted by theory. For subcooled conditions, the bubble formation' in water becomes independent of gravity when the subcooling is about 30°F. Again it is emphasized that these conclusions are based on limited tests of short duration and that longer testing times are a need for future work. For reduced gravity forced-convection boiling, little experimental information is available :

1. Isothermal two-phase flow visualization tests with air-water mixtures have shown that the bubble distribution over the cross section in a vertical tube at earth gravity is very similar to that in zero gravity. This points out the usefulness of the vertical orientation to simulate low gravity conditions. 2. Critical heat flux tests with upflow and downflow in an earth-bound vertical-tube experiment can define a fluid velocity range above which the critical flux is insensitive to flow orientation with respect to the gravity vector. Above this velocity range, performance should be insensitive to gravity reductions. For condensation without forced flow, no experimental results were found in the literature; hence, this would be another fruitful area for future work :

1. From laminar film condensation theory for a vertical plate at constant temperature, the heat-transfer coefficient depends on g114.For a turbulent condensate layer the exponential dependence increases to about g'l'. 2. The transient time to establish a laminar condensate layer on a vertical plate varies as g-'12 for a sudden reduction in plate temperature. For forced flow condensation in tubes, low gravity tests have been performed with nonwetting mercury :

1. The two-phase pressure drop was insensitive to gravity reductions for the conditions tested. 2. An indication of the stability of the vapor-liquid interface could be found from the Taylor type of instability theory. 3. Low gravity conditions can aid in the trapping of noncondensable gas.

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Further experimentation is needed to expand the range of condensing rates and to test fluids that wet the tube wall. A few combustion tests have been performed in reduced and zero gravity :

1. At zero gravity a candle continued to burn and the flame became

spherical with a light-blue color. 2. Reduced gravity decreased the rate of burning for fuel droplets. T h e luminous flame and hot-gas zones around a droplet became more spherical as gravity was reduced. 3. For several solid fuels the burning rate was substantially decreased by a change from earth to zero gravity. After ignition in zero gravity the flame reached a maximum size and brilliance ;then the flame receded and darkened.

NOMENCLATURE surface area acceleration relative to a fixed frame of reference constant in drag coefficient coefficient in boiling correlation, Eq. (9) specific heat at constant pressure bubble diameter [or droplet diameter in Eqs. (42) and (43)] bubble base diameter bubble departure diameter initial droplet size force buoyancy force defined by Eq. (28) buoyancy force defined by Eq. (27) drag force defined by Eq. (25) dynamic force defined by Eq. (31) inertial force defined by Eq. (30) pressure force defined by Eq. (29) surface-tension force defined by Eq. (24) functional dependence Grashof number based on length L gravity field of heat-transfer system being studied earth gravity field (also used in some equations to designate the conversion between Ib mass and lb force) effective gravity field caused by rotation

height of vertical surface heat-transfer coefficient local heat-transfer coefficient at position x average heat-transfer coefficient thermal conductivity characteristic length total condensing length mass average Nusselt number, i L / k Prandtl number, c p p / k static pressure pressure inside bubble static pressure at local position along tube static pressure at beginning of condensing length, or pressure outside bubble heat transfer per unit time critical heat transfer rate in boiling heat transfer per unit time and area heat flus from vapor in bubble to liquid radius of curvature modified characteristic length, Eq. (14) Rayleigh number based on length L , [gP(T,, - TdL3/v21Pr Rayleigh number based on length x, rgP(T,"- T b w / v 2 I p r

ROBERTSIEGEL

222 RaL* Ra,' Re, S

T Tb

t

urn

-U V Vb 2,

X

a

B Ycr

s

modified Rayleigh number based on length L , (g,G&"/kv2)Pr modified Rayleigh number based on length x, (g/?qx"/kv2)Pr Reynolds number based on length 6 height of drop tower temperature fluid bulk temperature time terminal velocity of bubble rising value in in a pool of liquid; earth gravity average velocity total volume of bubble portion of bubble volume lying above bubble base velocity length coordinate thermal diffusivity coefficient of volume expansion critical wavelength, Eq. (41) boundary-layer thickness or thickness of condensate layer

e bubble contact angle Ke v

evaporation coefficient

h latent heat of vaporization ; A' = h BcJT,,, - T,)

+

CL V

P U

T

absolute fluid viscosity kinematic fluid viscosity fluid density surface tension dimensionless time, t a / U

SUBSCRIPTS conduction regime (except in Q,) fuel 1 liquid phase (except in p,) min minimum flux between transition and film boiling steady state (except in F,) S sat saturation condition t top of bubble u vapor phase vf vapor film wall W C

f

REFERENCES l a . W. Unterberg and J. T. Congelliere, Zero gravity problems in space powerplants: A status survey. A R S (Am. Rocket SOC.) J . 32, 862 (1962). 1. I,. M. Habip, On the mechanics of liquids in subgravity, Astrotiuuticu Actu 11,401 (1965). 2. D . A. Petrash, R. C. Nussle, and E. W. Otto, Effect of contact angle and tank geometry on the configuration of the liquid-vapor interface during weightlessness. N A S A Tech. Note T N D-2075, October 1963. 3. M. Delattre, Installation d'Essais en impesanteur de L'0.N.E.R.A. 5th European Space Symp., Munich, Germany, July 1965, ONERA T.P. No. 258 (1965). 4. E. T . Benedikt and R. Lepper, Experimental production of a zero or near-zero gravity environment, in "Weightlessness-Physical Phenomena and Biological Effects" (E. T. Benedikt, ed.), p. 56. Plenum, New York, 1961. 5. R. Siege1 and E. G. Keshock, Nucleate and film boiling in reduced gravity from horizontal and vertical wires. N A S A Tech. Rept. TR-R-216, February 1965. 6a. J. W. Useller, J . H. Enders, and F. W. Haise, Jr., Use of aircraft for zero-gravity environment, N A S A Tech. Note T N D-3380, May 1966. 6. C . C. Crabs, AJ-2 zero-gravity flight facility. N A S A Tech. Note T N D-2838, Appendix A, June 1965. 7. R. R. Nunamaker, E. L. Corpas, and J. G. McArdle, Weightlessness experiments with liquid hydrogen in aerobee sounding rockets : uniform radiant heat addition-Flight 3. NASA T M X-872, December 1963 (confidential).

REDUCEDGRAVITY ON HEATTRANSFER

223

8. D. A. Petrash, R. C. Nussle, and E. W. Otto, Effect of the acceleration disturbances encountered in the MA-7 spacecraft on the liquid-vapor interface in a baWed tank during weightlessness. N A S A Tech. Note T N D-1577, January 1963. 9. M . E. Kirkpatrick, A study of space propulsion system experiments utilizing environmental research satellites. AFRPL-TR-65-73 (DDC No. AD464581) T R W Space Techno]. Labs., p. 60, 1965. 9a. M. E. Nein and C. D. Arnett, Program plan for earth-orbital low g heat transfer and fluid mechanics experiments, NASA T M X-53395, February 1966. 10. D. A. Kirk, Means of creating simulated variable gravity fields for the study of free convective heat transfer, in Proc. 1st Space Vehicle Thermal Atmospheric Control Symp. February 1963, ASD-TDR-63-260, April 1963. 11. D. A. Kirk, T h e effect of gravity on free convection heat transfer, I : The feasibility of using an electromagnetic body force. WADD T R 60-303, August 1960. 12. S. Ostrach, Laminar flows with body forces, in “Theory of Laminar Flows,” (F. K. Moore, ed.), Vol. IV: High Speed Aerodynamics and Jet Propulsion, Princeton Univ. Press, Princeton, New Jersey, 1964. 13. J. T. Stuart, Hydrodynamic stability, in “Laminar Boundary Layers” (L. Rosenhead, ed.), Chapter 9, p. 492. Oxford Univ. Press (Clarendon) London and New York, 1963. 14. H. Grober, S. Erk, and U. Grigull, “Fundamentals of Heat Transfer,” p. 313 (transl. by J. R. Moszyenski). McGraw-Hill, New York, 1961. 15. H. Jeffreys, Some cases of instabilityin fluid motion. Proc. Roy. Sac. A118,195 (1928). 16. A. R. Low, On the criterion for stability of a layer of viscous fluid heated from below. Proc. Roy. Sac. A125, 180 (1929). 17. R. J. Schmidt and 0. A. Saunders, On the motion of a fluid heated from below. Proc. Roy. SOC.A165, 216 (1938). 18. W. V. R. Malkus, Discrete transitions in turbulent convection. Proc. Roy. Sac. A225, 185 (1954). 19. A. Pellew and R. V. Southwell, On maintained convective motion in a fluid heated from below. Proc. Roy. Sac. A176, 312 (1940). 20. E. R. G. Eckert and W. 0. Carlson, Natural convection in an air layer enclosed between two vertical plates with different temperatures. Intern. J. Heat Mass Transfer 2, 106 (1961). 21. G. K. Batchelor, Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quart. Appl. Math. 12, 209 (1954). 22. A. Emery and N. C. Chu, Heat transfer across vertical layers. J. Heat Transfer 87,110 (1965). 23. E. R. G. Eckert and R. M. Drake, Jr., “Heat and Mass Transfer,” 2nd ed., p. 314. McGraw-Hill, New York, 1959. 24. J. H. Chin, J. 0. Donaldson, L. W. Gallagher, E. Y. Harper, S. E. Hurd, and H. M. Satterlee, Analytical and experimental study of liquid orientation and stratification in standard and reduced gravity fields. Rept. 2-05-64-1, Lockheed Missiles and Space Co., July 1964. 25. E. H. W. Schmidt, Heat transmission by natural convection at high centrifugal acceleration in water-cooled gas-turbine blades. Proc. Gen. Discussion Heat Transfer ASME-IME, London, 1951, p. 361. Inst. Mech. Engrs, London, 1951. 26. W. H. McAdams, “Heat Transmission,” 3rd ed., p. 173. McGraw-Hill, New York, 1954. 27. M. Jakob, “Heat Transfer,” Val. 1. Wiley, New York, 1949. 28. H. Senftleben, Die Warmeabgabe von Korpern verschiedener Form in Flussigkeiten und Gasen bei freier Stromung. 2. Angew. Phys. 3, 361 (1951).

224

ROBERTSIEGEL

29. J. J. Mahony, Heat transfer at small Grashof numbers. Proc. Roy. Sac. A238, 412 (1957). 30. S. H. Schwartz and M. Adelberg, Some thermal aspects of a contained fluid in a reduced gravity environment. Douglas Missile and Space Systems Div. Paper No. 3394, Symp. Fluid Mechs. Heat Transfer under Low Gravitational Conditions. USAF Office of Sci. Res. and Lockheed Missiles and Space Co., Palo Alto, California, 1965. 31. M. Adelberg and S. H. Schwartz, Heat transfer domains for fluids in a variable gravity field with some applications to storage of cryogens in space. Douglas Missile and Space Systems Div. Paper No. 3516, Cryogenic Eng. Conf. Houston, Texas, August 1965. 32. R. Siegel, Transient free convection from a vertical flat plate. Trans. A S M E 80, 347 (1958). 33. R. J. Goldstein and E. R. G. Eckert, The steady and transient free convection boundary layer on a uniformly heated vertical plate. Intern. J . Heat Mass Transfer 1,208 (1960). 34. W. M. Rohsenow, A method of correlating heat-transfer data for surface boiling of liquids. Trans. A S M E 74, 969 (1952). 35. J. W. Westwater, Boiling of liquids. Advan. Chem. Eng. 1, 1 (1956). 36. R. Siegel and C. Usiskin, A photographic study of boiling in the absence of gravity. J . Heat Transfer 81, 230 (1959). 37. C. M. Usiskin and R. Siegel, An experimental study of boiling in reduced and zero gravity fields. J . Heat Transfer 83, 243 (1961). 38. J. E. Sherley, Nucleate boiling heat-transfer data for liquid hydrogen at standard and zero gravity, in “Advances in Cryogenic Engineering,” (K. D. Timmerhaus, ed.), Vol. 8, p. 495. Plenum, New York, 1963. 39. H. Merte, Jr. and J. A. Clark, Boiling heat transfer with cryogenic fluids at standard, fractional, and near-zero gravity. J . Heat Transfer 86, 351 (1964). 40. R. G. Clodfelter, Low-gravity pool-boiling heat transfer. APL-TDR-64-19 (DDC No. AD-437803), 1964. 41. E. W. Lewis, H. Merte, Jr., and J. A. Clark, Heat transfer at zero gravity. AZChE55th Natl. Meeting, Houston, Texas, February 1965. Preprint 24e. 42. L. M. Hedgepeth and E. A. Zara, Zero gravity pool boiling. ASD-TDR-63-706 (DDC No. AD-431810), September 1963. 43. S. H. Schwartz and R. L. Mannes, A study of nucleate boiling with water in a low gravity field. Submitted for publication. 44. J. Rex and B. A. Knight, An experimental assessment of the heat transfer properties of propane in a near-zero gravity environment. Roy. Aircraft Estab. Tech. Note No. Space 69, August 1964. 45. S . S. Papell and 0. C. Faber, Jr., Zero and reduced gravity simulation on a magnetic colloid pool-boiling system. N A S A Tech. Note T N D-3288. February 1966. 46. M. T. Cichelli and C. F. Bonilla, Heat transfer to liquids boiling under pressure. Trans. A m . Inst. Chem. Engrs. 41, 755 (1945). 47. S. S. Kutateladze, A hydrodynamic theory of changes in the boiling process under free convection conditions. Zzv. Akad. Nauk S S S R , Otd. Tekhn. Nauk No. 4, 529 (1951) (transl. available as AEC-TR-1441). 48. N. Zuber, “Hydrodynamic Aspects of Boiling Heat Transfer,” Ph.D. Thesis, Univ. of California, Los Angeles, California, 1959 (also available as AECU-4439). 49. R. C. Noyes, An experimental study of sodium pool boiling heat transfer. .I. Heat Transfer 85, 125 (1963). 50. J . H. Lienhard and K. Watanabe, On correlating the peak and minimum boiling heat fluxes with pressure and heater configuration. ,I. Heat Transfer 88, 94 (1966).

REDUCED GRAVITY ON HEATTRANSFER

225

51. H. F. Steinle, An experimental study of the transition from nucleate to film boiling under zero-gravity conditions, in “Weightlessness-Physical Phenomena and Biological Effects” (E. T. Benedikt, ed.), p. 111. Plenum, New York, 1961 [see alternately Proc. 1960 Heat Transfer and Fluid Mechanics Institute (D. M. Mason, W. C. Reynolds, and W. G. Vincenti, eds.), p. 208. Stanford Univ. Press, Stanford, California, 19601. 52. R. Siegel and J. R. Howell, Critical heat flux for saturated pool boiling from horizontal and vertical wires in reduced gravity. NASA Tech. Note T N D-3123, December 1965. 53. D. N. Lyon, M. C. Jones, G. L. Ritter, C. Chiladakis, and P. G. Kosky, Peaknucleate boiling fluxes for liquid oxygen on a flat horizontal platinum surface at buoyancies corresponding to accelerations between -0.03 and lg,, A.Z.Ch.E. ( A m . Znst. Chem. Engrs.) J . 11, 773 (1965). 54. P. J. Berenson, Film-boiling heat transfer from a horizontal surface. J . Heat Transfer 83, 351 (1961). 55. J. H. Lienhard, Interacting effects of geometry and gravity upon the extreme boiling heat fluxes. Submitted for publication. 56. L. A. Bromley, Effect of heat capacity of condensate. Ind. Eng. Chem. 44, 2966 (195 2). 57. Y. Y. Hsu and J. W. Westwater, Approximate theory for film boiling on vertical surfaces. Chem. Eng. Progr. Symp. Ser. 56, No. 30, 15 (1960). 58. T. H. K. Frederking and J. A. Clark, Natural convection film boiling on a sphere, in “Advances in Cryogenic Engineering” (K. D. Timmerhaus, ed.), Vol. 8, p. 501. Plenum, New York, 1963. 59. R. Siegel and E. G. Keshock, Effects of reduced gravity on nucleate boiling bubble dynamics in saturated water. AZChE ( A m . Znst. Chem. Engrs.) J . 10, 509 (1964). 60. E. G. Keshock and R. Siegel, Forces acting on bubbles in nucleate boiling under normal and reduced gravity conditions. N A S A Tech. Note T N D-2299, August 1964. 61. W. Fritz and W. Ende, The vaporization process according to cinematographic pictures of vapor bubbles. Z . Physik. 37, 391 (1936). 62. M. S. Plesset and S. A. Zwick, The growth of vapor bubbles in superheated liquids. 1. Appl. Phys. 25,493 (1954). 63. H. K. Forster and N. Zuber, Growth of a vapor bubble in a superheated liquid. .I. Appl. Phys. 25, 474 (1954). 64. N. Zuber, T h e Dynamics of vapor bubbles in nonuniform temperature fields. Intern. J . Heat Mass Transfer 2, 83 (1961). 65. W. Fritz, Maximum volume of vapor bubbles. Z . Physik. 36, 379 (1935). 66. T. H. Cochran, J. C. Aydelott, and T. C. Frysinger, The effect of subcooling and gravity level on boiling in the discrete bubble region. N A S A Tech. Note T N D-3449, September 1966. 67. W. M. Rohsenow, Heat transfer with boiling in “Modern Developments in Heat Transfer” (W. E. Ibele, ed.), p. 111. Academic Press, New York, 1963. 6 8 . T. Z. Harmathy, Velocity of large drops and bubbles in media of infinite or restricted extent. AZChE ( A m . Inst. Chem. Engrs.) ,I. 6, 281 (1961). 69. J. L. McGrew, D. W. Murphy, and R. V. Bailey, Boiling heat transfer in a zero gravity environment. S A E - A S M E Air Transport and Space Meeting, New York, April 1964. SAE Paper 862C. 70. T. R. Rehm, Subcooled boiling in a negligible gravity field. Denver Res. Inst. Rep. DRI-2227, 1965. 71. J. H. Lienhard and P. T. Y. Wong, The dominant unstable wavelength and minimum heat flux during film boiling on a horizontal cylinder. J . Heat Transfer 86, 220 (1964).

226

ROBERTSIEGEL

72. D. G. Evans, Visual study of swirling and nonswirling two-phase two-component flow at 1 and 0 gravity. NASA T M X-725, August 1963. 73. Y. Y.Hsu and R. W. Graham, A visual study of two-phase flow in a vertical tube with heat addition. NASA Tech. Note T N D-1564, January 1963. 74. M. Adelberg, Effect of gravity upon nucleate boiling, in “Physical and Biological Phenomena in a Weightless State,” Advances in the Astronautical Sciences, (E. T. Benedikt and R. W. Halliburton, eds.), VoI. 14, p. 196. Western Periodicals, North Hollywood, California, 1963. 75. M. Adelberg, Boiling, condensation and convection in a gravitational field. A I C h E 55th N a t l . Meeting, S y m p . Effects Zero Gravity on Fluid Dynamics and Heat Transfer Houston, Texas, February 1965, Pt. 11, Paper 24d. 76. R. J. Simoneau and F. F. Simon, A visual study of velocity and buoyancy effects on boiling nitrogen. N A S A Tech. Note T N D-3354, September 1966. 76a. R. J. Simoneau and F. F. Simon, Personal communication, 1966. 77. S. S. Papell, R. J. Simoneau, and D. D. Brown, Buoyancy effects on the critical heat flux of forced convective boiling in vertical flow. N A S A Tech. Note T N 1)-3672, October 1966. 78. R. V. Macbeth, Burn-out analysis, 5: Examination of published world data for rod bundles. AEEW-R-358, 1964. 79. S. S. Papell, An instability effect on two-phase heat transfer for subcooled water flowing under conditions of zero gravity. A m . Rocket SOC.Space Power Systems Conf. Santa Monica, California, September 1962. Paper 2548-62. 79a. C. J. Feldmanis, Performance of boiling and condensing equipment under simulated outer space conditions, ASD-TDR-63-862, November 1963. 79b. C. J. Feldmanis, Pressure and temperature changes in closed loop forced convection boiling and condensing processes under zero gravity conditions. Proc. Inst. Enwiron. Sci. 1966 Ann. Tech. M t g . , Sun Diego, California, April 1966. 80. K. Ginwala, T. Blatt, W. Jansen, G. Khabbaz, and S. Shenkman, Engineering study of vapor cycle cooling equipment for zero-gravity environment. WADD T R 60-776. Northern Res. Engr. Corp., January 1961. 81. W. R. Gambill and N. D. Greene, Boiling burnout with water in vortex flow. Chem. Eng. Progr. 54, 68 (1958). 82. W. R. Gambill, R. D. Bundy, and R. W. Wansbrough, Heat transfer burnout, and pressure drop for water in swirl flow through tubes with internal twisted tapes. ORNL-2911, April 1960. 83. H. J. Ivey, Acceleration and the critical heat flux in pool boiling heat transfer. Proc. Znst. Mech. Engrs. (London) 177,15 (1963). 84. V. I . Morozkin, A. N. Amenitskii, and I. T. Alad’ev, An experimental investigation of the effect of acceleration on the boiling crisis in subcooled water. High Temp. (English Trans.) 2, 104 (1964). 85. C. P. Costello and J. M. Adams, Burnout heat fluxes in pool boilingat high accelerations, in “International Developmentsin Heat Transfer,” Proc. Heat Transfer Conf., Boulder, Colorado, and London, 1961-1962, p. 255. Am. SOC.Mech. Engrs. New York, 1963. 86. R. W. Graham, R. C. Hendricks, and R. C . Ehlers, Analytical and experimental study of pool heating of liquid hydrogen over a range of accelerations. NASA Tech. Note T N D-1883, February 1964. 87. M. L. Pomerantz, Film boiling on a horizontal tube in increased gravity fields. ./. Heat Transfer 86, 2 1 3 (1964). 88. H. Merte, Jr. and J. A. Clark, Pool boiling in an accelerating system. ./. Heat Transfer 83, 233 (1961).

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89. W. D. Allingham and J. A. McEntire, Determination of boiling film coefficient for a Heated horizontal tube in water-saturated wick material. J . Heat Transfer 83, 71 (1961). 90. C. P. Costello and W. J. Frea, The roles of capillary wicking and surface deposits in the attainment of high pool boiling burnout heat fluxes. AZChE ( A m . Inst. Chem. Engrs.) J . 10, 393 (1964). 91. W. M. Rohsenow and H. Y. Choi, “Heat, Mass, and Momentum Transfer,” p. 239. Prentice-Hall, Englewood Cliffs, New Jersey, 1961. 92. E. M. Sparrow and R. Siegel, Transient film condensation. ,I. Appl. Mech. 26, 120 (1959). 93. R. L. Cummings, P. E. Grevstad, and J. G. Reitz, Orbital force field boiling and condensing experiment, in “Weightlessness-Physical Phenomena and Biological Effects” (E. T. Benedikt, ed.), p. 121. Plenum, New York, 1961. 94. R. A. Trusela and R. G. Clodfelter, Heat transfer problems of space vehicle power systems. S A E Natl. Aeron. Meeting, New York, April 1960. Paper No. 154c. 95. J. A. Albers and R. P. Macosko, Experimental pressure-drop investigation of nonwetting, condensing flow of mercury vapor in a constant-diameter tube in 1-G and zero-gravity environments. N A S A Tech. Note T N D-2838, June 1965. 96. J. A. Albers and D. Namkoong, Jr., Single-tube mercury vapor condensing characteristics in one G and zero G environments. AZAA Rankine Cycle Space Power System Specialists Conf., Cleveland, Ohio, October 1965, AEC C O N F 65 1026, 1966. 96a. J. I. Sturas, C. C. Crabs, and S. H. Gorland, Photographic study of mercury droplet parameters including effects of gravity, N A S A Tech. Note T N D-3705, November 1966. 97. J. A. Albers and R. P. Macosko, Condensation pressure drop of nonwetting mercury in a uniformly tapered tube in 1-G and zero-gravity flight environments. N A S A Tech. Note T N D-3185, January 1966. 98. R. T. Lancet, P. Abramson, and R. P. Forslund, T h e fluid mechanics of condensing mercury in a low-gravity environment. Symp. Fluid Mech. Heat Transfer Under Low Gradational Conditions. USAF Office of Sci. Res. and Lockheed Missiles and Space Co., Palo Alto, California, 1965. 98a. R. J. Denington, A. Koestel, A. V. Saule, R. I. Shure, G. T. Stevens, and R. B. Taylor, Space radiator study. ASD-TDR-61-697 (DDC No. AD-424419), October 1963. 99. J. G. Reitz, Zero gravity mercury condensing research. Aerospace Eng. 19, 18 (1960). 100. A. L. Hall, Observations on the burning of a candle at zero gravity. School of Aviation Med., Rept. No. 5 (DDC No. AD-436897), February 1964. 101. S. Kumagai, Combustion of fuel droplets in a falling chamber with special reference to the effect of natural convection. Jet Propulsion 26, 786 (1956). 102. S. Kumagai and H. Isoda, Combustion of fuel droplets in a falling chamber. 6th Symp. Combustion, p. 726. Reinhold, New York, 1957. 103. H. Isoda and S. Kumagai, New aspects of droplet combustion. 7th Symp. Combustion, London, Oxford 1958, p. 523. Butterworths, London, 1959. 104. J. H. Kimzey, Flammability during weightlessness, Proc. Inst. Environ. Sci. 1966 Ann. Tech. Mtg., San Diego, Cal$ornia, April 1966, p. 433.

ADDITIONAL REFERENCES J. C. Aydelott, E. L. Corpas, and R. P. Gruber, Comparison of pressure rise in a hydrogen dewar for homogeneous, normal-gravity quiescent, and zero-gravity conditions-Flight 9. NASA T M X-1052, February 1965 (confidential).

ROBERTSIEGEL K. L . Abdalla, R. A. Flage, and R. G. Jackson, Zero-gravity performance of ullage control

surface with liquid hydrogen while subjected to unsymmetrical radiant heating. NASA T M X-1001, August 1964 (confidential). J. C. Aydelott, E. L. Corpas, and R. P. Gruber, Comparison of pressure rise in a hydrogen dewar for homogeneous, normal-gravity quiescent, and zero-gravity conditions-Flight 7. NASA T M X-1006, September 1964 (confidential). K. L. Abdalla, T. C. Frysinger, and C. R. Andracchio, Pressure-rise characteristics for a liquid-hydrogen dewar for homogeneous, normal-gravity quiescent, and zerogravity tests. NASA T M X-1134, September 1965 (confidential). M . Adelberg and S. H. Schwartz, Scaling of fluids for studying the effect of gravity upon Inst. Environ. Sci. 1966 Ann. Tech. Mtg., Sun Diego, Califortiia, nucleate boiling, PYOC. April 1966. C. S. Boyd, An investigation of boiling water in a porous material as a means of separating liquid and vapour in zero gravity, M.S. Thesis, Air Force Inst. of Tech., WrightPatterson AFB, Ohio, March 1965 ( D D C No. AD-618022). R. D. Cess, Laminar-film condensation on a flat plate in the absence of a body force, %. Angew. Math. Phys. 11, 426 (1960). J. T. Congelliere, W.Unterberg, and E. B. Zwick, The zero G flow loop: steady-flow, zerogravity simulation for investigation of two-phase phenomena, in “Physical and Biological Phenomena in a Weightless State,” Advances in the Astronautical Sciences, (E. T. Benedikt and R. W. Halliburton, eds.), Vol. 14, p. 223. Western Periodicals, North Hollywood, California, 1963. R . L . Hammel, J. M. Robinson, and H. T. Sliff, Environmental research satellites for space propulsion systems experiments, Quart. Progr. Repts. AFRPL-TR-66-64, AFRPL-TR66-178, T R W Systems, March and June 1966. R. H. Knoll, G. R. Smolak, and R. P. Nunamaker, Weightlessness experiments with liquid hydrogen in aerobee sounding rockets: uniform radiant heat addition-flight 1, KASA T M X-484, June 1962 (confidential). P. P. Mader, G. V. Colombo, and E. S. Mills, Summary report of tests conducted to determine effects of atmospheric compositions and gravity on burning properties, Ilouglas Missile and Space Systems Div. Rept. SM-48862, October 1965. J. G . McArdle, R. C. Dillon, and D. A. Altmos, if’eightlessness experiments with liquid hydrogen in aerobee sounding rockets: uniform radiant heat addition-flight 2, NASA T M X-718, December 1962 (confidential). R . M. Singer, T h e control of condensation heat transfer rates using an electromagnetic field, Argonne National Lab., ANL-6861, July 1964. E. A. Zara, Boiling heat transfer in zero gravity, R T D Technol. Briefs, Headquarters Res. and Tech. Div., Bolling AFB, D. C. 11, January 1964.

Advances in Plasma Heat Transfer E. R. G. ECKERT Department of Mechanical Engineering University of Minnesota, Minneapolis, Minnesota

E. PFENDER Department of Mechanical Engineering University of Minnesota, Minneapolis, Minnesota

I. Introduction . . . . . . . . . . . . . . . 11. T h e Plasma State and Its Influence on Heat Transfer . . . A. Definition and Characteristic Properties of a Gaseous Plasma B. Survey of Steady, Dense Plasmas . . . . . . . . C. T h e Approach to Thermodynamic Equilibrium in a Plasma . D . Plasma-Wall Boundaries and Plasma Sheaths . . . . . E. Thermodynamic and Transport Properties of Plasmas . . F. Increase of Plasma Temperatures . . . . . . . . 111. Plasma Heat Transfer in Absence of an Externally Applied Electric or Magnetic Field . . . . . . . . . . . . . A. Qualitative Considerations . . . . . . . . . . B. Basic Transport Equations . . . . . . . . . . C. Laminar Boundary Layer Equations . . . . . . . D. Results of Reentry Studies . . . . . . . . . . IV. Heat Transfer in the Presence of an Electric Current . . . . A. Electrically Insulating Surface , . . . . . . . . B. Electrically Conducting Surface (Electrode) . . . . . V. Heat Transfer in the Presence of a Magnetic Field . . . . A. Electrode Heat Transfer . . . . . . . . . . . B. Electrically Insulating Surface . . . . . . . . . Nomenclature . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .

229 23 1 23 1 242 244 252 256 262 270 270 272 276 278 282 282 289 304 305 307 311 313

I. Introduction Serious heat transfer problems are encountered in the development of various engineering devices which have to deal with gases heated to such temperatures that they are partially ionized. T h e term plasma has been 229

230

E. R. G. ECKERTAND E. PFENDER

adopted to describe such gases, and the present paper surveys our knowledge of plasma heat transfer processes encountered in various engineering applications, especially in the reentry problem in space flight, in arc technology, and in magnetohydrodynamic acceleration and energy conversion processes. Certain conditions are common to heat transfer in these applications and the present discussion will be restricted to these. The pressure, for example, is usually such that the gases involved can be described as continua. An exception to this is possibly the plasma sheath, a very thin layer in the immediate proximity of surfaces. The cooling effect of the walls to which heat is transferred extends generally into the plasma in a thin layer only which can, in the usual terminology, be described as a boundary layer. Analytical investigations on plasma heat transfer are almost completely restricted to laminar flow. In experiments, the degree and the nature of turbulence in the plasma streams is generally only poorly known. On the other hand, certain conditions are characteristic of the individual application, and research on plasma heat transfer has also been guided by these. For example, in the reentry problem, electric and magnetic fields are generally absent. Analytical as well as experimental investigations have primarily considered air and in second place gases as they are encountered in the entry of space vehicles to other planets. Considerable attention has been paid in these studies to the effects of a departure from local chemical equilibrium (composition of the gas with regard to molecules, atoms, ions, and electrons). Some discussion will be devoted in the following section to these effects. For a more extensive survey with respect to chemical equilibrium, the reader is referred to the contribution by Chung (1). Departures from thermodynamic equilibrium in the distribution of energy to the various degrees of freedom, however, has largely been disregarded. Some studies have been concerned with the possibility of influencing drag and of reducing heat transfer on a space vehicle during reentry by a magnetic field. The results of these have been thoroughly discussed in the contribution by Romig (2),and will not be included in this paper. Arc technology utilizes gases which have been heated by an electric arc to very high temperatures. Heat transfer to the electrically insulated structural elements of arc devices is essentially of the same nature as the heat transfer encountered in the reentry process. Conditions are, however, widely different on the electrodes as a consequence of the fact that electrons pass through their surfaces. This process generates a strong deviation from thermodynamic equilibrium with regard to composition as well as to energy distribution to the various degrees of freedom. To a first approximation, the local state of a plasma can now be described by two kinetic temperatures : an electron temperature and a heavy-particle temperature. Electric arcs have also the tendency to attach to the electrodes at localized areas. This causes extremely

PLASMA HEATTRANSFER

23 1

high energy flux densities, not infrequently of such a magnitude that the electrodes are destroyed. In MHD applications, finally, heat transfer to the electrically insulated as well as to the current-carrying walls is strongly influenced by the simultaneous presence of electric and magnetic fields. The gases involved are at a relatively low temperature and are made conducting by seeding with a material which easily ionizes. In the first part of this survey some general plasma properties will be reviewed which are relevant to plasma heat transfer. The discussion will then, in sequence, deal with the conditions as they have been described for the various engineering applications, starting with heat transfer in the absence of electric or magnetic fields, proceeding to heat transfer encountered on electrodes in arc devices, and ending with heat transfer to the walls of MHD devices. Throughout this review the MKSA system of units will be used.

11. The Plasma State and Its Influence on Heat Transfer

A. DEFINITION AND CHARACTERISTIC PROPERTIES OF A GASEOUS PLASMA T h e constituents of a plasma are neutral particles, electrons, and ions among which continuous and vigorous interactions exist. It is conceivable to generate such a plasma by adding energy to an enclosed gas volume (heat or electrical energy). As soon as the gas emits light, it is called a plasma. The emitted light indicates that excitation and ionization processes occur. The originally nonconducting gas is now converted into a more or less electrically conducting plasma. Occasionally a plasma is designated as the fourth state of matter in the sequence solid, liquid, gas, plasma. This designation is justified by the fact that more than 9 9 O 4 , of the universe consists of gaseous plasma. A significant example is the sun which appears as a huge plasma sphere. There are a number of obvious plasma features which permit an unambiguous distinction from an ordinary gas. The electrical conductivity which almost vanishes in an ordinary gas turns out to be a function of the temperature in a plasma. A hydrogen plasma, for example, which is kept at a pressure of 1 atm and heated up to temperatures of 107"Khas the same electrical conductivity as copper at room temperature. This review does not deal with plasmas at such high temperatures and, therefore, the electrical conductivities will be appreciably lower. The composition of a plasma owing to the presence of neutral particles, electrons, and ions is again a function of the temperature and represents another important distinction. Figure 1 shows the constituents of a nitrogen

232

E. R. G. ECKERT AND E. PFENDER

plasma in thermodynamic equilibrium as a function of the temperature at 1 atm. At rather low temperatures (< 1000°K) nitrogen consists almost entirely of N2 molecules which dissociate at higher temperatures (> SOOO°K) into nitrogen atoms. The number of molecular ions (N2+)which are already formed at low temperatures is negligible. With increasing temperature, beyond about 8000”K, an appreciable degree of ionization is found which reaches finally almost looo/, in the neighborhood of 20,000”K. At this

TEMPERATURE f K )

FIG.1. Composition of a nitrogen plasma in thermodynamic equilibrium ( 3 ) .

temperature the plasma is almost fully ionized. A further increase in temperature yields higher ionized species. At 30,00OoK, for example, the plasma consists of about equal numbers of N+ and N++ ions and a three times higher number of electrons. T h e plasma remains electrically neutral which, however, is only true for plasma volumes which are large compared to a Debye sphere. This property of quasineutrality prevails for all plasmas discussed in this paper. Appreciable charge imbalances, however, exist in plasma sheaths which are formed in the “contact zone” between a plasma and a wall. ‘The thickness of such a plasma sheath is in the order of a Debye length AD = (eokoT/e2n,)’’z(kB is the Boltzmann constant, T the plasma temperature, e the electronic charge, and n, the electron density).

PLASMA HEATTRANSFER

233

Besides the electrical conductivity, the heat conductivity of a plasma differs in a characteristic manner from that of an ordinary gas. As an example, a comparison shall be made between the energy exchange processes of a nitrogen gas column heated to moderate temperatures and those of a nitrogen plasma column with asurrounding at the same pressure. The heat conduction

FIG.2. Heat conductivity of a nitrogen plasma ( 3 ) ;k,, heat conductivity of nitrogen molecules ; k, , heat conductivity of nitrogen atoms ; k,, heat conductivity of electrons ; h i , heat conductivity of ions ; k, , heat conductivity by transport of dissociation energy ; kI,heat conductivity by transport'of ionization energy ; k, total heat conductivity.

process of a plasma turns out to be much more complicated than that of an ordinary gas. For the sake of simplicity, thermodiffusion effects shall be neglected in this discussion. T h e heated gas column loses energy only by atomic conduction (atom-atom collisions) to the surroundings whereas the plasma column loses additional energy by diffusion of chemical energy and heat conduction processes of the other plasma constituents (electrons and ions). Diffusion of chemical energy is comprised of the flux of dissociation

234

E. R. G. ECKERT AND E. PFENDER

and ionization energy to the surroundings. According to the high temperature in the plasma column, nitrogen molecules are dissociated and a nitrogen atom density gradient from the surroundings to the center of the plasma column exists, causing a continuous flow of nitrogen atoms to the surroundings and an opposite flow of nitrogen molecules into the plasma column. The nitrogen atoms transfer their energy of dissociation in the form of potential energy to the surroundings where they recombine causing a continuous energy flux out of the plasma column. A similar situation exists for the diffusion of ionization energy. However, the electrons and ions in a plasma are always under the influence of their mutual Coulomb fields and, therefore, leave the plasma column in pairs only, an effect which is called ambipolar diffusion. Electrons and ions recombine in the surroundings or on catalytic surfaces releasing their ionization energy and causing again a continuous energy flux from the plasma to the surroundings. Figure 2 represents a diagram of the heat conductivity of a nitrogen plasma versus temperature. The most important contribution to the total heat conductivity in the temperature interval 5000°K < T < 10,000"K is obviously due to the diflusion of chemical energy (dissociation energy). At higher temperatures (fully ionized plasmas) the heat conductivity of the electrons becomes important and, finally, at extremely high temperatures the heat conductivity of the ions predominates. In addition self-emitted radiation from gases or plasmas may represent a significant contribution to the described heat transfer processes. T h e radiation from an ordinary, moderately heated atomic gas is usually negligible in comparison with the other heat transfer mechanisms. This situation is entirely different in a plasma, especially at high temperatures and densities (3).The total radiation emitted from plasmas which is of interest in this review may be attributed to four essential radiation mechanisms which will be briefly discussed.

1. Line Radiation (Bound-Bound Radiation) Excited neutral atoms or ions may return to the ground state by spontaneous emission. The frequency of the emitted light quantum for a transition from an energy state xs to a lower state xt is, according to Bohr's frequency relation, x s - X I = hv (1) h denotes Planck's constant and v the frequency of the emitted photon. The emission coefficient of a spectral line which is emitted from an optically thin homogeneous plasma is given by EL=

(1/444::nr*,hv

(2)

PLASMA HEATTRANSFER

235

The transition probability A:;;refers to a spontaneous transition within a r-times ionized atom from a quantum level s to a lower quantum level t ; nr,r represents the number of r-times ionized atoms in the sth quantum state (Y = 0 designates the neutral atoms). For the following considerations thermal equilibrium, which will be discussed in the next paragraph, will always prevail in the plasma. Therefore, the number density of excited atoms or ions may be expressed by a Boltzmann distribution n r , s = n r ( g r , s / z r ) exp (- X r , s / k , T ) (3) n, is the total number density of ions of species Y, g,,s the statistical weight of the sth quantum level, 2, the partition function (or sum over all states) of particles of species r, x,,~the energy of the sth quantum state, K B the Boltzmann constant, and T the temperature. The line emission coefficient eL = SAX evdv is an already integrated value over the natural width A x of a spectral line and refers to the emission from an optically thin homogeneous plasma slab with a thickness of unity (ev is the emission coefficient). From Eqs. (2) and (3) follows for the line emission coefficient The energy transfer rate by line radiation depends for a particular spectral line and a given plasma configuration only on the temperature. In Eq. (4) the temperature appears explicitly only in the exponential term, but n, also depends strongly on the temperature whereas& is a rather weak function of the temperature. For constant pressure cL(T)assumes a maximum at a certain temperature T* because the increasing exponential term beyond T* is overbalanced by the decreasing tendency of n, owing to the depletion of particles of species r by increasing ionization to species Y + 1 and the effect of the perfect gas law (n,+ n, = p / k B T ,wherep is the total pressure). As an example, Fig. 3 illustrates the emission coefficients of two argon lines (Y = 0 and r = 1) on a relative scale. The total energy transfer by line radiation from a plasma, which is frequently only a small fraction of the total radiated energy, depends on the number and wavelength of the emitted lines, which, in turn, depend on the kind of gas and the number of possible species Y for a given temperature. A plasma based on a particular gas may be a “strong” or “weak” line radiator depending on the plasma composition which is a function of the temperature (Fig. 1). 2. Recombination Radiation (Free-Bound Radiation) In the process of radiative recombination a free electron is captured by a positive ion into a certain bound energy state and the excess energy is converted into radiation according to the relation

E. R. G. ECKERT AND E. PFENDER

236

+ x r , s = hv

(me ue2/2)

(5)

In this equation me represents the electron mass, u, the electron velocity, and the energy level s of an r-times ionized atom to which the electron is trapped. Since the captured free electrons possess a continuous kinetic energy spectrum according to a Maxwellian distribution, the emitted radiation will also be continuously distributed, but with a threshold value (or

0

5

10

IS

20

25

3OXlOs

TEMPERATURE (K)

FIG.3. Relative emission coefficients of two argon lines.

series limit), , ,A

due to the trapping of electrons with near zero velocity:

hmax hc/Xr,s (6) designates the light velocity. Recombinations may occur into all possible energy levels xr,s so that the number of continuous spectra will coincide with the number s of energy states of these particular ions of species r. T h e entire free-bound continuum will then consist of a superposition of all continuous spectra emitted from the different species r which are present in the plasma. An exact calculation 1

c

PLASMA HEATTRANSFER

237

of the total free-bound continuum over the entire wavelength range is only possible for hydrogen ( 4 5 ) .Since the rate with which radiative recombination processes occur is proportional to the electron as well as the ion density, the emitted free-bound radiation intensity is proportional to the product of both. In the following section the contribution of free-bound radiation to the total continuum will be discussed.

3. Bremsstrahlung (Free-Free Radiation) Free electrons in a plasma may lose kinetic energy in the Coulomb field of positive ions and this energy is readily converted into radiation. Since the initial as well as the final states of the electrons are free states in which the electrons may assume arbitrary energies within the Maxwellian distribution, the emitted radiation is of the continuum type. Radiation in a frequency interval between v and v + dv will be emitted by free electrons which have a kinetic energy me ve2/22 hv (7) i.e., the spectrum of the emitted radiation depends on the kinetic energy or temperature of the electrons. In the elementary process of bremsstrahlung one electron and one ion are involved. Therefore, the intensity of this radiation is expected to be proportional to the product of electron and ion density. T h e actual emission coefficient for a free-free continuum is given by

where C=

32n2e6 3~ % ~ ( 2 n m , ) ~ / ~

(9)

e designates the elementary charge, n, and ni the electron and ion density, and Z ' e is the ion charge. T h e number s accounts for the fact that fast electrons may penetrate some of the outer electron shells of ions with higher nuclear charge (s = 0 for hydrogen ions). Therefore, such electrons will be exposed to a higher positive charge than the ionic charge Z'e. Considering the total radiation continuum consisting of free-free and free-bound radiation the emission coefficient turns out to be independent of the frequency for v < vg E, =

C(Z' + S)Z-

?l,?li

(kBT ) 1 / 2

The energy hv, is taken between the ionization level and the energy level corresponding to the limiting frequency vg. Since Eq. (10) includes only

E. R. G. ECKERTAND E. PFENDER

23 8

free-bound radiation of closely neighboring energy levels (quasi-continuum), which are usually found in the vicinity of the ionization level, the validity of Eq. (10) breaks off for v > vg. Free-bound radiation stemming from trapping of electrons to lower energy levels which are farther separated from each other leads to pronounced series limits. For a plasma in thermal equilibrium the product Z e n i may be expressed by the Saha-Eggert equation, which assumes the following form in a singly ionized gas : n, ni _ -2 Z j ( 2 ~ ~kB 2 ,7')3/2exp No Z" h3

(- XT ) kB

no designates the number density of neutral atoms, Zi and Z , the partition functions for ions and neutral atoms, respectively, and xi the ionization energy. With Eq. ( 1 1 ) the emission coefficient for the combined continuum radiation may be written as E, =

6 4 T 2 ( me6Zi - nok3 Texp 3 d h 3 c 3 2,

(x) 1' k3

In this equation nok3 T may be replaced by the total pressure p provided that the degree of ionization is small (f < 10%). Continuous radiation from electric high intensity arcs may amount to an appreciable fraction of the total power input as shown, for example, by Barzelay ( 6 )for an argon arc.

4. Blackbody Radiation The radiation intensities or emission coefficients described in Section II,A,l-3 were all based on the assumption that the plasma is optically thin or, in other words, that there is no appreciable absorption of radiation in the plasma itself. This assumption may fail for line as well as for continuum radiation. T h e intensity loss which radiation of the intensity I , suffers by penetrating a plasma layer of thickness dL is given by

(13) with K,' as absorption coefficient. This coefficient is a function of the frequency v, the properties of the penetrated layer, and sometimes also of the direction of dL. Integration of Eq. (13) yields dI,ldL

= -K,,'

I , = I",,,exp (=

1

zv,oexp (-7,)

I,

K,,'

dl,) (14)

PLASMA HEATTRANSFER

239

is the intensity of the initial radiation and 7,= JL K,'dL represents the optical depth of the layer. If the layer is optically homogeneous, this relation may be written as T , = K,,' L. Under perfect thermal equilibrium conditions Kirchhoff's law may be applied, which gives a correlation between emission and absorption coefficient : where

designates the Planck function for blackbody or cavity radiation. Kirchhoffs law may still be applied for conditions I,, < B , as long as the number of atoms which are able to absorb radiation is maintained; i.e., as long as excitation processes follow a Boltzmann distribution. E,' represents in Eq. (15) the total emission coefficient which consists of the coefficient E , for spontaneous emission and the contribution of induced or forced emission caused by the radiation field. This fact can be expressed by the Einstein relation E,'

or with Eq. (15), E,, = K,'[

=

EV

1 - exp (-hv/k, T )

1 - exp (-hv/kB T ) ]B, = K , B,

designates the absorption coefficient which corresponds to spontaneous emission. A radiation balance in a homogeneous finite plasma volume leads to the following local change of the radiation intensity : K,

dI,/dL

= E,' - K,'

I , = K,'(B,- I,)

(19)

or in integrated form with the boundary condition I , = 0 for L = 0:

I,= B,[l

- exp(-Ky'L)]

From the relation follows for an optically thin medium

I , = K,,' LB,

(K,'

L e 1) (21)

or without the contribution of the induced emission

I , = K , LB, = E , L

(22)

For an optically thin plasma, the radiation intensity increases proportionally with the thickness L of the layer.

240

E. R. G. ECKERTAND E. PFENDER

For large optical depths (K,’ L 9 1) follows from Eq. (20)

I, = B,

(23)

i.e., the highest possible radiation intensity which a plasma is able to emit is the blackbody radiation for the respective plasma temperature. I n laboratory plasmas the emitted radiation is usually several orders of magnitude less than the corresponding cavity radiation. For such plasmas most spectral lines as well as the different continua are emitted from an optically thin layer (K,’L4 1).

FIG.4. Blackbody radiation of plasmas (4).(For a layer thickness >2 mm, the indicated plasmas emit blackbody radiation in the visible range of the spectrum.)

Very strong absorption (K,’ > 1) occurs for resonance lines for which the absorption coefficient K,‘ is so high that a layer thickness L of a fraction of a millimeter is already sufficient for complete absorption. In the immediate neighborhood of such a resonance line, the absorption coefficient may be a factor of 10* smaller so that layer thicknesses of lo6 cm or more are required for complete absorption. Finkelnburg and Peters ( 4 ) calculated the conditions for which the continuous radiation of a laboratory plasma would approach a blackbody radiator ( I , 2 0.9BV).Figure 4 shows the result of their calculation for five different gases. By considering only singly ionized species of these gases, all curves merge into a common curve which corresponds to an ionization degree of 100n;,. Above this common curve a laboratory plasma with a layer thickness of 2 mm or larger would be a cavity radiator at the plasma temperature. Argon, for example, with an ionization potential of 15.8 V would fall

PLASMA HEATTRANSFER

24 1

between the curves for helium and hydrogen and become a blackbody radiator for temperatures T > 2 x 104”K and pressures p > 100 atm. Cesium with the lowest ionization potential would require a minimum pressure of about 5 atm to become a cavity radiator at 5000°K. Conceivably, there are other physical processes in plasmas which also lead to the emission of continuous radiation. Neutral atoms or molecules of certain elements, for example, may have an affinity for electrons. This causes free-free and free-bound radiation by similar mechanisms as described for the interaction of positive ions with electrons. The recombination process corresponds in this case to the formation of negative ions. Another process which may be responsible for the generation of continuous spectra is the chemical reaction between neutral particles in the plasma. Such a reaction may be considered as a “recombination” process with the corresponding “recombination” continuum. This type of chemical reaction plays an important role in reentry plasmas as well as in plasmas emanating from rocket exhausts. In the latter case the situation may become rather complex because of the numerous combustion products involved. Finally, in the presence of a magnetic field in the plasma, electrons which are forced into an orbital motion around the magnetic flux lines give rise to a continuous radiation which is called cyclotron radiation. However, the number of collisions which the electrons suffer in the plasma has to be small compared with the number of electron orbits. This requirement is usually expressed by the relation w e r e= Ac/rL

1

(24)

w e is the electron cyclotron frequency, T, the average time interval between two electron collisions, A, the mean free path length of the electrons, and rL the average Larmor radius. Since this review deals essentially with rather dense plasmas and moderate magnetic field intensities, cyclotron radiation will not be of importance. The contribution of radiation to the total heat transfer between a plasma and a bounding solid wall increases with increasing pressure, increasing scale, and especially with increasing temperature. Figure 5 presents, as an example, the total intensity of the radiant flux leaving an air plasma volume element per unit solid angle. The intensity is normalized with the dimensionless density of the plasma on the basis that the intensity increases with the power 1.7 of the density. The enthalpy and the equivalent flight velocity &fz are listed on the abscissa of the diagram. The figure has been taken from Morris P t al. (7). The points present the results of shock tube and ballistic experiments. The line indicates the total continuum radiation generated by free-free and free-bound transitions of electrons. It has been obtained by extrapolation of measured values.

E. R. G. ECKERT AND E. PFENDER

242

In Sections I11 and IV of this review, radiation is not included in the discussions on boundary layer flow. It has to be added to the convective heat transfer as long as the photon mean free path is large compared to the boundary layer thickness. For the situation that the photon mean free path is of the same order as the boundary layer thickness, the radiative and the convective heat transfer interact mutually and have to be considered together. For ENTHALPY ( kJ/kg 1

.5

.I

5

I

0x10~

( M I S PROORAM A > Z o 0 0 A )

- 2 -

L

SHOCK TUBE WTA

t Id-

0

x

A NEREM.R.H. 0 HOSHIZAKI, H.

n

c3 5 -

e-

2

J-

fld

5

-

-

5-

0

FLAO5,R.F.

PAQE,W.A. (BALLISTIC OATA)

i

5 2-

pt Idc)

5-

W

y 2 -

9 lo5B 5 2 lo41

0

I 5

I 10

I 15xd

FLIGHT VELOCITY ( m/sec 1

FIG.5 . Comparison of normalized radiation intensity data of air (7).

information on methods to analyze this situation, the reader is referred to the literature, for instance, to the contribution by Cess (8). Regardless of whether or not radiation from a plasma constitutes an important heat transfer mechanism, it is frequently used for diagnostic purposes in a number of different spectroscopic measurements ( 5 ) . B. SURVEY OF STEADY, DENSE PLASMAS Before we proceed to a discussion of plasma properties and their influence on heat transfer, a classification of steady, dense (collision-dominated)

PLASMA HEATTRANSFER

243

plasmas will be undertaken. Transient plasmas will not be considered in this survey. There are essentially three different ways by which steady plasmas may be generated : either by electrical means, by combustion processes, or by shock waves. Plasmas generated by shock waves may sometimes be considered as quasi-steady plasmas, as, for example, in reentry simulation studies or in the reentry process itself. The generation of steady high-temperature plasmas is restricted to electrical methods which may be subdivided into electrical discharges with electrodes and electrodeless discharges. The commonly used discharge type with electrodes, which represents a rather simple means for producing high-temperature, high-density plasmas, is the electric arc, which attracted increasing interest during the past 20 years. Electric arcs are used for basic research as well as for many applications. Figure 6 shows a number of various arcs in a temperature sequence. The temperatures in this diagram should be understood as approximate maximum values. The arc types which fall below a maximum temperature of 104"K in Fig. 6 are designated as low-intensity arcs and they are, in general, operated at current levels I < 50 A. High-intensity arcs with axis temperatures T > 104"K ( I > 100 A) fill a rather wide gap in the temperature scale. Among these arc types, the cascaded arc, first described by Maecker (9),found widespread interest as a research tool for the experimental determination of plasma transport properties, and for flow and heat transfer studies. The highest steady-state temperature level of about 100 x lo3O K was recently achieved by Mahn et al. (10)in a magnetically confined hydrogen arc at a pressure of about & atm. Even though high-frequency (or rf-) discharges have the advantage of being free from possible electrode contaminations, this discharge type has not yet found as widespread an application as the electric arc, although it seems feasible to generate with rf-discharges atmospheric pressure plasmas which are in the same temperature range as high-intensity arc plasmas. However, several disadvantages of the rf-discharge, such as poor coupling to the power source, restriction of the heating process by the skin effect, stability of the plasma, etc., may be responsible for its limited adoption in the laboratory and for applications. Combustion-generated plasmas became particularly interesting for magnetohydrodynamic (MHD) applications. Because of their relatively low temperature, these plasmas are frequently seeded with alkaline metals or their compounds which have a rather low ionization potential. In this way reasonable electrical conductivities can be achieved in spite of the relatively low temperatures. Shock tubes of different designs are used to generate high-temperature, high-density plasmas which are especially interesting for basic studies. The

E. R. G. ECKERT AND E. PFENDER

244

temperatures observed behind the shock front are in the same range as in high-intensity arcs. A comparison of shock heated with arc-generated plasmas reveals a considerable advantage of the former in that they are almost homogeneous. In reentry simulation studies and during the actual reentry phase of a space vehicle, plasmas are generated in a similar manner, covering a rather wide pressure range (0.01 < p < 100 atm). T CKK)

loo xid

50 40

30

20

f-

MAGNETicALLY CONFINED ARCS (HYDROGEN)

-WATER

-

-GAS

WHIRL STABILIZED ARCS

WHIRL STABILIZED SHOCK WAVE GENERATED

ARCS PLASMAS

CASCADED ARCS

-HIGH

FREQUENCY DISCHARGES

-HIGH

CURRENT ARCS IN AIR

10

7 5

4 3

-LOW -HG

CURRENT ARCS IN AIR OR NOBLE ARCS GASES

-

-ALKALINE METAL ARCS CHEMICAL ROCKET EXHAUSTS SEEDED FLAME GENERATED PLASMAS

FIG.6. Placement of various plasmas on the temperature scale.

Plasmas which are of interest in this review are frequently referred to as thermal plasmas, indicating that thermalprocesses govern the plasma state in contrast to cold plasmas (for example, glow discharges) in which thermal effects of the heavy plasma constituents are not significant. The question whether such a thermal plasma is in a thermodynamic equilibrium state or not is of importance for the correct interpretation of the observed plasma phenomena. Therefore, the next section will deal with the approach to thermodynamic equilibrium in a plasma.

C. THEAPPROACH TO THERMODYNAMIC EQUILIBRIUM I N A PLASMA Many considerations in plasma physics and technology are based on the assumption of thermodynamic equilibrium in the plasma. Because of the

PLASMAHEATTRANSFER

245

fundamental importance of this concept and its bearings on plasma heat transfer, the main facts will be discussed in this review. A more comprehensive treatment of this subject may be found in the work of Finkelnburg and Maecker (3),Griem (5), Unsold (ZI), and Drawin and Felenbok (12).

1. The Plasma in Perfect Thermodynamic Equilibrium Thermodynamic equilibrium prevails in a uniform, homogeneous plasma volume if kinetic and chemical equilibria as well as every conceivable plasma property are unambiguous functions of the temperature which in turn is the same for all plasma constituents and their possible reactions. More specifically, the following important conditions must be fulfilled. (a) The velocity distribution functions for particles of every species r which exists in the plasma, including the electrons, follows a MaxwellBoltzmann distribution :

vr is the velocity of particles of species r, m, is their mass, and T is their temperature, which is the same for every species r , and which is, in particular, identical with the plasma temperature. (b) The population density of the excited states of every species r follows a Boltzmann distribution [see Eq. (3)] :

nr(gr,r/zr)exp ( - X r . s l k B (26) T h e excitation temperature T which appears explicitly in the exponential term and implicitly in the terms n, and 2, of this relation is identical with the plasma temperature. (c) T h e particle densities (neutrals, electrons, ions) are described by the Saha-Eggert equation which may also be considered as a mass action law : %,r =

x r f l represents

the energy which is required to produce a ( r + 1)-times ionized particle from a r-times ionized particle. Equation (11) is identical with Eq. (27) for r = 0. The ionization temperature T in this equation is identical with the plasma temperature. A similar relation holds for the dissociation process, which is of importance for plasmas which contain, in addition, molecular species. (d) The electromagnetic radiation field is that of blackbody radiation of the intensity B , as described by the Planck function [see Eq. (16)]

246

E. R. G. ECKERT AND E. PFENDER

B

2hV3 1 -~ ' c 2 exp(hv/ksT) - 1 =-..

~

The temperature of this blackbody radiation is again identical with the plasma temperature. I n order to generate a plasma which follows this ideal model as described by Eqs. (25) to (28), the plasma would have to dwell in a hypothetical cavity whose walls are kept at the plasma temperature or the plasma volume would have to be so large that the central part of this volume, in which thermodynamic equilibrium prevails, would not sense the plasma boundaries. In this way the plasma would be penetrated by blackbody radiation of the same temperature. An actual plasma will, of course, deviate from these ideal conditions. The observed plasma radiation, for example, will be much less than the blackbody radiation because most plasmas are optically thin over a wide wavelength range as pointed out in the first section of this review. Therefore, the radiation temperature of a plasma deviates appreciably from the kinetic temperature of the plasma constituents or the already mentioned excitation and ionization temperatures. In addition to radiation losses, plasmas suffer irreversible energy losses by conduction, convection, and diffusion which also disturb the thermodynamic equilibrium. Thus, laboratory plasmas as well as some of the natural plasmas cannot be in a perfect thermodynamic equilibrium state. In the following sections, deviations from thermal equilibrium and their significance will be discussed.

2. The Concept of Local Thermal Equilibrium Since an actual plasma does not exhibit a homogeneous distribution of its properties (for example, in temperature and density), equilibrium considerations can only be applied locally. For this reason the concept of local thermodynamic (or thermal) equilibrium (LTE) was introduced which is less restrictive than the definition of perfect thermodynamic equilibrium. In this sense L T E may be considered as a special case of the more general concept of thermal equilibrium. L T E requires that collision processes and not radiative processes govern transitions and reactions in the plasma and that there is a microreversibility among the collision processes ;in other words, a detailed equilibrium of each collision process with its reverse process is required. Steady state solutions of the respective collision rate equations will then yield the same energy distribution pertaining to a system in complete thermal equilibrium with exception of the rarefied radiation field. L T E requires further that local gradients of the plasma properties (temperature, density, heat conductivity, etc.) are sufficiently small so that a given particle which diffuses from one location to another in the plasma finds sufficient time to equilibrate, i.e., the

PLASMA HEATTRANSFER

247

diffusion time should be of the same order of magnitude as the equilibration time. From the equilibration time and the particle velocities an equilibration length may be derived which is smaller in regions of small plasma property

FIG.7. Temperature profiles at various cross sections of a high mass flux argon plasma jet (z:distance from nozzle exit). 10

8

6

2

0

0

I

2 3 RADIUS ( mm 1

4

FIG.8. Isotherms of a low mass flux argon plasma jet ( z : distance from nozzle exit).

gradients (for example, in the center of an electric arc). Therefore, with regard to spatial variations LTE is more probable in such regions. Heavyparticle diffusion and resonance radiation from the center of an inhomogeneous plasma source help to reduce the effective equilibration distance

248

E. R. G. ECKERT AND E. PFENDER

in the outskirts of the source. Figures 7 and 8 show a temperature profile and the isotherms of argon plasma jets (13)as examples of spatial variations of the temperature within a plasma. In the following a systematic discussion of the important assumptions for L T E will be undertaken based on actual plasmas among which the electric arc appears as the most appropriate source. a . Kinetic Equilibrium. It may be safely assumed that each species (electron gas, ion gas, neutral gas) in a dense high-temperature plasma will assume a Maxwellian distribution. However, the temperatures defined by these Maxwellian distributions may be different from species to species. Such a situation which leads to a two-temperature concept will be discussed for an arc plasma.

PRESSURE (mrnHg)

-

FIG.9. Electron and gas kinetic temperatures in an arc plasma (20).

T h e electric energy which is fed into an arc is dissipated in the following way. T h e electrons according to their high mobility pick up energy from the electric field which they partially transfer by collisions to the heavy plasma constituents. Because of this continuous energy flux from the electrons to the heavy particles, there must be a “temperature gradient” between these two species, so that Te T , T, is the electron temperature, and T , the temperature of the heavy species, assuming that ion and neutral gas temperatures are the same. In the two-fluid model of a plasma, defined in this manner, two distinct temperatures T , and T, may exist. The degree to which T , and T, deviate from each other will depend on the thermal coupling between the two species. The difference between these two temperatures can be expressed by the following relation (3):

’

PLASMA HEATTRANSFER

249

T,- Ta=4 m -_____ (A,eE)2 Te g m e (3/2kBTe)' ma is the mass of the heavy plasma constituents, A, the mean free path length of the electrons, and E the field intensity. Since the mass ratio ma/8m, is for hydrogen already about 230, the amount of (directed) energy (A,eE) which

the electrons pick up along one mean free path length has to be very small compared to the average thermal (random) enengy 3/2kBT, of the electrons. Low field intensities, high pressures (A, lip), and high temperature levels are favorable for a kinetic equilibrium a'mong the plasma constituents. At low pressures, for example, appreciable deviations from kinetic equilibrium may occur. Figure 9 shows in a semischematic diagram how electron and gas temperature separate in an electric arc with decreasing pressure. For an atmospheric argon high-intensity arc with E = 13 V/cm, A, = 3 x cm, mA/me= 7 x lo4, and T, = 30 x 103"K,the deviation between T, and Ta is only 2% (3). 6. Excitation Equilibrium. In order to determine the excitation equilibrium every conceivable process which mqy lead to excitation or deexcitation has to be considered. We will restrict ourselves in this discussion to the most prominent mechanisms which are collisional and radiative excitation and deexcitation.

-

Excitation (1) electron collision (2) photoabsorption

Deexcitation (1) collision of the second kind (2) photoemission

We saw that for the case of a perfect thermodynamic equilibrium microreversibilities have to exist for all prbcesses; i.e., in the above scheme excitation by electron collision will be balanced by the reverse process, namely, collisions of the second kind and excitation by photoabsorption processes will be balanced by photoemission processes which include spontaneous and induced emission. T h e population of excited states is given by a Boltzmann distribution [see Eq. (26)]. The microreversibility for the radiative processes holds only if the radiation field in the plasma reaches the intensity B, of blackbody radiation. However, actual plasmas are over most of the spectral range optically thin, so that the situation for excitation equilibrium seems to be hopeless, Fortunately, if collisional processes dominate, photoabsorption and photoemission processes do not have to balance; only the sum on the left-hand side and the right-hand side of the scheme above have to be equal. Since the contribution of the photoprocesses to the number of excited atoms is almost negligible under this condition, the excitation process is still close to LTE.

25 0

E. R. G. ECKERT AND E. PFENDER

c. Ionization Equilibrium. For the ionization equilibrium we will again only consider the most prominent mechanisms which lead to ionization and recombination.

Ionization (1) electron collision (2) photoabsorption

Recombination (1) three-body recombination (2) photorecombination

In a perfect thermodynamic equilibrium state with cavity radiation, a microreversibility among the collisional and radiative processes would exist and the particle densities would be described by the Saha-Eggert equation. Without cavity radiation the number of photoionizations is almost negligible, requiring instead of the microreversibility a total balance of all processes involved. Photorecombinations, especially at lower electron densities, are not negligible. Elwert (14) showed that the frequency of the three remaining elementary processes is only a function of the electron density leading, for n, = 7 x 1015 cmP3, to the same frequency of these elementary processes. The result is an appreciable deviation between actual and predicted values [from Eq. (27)]of the electron densities. Only for values n, > 7 x 1015 does the Saha-Eggert equation predict correct values. For smaller electron densities the Corona formula (14) has to be used which considers only ionization by electron impact and photorecombination :

In this equation a is Sommerfeld’s fine-structure constant, 5, the number of valence electrons, n the principal quantum number of the valence shell, the ionization energy of hydrogen, andg a constant with a value between 1.4 and 4 (14). T h e particle concentrations in low-intensity arcs at atmospheric pressure, for example, have to be calculated with this formula. Significant deviations from Saha-equilibrium may also occur in the fringes of high-intensity arcs and plasma jets generated by arcs. Another important chemical reaction in a plasma which is generated from a molecular gas is the dissociation process. The considerations for this reaction are very similar to ionization and will, therefore, not be reiterated here. In summary, we found that L T E exists in a steady optically thin plasma when the following conditions are simultaneously fulfilled : (u) The different species which form the plasma have a Maxwellian

distribution. (/3) Electric field effects are small enough, and the pressure and the temperature are sufficiently high so that T , = T,.

PLASMA HEATTRANSFER

25 1

( y ) Collisions are the dominating mechanism for excitation (Boltzmann

distribution) and ionization equilibrium (Saha-Eggert equation). (6) Spatial variations of the plasma properties are sufficiently small. Besides the conditions for the two extreme cases, namely L T E (based on Saha ionization equilibrium) and Corona equilibrium, we are also interested in the region between these two limiting cases. In this range three-body recombination as well as radiative recombination and deexcitation is significant. Several authors (14-18) present theories for ionization equilibrium over the entire range of radiative-collisional elementary processes. In particular, Bates et al. (15) report detailed calculations of optically thin and optically thick hydrogen plasmas. We will discuss their results for the optically thin case. If a is the combined collisional-radiative recombination coefficient and S the corresponding ionization coefficient, rate equations may be established which describe the effective rate of population and depopulation. The rate of population of the ground state is described by

(%c)

=

meni

POP

In this relation no,orepresents the number of neutral hydrogen atoms in the ground state ; n, and n iare electron and ion densities, respectively. The rate of depopulation of the ground state is given by

Under steady-state conditions

or Figure 10 shows a state diagram for values of S / a as a function of the electron density with pressure and electron temperatures as parameters mV3) . high electron densities [plotted from data of Bates et al. ( 1 5 ) ] At pairs of L T E and non-LTE curves plotted for the same electron temperature merge. At low electron densities mP3) the non-LTE curves merge into curves valid for Corona equilibrium. The divergence of the non-LTE curves from the L T E curves at lower pressures and/or lower electron temperatures and densities shows how large the deviation from L T E may become in such parameter ranges. Taking values for 1atm it can be seen that L T E is closely approached for electron temperatures in the interval 14,000 < T, < 28,000"K.

252

E. R. G. ECKERTAND E. PFENDER

Deviations of this kind from L T E may be found, for example, in arc plasma regions adjacent to walls where the electron density drops appreciably. T h e situation becomes even more complicated in the immediate vicinity of a wall where a thin layer, called a plasma sheath, separates the actual plasma from the wall. I n this sheath strong deviations from quasi neutrality may be found. Because of the general importance of plasma

Id8

lozo 102' loz2 ELECTRON DENSITY (I?) NON-LTE ----- -. LTE HYDROGEN 10"

loz3

loz4

~

FIG.10. State diagram for hydrogen in L T E and non-LTE (15).

sheaths for the interaction between a plasma and neighboring walls and in particular for plasma heat transfer, the next paragraph will be devoted to a brief discussion of plasma sheaths and their formation.

D. PLASMA-WALL BOUNDARIES AND PLASMA SHEATHS Every solid component (electrodes and walls) of a plasma device which is exposed to a hot plasma has to be protected by appropriate cooling. In

PLASMA HEATTRANSFER

253

engineering and laboratory devices the plasma is frequently constricted or enclosed by walls and/or electrodes which are water-, transpiration-, or radiation-cooled. In steady-state operation the walls and electrodes carry a continuous heat flux driven by the temperature difference between the plasma and these cooled components which extend their cooling effect into the plasma in a thin layer which may be described as a temperature boundary layer. T h e thickness of this boundary layer comprises many free path lengths of the electrons or heavy particles. At the bottom of this boundary layer, overlying the solid wall, a plasma sheath is formed with a thickness in the order of a Debye length.

Sheath Formation For the following consideration, we assume an electrically conducting but insulated wall “in contact” with a dense plasma, a situation found, for example, in a wall-stabilized high intensity arc at atmospheric pressure. Since there is no net current flow to an electrically insulating wall, electron and ion currents reaching the wall have to compensate each other. Recombination of the electrons and ions occurs at the wall surface and, for a cold wall, also to a certain degree in the plasma close to the wall surface. T h e corresponding concentration gradient of the electrons and ions causes these particles to diffuse toward the wall. The diffusion coefficient of the electrons is much higher than that of the ions, which would mean an electron current larger than the ion current. Such a situation which cannot exist on an insulated wall is prevented by the fact that the wall assumes a negative potential which retards the electrons and accelerates the ions until both fluxes are of equal magnitude. This requires that quasi-neutrality can no longer exist close to the wall. Deviations from quasi-neutrality, however, are restricted to a very thin layer which is called a sheath (Fig. 11).The fact that there is no net charge carrier flow to the surface can be expressed mathematically in the following way, assuming a one-dimensional situation with many collisions occurring in the sheath :

i = j , + j i = (eneve+ enipi)Ex + eD,dn , dx

-

dni dx

EDi - = 0

(36)

In this relation, n, indicates the particle density of the electrons, ni the particle density of the ions, pe and p i are the electron and ion mobility, respectively, Ex is the electric field intensity in the direction normal to the surface, and D, and Di are the electron and ion diffusion coefficients into the predominantly neutral gas. The electron mobility and diffusion coefficients are much larger than the corresponding values for the ions. The gradient of the electric field intensity is connected with the space charge c(ni - n,) by the Poisson equation

E. R. G. ECKERT AND E. PFENDER

254

Since charge imbalances can only occur over a distance in the order of a Debye shielding length, AD, the thickness of the sheath will also assume a value in this order of magnitude.

WALL

0

v I

7

L

7

L

1 I

I

II I

I I

-c

FIG.11. Temperature boundary layer and sheath formation.

*

Assuming that T , T, and considering shielding by electrons only in a singly ionized gas, the Debye shielding length may be expressed by

This shielding length depends on the heavy-particle temperature and the electron density. Any relation between the electron density and the temperature T, will depend on the thermodynamic state of the plasma. For an estimate of the order of magnitude of the Debye length in an atmospheric argon plasma at a temperature of 104"K,LTE will be assumed in spite of the fact that Eq. (38) does not hold for LTE. The electron density in such a plasma is approximately 3 x 1 O I 6 cmp3 and the corresponding Debye length is about cm. T h e Debye length is, in this case, small compared to the mean 4x free path length of the electrons (about 3 x lo-' cm) and the atoms (about 5 x lop5cm). The buildup of a space charge can, therefore, not occur in the

PLASMA HEATTRANSFER

255

plasma itself, since Coulomb forces effectively prevent the separation of ions and electrons and keep the plasma neutral (Fig. 11). Near an electrically insulated surface the electron density will be much smaller and the Debye length as well as the sheath thickness, according to Eq. (38), much larger. T h e Saha equation certainly does not apply in this region and too little is known about the electron density to make a quantitative prediction of the sheath thickness possible. In most weakly ionized high-pressure plasmas, the sheath is, therefore, probably collision dominated and Eq. (36) can be applied. Sheaths in highly ionized plasmas at moderate or low pressures may, on the other hand, be considered as collisionless and the following relations exist for this condition. An estimate of the thickness of a collisionless sheath yields (29)

T h e potential drop across the sheath, assuming that there are no collisions in the sheath, can be found from the requirement that electron and ion currents balance (29):

T , is the electron temperature in the plasma and ma is the mass of the ions (or atoms). For a net current flow to the originally insulated surface, the thickness d, of the collisionless sheath will depend on the potential difference across the sheath and on the net current drawn to the surface. A net current flow to the surface may be achieved by biasing the surface with respect to the plasma. With a positively biased surface the situation may be compared with that of a space-charge-limited thermionic vacuum diode by identifying the edge of the plasma sheath with the thermionically emitting cathode. The random electron current reaching the sheath from the plasma corresponds in this analogy to the electron current emitted from the cathode. As in a diode, the current to the biased surface depends on the applied potential across the collisionless sheath and on the thickness of the sheath (20) :

T,is the electron temperature at the edge of the sheath. The last term in the bracket of Eq. (41) represents a correction term for the common spacecharge equation which accounts for the initial velocities of the electrons

256

E. R. G. ECKERT AND E. PFENDER

entering the sheath. A very similar relation holds for a negatively biased surface which draws an ion current of the density

where Tirepresents the ion temperature at the edge of the sheath. T h e potential imposed on the surface is essentially confined to the sheath in which a space charge of the opposite sign absorbs the electric field. This fact has a very important consequence for the applicability of Langmuir probes in plasmas (21). T o what degree the existence of the sheath, whether collisionless or collision dominated, influences the heat transfer to a wall depends on the value of the ratio of the sheath thickness d, to the boundary layer thickness 6. T h e influence of the sheath will be small as long as the ratio hD/S is small in order of magnitude. In the analyses discussed in Sections I11 and IV, the influence of the sheath on heat transfer is assumed to be negligible.

E. THERMODYNAMIC AND TRANSPORT PROPERTIES OF PLASMAS For plasma heat transfer a number of thermophysical properties of the plasma are of importance which may be subdivided into thermodynamic and transport properties. In this review these properties will be briefly discussed for argon plasmas.

1. Plasma Composition and Thermodynamic Properties For the calculation of the chemical composition of a plasma in thermodynamic equilibrium (see Fig. 1) as a function of the temperature, a system of simultaneous nonlinear equations has to be solved. This system of equations remains essentially the same for any given gas or gas mixture from which a plasma may be generated. The appropriate equations are obtained from the conservation-of-mass law na = C nua r

(43)

where nu is the total number density of particles of a given species ( a ) , n/ represents the number density of atoms or ions of the same species ( a ) , and r indicates the ionization stage ;the condition for quasi neutrality n, = C rn/ r

(44)

PLASMA HEATTRANSFER

257

and the mass-action law which is, in this case, expressed by the Saha-Eggert relations [Eq. (27)] which connect the particle densities of the different ionization stages with the electron densities

(45)

'

TEMPERATURE

fi)

i x 10'

FIG.12. Composition of an argon plasma in thermodynamic equilibrium (22).

If the plasma contains r different ionization stages of a certain species ( a ) and if there are N different species in the plasma, then the number of unknown particle densities is rN + 1 where the electron density is the additional unknown. Since there are ( Y - 1) Saha-Eggert equations for each species and one conservation-of-mass relation, the total number of equations is r N . The last required relation is provided by Eq. (44).From this complete set of equations the equilibrium composition of any given plasma may be calculated. As an example, Fig. 12 shows the equilibrium composition of an

258

E. R. G. ECKERT AND E. PFENDER

argon plasma at atmospheric pressure. As pointed out in the preceding section, actual plasmas deviate more or less from a perfect thermodynamic equilibrium which, of course, will also cause deviations from the equilibrium plasma composition.

TEMPERATURE (K)

FIG.1 3 . Mass density of an argon plasma (22).

Having the composition of a plasma calculated, the thermodynamic properties of the plasma (mass density, internal energy, enthalpy, entropy, and specific heat) may be found from the contribution of the different plasma constituents to the overall properties. Figures 13-17, taken from Cambel (22), demonstrate examples of such equilibrium properties for argon plasmas as a function of the temperature. Another frequently used method for illustration of the most important thermodynamic properties in a single diagram is by means of enthalpy-entropy charts (Mollier diagrams). Figure 18 shows

PLASMA HEATTRANSFER

259

such a Mollier diagram for an argon plasma in perfect thermodynamic equilibrium whereas Fig. 19 refers to an argon plasma based on an ionization equilibrium according to Elwert (14).Both diagrams are valid for a temperature range lo4< T < lo5OK and are plotted with temperature, mass density, and total pressure as parameters (23).At high pressures there is a reasonable

r

5x18

2-

'01

-

I

X 0

\ X 7

>

5-

W

a W

2 W

d 2z

E f

5E 0

10

15

20

TEMPERATURE

25

30

-.

35xld

fK 1

FIG.14. Internal energy of an argon plasma (22).

agreement of the different thermodynamic properties of Fig. 19 with the corresponding values in Fig. 18 up to temperatures of about 5 x 104"K. As discussed in the preceding section, there is a predomination of collisional processes in this parameter range so that even without blackbody radiation (photo processes neglected) thermodynamic equilibrium may be closely approached. At pressuresp < 1 bar, Corona equilibrium prevails causing the isotherms to be parallel to the abscissa. For the temperature range T < 2 x 104"K,which is of particular interest in plasma applications, Fig. 20

E. R. G. ECKERT AND E. PFENDER

260

shows thermodynamic properties for an argon plasma in perfect thermodynamic equilibrium (24). 2. Plasma Transport Properties

Transport phenomena in plasmas encompass the flow situation of every plasma constituent, namely electrons, ions, and neutrals including radiation fluxes under the influence of driving “forces” as, for example, electric fields, 5

2

loB

a

X

5

\

X 7

t J

9!-

2

z

w

10‘

5

0

10

15 20 25 TEMPERATURE fK)

30

35x10~

FIG.15. Enthalpy of an argon plasma (22).

temperature-, pressure-, density-, and velocity-gradients. In order to describe the transfer of electrical charge, mass, momentum, and energy within the plasma and from the plasma to its surroundings, characteristic transport properties have been defined as, for example, electrical conductivity, heat conductivity, viscosity, and diffusivity. For the calculation of these transport properties different methods have been proposed. For further information, the reader is referred to the work of Finkelnburg and Maecker (3) and Cambel(22).

26 1

PLASMA HEATTRANSFER

---l--/

3 / / -

0

10

15

20

25

30 x 10’

TEMPERATURE CK)

FIG. 16. Entropy of an argon plasma (22).

TEMPERATURE
FIG.17. Specific heat of an argon plasma (22).

262

AND E. PFENDER E. R. G. ECKERT

Electrical and thermal properties of argon plasmas were determined by several authors and results reported by Cambel(22) are shown in Figs. 21 and 22. He also reports data for the viscosity of argon plasmas as a function of the temperature (Fig. 23). T h e agreement of these property data from

5

10

15 ENTROPY

20

25

30

(kJ/kq*K)

FIG.18. Mollier diagram of an argon plasma in thermodynamic equilibrium (23).

author to author is in most cases not very satisfactory which seems to be due to the different assumptions made for the calculations and to the lack of experimental data at higher temperatures.

F. INCREASEOF PLASMA TEMPERATURES One of the main obstacles in the generation of steady plasmas with extremely high temperatures (>lo5OK) is the excessive specific heat flux to the walls of the container including the electrodes. From the survey in Fig. 6 it follows, for example, that a water-cooled constricted arc provides higher maximum temperatures than a free-burning arc because this maximum depends on the power input ZE per unit length of the arc column (25). For a given current I , the field intensity E and therefore the power input I E per unit length is a function of the heat flux to the surroundings. This flux is appreciably higher for a constricted arc column.

PLASMA HEATTRANSFER

263

T h e difficulty in increasing the temperature can be seen by a simple consideration of the conditions in a fully developed rotationally symmetric arc column within a water-cooled constrictor. Radiation from the arc column will be neglected so that energy losses occur by radial conduction only. Balancing the power input to the arc per unit volume with the radial heat flux, we obtain the Elenbaas-Heller differential equation

j E + div (kgrad 2')

5

10

15

=

0

20

(46)

25

30

ENTROPY (kJ/k
FIG.19. Mollier diagram of an argon plasma in Corona equilibrium (23).

which may be written in cylindrical coordinates as

This differential equation will be solved with the simplified assumption of a uniformly conducting core with radius yo assuming that for r > Y,, the electrical conductivity vanishes. T h e temperature at r = yo may be To. Double integration of Eq. (47) with the condition that the temperature gradient is equal to zero at 1' = 0 results in the relation

264

E. R. G. ECKERT AND E. PFENDER

T o solve the right-hand integral, a relation k = f ( T ) is required. Since we are interested in extremely high temperatures, the conducting core may be considered as fully ionized and Spitzer’s formula k T5/*may be used as

-

DENSITY (hg/m’l

FIG.20. Enthalpy-density diagram of an argon plasma in thermodynamic equilibrium (24.

an approximation (26). Assuming, additionally, the temperature To to be small compared to T,,, results in the relation with

IE

- Td!:

I = n r o2 ’

(49)

(50) To raise the maximum temperature by a factor of 3 requires about a 50 times higher power input according to Eq. (49). Radiation, which has been neglected, makes this factor even larger. Calculations by Maecker (25)for

PLASMA HEATTRANSFER

265

hydrogen and gases similar to hydrogen indicate that temperatures T > 3 x 104"Kare not feasible in this way even if one provides this power because there is no material which is able to withstand the high specific wall heat fluxes. The highest permissible specific heat flux to a water-cooled

1

/

I

ARGON I ATM

3 10

5 TEMPERATURE fK )

FIG.21. Electrical conductivity of an argon plasma (22).

copper wall is about 19 kW/cm2.Steady plasmas at higher temperature levels will only be feasible if appropriate means can be found to reduce the heat flux to the walls. A reduction of the wall heat flux can be achieved either by a reduction of the heat conductivity or by reducing the temperature gradient at the wall which determines the heat flux into the wall. Two possibilities have been proposed by which the heat conductivity can be reduced (10). One of these possibilities considers the fact that k V$ for a Knudsen number Kn < 1 with the tube radius as reference length. A plasma maintained in a low-

-

E. R. G. ECKERT AND E. PFENDER

266

-

ARGON IATM

5

L

//I

10 TEMPERATURE CK)

15x10’

FIG.22. Thermal conductivity of an argon plasma (22).

TEMPERATURE

(K)

FIG.23. Viscosity of an argon plasma (22).

PLASMA HEATTRANSFER

267

pressure environment would meet this condition. T h e second method uses a superimposed magnetic field parallel to the axis of the arc which reduces the heat conductivity in the radial direction. A reduction of the wall heat fluxes by reducing the temperature gradient at the wall can also be implemented with a transpiration-cooled wall. T h e last two methods will be briefly discussed based on some recent publications. 0.5 0.4

W

a

2 0.2 W m h

0.I TEMPERATURE

(.K)

FIG.24. Pressure increase in a magnetic field (28).

1. Reduction of the Heat Conductivity by M e a m of a Magnetic Field For the following discussion, a rotationally symmetric steady highpressure plasma column will be considered which is enclosed in a watercooled tube. Energy losses are compensated by ohmic heating which is provided by an axial electric current. Such a column exchanges steady diffusion currents in the radial direction, consisting of electrons and ions flowing toward the wall, and the neutral gas particles flowing toward the center of the plasma column as described in Section I1,A. For steady-state conditions there is no net electric current in the radial direction and the total mass flux in the radial direction has also to vanish. By superimposing an axial magnetic field, the diffusion of the charged particles will initially be reduced whereas the diffusion of the neutral particles is not influenced until again a steady state is reached with a somewhat higher pressure in the plasma column. Wienecke (27) and Witkowski (28) calculated the pressure increase in the column of a high-pressure arc under the influence of an axial magnetic field. Wienecke assumed LTE in the plasma whereas Witkowski based his

268

E. R. G. ECKERT AND E. PFENDER

calculations on Corona equilibrium. Figure 24 shows a typical result of their calculations. For sufficiently high temperatures (fully ionized plasma) the heat conductivity perpendicular to the magnetic field lines becomes rather small (k T-''') as shown by Braginskii (29) and Feneberg (30). Not only the

-

L

15x10~

HYDROGEN

5x10' BAR

0' 0

I

1

2

4

6

'-f

8x10'

TEMPERATURE (OK)

FIG.25. Influence of a magnetic field on the heat conductivity (30).

electron heat conductivity k, and the ion heat conductivity kiare reduced but also kI (see Fig. 2), which represents the diffusion transport of ionization energy, is appreciably influenced by the magnetic field (31). T h e strong effect of a magnetic field on the heat conductivity is demonstrated in Fig. 25 which was taken from Feneberg (30). Estimates for such favourable conditions show that plasma temperatures of lo5"K are feasible with a power input of only 3 kW/cm using hydrogen at 0.1 atm and applying a magnetic field of 2 W/m2 (10).

PLASMA HEATTRANSFER

269

2. Reduction of Wall Heat Fluxes by Transpiration Cooling In a cylindrical plasma column enclosed in a nonconducting porous transpiration-cooled tube, the temperature profile assumes an entirely different shape compared with that obtained in a water-cooled constrictor (Fig. 26). The temperature gradient at the wall becomes much smaller

HYDROGEN

-TRANSPIRATION-COOLED

-

.---WATER-COOLED

rr

0.25 cm

= 60.000%

,-Tmoi

-

-

-

0

.2

.4

.6

NORMALIZED

.8

10 .

RADIUS

FIG.26. Temperature profiles of a water-cooled and transpiration-cooled, constricted arc (32).

because the cold gas transpiring through the wall into the tube causes a convective energy transport in a direction opposite to that of the conduction heat flux. The energy which would, without transpiration cooling, flow to the wall is now intercepted and used to heat the transpiring gas. Anderson and Eckert (32) show that maximum temperatures of about 60 x lo3"K can be reached in the axis of a transpiration-cooled arc using a power input of about 50 kW/cm in atmospheric hydrogen. The power input is, of course, very high because the gas transpiring through the wall into the arc has to be heated, as mentioned before. However, the wall heat fluxes

270

E. R. G. ECKERT AND E. PFENDER

which constitute the primary restriction for the maximum temperatures obtainable in constricted arcs with water-cooled walls are in this case almost negligible. A more detailed discussion of the transpiration-cooled arc follows in Section IV,A.

111. Plasma Heat Transfer in the Absence of an Externally Applied Electric or Magnetic Field

A. QUALITATIVE CONSIDERATIONS This area of heat transfer has found attention mainly in connection with the reentry problem as was mentioned in the Introduction. We will first attempt to evaluate qualitatively how much such a heat transfer in an ionized gas is expected to differ from heat transfer in an ordinary gas at low temperature and we will do this for the example of rotationally symmetric stagnation flow. Heat transfer of a fluid with constant properties flowing with a rotationally symmetric laminar boundary layer over a surface near a stagnation point can be calculated from the following equation (33): Nu = 0.76 Re'/* Prn.4

(51) (for an explanation of the symbols, see the Nomenclature). The Nusselt number, Nu, is a dimensionless expression for the heat transfer coefficient. The symbol Re denotes the Reynolds number and Pr the Prandtl number. Relation (51) also holds for a gas as long as the product pp of density p and viscosity p as well as the Prandtl number are constants. The specific heat cp of the gas is allowed to vary arbitrarily with temperature when the heat transfer coefficient hi,combined in the Nusselt number Nu = hicpL/k with the specific heat cp, the thermal conductivity k, and a characteristic length L, is defined with an enthalpy difference as driving potential according to the equation (33): q r u = hi(G - j z u ) (52) qw, in this equation, denotes the heat flux to the wall surface per unit area and time, ie the stagnation enthalpy of the gas outside the boundary layer, and i, the enthalpy which the gas has in immediate proximity of the wall surface. It is easily checked that for a gas as defined above Eq. (51) leads to the same result regardless of at which temperature (or enthalpy) the properties are introduced into the dimensionless parameters Nu, Re, or Pr. In a two-component gas mixture, energy is transported not only as heat by thermal conduction, but also as enthalpy carried along by the individual components in their interdiffusion process. The relative intensity of the two transport mechanisms is described by the Lewis number, Le, defined as

PLASMA HEATTRANSFER

27 1

Le = pep Dlk (53) with p indicating the density, cp the specific heat at constant pressure, k the thermal conductivity of the mixture, and D the mass diffusion coefficient for the two components. For a gas mixture with a Lewis number equal to one, Eqs. (51) and (52) describe the total heat transfer process including diffusional and conductive energy transport. This applies even when chemical reactions between the components occur within the boundary layer, provided the enthalpies in Eq. (52) include the reaction enthalpy, and it holds regardless of whether or not local composition equilibrium exists within the boundary layer, and whether or not some relaxation process is involved which delays the reactions, as long as the proper enthalpies outside the boundary layer and at the wall surface are introduced in Eq. (52). Equation (51) can still be used to obtain approximate heat transfer coefficients for a gas with property variations different from the ones mentioned above, provided the thermodynamic and transport properties are introduced at a properly selected enthalpy with a value between the extremes occurring within the boundary layer. The arithmetic mean between i, and iw has been found to be a fairly good approximation. For a gas with a Lewis number different from one, corrections have been established by various authors. Fay and Riddell (34),for instance, found that the heat flux increases when the gas under consideration has a Lewis number larger than one and that this increase can, with good approximation, be accounted for in dissociated air by adding the term qc =

k (Len- 1)iD c* L

(54)

to the heat flux calculated with Eqs. (51) and (52). The exponent n was found to be 0.52 for equilibrium composition and 5 for frozen flow within the boundary layer. iD indicates the dissociation enthalpy. When the temperature is sufficiently high for thermal ionization, electrons and ions appear in addition to neutral particles in the gas. The Prandtl number Pr = p c p / k is now strongly reduced because of electron conduction and may drop at sufficiently high temperatures to values as low as 0.01 which are comparable to those of metals. The thermal conductivity is correspondingly increased. One might assume that the heat transfer coefficient, which according to Eq. (51) is proportional to k".6,increases drastically for such a situation. This, however, is not the case for a cold catalytic wall because the surface will always be separated by a nonionized or poorly ionized layer from the hotter part of the boundary layer. This layer will extend through the major portion of the boundary layer when equilibrium composition between charged and neutral particles is established within the boundary layer, because the temperature at which ionization starts will be, in

272

E. R. G. ECKERT AND E. PFENDER

general, closer to the temperature in the hot plasma outside the boundary layer than to the wall surface temperature. This nonionized layer will have a relatively low conductivity and will act as a thermally insulating layer between the ionized gas and the wall surface. In a chemically frozen boundary layer, conditions will be somewhat different. The concentration of ionized particles within the boundary layer is then determined by the diffusion process and will drop in a more or less linear way from the value outside the boundary layer to the value zero at the catalytic wall surface. One has therefore to expect that heat transfer will be somewhat larger for a frozen boundary layer than for an equilibrium boundary layer. Only a noncatalytic surface together with a nearly frozen condition within the boundary layer can establish an electron density with finite values through the whole width of the boundary layer and electronic conduction is then expected to increase the heat transfer to the wall surface by an order of magnitude. There is also another factor involved in heat transfer from an ionized gas which tends to reduce the heat flux. In the interdiffusion of electrically charged and uncharged particles in an ionized gas, the electrons and ions move as if they were joined together. This diffusion process is called ambipolar diffusion. The diffusion coefficient D for this process is assumed to be approximately by a factor of two larger than the diffusion coefficient for the corresponding neutral particles. It therefore maintains its order of magnitude in the Lewis number [Eq. (53)], whereas the thermal conductivity increases, possibly by an order of magnitude or more. This causes the Lewis number to assume values smaller than one. Correspondingly the correction term qc given by Eq. (54) is negative and reduces the Nusselt number. In summary, it may therefore be expected that the Nusselt number and the heat flux to the surface are not strongly influenced by the ionization process and that the heat flux in a boundary layer at composition equilibrium will be somewhat smaller than in a frozen boundary layer. More detailed analyses and experiments which verify our conclusions will be discussed in a later section. Before doing so, the basic equations describing combined diffusion and conduction processes will be developed.

B. BASICTRANSPORT EQUATIONS In this section an ionized gas will be considered which is macroscopically at rest. Such a gas consists, in general, of molecules, atoms, ions, and electrons, each again possibly comprising several species. Diffusion processes will set in when the concentration of the components varies locally. T h e analysis of this diffusion process is extremely involved and simplifications are therefore usually introduced. Such simplifications have been discussed,

PLASMA HEATTRANSFER

273

for instance, in the review paper by Chung (I). In the papers which will be discussed in the next section, the “binary diffusion model” is utilized. This model replaces, with regard to diffusion, the actual gas by a two-component mixture, one component comprised of the molecules and the other of the atoms, ions, and electrons. T h e justification for the use of such a model is the fact that at sufficiently high densities Coulomb forces prevent the ions and electrons from being separated by a diffusion process. The atoms and ions, on the other hand, differ less in their molecular weight than both differ from the molecular weight of the molecules. The binary diffusion model for this reason was found to give useful results and will be used in the basic equations describing mass and heat fluxes. A diffusional mass flux can be generated by a concentration gradient, by a pressure gradient, and by temperature gradients. In a boundary layer, diffusion occurs in a direction in which pressure gradients are negligibly small. For this reason pressure diffusion will be disregarded. Diffusion caused by an electric field tends to separate the electrons and the ions. Coulomb forces acting between the particles themselves, however, counteract this tendency and diffusion by body forces will therefore also be neglected. T h e remaining diffusion by concentration gradients and by temperature gradients is described in the two-component gas mixture by the following equation (35):

In this equation it is assumed that a concentration and temperature gradient exists in y direction. j D l denotes the mass flux per unit time and area of component one in this direction. w 1 is the mass fraction of component one, w 2the mass fraction of component two. T denotes the temperature, p the density of the mixture, D the binary mass diffusion coefficient, and CL the thermal diffusion ratio. The mass flux j D 2of component 2 is by definition equal to -jDl. An energy flux E per unit time and area will also be present. Its magnitude is described by the following equation : E =

-k

aT -

aY

+ (i, - i2)jD1- CLRTMM2 lM p 1

The first term in this equation describes energy transport by heat conduction. The second term gives the energy transport by enthalpy interdiffusion and the third term describes energy transport by the diffusion thermo effect. h denotes the thermal conductivity, il and i2 are the enthalpies of component 1 and component 2, respectively. R is the gas constant and M the molecular weight of the mixture. M I and M 2 denote the molecular weights of the two

E. R. G. ECKERT AND E. PFENDER

274

components, respectively. Equation (56) can also be written in a different way by introducing the mass flux given by Eq. (55),

T h e two equations (56) and (57) indicate that care is to be taken in specifying a thermal conductivity for a two-component mixture. The transport property k used in Eq. (56) describes the energy transport under the condition that the mass fluxjD1is zero. One can, however, also interpret the term in the square bracket of Eq. (57) which is multiplied by the temperature gradient as a thermal conductivity describing the energy transport under the condition that no concentration gradient exists. The analyses which will be discussed in the following section neglect the thermal diffusion and diffusion thermoprocesses assuming that they contribute little to the overall mass and energy transport.' Equations (55) and (56) or (57), respectively, simplify thus to

The difference between the two conductivities discussed above has now disappeared. However, an index f is added to the symbol k , indicating that the first term on the right-hand side of Eq. (59) considers energy transport due to heat conduction only, and does not include the energy transport by enthalpy interdiffusion described by the right-hand term of the equation. In various analyses the energy flux is written differently, replacing the temperature T in Eq. (59) by the mixture enthalpy i in the following way. The enthalpy of component one is described by the equation

il =

I"

rpl dT

+ilo

(60)

'The zero enthalpy i l o has to be included when chemical reactions occur between the two components. The enthalpy of the mixture is correspondingly

i = w i, + w2i2 = w

jcDldT + w2

and its differential

+

d i = w 1cpldT + w 2 r p Z d T dwl 1

cp2dT

+w

il

+ w 2iZ0

(61)

jrpl dT + dw, 1 rp2dT+ i l o d w l+ izodw2

Eckert (36) and Sparrow et al. (37) show that this is not always the case.

PLASMA

HEATTRANSFER

275

With the specific heat of the mixture and utilizing Eq. (60), the last equation becomes di = c p f d T (il- i2)dw I (62)

+

Introducing this enthalpy differential changes Eq. (59) to

T h e Lewis number in this equation is defined as

Indices f are again added to the thermal conductivity and to the specific heat, indicating that they describe the enthalpy and the energy transport, respectively, measured while the temperature changes, with the concentration, however, held constant. They are referred to as frozen properties. The last equation can also be written as

with the frozen Prandtl number Prf= pcpjp/k,. This equation is of advantage because in many cases the Lewis number of a gas mixture differs but little from the value one. An analysis based on the assumption of a Lewis number equal to one serves then as a good approximation to actual conditions. For such a gas, the second term in Eq. (63) is equal to zero, simplifying the equation for the energy flux considerably. Many analyses of reentry heat transfer assume local chemical equilibrium between the various species within the boundary layer. This means that the local mass fraction is, at a prescribed pressure, a function of temperature and consequently of enthalpy. Equation (63) can then be written in the following form :

T h e whole term within the brackets is now a parameter, depending on the local state in the gas, and can be used to define a Prandtl number, Pr,, according to the equation

This equation now describes, together with the equilibrium Prandtl number, the total energy flux in a gas under the condition of local chemical

E. R. G. ECKERT AND E. PFENDER

276

equilibrium and is useful in heat transfer calculations which are based on that condition. A comparison of Eqs. (63) and (64) shows that the difference between the frozen Prandtl number and the equilibrium Prandtl number vanishes for a gas with a Lewis number equal to one. T h e equations for the mass flux and energy flux in this section will now be utilized to obtain the boundary layer equations for a two-component gas mixture.

FIG.27. Rotationally symmetric boundary layer coordinates.

C. LAMINAR BOUNDARY LAYER EQUATIONS T h e equations describing conservation of mass, momentum, and energy for plane and rotationally symmetric, laminar boundary layers of a twocomponent gas mixture can be written in the following form :

a (py”u) +

ax

a (py”v) = 0

aY

PLASMA HEATTRANSFER

277

T h e coordinates x and y as well as the length r are explained in Fig. 27. T h e exponent n has to be set equal to zero for plane flow and equal to one for rotationally symmetric flow. The velocity components u and + are also indicated in Fig. 27. Parameters without an index refer to the mixture. The index one refers to component one. The symbol K ~ in, Eqs. (66) and (68), denotes the rate of mass generation of component one per unit volume and time by a chemical reaction. The differential quotient dpldx is written in total differentials because the pressure variation is negligibly small within the boundary layer in a directionnormal to the surface, according to boundary layer theory. The mass flux ratej,, is now introduced into Eq. (66) leading to

Two energy equations can be written depending on which of the equations (59) or (63) is introduced into Eq. (68) :

The symbol io in Eq. (70a) indicates the total enthalpy which includes kinetic energy. One of these equations is used in analyzing heat transfer processes in a gas which is not in composition equilibrium. For equilibrium, the energy equations can again be simplified to

These are the equations used in the analyses discussed in the next section. They will have to be modified in the presence of a magnetic field and when electrons enter or leave the surface on which the boundary layer borders.

278

E. R. G. ECKERTAND E. PFENDER

Boundary conditions have to be described at the wall surface and at the outer border of the boundary layer. At the outer border they are prescribed by the state of the main body of the gas. At the wall surface, both velocity components are prescribed to have a value zero in the following section. This excludes, for instance, ablation of wall material. The temperature T is in this section set equal to the wall surface temperature and a catalytic wall is assumed, so that the gas composition is the equilibrium composition at the wall surface temperature.

D. RESULTS OF REENTRY STUDIES Analyses and experiments on heat transfer in the reentry problem have been primarily concerned with rotationally symmetric flow near the stagnation point on a blunt body. This is the location where, in the range of laminar boundary layers, heat transfer coefficients reach the highest values. On the other hand, it is also a situation for which the boundary layer equations can be solved in a comparatively simple way by transforming them to total differential equations. T h e terms udp/dx and ~ ( d u / d y )or~ @jay) [(l - 1/Pr)p (ajay)(u2/2)] can be neglected near the stagnation point in the energy equations (68) or (69) and no difference exists between total and static enthalpies or temperatures. Results of such calculations are presented in Fig. 28. T h e ordinate of the figure is the parameter Nu/(Re)’” defined in the following way :

This parameter is plotted over the approach velocity of the air stream in meters per second defined by the equation V m= (2ieo)1’2. Hoshizaki (38), whose results are shown in the figure as the dashed line, performed his analysis for an equilibrium boundary layer of air and CO,, respectively. Correspondingly, he used the energy equation (71b) with properties reported by C. F. Hansen for air and by J. L. Raymond and M. Thomas for CO-,. His analysis for air covered a pressure range from 0.001 to 100 atm and wall temperatures between 300 and 3000°K. T h e results could be represented within +6”,, by the single line shown in Fig. 28. The results for CO-, correlated on the same line for wall temperatures higher than 500%. Fay and Kemp (39) performed their calculations for nitrogen. They considered a frozen condition as well as an equilibrium condition within the boundary layer. In the first case they used the energy equation (70b) and in

PLASMA HEATTRANSFER

279

the second case (71b). With regard to diffusion, they simplified the actual situation by utilizing the binary diffusion model. One component in this model is comprised of the molecules and the other component of the atoms, ions, and electrons. The thermodynamic and transport properties were calculated on the basis of kinetic theory. The Lewis number was in this way determined to be 0.6 in the temperature range between lo3 and 104"K.A temperature of 300°K was assumed for the wall surface. T h e results are

O.* 0.6

f

OS4

2

0.2

0.I

1

4

6

8

10

12

FLIGHT VELOCITY, V,

14

16

18x10'

(m/sec)

FIG.28. Calculated heat transfer parameter for rotationally symmetric stagnation point

flow: a, air and C O , , after Hoshizaki (38);b, N2, after Pallone and van Tassel (40);c , air, after Pallone and van Tassel (40);d, air [after Cohen (4Oa)l.The point symbols are for N2, [after Fay and Kemp (39)l.

plotted in Fig. 28 with the open symbols indicating an equilibrium condition and the full symbols denoting a frozen condition. In agreement with the qualitative discussions presented in the preceding section, it is observed that the heat transfer in the frozen boundary layer is larger than in the equilibrium boundary layer, with the difference increasing with increasing flight velocity. Pallone and van Tassel1 (40)also made calculations on heat transfer in the region of a rotationally symmetric stagnation point with boundary layer flow

280

AND E. PFENDER E. R. G. ECKERT

of nitrogen and air and with the assumption of equilibrium dissociation and ionization. They found that they could represent their results for nitrogen with an accuracy of &,5% by the following equation :

where Nu indicates the Nusselt number, Re the Reynolds number, Pr the Prandtl number, p the density, p the viscosity, V , the upstream velocity (flight velocity), p the upstream pressure, andp, = 1 atm. The index w refers to conditions at the wall surface and e to conditions outside the boundary layer. T h e dimensionless parameters Nu, Re, and Pr are based on properties at the wall surface. For a velocity V , below 11,300 m/sec, the last term in the above equation has to be replaced by one. Thermodynamic and transport properties published by Yos were used in these calculations. Corresponding calculations based on properties by Hansen resulted in heat transfer parameters for air which are again represented with good accuracy by the equation

The dependence on flight velocity V , was found to exist only for values V , > 9900 m/sec. Correspondingly, the last term in the above equation has again to be replaced by one for velocities below this value. The heat transfer parameters represented by these equations are also presented in Fig. 28. Experiments to measure heat transfer in the stagnation region of a blunt object under conditions simulating reentry were performed, mainly in shock tubes, by a number of investigators. Figure 29 summarizes earlier results and compares them with the results of analyses. The experimental difficulties made a certain amount of scatter unavoidable. In general, it can be stated, however, that agreement exists between the results of analyses and of experiments. Some disagreement among the analytical results obtained by various investigators is mainly due to the uncertainty in the transport properties. An analysis by Scala (41)was not included in Fig. 28 or 29 because it appears today that some of the transport properties used were wrong. This analysis results in heat transfer parameters up to twice as large as the ones shown in Fig. 28. They were originally supported by experimental results published by Warren and associates. Later refined measurements by Gruszcynski and Warren (421, however, resulted in values which agree with those in Fig. 29. Other recent measurements are also in agreement with the values in this figure. Park (43)calculated heat transfer of an ionized argon boundary layer for axisymmetric stagnation flow and laminar flat plate flow. He considered

PLASMA HEATTRANSFER

28 1

equilibrium as well as frozen conditions, the latter one for a catalytic wall. The energy equations (70a) and (71a), respectively, were used. The heat transfer parameter Nu/Re'i2 was again defined by Eq. (72). This parameter is presented as a function of the equivalent flight velocity V , [defined by V , = (2ie0)'/2]in Fig. 30 for the rotationally symmetric stagnation flow. Inserted in Fig. 30 are also the results of experiments which Park performed in a plasma jet wind tunnel. T h e experimental results indicate that the boundary layer was in a frozen condition. Inserted, in addition, as dashed lines, are the results of the analysis by Fay and Kemp (39)and as shaded area experimental results by Rose and Stankevics (44).Both of these results apply 1.0 0.9 0.8-

0.7 0s 0.5 -

-

I

-

-

-

-

0

ammo

to air. T h e figure shows that Nusselt numbers are not too different for argon and air. Ionization does not change the Nusselt number very much and the Nusselt number for a frozen condition is in all gases which were investigated larger than for an equilibrium condition, with the difference increasing with increasing flight velocity. T h e flight velocity as it has been used in the preceding figures is defined of the approaching gas stream, according to the through the total enthalpy ien equation i,O =

v:/2

and, is therefore, actually a statement of the gas enthalpy in the free stream. T h e enthalpy of the gas corresponding to the wall surface temperature is in all cases negligibly small compared to the total gas enthalpy so that the

E. R. G. ECKERT AND E. PFENDER

282

abscissa in these figures presents with good approximation also the difference

i, - i, which is used in Eq. (52)as the driving potential. T h e abscissa should be interpreted in this way if the figures are used to determine Nusselt numbers for stationary applications, for instance, in arc technology. Figure 31 presents as an example the frozen as well as the equilibrium Prandtl number of argon at a pressure of 0.1 atm. Both of these parameters have been used in the integration of the energy equations for frozen and equilibrium flow, respectively.

I

A

0.8

A

A

0

&

I 0

0.6

5P

---

/FROZEN

0'4

-

0.2

0.I

8

10

12

14

F L I G H T VELOCITY,V,

16

(m/sec)

18x lo3

FIG.30. Calculated and measured heat transfer parameter for rotationally symmetric stagnation point flow of argon [after Park (43)].

IV. Heat Transfer in the Presence of an Electric Current

A. ELECTRICALLY INSULATING SURFACE An ionized gas is electrically conducting and permits, therefore, a current to flow. Heat transfer to the wall adjacent to such a gas depends strongly on whether or not the wall is electrically insulating or carrying an electric current. On an insulating surface, heat transfer is mainly influenced by the electric current in the plasma owing to the fact that ohmic heat is generated and that a corresponding heat source termj2/a or o E has to be added to the energy equations (70) and (71). The existence of local kinetic equilibrium between the energy of the electrons and the heavy particles is generally assumed even though there is some question as to how well this condition

PLASMA HEATTRANSFER

283

is approximated in the regions of lower temperature near the wall. Heat fluxes can become quite large and in this way pose serious design problems. Such a situation will be discussed for the example of a constricted arc.

The Constricted Arc T h e constricted arc has become a very useful tool in research especially for the determination of properties of plasmas and it is also used as a means of generating high-temperature ionized gases for hypersonic wind tunnels

0.I

4poo

woo

~000

lop00

TEMPERATURE (k)

12pOo

FIG.31. Frozen and equilibrium Prandtl number for argon at 0.1 atm pressure [after Penski (44a)l.

and for space propulsion devices. The identifying characteristic of this arc is the achievement of high temperatures by constricting the arc in a tube, thereby increasing the electric energy dissipation per unit volume. T h e temperature achieved in such an arc is limited by the heat transfer to the constrictor walls. Usually water cooling is used for this purpose. Recently, however, attention has also been directed towards the application of transpiration cooling. Figure 32 sketches such arcs with a water-cooled and transpiration-cooled constrictor. The constrictor consists in the first case of a stack of water-cooled segments insulated electrically from each other. In the second case the constrictor is a porous tube through which the working gas is injected. A number of analyses have been reported (45-50) which investigate the performance characteristics of the water-cooled constricted

284

E. R. G. ECKERTAND E. PFENDER

arc. A few analyses (32,51,52)are also available for the transpiration-cooled arc. With sufficient constrictor length, the temperature field becomes developed in the downstream section of the constrictor in the sense that the temperature varies only radially maintaining its values in the axial direction. This is the case in the water-cooled constrictor when all of the electric energy dissipated within this section is transferred radially to the constrictor wall. In the transpiration-cooled constrictor, the developed stage is achieved ANODE (+)

WATER

WATER

TRANSPIRING GAS

FIG.32. Constricted arc with water cooling and transpiration cooling.

when all of the electric energy is used to heat the gas injected through the porous wall to the temperature of the axially moving gas stream. The velocity field also becomes developed in the sense that the velocity profiles at various cross sections are similar to each other. Most of the analyses are restricted to the developed condition. This will also be the case in the following discussion which is essentially based on the work of Anderson and Eckert (32). T h e existence of complete local thermodynamic equilibrium, of rotationally symmetric laminar flow, and of negligible energy dissipation by internal friction is also postulated. The gas properties entering the analysis are assumed to be functions of temperature only, which implies that the pressure differences in the flow field are moderate. ‘The following equations describe, under these assumptions, conservation of mass, of momentum in axial direction, and of energy for both constricted arc types :

PLASMA HEATTRANSFER

a@,

l a

-r-ar (YPc,)+pz

ac, aZ

ac ap aT az

pvz-+/Jur-2+-=

=o

(75)

1a ac, -- Yp-

+aE2-P,.=pz',-

285

al. ai

ar

1 (77)

T h e coordinate system is indicated in Fig. 32 and the symbols are explained in the List of Nomenclature. The conservation equation for momentum in the radial direction was also included in the analysis of Anderson and Eckert (32). It was, however, found that the radial pressure variation determined by this equation is small in the cases which will be discussed later on. 'The term P, on the left-hand side of Eq. (77) describes the energy loss of the gas by radiation, assuming the gas to be optically thin. This term also was comparatively small for the cases which were investigated. For the water-cooled constrictor, the system of equations simplifies because in the developed regime the radial velocities z-,. are equal to zero. This has the important consequence that the convective term on the righthand side of Eq. (77) is zero, which means that the energy equation is completely decoupled from Eqs. (75) and (76). T h e resulting equation becomes the well-known Elenbaas-Heller equation when the radiative term is neglected. T h e temperature field can therefore be calculated without knowledge of the velocity field. This calculation is usually performed by introduction of the heat conduction potential '2 = J kdT as the independent variable instead of the temperature. I t has been found that the inner wall surface temperature T,, and the magnitude of the heat flux 9,' per unit area which can be absorbed by the water-cooled constrictor wall are the parameters which limit the performance of this arc type. T h e proper boundary conditions are therefore

Figure 33 presents some results of the described analysis considering hydrogen at 1 atm pressure as the working gas. T h e properties of hydrogen have been obtained from the literature. Two heat transfer processes determine the maximum allowable heat flux, the heat conduction through the constrictor wall and the heat transfer from the wall to the cooling water. T h e temperature drop in the constrictor wall is appreciable. For a heat flux of 10 kW/cm' ( 3 x 10' Btu/ft? hr), for example, the temperature drop in a l-mm thick wall of pure copper or silver is 250°C. For this reason, only metals of very high conductivity can be used for the constrictor. Local boiling occurs on the water side of the constrictor wall and the permissible heat flux in this process

E. R. G. ECKERT AND E. PFENDER

286

is determined by the burn-out heat flux. A heat flux density of order 10 kW/cm2 is the maximum so far achieved. A limit for the wall temperature is, on the other hand, prescribed by the melting point of the material of which the constrictor wall is formed. T h e melting point of copper is, for instance, 1083°C. From Fig. 33 it can be seen that for these values a maximum gas temperature at the axis of the constrictor tube of approximately 40,000"K can be reached.

:----r--

36

I

35

1

!

[r,.0.25cm/

i

1

v)

z 33 w

t-

z

n J

w LL

32

;;

31

45poO'K

---

30

~

,

I

I

FIG.3 3 . Performance data of water-cooled constricted arc in hydrogen, 1 at m [after Anderson and Eckert (.??)I.

‘The transpiration-cooled constrictor operates with a finite radial flow velocity -cr leaving the constrictor wall. Therefore the right-hand term in Eq. (77) has to be maintained and the three conservation equations (75)-(77) are interrelated. Figure 34 presents numerical solutions of these equations for the boundary condition of a locally uniform velocity -cr at the constrictor surface and for the situation that all of the heat flowing toward the constrictor wall is returned to the gas in the interior of the constrictor tube by the gas injected through the porous wall. Again, hydrogen at 1 atm pressure is the working gas. A limitation to the performance of the transpiration-cooled

PLASMA HEATTRANSFER

287

constricted arc is set by the permissible inner wall surface temperature. Use of a ceramic material will allow fairly high values for this temperature. Another restriction is given by the rate with which gas is injected through the porous constrictor wall. Figure 35 shows how the inner constrictor surface 56

I

1

I

I

54

-

I

I

52

1

E

V

F

k

40 z W

V,

c

i 44

~

1

42

I

I

40 0

500

I

000

I CURRENT (AMP)

I

woo

2300

FIG.34. Performance data of transpiration-cooled constricted arc in hydrogen, 1 atni [after Anderson and Eckert (32)l.

temperature and the electric power input per unit length determine the rate of fluid injection per unit length. The average axial mass velocity Fz increases linearly with the coordinate z.This determines the tube length at which sonic velocities are reached. On the other hand, a certain tube length is required to achieve a developed temperature field and the balance of both factors determines the maximum allowable injection rate. Other factors like transition to turbulence and magnetic pressure do not appear to impose more restricting limitations.

E. R. G. ECKERT AND E. PFENDER

288

Figure 36 compares normalized enthalpy and mass velocity profiles for the two types of arcs. T h e enthalpy profile of the transpiration-cooled arc is characterized by a small value of the gradient at the constrictor wall. This means that the energy flux away from the tube axis is, for the most part, arrested by the radial mass flow towards the tube axis. T h e heat flux into the constrictor wall is therefore much smaller than for the water-cooled arc. T h e mass velocity profiles, on the other hand, indicate that the majority of the gas moves in the transpiration-cooled constrictor through a ring-shaped cross section adjacent to the tube wall. T h e mass averaged enthalpy flux, 0.12.ld'

I

1 7

0.10

8 0.08

.-sI

'€ 0.06

!i

FIG.35. Performance data of transpiration-cooled constricted arc in hydrogen, 1 atm [after Anderson and Eckert ( 3 2 ) ] .

therefore, is, at the exit of the transpiration-cooled arc, not larger than for the water-cooled arc, even when the axial temperature is much higher. T h e advantage of the first system lies in the high temperature in the region close to the tube axis. The results in Figs. 33-36 apply to a constrictor diameter of 0.5 cm. They hold for other diameters as well, as long as the radiative heat transfer is negligible, if the electric field strength E is scaled proportionally to the

PLASMA HEATTRANSFER

289

3.8X I 0 0 3.6 3.4 3.2

3.0 2.6 2.6 2.4

.

2 2.2

3

n

t

-1

2.0

4 3 x a

a

2I 1.6 t5 1.6 14 12 I .o

08 0.6 0.4 0.2 0

0

0.2

0.4

06

NORMALIZED RADIUS

Od

I.0

FIG.36. Comparison of enthalpy profiles and axial mass velocity protiles for watercooled and transpiration-cooled arcs in hydrogen, 1 atm [after Anderson and Eckert (.??)].

constrictor radius r7L,and the mass injection rate m per unit length inversely proportional to r7c.This scaling law follows readily from the conservation equations.

B. ELECTRICALLY CONDUCTING SURFACE (ELECTRODE)

1. Basic Considerations T h e heat transfer process becomes very involved to an electrically conducting surface, especially when an appreciable gradient of the electric

290

E. R. G. ECKERTAND E. PFENDER

potential and a flow of electrons exists normal to the surface as in the case of electrodes. A fall space with a thickness of the order of a mean molecular path length exists close to the surface and the ionized gas which is neutral in the bulk flow loses its neutrality within the sheath. T h e corresponding space charges create very strong gradients of the electric potential in this region, and the energy distribution among the various particles is far from equilibrium. However, even outside of the sheath within the temperature boundary layer, thermal equilibrium does not exist. The gradient of the electric potential is large enough to add, between collisions, considerable energy to the electrons and as a consequence the average energy of the electrons is higher than the energy of the heavy particles. T h e energy distribution among the electrons and among the heavy particles is still fairly close to a Maxwell-Boltzmann distribution ; as a consequence, one can define an electron kinetic temperature T, and a kinetic heavy particle temperature T . A model which describes conditions in the neighborhood of an electrically conducting surface with current flow into or out of the surface consists then of a binary gas mixture, namely, an electron gas and a heavy particle gas. One has then two sets of conservation equations with coefficients which describe the mutual interactions of the two gases. A simpler model was developed by Kerrebrock (53).H e starts out with an energy balance determining in an approximate way the electron temperature :

T h e definition of the symbols in this equation is contained in the List of Nomenclature. a indicates a dimensionless constant, the value of which is estimated by Kerrebrock to be of order 2. T h e drift velocity u,of the electrons is connected with the current density j by the equation j = en,u, neglecting the contribution of the ions to the current density. T h e collision frequency v, between electrons and heavy particles is given by the relation v, = e2n,/m,a. Introducing these expressions into Eq. (78) and solving for the electron temperature results in the equation

in which m, denotes the mass of the heavy particles, kB Boltzmann’s constant, e the electron charge, and n, the electron number density. Equation (79) which describes the difference between electron and gas temperature as a function of the current density, is identical with Eq. (30) of Section 11. Kerrebrock also assumes that the electron number density n, is the same as for a plasma in thermodynamic equilibrium at the temperature T, and is therefore given by Saha’s equation [see Eq. (27)]. T o describe with these

PLASMA HEATTRANSFER

29 1

equations the energy exchange within a boundary layer, the continuity equation (65) and the momentum equation (67) are maintained. T h e energy equation is

T h e two new terms in this equation are the last ones on the right-hand side.

I

2

3

RADIUS ( m m )

4

5

FIG.37. Profiles of heavy-particle temperature (7’)and electron temperature ( T , )for constricted arc in helium [after Pytte and \Villiams (541.

j 2 / u denotes Ohm’s energy dissipation and the last term the convective transport of the enthalpy of the electron gas. With Eqs. (79) and (27), the electron temperature can be determined as a function of the current density j and the heat transfer through the boundary layer can be calculated as a function of j . Kerrebrock performed such an analysis for a boundary layer on a flat plate by the integral technique. A similar calculation with a somewhat extended Eq. (78) was performed by Pytte and Williams (54) for a water-cooled constricted helium arc with a I-cm constrictor diameter, assuming that the constrictor wall is slightly biased and that therefore a flux

292

E. R. G. ECKERT AND E. PFENDER

of electrons enters the constrictor surface. From the results of this onedimensional calculation, Fig. 37 has been selected in which the variation of the gas temperature T and the electron temperature T, is plotted over the radial distance r from the arc axis. T h e parameter on the curve is the current density j , at the constrictor wall. I t is interesting to observe how, with increasing current density, the electron temperature exceeds the gas temperature more and more over an appreciable part of the tube cross section. T h e discussion up to now was concerned with the energy flux within the ionized gas close to the electrically conducting surface. An analysis of heat transfer to the surface itself has, in addition, to consider the energy conditions within the sheath and the energy which is freed when the electrons enter the solid surface, an energy which is given by the “work function” and which is analogous to a condensation energy. This will be discussed in connection with the experimental investigations.

2. Experimental Studies ‘Two problems arise in the development of engineering devices utilizing electrically generated plasmas (electric arcs). Firstly, the total heat flux to a water-cooled current-carrying surface represents in many applications the predominant energy loss which diminishes the efficiency of the particular plasma device. Therefore, efforts have been made to reduce such losses. Transpiration cooling has been considered because part of the heat flux to a surface is thus recovered by the gas transpiring through this surface. The second problem is connected with the specific heat flux to a currentcarrying surface which may become so high that local melting occurs, in spite of applying the most efficient water-cooling system which is able to remove about 19 kW/cm2.This situation has been experienced in high-pressure arcs even at moderate currents using working fluids with high heat conductivity (for example, hydrogen). The high pressure as well as the high heat conductivity of the working fluid lead to a constricted anode attachment resulting frequently in anode failure. The properties of the anode fall space and of the boundary layer apparently have a decisive influence on the current transition and therefore also on the local heat transfer to the anode. A cold surface favors a constricted attachment in order to keep the electron temperature and with it the electric conductivity sufficiently high in spite of the intense cooling of the boundary layer. High-temperature surfaces may permit a more diffuse current transition. In order to prevent too high specific heat loads at the surface, the anode arc terminus may be moved with high velocities over this surface distributing the heat flux to a larger area. Heated anodes offer another possibility to keep specific heat fluxes within reasonable limits. In the following sections anode heat transfer studies will be reviewed

PLASMA HEATTRANSFER

293

which have been carried out on various arc geometries and under various test conditions. For many of these investigations, energy transfer models have been proposed in order to determine the contribution of the different energy transfer modes to the total anode heat flux. Detailed anode heat transfer studies are available for arcs operated in an argon atmosphere and for water-cooled copper anodes which permit rather accurate overall and local heat transfer measurements. Therefore, this review will only deal with results obtained from such anodes. a. Free-Burning Arcs with Plane Anodes. The characteristic arc geometry of the free-burning, high-intensity arc shows a well-known bell-shaped stationary arc column (3, 55). T h e reason for the steadiness of this arc column is the strong cathode jet generated by magnetohydrodynamic forces in the cathode region (56). This region acts as an electromagnetic pump

q,e-o

qro

h,(Ie--lw)

Ma

q,

q,-o

FIG.38. Energy balance of anode surface element.

drawing gas from the surrounding and ejecting it towards the anode in the form of a jet. T h e cathode jet has an important influence on the convective part of the anode heat flux, especially at small arc lengths and high currents. T h e energy transfer to the anode involves a variety of transfer mechanisms which can be described by an energy transfer model (57-59) indicated in the scheme of Fig. 38. T h e energy transferred to the anode consists of: ( a ) Thermal and kinetic energy of the electrons comprising the arc current and penetrating the anode surface,

+

q, = ( j/e) ( i k B T, e U0) (81) q, is the energy flux per unit area connected with the current flow into the surface, j the current density, e the electronic charge, kB the Boltzmann constant, 7,the electron temperature at the border of the arc column, and U , the anode fall voltage. ( p ) Heat flux by condensation of the electrons which is proportional to the anode work function @*, w, .Pa (82) ( y ) Convective heat transfer from the hot plasma through the boundary layer, = U i , 4) (82a) L -

(IL"",

~

E. R. G. ECKERT AND E. PFENDER

294

(6) Radiative heat transfer from the arc plasma qra. Electrons suffer elastic collisions with heavy particles in the boundary layer and in the anode fall space, a process which will reduce that part of the electron current stagnation enthalpy which is transferred to the anode. However, the number of elastic collisions in the thin fall region is not large enough to transfer an appreciable amount of energy to the heavy particles since the energy transfer per collision is very small. The energy loss of the electrons to the heavy particles will therefore be essentially only the amount required for ion generation in this region. But the ion current density, in turn, is only on the order of one per cent of the total current density, so that the energy loss of the electrons traveling through the anode fall space may be neglected.

-

I

I

1

I

1

3

4

5

6

7

5 d

0

I

2

RADIUS ( m m )

FIG.39. Radial distribution of energy transfer at anode surface (59).

T h e energy carried away from the anode surface consists of: ( E ) T h e heat conducted away from the anode surface by the cooling water denoted by q. (Z;) T h e heat radiated away from the anode surface to the environment denoted by qye. Under steady state conditions the energy balance for a surface element of the anode can be written as

+ qre + q a b l =

qj

+ qo,,+ + Qva

qcon,.

(83)

In normal operation, the anode surface is kept well below the melting point ; therefore, qreand qablare negligible. All the other quantities besides U , and @, can be experimentally determined. T h e results of such measurements on a water-cooled anode in a current range 50-150 A (57-59) indicate that the heat flux due to the electron flow into the anode is the predominant heat transfer mechanism. A typical diagram of such measurements is shown in

PLASMA HEATTRANSFER

295

Fig. 39. The convective heat transfer was calculated with Eq. (52) assuming that this equation holds for a current-carrying surface. Measurements by Nestor (60) for currents up to 300 A show essentially the same results. Overall energy balances demonstrate that up to 85 yoof the arc power input is transferred to the anode (with electrode gaps up to 20 mm) and that the percentage of energy transfer to the anode is almost independent of the current. Another interesting finding of these studies is the behavior of the current density which maintains its maximum value at the anode center regardless of the total current (100 A < I < 400 A). T h e arc column and especially the anode attachment region spread correspondingly out with increasing current, and it appears that the arc adjusts its attachment area to that current density which provides a sufficiently high electron temperature and electric conductivity. T h e corresponding anode heat fluxes by Schoeck (57, 59) and Schoeck and Eckert (58) are in the range of 4 to 6 kW/cm2. T h e convective heat flux density increases with increasing current corresponding to a nearly linear increase of the velocity with which the cathode jet approaches the anode reaching a value of 180 mjsec at 150 A and 6 mm cathode to anode distance. I n a more recent thesis by Eberhart (64, the current range has been extended to 1100 A. T h e percentage of energy transferred to the anode is, up to 1100 A, again independent of the current. However, the electron heating of the anode which predominates at lower currents is surpassed by gaseous convection at very high currents depending upon the electrode gap. This fact is at least in part a consequence of another finding which indicates that the local current density at the stagnation point even decreases with increasing current. b. Cathode Axis Parallel to a Plane Anode. When the cathode is oriented parallel to the anode surface, one obtains quite a different shape of the plasma column as shown in Fig. 40. The cathode jet now impinges against an anode jet which is generated by a current density gradient toward, and in the immediate vicinity of, the anode. T h e reason why the arc column tends to attach to the anode in a rather contracted manner is probably that a lowtemperature gas layer is present on the anode surface. This constriction is usually not as strong as in the cathode region (62). The heat flux per unit area is expected to have a distribution curve similar to the one shown in Fig. 39, with, however, a smaller attachment area and a correspondingly larger maximum. From the photos it appears that it may be 100 times as high. 'I'he local heat flux at the anode may, under such conditions, surpass the highest permissible value and it is, therefore, fortunate that a superimposed axial gas flow frequently causes the arc to fluctuate, leading to a moving anode arc terminus. This spreads the anode heat flux over a wider surface area a n d reduces the maximum temperature of the anode appreciably.

E. R. G. ECKERTAND E. PFENDER

296

An estimate of the reduction in the maximum temperature which is obtained in this way can be found from the following model. Consider a semi-infinite solid with a point heat source moving with constant velocity ZI along a straight path on its surface. The temperature field created in this way

PLASMA COLUMN CATHODE JET

FIG.40. Scheme and photograph of impinging jets in argon (64).

in the solid material is steady when viewed by an observer who moves along with the heat source. Surfaces of constant temperature have a shape similar to half an egg shell and are described by the following equation (63):

e=-8=

[ p>(pP2 -+ 1)]

1 exp -27rPe”

( T - T,)k2 PCPVQ

VY

’

Per=-,

ff

Pe,=-

vx ff

in which 6 denotes a temperature parameter and Pe a dimensionless parameter which is frequently used in heat transfer analysis and is called the Peclet number. T is the local temperature in the solid at the radial distance r from the heat source, T , the temperature at r = 00, k the thermal conductivity, p the density, and cp the specific heat of the solid. q denotes the heat flux released by the point source per unit time, x the projection of r onto a straight line through the heat source in the direction of its movement, and CL

PLASMA HEATTRANSFER

297

the thermal diffusivity of the material. The curves in Fig. 41 present the intersections of the isothermal surfaces with the plane surface of the solid. Equation (84) can also be interpreted as describing the temperature of the material when, instead of the point source, a situation is considered in which ,

aw

e =

(T-Te)kz Pcpvq

\

,

pe

r

vx ;x, pe, = a

FIG.41. Temperature field of a moving point heat source on the surface of a semi-infinite

body.

the heat flux q is distributed over any of the isothermal surfaces. The equation describing the temperature field 8, of a stationary heat source can be brought into the following form :

8, = 1/2~rPe, (85) which contains the same parameters as Eq. (84). T o compare the temperature increase by a heat source of radius r moving with the velocity v with the temperature increase created by a stationary heat source of the same size, one can now calculate the parameter Pe, and determine with it 8, from Eq. (85) and 8 from Fig. 41, respectively. In this way the values 8/8, in Table I TABLE I Per

sje,

1.1 0.69

2.25

0.56

3.9 0.49

4.82 0.45

6.7 0.42

10

0.38

were obtained. It can be seen that the maximum temperature increase in the solid material is reduced to about one-third of the value created by a stationary heat source of equal strength when the Peclet number Pe, has a value of 10. The fluctuating arc repeats its movement down the anode almost periodically and the maximum temperature in the solid material will be somewhat

298

E. R. G. ECKERT AND E. PFENDER

larger than the value obtained by the estimate described above. The model suggested here can certainly be refined, taking into account the fact that the anode usually is a thin wall cooled on the back side by water and considering a heat flux distribution as presented in Fig. 41. I t is, however, felt that it 0.6

0.4

?* 0“

0.2

0

0

2.5

!j.o

7.5

10

0

2.5

X (cm)

(01 VELOCITY DEPENDENCE

5A X (cm)

7.5

10

( b ) PRESSURE DEPENDENCE

I = K)OAmp,S -7mm,V= 5Om/sec

I =IOOAmp,S =7mm,P*IOOmmHq

o

0.4

S=6mm

?*

0”

0.2

0

0

2.5

5.0

7.5

10

X (cm)

(c) CURRENT DEPENDENCE S= 7 mm, P*38OmmHg, V= 100m/rrc

0

2.5

!LO 7.5 X tern)

10

(d) ELECTRODE GAP DEPENDENCE

I =100omp,P=380mmHg,V=100m/rrc

FIG.42. “Local” heat flux distribution at plane segmented anode (argon).

already correctly describes, in its present form, the order of magnitude. No measurements of the distribution curve of the heat flux or of the temperature distribution in the anode have been performed to date with which the above analysis can be compared. Some indication of the relief provided by the arc movement is obtained by the results of some tests in which the anode was composed of segments with a width of 1.25 cm measured in the flow direction. The results are presented in Fig. 42 in which Qs presents the flux into an individual segment and Q A the total anode heat flux. Figure 43 shows the

PLASMA HEATTRANSFER

,

n <-

299

300

E. R. G. ECKERT AND E. PFENDER

time dependence of the current flux to the segments with different parameter settings. The six traces on the oscillograms, starting at the top, show the individual currents through four consecutive segments and the total current as a dc signal, and the overall voltage as an ac signal. In the third oscillogram of the second row a complete cycle of the so-called “fluctuating arc” is clearly demonstrated. The first cycle shows a restriking of the arc at

GAS FLOW

-

5 2

4

FIG.44. Frames from high-speed movies of fluctuating arc mode in argon ( 6 4 .

the first segment, transition to segment two, three, and four, and decay of the arc at segment four accompanied by restriking again at the first segment. ‘The following cycles show essentially the same behavior. In a sequence of frames taken from a high-speed movie (Fig. 44) the actual anode attachment and its movement is shown in a complete cycle (frames 2-12). Analysis as well as experiments indicate that the maximum temperature in the anode can be considerably reduced by the movement of the arc, and it is therefore of interest to know under which conditions the arc fluctuates and under which conditions a stationary mode prevails. Figure 45 presents the results of

PLASMA HEATTRANSFER

301

pertinent experiments. The curves in this figure separate the domains of the two modes of arc operation. T h e figure indicates that the electric current shifts the curve separating the two modes somewhat which leads to the conclusion that the flow and the energy transfer process have a decisive influence on the mode of operation. It is reasonable to assume that a temperature field similar to the one in Fig. 41, however, for a line heat source, describes the conditions in the neighborhood of an arc in that part of the column which is normal to the flow direction. It will, of course, be modified by the large density variations. T h e previous analysis showed that a Peclet number Per = VY/E = pcpvr/k which indicates the ratio of the convective to the conductive transfer process determines the temperature field. A check on the curves in Fig. 45 shows that they are quite well represented by hyperbolas

10

20

30 40 50 VELOCITY ( m I SIC 1

60

70

80

FIG.45. Steady and fluctuating mode of an arc in argon atmosphere ( 6 4 .

pv = const which for approximately constant temperature leads topv = const. This suggests the conclusion that a characteristic Peclet number may be the main parameter determining the arc mode. On the other hand, viscous effects may also be of influence since the attachment of the arc occurs through a boundary layer which suggests an influence of the Reynolds number. Further experiments with different gases may shed some light on these questions. T h e fluctuating mode has also been achieved in engineering arc plasma devices by magnetically rotating the anode arc terminus or by using segmented anodes where each segment has been connected over a small resistor to the power source. This forces the anode arc terminus either to rotate or to split into a number of anode arc attachment spots corresponding to the number of segments. A comparison of the overall energy balance results obtained from this arc with those obtained from the classical free-burning arc shows a drastic

E. R. G. ECKERT AND E. PFENDER

302

reduction in the anode losses. I n the latter case, between 75 and 8 5 O , , of the electrical input energy was consumed by the water-cooled anode, while in the present case only 2O-6Ooj,of the input energy is transferred to the anode. There are essentially two reasons for the drastic change in the anode losses, namely, the change of the electrode configuration and the adding of a gas flow. In the present case, the cathode jet is oriented parallel to the anode surface which results in a longer arc column and, therefore, at constant current, also in a higher power input reducing the fraction of the total energy which is transferred to the anode. At the same time the convective part of the

- ANODE

4-

-COOLANT CHPiNNELS

D.C. SOURCE

=

CONDUCTING MATERIAL ELECTRICAL

INSULATOR

FIG.46. Scheme of water-cooled cascaded arc.

anode energy transfer caused by the cathode jet is almost eliminated. Experiments demonstrated, however, that the main reduction occurs when a gas flow parallel to the anode is added, causing a large amount of the heat generated in the arc to be carried away by bulk convection and reducing the anode energy flux considerably in this way. c. Cylinder Geometry with Annular Anode. The wall-stabilized cascaded arc arrangement ( 9 ) exhibits well-defined arc parameters and represents a close approach to the gas flow, current transition, and heat transfer situation found in actual arc plasma generators which are used in numerous applications. On the other hand, it offers less possibilities for detailed local studies than the geometries discussed before and experiments are mainly confined to overall mean values. A schematic diagram showing the type of apparatus

PLASMA HEATTRANSFER

303

used in these experiments is presented in Fig. 46. T h e arc is operated with a relatively high axial flow from the cathode end of the constrictor tube. T h e flow and the current transitions at the anode are similar to that described in the preceding section. However, the arc column is more constricted in the prevailing case which leads to higher temperatures in the plasma column. Corresponding to the small length of the anode, the longitudinal movement of the anode arc terminus is limited. Nevertheless, the same characteristic behavior of the arc voltage which is symptomatic for the fluctuating mode of the arc and which has been clearly demonstrated with plane anodes (64)may also be found in cylindrical geometries (65). Therefore, the basic heat transfer situation at the anode will remain the same. I n order to distinguish between the various energy transfer modes, especially between the contribution of the current flux and the other heat transfer mechanisms, an anode energy transfer model is established but in this case for the overall energy fluxes. The single energy transfer modes are the same as discussed before. Combining the convective and radiative energy transfer into one term, the over-all energy balance may be written as

L? +

Qre

+ Qabl=

QI

+ QQ~" + Qrai-c

where the designation of the terms is the same as in Eq. (83) with exception of the last term Qra+(which is

T h e terms Qabland Qremay be neglected because the anode surface temperature is kept well below the melting point. Q is determined calorimetriccan be calculated from the anode work function. The values for ally and Q@,< the work function of metals found in tables refer usually to vacuum conditions. The surface conditions which are doubtless quite different at the anode of an arc discharge may cause a drastic change of the work function. Therefore QG,,may be appreciably in error by using tabulated values for the work function. is determined in the following way. Over the length of the T h e term QrnfC constrictor tube where the flow is fully developed, the heat transfer per unit area should be constant. Therefore, Qro+rcan be determined from the heat flux to a segment situated in the thermally fully developed region because the only heat flux to such a constrictor segment is the convective and radiative transfer from the plasma. Two typical diagrams for two different flow rates nic through the constrictor tube are shown in Fig. 47. From this figure, the energy acquired by the electrons in the anode fall region is seen to be the most important single factor in determining heat transfer to the anode.

E. R. G. ECKERT AND E. PFENDER

304

Convection and radiation are more than an order of magnitude smaller than the total contribution of the current to the anode energy flux. More details about the experimental procedure may be found in the work of Pfender et al. (66).

i

20

40

I

IO,+C/O’-\,

+ =

20

t

40

80 100 120 CURRENT (amps 1

60

140

I

I

I

I

I

60

80

100

120

140

CURRENT ( amps )

FIG.47. Contribution of different heat transfer mechanisms to the total anode heat flux (66).

V. Heat Transfer in the Presence of a Magnetic Field

The presence of a magnetic field further influences heat transfer. This situation occurs, for instance, in magnetohydrodynamic power generation and the following discussion will be made with this application in mind. Gases involved are then at a temperature of order 2000°K and are seeded to obtain sufficient electric conductivity. T h e geometry involved consists essentially of a channel with rectangular cross section as indicated in Fig. 48. Two opposite walls of the channel serving as electrodes are electrically conducting and a current of density j flows from one wall to the other under the influence of an electric field with the intensity E. The other two walls are constructed of an electrically insulating material. A magnetic field of intensity B is applied in a direction normal to the electric field. The cross

PLASMA HEATTRANSFER

305

section of the channel in flow direction (x) is either constant or varying. T h e flow can be considered as consisting of an internal main stream and of boundary layers along the channel walls. A. ELECTRODE HEATTRANSFER Kerrebrock (67) has analyzed heat transfer in the laminar boundary layers along the electrodes for the situation that in the main stream (indicated by subscript a) the temperature T,, the electric field strength E, and the electric conductivity urnare constant. T h e velocity urnin the main stream is /-INSULATOR

-ELECTRODE

MHD DUC

EL

FIG.48. Channel of magnetohydrodynamic power generator.

assumed to vary according to the relation u, = Cx”’. This leads, for values of m between 5 and 4, to a channel with a decreasing area in flow direction, and, for m larger than $, to a channel with increasing area. For m = i, the cross section is constant in flow direction. Local thermodynamic equilibrium was assumed in the main stream as well as in the boundary layer. T h e boundary layer equations have to be changed to account for the action of the magnetic field. The continuity equation (65) remains unchanged (with n = 0). A term j B has to be added to the momentum equation (67). T h e accelerating force j B has the same value in the main stream. As a consequence, it influences the velocity field in the boundary layer in the same way as a pressure gradient. The energy equation (80) applies also to the present situation.

E. R. G. ECKERT AND E. PFENDER

3 06

T h e system of these boundary layer equations was transformed to the new coordinates

T h e subscript 0 indicates reference values measured in the main stream at an arbitrary cross section at a distance x,, from the channel entrance. Kerrebrock concluded that for finite Mach numbers no conditions exist which lead to self-similar boundary layers. T h e terms containing functions or derivatives of x,however, vary only slowly and one can therefore

0.5

0.6

0.7 0.8 0.9 NORMALIZED TEMPERATURE T/T,

1.0

FIG.49. Temperature profiles on electrode walls of magnetohydrodynamic channel [after Kerrebrock (67)].

get an approximate solution by neglecting terms containing such functions. This procedure has been used occasionally before and is referred to as “ local similarity.’’ The equations are then total differential equations in the independent variable 7 and were solved on a digital computer. As an example of the results, temperature profiles are presented in Fig. 49 for the following condition :

Pr = 1,

Ma,

= 1,

TJT,

= 0.5

It can be observed that the thermal boundary layer thickness gets smaller as the parameter m increases and that, in addition, a peak appears in the temperature profiles. An increasing value of the parameter m means an increasing electric current density according to the equation

PLASMA HEATTRANSFER

3 07

. 5m-1umE, I = - - m yMam2 ___ T h e ratio y of specific heat at constant pressure and constant volume was selected as $. A dashed curve indicates the temperature profile for laminar boundary layer flow over a flat plate (at constant pressure). The difference between the profiles for the electromagnetically influenced boundary layers and the flat boundary layer is to a minor degree due to the pressure gradient connected with the first group of profiles ;to a major degree, however, due to ohmic heat dissipation. The velocity profile is influenced only to a small degree by the parameter m. Results for Mach numbers larger than one are found to differ very markedly from the ones in Fig. 49. This is mainly due to the additional effect of aerodynamic heating. T h e temperature gradient and the electric current density at the surface determines the heat flux into the wall surface according to the equation

A NusseIt number is defined with this heat flux by the following relation :

One finds then that, at the reference location xo, for m = $, and Ma, following relation is obtained :

=

1, the

T h e constant, in this equation, is more than 10 times larger than the constant 0.332 for a laminar boundary layer without magnetoelectric effects. I n reality, one has to expect an even larger heat transfer coefficient because the electron temperature in the boundary layer is larger than the gas temperature. Correspondingly, the temperature T, in Eq. (89) should be replaced by the electron temperature at the wall surface and the heat conduction will also be altered. In addition, the “heat of condensation” of the electrons determined by the work function of the wall material has to be included.

B. ELECTRICALLY INSULATING SURFACE Hale and Kerrebrock (68) have also investigated heat transfer to the insulating walls of a magnetohydrodynamic channel. The model on which the analysis is based is somewhat different from the one used in the preceding section. The electric field strength E is postulated constant in z direction. Both, however, are allowed to vary in x direction. T h e temperature T ,

308

E. R. G. ECKERT AND E. PFENDER

and the wall temperature T, are postulated constant. Included in the analysis is the Hall effect. The Hall parameter b , = u,B/n,,e is also assumed constant in the main stream. A Hall effect of finite magnitude generates an electric current density js in flow direction within the boundary layer. An electromagnetic loading factor K = E3,/umB is postulated constant throughout the field. The boundary layer equations now take on the following form. T h e continuity Eq. (65) remains unchanged (for rn = 0). Local chemical equilibrium was assumed so that Eq. (66) can be discarded. T h e momentum Eq. (67) has to be enlarged by a term j,B added to the right-hand side. The force term j s B creates within the boundary layer a flow in y direction. The analysis indicates, however, that the corresponding velocities are very small compared to the velocities in x direction. T h e direction normal to the wall surface is now, according to Fig. 48, the direction of the coordinate z . Correspondingly, the symboly in Eqs. (65) and (67) has to be changed to z and the velocity component u to w.T h e energy equation written for the total enthalpy reads

+jsEs +j , h',

(92) The Prandtl number Pr has been assumed constant. T h e current density j and the electric and magnetic fields are interconnected through the vector equation j=u[E+(VxB)-(l/n,e)(j xB)] (93) The scalar electric conductivity u was postulated to be either the equilibrium value at the gas temperature T or the equilibrium value at the electron temperature T , determined by Eqs. (79) and (27). The first case then applies to a situation where the electron temperature is equal to the gas temperature ( T ,= T ) ,the second one to a condition with elevated electron temperature ( T ,> T ) . T h e boundary layer equations are again transformed to a new coordinate system defined by the Eq. (88). Stipulation of a constant temperature T , and electromagnetic loading factor K in the main stream lead to a velocity u which varies according to the following relation in x direction : ~~

T h e subscript zero indicates again conditions at an arbitrary reference cross section. T h e last equation together with the condition (MaO/Ma,)3 < 1 leads to the relation

PLASMA HEATTRANSFER

309

This condition, neglecting slow variations in x direction of some parameters involved, leads to a system of total differential equations in 7, an approximation called “local similarity”. T h e resulting total differential equations were solved on an electronic computer for a gas with the following properties: p = pRT, Pr = 0.7, cp = const, p T. Some profiles obtained as a result of a numerical solution of these boundary layer equations are presented in Fig. 50 for a boundary layer with T, = T and in Fig. 5 1 for a boundary layer with T, > T. Characteristic is the fact that

-

“0

05

u /urn

10

bm= 0

K =2

u/u, and j / j

Y m

FIG.50. Profiles of velocity, temperature, electric conductivity, and electric current density on insulated walls of magnetohydrodynamic channel for thermodynamic equilibrium in the plasma (T, = T ) [after Hale and Kerrebrock ( 6 8 ) ] .

the temperature profiles peak within the boundary layer especially when the electron temperature is higher than the gas temperature. I n this case, however, the reason for this phenomenon is different from the one which caused the peaking in an electrode boundary layer. It is connected with a process occasionally referred to as short circuiting. T h e electric potential applied externally is in the main stream partially compensated by the counter electromotive force u,B. In the boundary layer, the velocity u is smaller and the compensating effect of the magnetic field is therefore reduced. This causes a larger current densityj, and, correspondingly, a larger ohmic energy dissipation. T h e effect of the Hall parameter was also investigated and was

E. R. G. ECKERT AND E. PFENDER

3 10

found to be comparatively small for values between 0 and 2. Mach numbers beyond a value one, however, have quite a strong effect on the velocity and the temperature profiles. The heat flux to the wall is given by the relation qu=--k(W/&z),. A Nusselt number Nu has been defined by the relation Nu = qwx/k,(T, - T,) which differs from the one in the preceding section by the use of the thermal conductivity at the wall temperature. This results in the relation

(96)

N u = CRei/2Ma"1'2

7 4~~

OO

20

~~

10

05

u /uw

05 T/Tw

OO

10

K = I33 bw=I

7

Marn= I lm

-

10

-

20

u/ua and j,,/lW

FIG.51. Profiles of velocity, temperature, electric conductivity, and electric current density o n insulated walls of magnetohydrodynamic channel for elevated electron temperature ( T , > T )[after Hale an d Kerrebrock ( 6 8 ) ] .

For the conditions b , = 0, K = 2, and Ma, = 1, the constant in this equation assumes the value 0.33 for flat-plate flow without electromagnetic influence, 1.84 for the equilibrium boundary layer, and 3.58 for the nonequilibrium boundary layer. This indicates that heat fluxes in a magnetohydrodynamic power generator will be of considerable magnitude on the insulator walls as well as on the electrodes. Figure 52 demonstrates a typical measured heat flux distiibution to the walls of the Mark I1 MHD channel (69)which was operated at a mass flow rate of about 2.3 kg/sec, a magnetic field intensity of 3.3 Wb/m2and a channel area ratio (outlet to inlet) of 1.45. T h e channel consisted of a nonablating

PLASMA HEATTRANSFER TRANSFER PLASMA HEAT

311 311

material with graphite electrodes. T h e indicated heat fluxes in Fig. 52 represent average values of the heat transfer to the electrodes and to the insulating walls. A mixture of gaseous oxygen and a fuel consisting of methylcyclohexane with a solution of KOH and ethyl alcohol was fed into the combustion chamber. T h e electrical conductivities of the flame approached those of kerosene-oxygen flames.

-

500 -

“5400 \

c

s3

-300

-‘

f ! I

-

-~

,

~~~~~ + _I -

I I

w

LL

3 200

_______

-

~

1 l

a c

9

I

~ I

-

Y L

v)

t

_

loo-

-

-

I

~

I

0

NOMENCLATURE probability for spontaneous transition within a r-times ionized atom from a quantum level s to a lower quantum level t B magnetic field strength B” intensity of blackbody radiation b Hall parameter C constant C light velocity (3 x lo8 mjsec) specific heat at constant pressure CP D mass diffusion coefficient 4 sheath thickness E electric field intensity e electronic charge (1.6 x A sec) statistical weight, constant

Planck’s constant, heat transfer coefficient Z total electric current, radiation intensity spectral intensity initial spectral intensity enthalpy electric current density diffusional mass flux electromagnetic loading factor thermal conductivity Boltzmann constant layer thickness, length Lewis number = p D c p /k molecular weight Mach number mass of particles

h

E. R. G. ECKERT AND E. PFENDER

3 12 Nu n Pe Pr

pr

P

Q 9

R

Re TL y,

X,Y,2

S

S

T t u o

u, v

V V W

z Z’e a

Y Ax 6 E QL

€0

Qv

eV’

5

v e K Kv

Kv’

h AD

Nusselt number = hiLc,/k number density of particles, principal quantum number Peclet number = vL/a Prandtl number = p c , / k radiative coefficient of emission pressure total heat flux heat flux per unit area gas constant Reynolds number = u, Llv average Larmor radius coordinates ionization coefficient correction for the penetration of the outer electron shells by fast electrons temperature time anode fall voltage velocity components electric potential thermal velocity (see text) mass fraction partition function ionic charge combined collisional-radiative recombination coefficient, Sommerfeld’s fine-structure constant, thermal diffusion ratio ratio of specific heats natural width of a spectral line boundary layer thickness energy flux spectral line emission coefficient permittivity of vacuum (8.86 x A sec/V m) spontaneous spectral emission coefficient total spectral emission coefficient number of valence electrons dimensionless coordinate temperature parameter mass generation rate absorption coefficient corresponding to c,, total absorption coefficient wavelength Debye length

mobility, viscosity frequency, kinematic viscosity collision frequency degree of ionization density electric conductivity optical depth average time interval between two electron collisions anode work function energy ionization energy ionization energy of hydrogen electron cyclotron frequency

SUBSCRIPTS A argon a heavy particle abl

referred to ablative cooling of the anode conv referred to convective heat transfer e equilibrium, electron J frozen R limiting value i referred to enthalpy, ion j referred to current density flow into a surface max maximum value P constant pressure I ionization stage ra referred to radiative heat transfer to the anode re referred to radiative heat transfer from the anode S energy state t energy state, total value W at wall surface u3 in free stream, outside boundary layer 0 neutral particles, reference state 0 at zero temperature 1,2 components in a mixture

SUPERSCRIPTS a atoms of species a

*

0

critical value total energy (including kinetic energy)

PLASMA HEATTRANSFER

313

REFERENCES 1. P. Chung, Chemically reacting nonequilibrium boundary layers. Advan. Heat Transfer 2, 109 (1965). 2. M. Romig, The influence of electric and magnetic fields on heat transfer to electrically conducting fluids. Advan. Heat Transfer 1, 267 (1964). 3. W. Finkelnburg and H. Maecker, Elektrische Bogen und thermisches Plasma. In “Encyclopedia of Physics” (S. Flugge, ed.) Vol. XXII, Springer, Berlin, 1956. 4. W. Finkelnburg and Th. Peters, Kontinuierliche Spektren, “Encyclopedia of Physics,” “Spectroscopy 11” (S. Fliigge, ed.) Vol. XXVIII. Springer, Berlin, 1957. 5 . H. R. Griem, “Plasma Spectroscopy.” McGraw-Hill, New York, 1964. 6. M. E. Barzelay, Continuum radiation from partially ionized Argon. AZAA J . 4, 815 (1966). 7. J. C. Morris, G. R. Bach, R. U. Krey, R. W. Liebermann, and J. M. Yos, Continuum radiated power for high-temperature air and its components. A I A A J. 4, 1223 (1966). 8. R. D. Cess, The interaction of thermal radiation with conduction and convection heat transfer. Advan. Heat Transfer 1, 1 (1964). 9. H. Maecker, Messung und Auswertung von Bogencharakteristiken (Ar, N2). Z . Physik 158, 392 (1960). 10. C. Mahn, H. Ringler, R. Wienecke, S. Witkowski, and G. Zankl, Experimente zur Erhohung der Lichbogentemperatur durch Reduktion der Warmeleitfahigkeit in einem Magnetfeld. Z . Naturforsch. 19a, 1202 (1964). 1 1 . A. Unsold, “Physik der Sternatmospharen,” 2nd ed. Springer, Berlin, 1955. 12. H. W. Drawin and P. Felenbok, “Data for Plasmas in Local Thermodynamic Equilihrium,” Gauthier-Villars, Paris, 1965. 13. C. J. Cremers and E. Pfender, Thermal characteristics of a high and low mass flux Argon plasma jet, Univ. of Minnesota, Minneapolis, ARL Rept. 64-191, 1964. 14. G. Elwert, Uber die Ionisation- und Rekombinationsprozesse in einem Plasma und die Ionisationsformel der Sonnenkorona, Z . Naturforsch. 7a, 432 (1952) ; Verallgemeinerte Ionisationsformel eines Plasmas, Z . Naturforsch. 7a, 703 (1952). 1 5 . D. R. Bates, A. E. Kingston, and R. W. P. McWhirter, Recombination between electrons and atomic ions, I. Optically thin plasmas. Proc. Roy. Sac. A267, 297 (1962); 11. Optically thick plasmas. Proc. Roy. Sac. A270, 1 5 5 (1962). 16. D. R. Bates and A. E. Kingston, Properties of a decaying plasma. Planetary Space Sci. 11, l(1963). 17. N. D’Angelo, Recombination of Ions and Electrons. Phys. Rev. 121, 505 (1961). 18. H. W. Drawin, Besetzungsdichten angeregter Wasserstoffniveaus in stationaren Plasmen unter Berucksichtigung der Diffusion von Neutralteilchen. Z . Physik 186, 99 (1965). 19. W. D. Thompson, “An Introduction to Plasma Physics.” Pergamon Press, Oxford, 1962. 20. J. D. Cobine, “Gaseous Conductors.” Dover, New York, 1958. 21. F. F. Chen, Electric probes. In “Plasma Diagnostic Techniques” (R. H. Huddlestone and S . L. Leonard, eds.). Academic Press, New York, 1965. 22. A. B. Cambel, “Plasma Physics and Magnetofluidmechanics.” McGraw-Hill, New York, 1963. 23. F. BobnjakoviC, W. Springe, and K. F. Knoche, Mollier-Enthalpie-Entropie-Diagramme fur Hochtemperaturplasmen. Z . Flugwiss. 10,413 (1962). 24. F. BobnjakoviC et al., Unpublished data, Inst. f. Thermodynamik d. Luft-und Ranmfahrt, Stuttgart, Germany. ~~

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E. R. G. ECKERT AND E. PFENDER

25. H . Maecker, Uber die Charakteristiken zylindrischer Bogen. 2. Physik 157,l (1959). 26. L. Spitzer, “Physics of Fully Ionized Gases.” Wiley (Interscience), New York, 1956. 27. R. Wienecke, Druckerhohung in der zylindersymmetrischen Lichtbogensaule hei uberlagertem axialem Magnetfeld. 2. Naturforsch. 18a, 1151 (1963). 28. S. Witkowski, Druckerhohung in der zylindersymmetrischen Lichtbogensaule bei hberlagertem axialem Magnetfeld. 2. Naturforsch. 20a, 463 (1965). 29. S. I. Braginskii, T h e behavior of a completely ionized plasma in a strong magnetic field. Soviet Phys. J E T P (English transl.) 6(33), 494 (1958). 30. W. Feneberg, Transport properties of a partially ionized plasma in a magnetic field. Proc. Intern. Conf. Ionization Phenomena Gases, 6th, Paris, 1963, 1, 357. North-Holland Puhl., Amsterdam, 1963. 3 1. R. Wienecke, Reaktionswarmeleitfahigkeit von Wasserstoff und einfach ionisiertem Helium in einer zylinder-symmetrischen Entladung mit uberlagertem axialem Magnetfeld. 2. Naturforsch. 19a, 675 (1964). 32. J. E. Anderson and E. R. G . Eckert, Transpiration Cooling of a constricted Electric Arc Heater. A Z A A J . 5, 699 (1967). 33. E. R. G . Eckert and R. M. Drake, Jr., “Heat and Mass Transfer,” 2nd ed. McGrawHill, New York, 1959. 34. J. A. Fay and F. R. Riddell, Theory of stagnation point heat transfer in dissociated air. J . Aerospace Sci. 25, 73 (1958). 35. S. Chapman and T . G . Cowling, “ T h e Mathematical Theory of Non-Uniform Gases,” 2nd ed. Cambridge Univ. Press, London and New York, 1952. 36. E. R. G . Eckert, Diffusion thermo effects inmass transfer cooling. Proc. 5th Natl. Congr. Appl. Mech. Kew York, 1966, p. 639. Am. SOC.Mech. Engrs., New York, 1966. 37. E. M . Sparrow, C. J. Scott, R. J. Forstrom, and W. A. Ebert, Experiments on the diffusion thermo effect in a binary boundary layer with injection of various gases. J . Heat Transfer 87, 321 (1965). 38. H. Hoshizaki, Heat transfer in planetary atmospheres at super-satellite speeds. A R S ( A m . Rocket Soc.) J . 32, 1544 (1962). 39. J. A. Fay and N. H. Kemp, T h e theory of stagnation-point heat transfer in a partially ionized diatomic gas. A I A A J . 1, 2741 (1963). 40. A. Pallone and W. van Tassell, Effects of ionization on stagnation-point heat transfer in air and in nitrogen. Phys. Fluids 6, 983 (1963). 40a. N. B. Cohen, Boundary layer similar solutions and correlation equations for laminar heat transfer distribution in equilibrium air at velocities up to 41,100 feet per second. N A S A Tech. Rept. T r 118 (1961). 41. S. M. Scala and W. R. Warren, Hypervelocity stagnation point heat transfer. A R S ( A m . J . 32, 101 (1962). Rocket SOC.) 42. J. S. Gruszczynski and W.R. Warren, Experimental heat transfer studies of hypervelocity flight in planetary atmospheres. AZAA J . 2, 1542 (1964). 43. C. Park, Heat transfer from nonequilibrium ionized Argongas. A Z A A J . 2,169 (1964). 44. P. H . Rose and J. 0. Stankevics, Stagnation-point heat-transfer measurements in partially ionized air. AZAA J . 1, 2752 (1963). 44a. K. Penski, Zustands- itnd Transportgrossen yon Argonplasma. Chem. Ing.-Tech. 34, 84 (1962). 45. H. A. Stine and V. R. Watson, T h e theoretical enthalpy distribution of air in steady flow along the axis of a direct-current arc. N A S A Tech. Note TN-1331, 1962. 46. C. E. Shepard and V. R. Watson, Performance of a constricted-arc discharge in a supersonic nozzle. Proc. Bien. Gas Dyn. Symp. Sth, Evanston, Illinois, 1963. Northwestern Univ. Press, Evanston, Illinois, 1963.

PLASMA HEATTRANSFER

315

47. H . A. Stine, V. R. Watson, and C. E. Shepard, Effect of axial flow on the behavior of the wall-constricted arc. A G A R D Specialists’ Meeting, Rhode-Saint-Genese, Belgium, 1964. AGARDograph 84, Part I, 451 (1964). 48. G. L. Cann, R. D . Buhler, R. L. Harder, and R. A. Moore, Basic research on gas flows through electric arcs-hot gas containment limits. ARL Rept. 64-49, E.O.S., Pasadena, California, 1964. 49. R. R. John, H. Debolt, M. Hermann, A. Kusko, and R. Liebermann, Theoretical and experimental investigation of arc plasma-generation technology, ASD-TDR-62-729, AVCO Wilmington, Massachusetts, 1963. 50. H . E. Weber, Constricted arc column growth, Proc. Heat Transfer Fluid Mech. Inst., 1964, p. 245. Stanford Univ. Press, Stanford, California, 1964. 51. V. R. Watson, Comparison of detailed numerical solutions with simplified theories for the characteristics of the constricted-arc plasma generator. Proc. Heat Transfer Fluid Mech. Inst., 1965, p. 24. Univ. of California Press, Berkeley, California, 1965. 52. G. Schmitz, H. Druxes, and H . J. Patt, Zur Modelltheorie des zylindersymmetrischen Lichtbogens mit radialer Masseneinstromung. 2. Physik 187, 271 (1965). 53. J. L. Kerrebrock, Conduction in gases with elevated electron temperature. I n “Engineering Aspects of Magnetohydrodynamics, p. 327. Columbia Univ. Press, New York, 1962. 54. A. Pytte and A. R. Williams, On electric conduction in a nonuniform helium plasma. ARL Rept. 63-166, Dartmouth College, Hanover, New Hampshire, 1963. 5 5 . K. Goldman, Characteristics of the Argon arc. Proc. Intern. Conf. Ionization Phenomenu Gases, 5th Conf., Munich, 1961, l , 863. North-Holland Publ., Amsterdam, 1962. 56. H. Maecker, Plasmastromungen in Lichtbogen infolge eigenmagnet. Kompression. Z . Physik 141, 198 (1955). 57. P. A. Schoeck, An investigation of the energy transfer to the anode of high intensity arcs in Argon. Ph.D. Thesis. Univ. of Minnesota Press, Minneapolis, Minnesota, 1961. 58. P. A. Schoeck and E. R. G. Eckert, An investigation of anode heat transfer in high intensity arcs. Proc. Intern. Conf. Ionization Phenomena Gases, 5th Conf., Munich, 1961, 2, 1812. North-Holland Publ., Amsterdam, 1962. 59. P. A. Schoeck, An investigation of the anode energy balance of high intensity arcs in Argon. In “Modern Developments in Heat Transfer” (W. Ibele, ed.), p. 353. Academic Press, New York, 1963. 60. 0. H. Nestor, High intensity and current density distributions at the anode of high current inert gas arcs. J . Appl. Phys. 33, 1638 (1962). 61. R. C. Eberhart, T h e energy balance for the high current Argon arc. Ph.D. Thesis. Univ. of California, Berkeley, California, 1965. 62. G. Ecker, Electrode components of arc discharges. Ergeb. Exakt. Naturw. 33, 1 (1961). 63. D . Rosenthal, T h e theory of moving sources of heat and its application to metal treatments. Trans. Am. SOC.Mech. Engrs. (Trans. A S M E ) 68, 849 (1946). 64. E. Pfender and E. R. G. Eckert, Behavior of an electric arc with superimposed axial flow. Proc. Heat Transfer and Fluid Mech. Inst., 1965, p. 50. Univ. of California Press, Berkeley, California, 1965. 65. E. Pfender and C. J. Cremers, Steadiness of a plasmajet. A I A A J . 3, 1345 (1965). 66. E. Pfender, E. R. G. Eckert, and G. Raithby, Energy transfer studies in a wall-stabilized, cascaded arc. Proc. Intern. Conf. Ionization Phenomena Gases, 7 t h Conf. Belgrad, Yugodavia, 1965, 1, 691. Gwdevinska Knjiga Publishing House, Belgrad, 1966. 67. J. L Kerrebrock, Electrode boundary layers in direct current plasma accelerators. J . Aerospace Sci. 28, 631 (1961).

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68. F. J. Hale and J. L. Kerrebrock, Insulator boundary layer in magnetohydrodynamic channels. AZAAJ. 2,461 (1964). 69. T. R. Brogan, Recent developments in M H D power generation at the AVCOEVERETT research laboratory. In “Gas Discharges and the Electricity Supply Industry,” p. 571. Butterworths, London and Washington, D.C., 1962.

Exact Similar Solutions of the Laminar Boundary-Layer Equations .

C FORBES DEWEY. Jr

.

University of Colorado and Joint Institute for Laboratory Astrophysics Boulder. Colorado

.

JOSEPH F GROSS Department of Geophysics and Astronomy. R A N D Corporation. Santa Monica. California

I . Introduction . . . . . . . . . . . . . . . I1 . Similarity Equations for the Laminar Boundary Layer . . . A . General Equations . . . . . . . . . . . .

I11. IV . V.

B. Conditions for Similarity . . . . . . . . . . C . Equations for Solution . . . . . . . . . . . D . Special Classes of Reduced Equations . . . . . . . E . Computation of Boundary-Layer Properties . . . . . F . Range of Solutions and Parameters . . . . . . . Numerical Results . . . . . . . . . . . . . Numerical Integration Procedure . . . . . . . . . Applications of the Concept of Local Similarity . . . . . A . General Discussion . . . . . . . . . . . . B . Asymptotic Expansion of the Boundary-Layer Equations C . Determination offl@ 7) by Successive Approximations . Solutions for Large Values of the Pressure-Gradient Parameter /3 A . The Outer Limit Equations . . . . . . . . . B . The Inner Limit Equations . . . . . . . . . . Discussion . . . . . . . . . . . . . . . Keys to Tables . . . . . . . . . . . . . . Tables . . . . . . . . . . . . . . . . Symbols . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .

.

VI . VII .

317 319 319 322 324 327 328 330 330 331 333 333 335 337 341 341 344 345 357 370 442 443

.

I Introduction

During the last decade. many similar solutions to the laminar boundarylayer equations have been obtained . Most of these results have been for specific exact conditions. such as stagnation-point flow or flat-plate flow. and are based on assumptions regarding a linear temperature-viscosity law 317

318

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

and a Prandtl number, Pr, of unity. It is well recognized [cf. Kuerti (1); Hayes and Probstein (2); Beckwith ( 3 ) ;Gross and Dewey (4);Kemp (5)] however, that an accurate estimate of skin friction, heat transfer, and boundary-layer thickness in compressible flow usually requires the use of a realistic temperature-viscosity law and retention of the dissipation term which appears in the energy equation. The purpose of this chapter is to present comprehensive, systematic, and accurate tables of solutions to the laminar boundary-layer similarity equations for a perfect homogeneous gas. These solutions are applicable to the classical cases wherein the requirements for similarity are satisfied exactly. T h e solutions may be used also in applying local similarity methods to cases that do not meet the exact requirements for similar flows. In particular, the sensitivity of the numerical values of the wall derivatives (which govern heat transfer and skin friction) and the values of the integral thickness parameters to changes in the similarity variables may be accurately determined. For many physical situations, relaxing one or more of the conditions required for exact similarity will not lead to large errors when the local similar solutions are used to estimate boundary-layer properties. Solutions are included to illustrate the effects of the leading-edge sweep, the mass transfer at the surface, pressure gradient, the wall temperature, the free-stream Mach number, local external flow velocity, the Prandtl number, and the viscosity-temperature law of the fluid. The tables include most of the numerical solutions obtained by previous authors (recomputed to the accuracy of the present program) and also new solutions. These tables do not include values with /3 t O (decelerating flows) orfz, > 0 (suction). Section I1 presents the similarity equations in their general form, including the equation for species concentration. Transformations applicable to compressible flows are given, as are equations for the computation of skin friction, heat transfer, and boundary-layer displacement thickness. Classes of reduced equations are also discussed ;these classes represent cases where one or more of the parameters Pr, w , p, t,, t,,, U m 2 / 2 H eand , u,2/um2are either zero or unity. Section I11 gives a concise guide to the solutions of this chapter. Section IV presents the numerical method of computation used in this program. The concept of local similarity is discussed in Section V, where criteria are presented for judging the applicability of similar solutions in situations where the similarity parameters vary. The analysis is extended to large positive values of the pressure-gradient parameter p in Section VI. Our experience indicates that values of the skin friction and heat transfer derivatives and the boundary-layer thickness integrals may be estimated to high accuracy for intermediate values of the similarity parameters by careful

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

319

cross-plotting of the present exact solutions. The accuracy varies with the particular quantity and the specific values of the similarity parameters, but it is generally between 0.5 and 2.0%. In particular, use of the solutions for p + a, allows estimation of all quantities for all positive p. 11. Similarity Equations for the Laminar Boundary Layer

The purposes of this section are three. First, we shall display the laminar boundary-layer equations for a binary mixture of perfect gases in their most general form, listing the general conditions that lead to similar solutions. Second, the specific equations treated here are deduced from the general equations, and relations are given for computing heat transfer, skin fraction, and integral-thickness parameters. Finally, special reduced forms of the general equations (Pr = 1, etc.) are discussed.

A. GENERAL EQUATIONS The equations that describe the conservation of mass, momentum, and energy in a laminar boundary layer are well known. They can be expressed as follows :

a (purj) + a (pvr') = 0

--

ax

-

aY

j =0

two-dimensional flow

j=1

axisymmetric flow

320

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

FIG.1. Boundary-layer coordinate system.

The coordinates x, y, and r are defined in Fig. 1. Other symbols are defined

as follows:

ci = mass fraction of ithconstituent D I 2= binary-diffusion coefficient H = total enthalpy of mixture = h $(u' w') hi = chemical enthalpy of ithconstituent k = thermal conductivity of fluid Le = Lewis number, pDI2c,/k Pr = Prandtl number, cPp/k p = local fluid pressure Sc = Schmidt number, p / p D I 2 u, TI = flow velocities in the x,y directions, respectively w = flow velocity in direction normal to x,y plane p = viscosity coefficient of fluid p = local fluid density These equations neglect thermal diffusion and diffusion-thermo effects. If the fluid is a mixture of thermally perfect gases, Eq. (4)integrates to

+

+

P ( X , Y ) = Pe(x) = Pe RTe (8) A stream function Y ( x , y )is introduced that satisfies the continuity equation:

w / a x = - pvr] aYpy = + purl

(94 (9b)

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

321

and a transformation for the independent variables is introduced :

where subscript e denotes any property at the edge of the boundary layer and rk denotes a characteristic radius. The quantity C, by definition, is a combination of physical properties that are functions of x only. I n reducing the partial differential equations in 7 and 5 to ordinary differential equations in the single variable 7, all derivatives with respect to 5 within the boundary layer are neglected. The fluid velocity, the enthalpy, and the concentration are then defined in terms of the transformed similarity variable 7 : u/ue = df ( v ) / ~=vf ' ( ~ ) (12)

(13) H/He = G (7) (14) Cilciw = zi(7) (15) (The dependent variable f appearing in Eq. (12) is simply equal to (25)'12 times the stream function Y.) Then the boundary-layer equations (2), ( 3 ) , (S), and ( 6 ) for a two-component boundary layer may be written as a set of coupled ordinary nonlinear differential equations : w/we = g(7)

where A is the leading-edge sweep angle, A = (l/C)(pp/pepe), and, u, is the free-stream flow velocity in the x direction The above equations are applicable to the two-component boundary layer of an axisymmetric body or a swept two-dimensional surface. The boundary

322

C. FORBES DEWEY, JR.,AND JOSEPH F. GROSS

conditions for these equations are given at the surface of the body by the no-slip condition and the requirement of similarity for the wall concentration and enthalpy: f '(0) = 0 (204 d o ) =0 G(0) = G, (20c) Z,(O)= 1 (204 At the outer edge of the boundary layer, the velocity and enthalpy must match the values in the free stream. The concentration of any material injected at the surface is assumed to be zero in the external flow: f'(co) = 1

g(m) = 1 G(m) = 1

Zl(co) = 0 The final boundary condition is obtained by considering the 2 particles at the wall. This yields the well-known Eckert-Schneider condition represented by the equation

From the continuity equation, the injection parameter is related to the similarity coordinates by the equation

B. CONDITIONS FOR SIMILARITY In order that Eqs. (16)-(19) be similar, severe restrictions must be placed on some of the terms. These are

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

323

Le = Le(q)

G,, el,, and rW2/rk2 = const f ( 0 ) = const Equations (23) and (25)represent the most severe restrictions on the system because the external velocity distribution and the body surface must satisfy the requirements shown. It may be verified that Eq. (23) has the following form:

where O=(H - H,)/(He -H,), T ois the free-stream stagnation temperature, and t, = HwI/He= Tw/To (31) (adiabatic and calorically perfect inviscid flow) and

t, =

1 + +(y - 1)Mm2cos2A 1

+ +(y - 1)Mm2

It should be noted that either j = 0 and 0 G t, G 1, or j = 1 and t, = 1. If t, and t, are considered to be free parameters, Eq. (30) requires that

If we assume the relationship between ueand 5 to be given by ue

then

-

(Te/To)”2tm

(344

where p is the modified Falkner-Skan parameter. Using Eq. (34) to define the parameter /3, the equations assume the following similarity form :

(A$.g’)

‘Sfg‘

=

0

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C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

In Eq. (37), u = (Um2/2H,) and is given the designation of hypersonic parameter. Two other parameters can be defined as

and

The parameter u1 includes the shock-angle effect on the energy equation; it is u times the ratio of the velocity at the edge of the boundary layer to the free-stream velocity. The parameter u2 is a generalization of u1 including the effect of sweep angle. When u = 0, the ratio of kinetic energy in the flow to the stagnation enthalpy of the free stream is zero ;as a result the dissipation term in the energy equation is neglected. As CT increases, the dissipation term plays an ever-increasing role in the energy equation.

C. EQUATIONS FOR SOLUTION Subject to simplification regarding the flows to be considered, Eqs. (35)-(38) are those equations for which solutions are presented in this chapter. These simplifications are as follows : (1) The Chapman-Rubesin constant is defined as

C = pw PwlPe ~e so that the quantities 4 and h become

4=

1;

Pw P w u e

A = PPIPW P w

(2) Curvature effects are neglected: (r/r,J2= 1

r? dx

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

325

(3) The product (Le - l ) ( h 2- h i ) is assumed to be zero, and thermal diffusion is neglected. A heuristic justification of this simplification for equilibrium air is given by Beckwith and Cohen (9,Appendix A). As a final step, the concentration variable 2, is transformed to the dependent variable z=1 -Z1=(clw-CI)/clw Substitution of 0 for G and x for Z1yields the equations that were solved exactly to prepare the tables of solutions :

(y)’ +ff”

=

(hg’)’ +fg’

=0

p(f”

1

- - [(I ts

-

e

t w ) - (1 - ts)g2

+ tW]

(40)

This is a 9th-order nonlinear set of ordinary differential equations. Nine boundary conditions are required, five at the wall and four at the edge of the boundary layer. The boundary conditions to be satisfied are as follows : At the wall when 7 = 0:

fyo) = g(o) = e(o) = z(o) = o

(44) and the Eckert-Schneider condition which relates the conservation of 2 particles at the wall :

In obtaining Eq. (45) from Eq. (22a), we have made use of the fact that = 1. At the outer edge of the boundary layer, the conditions are : 7e)=1

(46) The value of ve is chosen to be large enough to insure that the boundary conditions given by Eq. (46) are satisfied asymptotically to a high degree of accuracy (see Section IV). The similarity conditions are satisfied by: f’(7e)

=d

q e ) =e

/3

( ~ e= ) 4

= const

= PP/Pw P W = A(rl)

either

Pr = 1

or ( u , / u , ) ~= const and Pr = Pr(7) Sc = Sc(7) [= Pr(7) for Le = 11

(47) (48) (49) (50)

326

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

t, and clw= const (51) f(0) = const =f w (52) The viscosity-temperature relationship may be characterized by the power-law expression: p = AT". This yields h T"-' . A value w = 0.7 corresponds to conventional wind-tunnel conditions, while w = 0.5 represents conditions encountered in hypersonic flight. In terms of the similarity variables, h may be written A = [(l/t,)(T/To)]"-'=[(l/ t w ) { ( l - t w ) 8 - (1- tJg2- oj C O S ~ A ~tw)]'"-'. '~ If w = I , then h = 1 and pp = const. In the literature, this latter assumption has often been made in order to simplify the boundary-layer equations. The authors have demonstrated in a previous paper ( 4 ) that boundarylayer characteristics calculated by means of a Sutherland-viscosity law can be approximated almost exactly using the power-law relation p T"., provided the empirical exponent w, is suitably chosen. Sutherland's law may be written p / p o = t3'2[(s.+ I)/($ t ) ] (53a) where t = T/Toand s = S/To.The Sutherland constant S is a characteristic temperature for the gas, and p o is the viscosity evaluated at the stagnation temperature To.The empirical equations for calculating w, are

-

-

9

+

t, = 0.5(te+ tw)+ 0.22( 1 - t,) (53c) The quantity t, is sometimes referred to as the Eckert reference temperature. Solution of Eqs. (40)-(43) subject to the boundary conditions of Eqs. (44)-(46) requires the specification of eight independent parameters. These are fw ~ f ( 0=) blowing parameter Pr = Prandtl number, taken to be constant = Sc t, = sweep parameter ; t, = normalized wall temperature (u,/u,)* = local streamwise velocity ratio /3 = pressure-gradient parameter (T U,'/2He = hypersonic parameter w or s = temperature-viscosity law All eight must be independent of the streamwise coordinate 5 for similarity to hold exactly. If Pr = 1, the parameters (T and ( u , / u , ) ~are not required. It should be emphasized that thermal diffusion has been neglected and the product (Le - l)(h, - h2) has assumed zero. I n this approximation, solution of the three equations forf,0, andg are independent of the diffusion equation and, consequently, are independent of the value of the Schmidt number.

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

327

D. SPECIAL CLASSES OF REDUCED EQUATIONS 1. Two-Dimensional Flow If r2j/rkzj= 1, the ordinary two-dimensional boundary-layer equations result : (hf”)’”p{f”-(~/~~)[(~ -t,>e-(1 (hg’)’ +fg’ = 0

- t s ) g 2 + t,]}

(54) (55)

Equations (54)-(57) are valid for axisymmetric flow if (1is set equal to zero and j is unity.

2. A = O When the free-stream-flow direction coincides with the x direction, then A = 0 and t, = 1. For this case, the boundary-layer equations reduce to

(y)’ +fy

(x;;)’+fZ’’

=

pv’2 - (1 - t w )e - t,]

(58)

=0

3 . No Mass Transfer I n the special case where no mass transfer takes place in the boundary layer, Z1= 0 and clw= 0, and the boundary conditions (including the Eckert-Schneider condition) are ignored’ :

(xf”)’ +fj”

8’ ) +fe’

(A Pr ’

=

=

p { j -~ ( 1 - t,) e - t w }

( (jr

- 1)

( 12.:,)f”f’

(61)

(

$:2)m

‘

(62)

1 Although the solution for the concentration ratio z1 is z~ = 0, the solution of Eq. (43) for the normalized concentration z has a nonvanishing solution and a nonzero gradient z’ at the wall.

328

C. FORBES DEWEY,JR., AND JOSEPHF. GROSS

Equations (61) and (62) are the ordinary two-dimensional Prandtl boundarylayer equation. 4. p - T If the viscosity is directly proportional to the temperature, h = 1, and Eqs. (61) and (62) reduce to

p +jf”= p p

-

(1 - tw)e + tw>

(63)

5. P r = l Equations (63) and (64) reduce to a form in which the dissipation term of the energy equation is zero :

f ” +fy= p { y 2 - (1 - t w )e + tw> el’

+jet

=

o

(65)

(66)

6. p = O Finally, if the pressure-gradient parameter is zero, the momentum equation is uncoupled from the energy equation; Eq. (65) becomes f‘“+ff”= 0

(67)

and by inspection, the solution of the energy equation is

e =ff

(68)

E. COMPUTATION OF BOUNDARYLAYER PROPERTIES Heat-transfer rates and skin friction are related by the similarity transformations to the derivatives O’(O), f ”(0),and g’(0). The heat transfer from the stream to the wall is given by 4w = + k w ( a T / a Y ) w (694 After the proper substitutions have been made, the expression for the heat transfer in similarity coordinates is

and the local heat transfer coefficient is ch = %~/[prn

-

Hw)]

(70)

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

329

The skin friction for the x direction can be described in similarity coordinates as follows :

The spanwise coefficient is

The total component of skin friction is the vector sum of the two perpendicular components T~ and 7,: 7=

PW

rkJ

(2W2 [{f”(0)12U,2 + {g’(0)l2w,z1112

~w ue ~~

(73)

and the local skin friction coefficient is defined by

c,= 2r/pmurn2

(74)

Several integral relations have also been tabulated. They are

Il = (1 I2=

-W l ( 1 ) -W ) 1

jOrnf’(l- f 0 4

-

(1 - W d 3 ) + W )

(75) (76)

where

These integrals can be related to the boundary-layer-thickness parameters expressed in similarity coordinates. Displacement thickness :

Momentum thickness :

330

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

F. RANGEOF SOLUTIONS AND PARAMETERS The parameters for the systems of equations under consideration here are O < p < 5 andp=m pressure-gradient parameter Pr = 0.5, 0.7, 1.0 Prandtl number Sc = Pr (i.e., Le = 1) Schmidt number o = 0.5, 0.7, 1.0, or temperature-viscosity-law parameter 0.01 < s < 0.3 O
This section provides a guide to the present solutions of the laminar boundary-layer equations. The available solutions are shown schematically in Keys I-VI (pp. 357-369). Each key shows the values of the eight independent parameters covered by the correspondingly numbered table (pp. 370-441). If the key indicates (by an “x” in the parameter matrix) that a solution is available, then the numerical values are given in the corresponding table. The key also provides the reader with a graphic display of the solutions available in the neighborhood of his range of interest. Inner solutions for the limit /3 + m are given in Table VII which is too short to warrant a separate key. Approximately a sixth of the solutions listed have been reported by other authors. All values listed have been computed, or recomputed, according to the numerical procedures described in Section IV. Some differences exist between the present values and those of earlier investigators, but the present values are believed to be accurate within the limits prescribed in the succeeding three sections (generally, correct to four significant figures). The most extensive of the earlier tabulations of solutions to the similarity equations for a laminar boundary layer of a perfect gas may be found in the works of Beckwith (3), Li and Nagamatsu ( 6 ) , Cohen and Reshotko (7), Reshotko and Beckwith (d), Beckwith and Cohen ( 9 ) , and Dewey (10).

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

33 1

Solutions of high accuracy have been obtained by Smith and Clutter (11), and others. Early contributions to the understanding of the role of fluid properties may be found in the works of Busemann (12), von Karman and Tsien (13),Crocco ( I d ) , Young and Janssen (15), and Van Driest (16).Emmons and Leigh (17) report a large number of solutions with the surface mass-transfer parameter fw other than zero. Several recent compendiums (18-21) on boundary-layer theory may be consulted for a more complete description of previous contributions. Since this chapter is restricted to positive values of the pressure-gradient parameter p and values fw < 0 corresponding to mass injection into the boundary layer, a list of sources is offered in which solutions for /3 < 0 and fw > 0 may be found. Emmons and Leigh (17) have obtained solutions for

/3=0andvaluesoffw.\/2equalto10,6,5,4,3,2.5,2,1.5,1.4,1.3,..., 0.5, 0.45,0.40,.. . , 0.1, and 0.05, assuming that Pr = 1 and the viscositytemperature law is linear. Suction results are reported also by Emmons and Leigh (17), Spalding and Evans (22), Pretsch (23),Watson (24), Eckert et al. (25), Thwaites (26), Schlichting and Bussman (27), Mangler (28), Schaefer (29), and Koh and Hartnett (30). Negative values of the pressure-gradient parameter /3 have been considered by Beckwith (3),Cohen and Reshotko (7), Smith (11), Stewartson (20, 31), Hartree (32),and Hufen and Wuest (33).Solutions for negative values of /3 differ from their counterparts for positive /3 in two ways. First, two sets of “proper” solutions exist for each value of p ; second, every value of the wall derivative fw” and its corresponding value of Ow’,etc., will satisfy the boundary conditions f’= 0 at r) = 0 and f’--f 1 as r ) 3 co. The “proper” solution must, therefore, be defined as that solution for which f’approaches unity most rapidly from above [see Beckwith (3),Cohen and Reshotko (7), Stewartson (ZO)].

IV. Numerical Integration Procedure The program for the numerical integration of the system of equations [Eqs. (40)-(43)]was written at RAND in FORTRAN for the IBM 7044. T h e system was treated as a two-point boundary-value problem, and the Runge-Kutta method was employed for the numerical integration. The sequence of operations is shown schematically in Fig. 2.Inputs to the program are values for the eight parameters and initial guesses for the four wall derivatives f ”(0),O’(O),g’(O), and ~ ’ ( 0 )T. h e solutions are then separated into two categories : ordinary boundary-layer solutions and adiabatic wall solutions. The latter requires a subprogram that searches for and obtains solutions in which the adiabatic wall condition is met. In either category, the Runge-Kutta method is used to integrate the equation to a given value of

332

C. FORBES DEWEY, JR., AND JOSEPH F. GROSS

vmax.On the basis of experience this value was selected to yield an acceptable asymptotic solution within the limits of the variable values at vmax.The integration stepsize was varied to insure the proper accuracy, and the values of the variables at qmaxwere held to within lop5.

COMPUTATION OF TW BASED ON INPUTS

4

YES EXPRESSIONS

~

1

RECOMPUTE

Tw

YES

RUNGE-KUTTA INTEGRATION

I

PUNCH CASE DATA a

PROFILE

IF THE INITIAL GUESSES ARE SUCH THAT THE INTEGRATION CANNOT BE COMPLETED THE PROGRAM ALTERS f ” ( 0 ) 6 BEGINS AGAIN

FIG.2. Program flow chart.

When the integration procedure is completed, the final values at ymaxare compared with the required values and the Newton-Raphson scheme is used to recalculate the initial condition. The procedure is as follows: Let 8 = S[f”(O), g‘(O), ~ ’ ( o )‘3’(0)] , =f’(vmax) - 1 (82)

9= F [ f ” ( O ) , g’(O), ~ ‘ ( o )e‘(O)] , =g(qmax) - 1 3 = gj-J”(O>, g’(O), ~’(o), e’(o)] = o(7max) - 1 3 = z [ f ” ( O ) , g’(O), ~ ’ ( o )e’(o)] , = Z(vmax) - 1

(83) (84) (85)

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

333

and

so thate = e ( S )= B(S)- 1 = d B . Itisrequiredthate = &(&,F, 9 , X )+ 0. Now de = dB = M dS, where

M=

T h e integration is performed using the initial guesses for S. This gives B and e0(S).T h e initial guesses are then perturbed as follows:

4 Y f ‘ ( O ) + df ”(O),

g’(O), e’(o), z’(0)I

(88) q f ” ( o ) , g’(0) + 4 ’(0 ), S’(O), z’(0)I (89) S[f”(O), g’(O), ef(o) + del(o>,z’(0)I (90) X[f ”(O), g’(O), B’(O), z’(0)+ A~’(0)l (91) These integrations give the columns of M . T h e new guesses for the initial condition are corrected by calculating: - d S = M-’eO (92) It was established that convergence was very difficult for cases involving high/? (/?2 4).I n these cases, the limit accuracy was relaxed, and this is indicated in each table as needed. V. Applications of the Concept of Local Similarity A. GENERAL DISCUSSION The numerical results tabulated in this chapter are rigorously applicable only when the numerous similarity requirements listed in Section I1 are satisfied. In considering the diverse applications of compressible laminar boundary-layer theory, it is a rare occurrence indeed when all of these conditions are met. A most important question then arises: What approach

334

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

should be used in predicting the behavior of the nonsimilar laminar boundary layer ? T h e copious literature relating to this question offers four basic types of approach. The first is to abandon the similar solutions entirely and adopt approximate techniques such as integral and series solutions containing free parameters. The most successful of the integral approaches appears to be that developed by Tani (34).The transcendental approximation proposed by Hanson and Richardson (35) and the “improved approximation ” technique of Yang (36)also appear very promising. Related to the integral methods is the powerful “strip method” proposed by Pallone (37),which follows closely the inviscid flow integral method of Belotserkovski and Chuskin (38). A second type of nonsimilar calculation employs a strictly numerical approach. The complete nonsimilar boundary-layer equations are used and a new set of calculations is performed for each particular problem. Examples of this approach may be found in the works of Smith and Clutter (39)and Flugge-Lotz and Baxter (40, 41). Although numerically satisfying, these calculations are extremely expensive and intractable to generalization. The third type of approach, that of Lees (42),has been most successful in capturing the spirit of the use of similar solutions in situations where exact similarity does not exist. He observed that under certain circumstances, notably when there is a highly cooled body in hypersonic flow, the local pressure-gradient parameter /3 had a negligible effect on the heat transfer to the surface. Many elaborations of this approach have been proposed to improve Lees’ simple result to provide more accurate numerical estimates of heat transfer, skin friction, and boundary-layer thickness. Moore (21) gives a lucid summary of one group of these results. Additional ideas for modifying local similarity not discussed by Moore may be found in the papers of Beckwith and Cohen (9),Smith (43),and Kemp et al. (44). This third approach is saddled with one difficulty: it is necessary to make one or more implicit ad hoc approximations regarding the contribution of the nonsimilar terms in the complete boundary-layer equations. In each of the papers cited in the previous paragraph, the question is not “Are similar solutions applicable ?” but rather “Which similar solution should be used ?” A number of methods have been proposed for choosing the similar solutions most appropriate to the local inviscid flow conditions, to wall temperature, and to boundary-layer history. In the context of the present discussion, this question necessitates the judicious choice of values for the eight similarity parameters. These values are usually determined [e.g. Beckwith and Cohen (9)] by satisfying one or more integral conservation equations exactly using assumed profiles obtained from similar solutions. Such procedures are closely related to the integral method of Thwaites (45) and to the more

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

335

recent use by Lees and Reeves (46, 47) of a family of similarity profiles generated by Stewartson (32). The fourth, and in many ways most satisfying, type of approach to nonsimilar boundary-layer calculations may be traced to the work of Meksyn (28). His underlying premise is very powerful: if the boundary layer is at all times very nearly described by a similar solution, then the direct effects of the nonsimilar terms may be calculated by asymptotically expanding the full boundary-layer equations in terms of small parameters which measure the departure of the solutions from similarity. I n this way, the accuracy of local similarity methods is explicitly determined by using the full nonsimilar equations. We shall demonstrate shortly that the linearized equations governing the departure from similarity depend only on the local similarity parameters and, consequently, need be computed only once.

OF THE BOUNDARY-LAYER EQUATIONS B. ASYMPTOTIC EXPANSION

For purposes of illustration, we shall limit our consideration to the “incompressible” momentum equation

f,,,+ff,, + P(5)

-f,’I

251f,fE, -fEf,,l

(93) where 5, 7, and /3(5) are as defined in Section I1 and the subscripts 71 and 4 denote derivatives. Equation (93) may be obtained from the general equations by assuming Pr = w = t, = t, = l. We also assume thatf, = 0, making the three boundary conditions for Eq. (93) =

Merk (48)was the first to expand the complete nonsimilar momentum equation in terms of a small parameter. Subsequently Bush (49) pointed out that Merk’s derivation neglected important terms in the correction equation. The remainder of this section will be concerned with Bush’s equations and their approximate solution. In the spirit of Meksyn’s approximation, we look for solutions to Eq. (93) when the right-hand side is small. The key to an appropriate expansion is the inversion introduced by Merk. We change variables from [ t , ~ /3(5)] ; to [p, 7 ; 5(P)] so that the streamwise momentum equation becomes

f,,,+ff,, + P[1 -#I

where

f(P, C) =f,(P,

0)

4P)[f,fp, -fpf,,I = 0; f,(P, = 1 =

4 P ) = 25PY5) = 2t(P)/t’(P)

(96) (97) (98)

336

C. FORBES DEWEY,JR., AND JOSEPH F. GROSS

If E is zero, the momentum equation reduces to the Falkner-Skan similarity equation with /3 as a single parameter. For small E we may perform an asymptotic expansion off(/3,7) of the form

f(P9 7) =fo(B, d + E(B>fl(B,rl) +

(99)

* * *

Then the derivativesf9 andfp are

+

+ ‘’’

(100) (101) As Bush pointed out, the term ~’(jl)is, in general, of order unity and may be expressed as f 9 = (f0)T

(fl),

fp = (fo)p+ 48)(fdp + - + E ’ ( P ) f t + * *

1

d

4 3 = B’(5)@”)I where

* *

= 2[1 +

E(P) = 5/3”(5)//3’(0= - 5(/3)5”(/3)/[4’(/3>12

(102) (103)

Substitution of the asymptotic sequence [Eq. (99)] into the momentum equation and boundary conditions produces a hierarchy of equations, the first two of which are (primes onfo andfl denote differentiation with respect to 7). Order unity =0 fo”’ +fofo” /3[1 (104) fo(8, 0) =fo’(P, 0) = 0 ; f O ’ ( B , a)= 1 Order E fl”’ + f O f l ” - AlfO’fl’ + A2fO”fl = @(Arl) (105) =0 fl(8, 0) = f t ’ ( B , 0)

+

=f*’(B,

In Eqs. (105) the terms (Al, A2,@) are A1=2+2/3+2E A2=3+2E (106) @(B, 7) =fo’(fo’>p - (f0)pfo” Note that two independent parameters (8, E ) appear in the first-order equation. Equations (104) represent the similar solutions of Falkner and Skan (50). Equations (105) represent the first correctionf, to the velocity profile which arises from nonsimilar terms. For example, the skin-friction derivative f”(/3, 0) is expressed as

0) + .fl”(B, 0) + * ’ * (107) where fo”(/3, 0) is the local similarity solution corresponding to the local value of /3, andfl”(/3, 0) is the correction obtained as a solution of Eqs. (10.5).

f”(B,0) =fo”(B,

SOLUTIONS OF

BOUNDARY-LAYER EQUATIONS

337

It is apparent that the correction will be of order E as long asfl”(/?, 0) is of the same order asfo”(p, 0) and E is small.

C. DETERMINATION 0 ~ f ~ ( /), 3 ,BY SUCCESSIVE APPROXIMATIONS We proceed to a consideration of the first-order equations for fl(p,v), Eqs. (105). The differential equation is linear with homogeneous boundary conditions and may be solved numerically. The primary difficulty arises in computing the inhomogeneous term @(p,7) which contains derivatives of fowith respect to both77 andp. This is a difficultterm to calculate numerically because a number of similarity solutions in the neighborhood of p must be known with high precision. The exact numerical calculation off, appears possible but has not been attempted. T h e technique adopted here is to substitute a transcendental approximation forfo(/3, 7) which allows the coefficients of the differential equation and to be expressed in terms of known functions. the forcing function @(p,~) Following an earlier paper by Bush (52),we representfo by the relation

This relation is found to be in excellent agreement with exact solutions forfo and appears quite adequate for our present purposes. I t is convenient to ) ( u , t ) and define a new dependent transform coordinates from ( 8 , ~to variable g(u, t ) according to the relations

a a,

= a -a

at,

[ 2 t L]

-a= a ’ @ ) - - + -

ap

2

0.23084

Substitution of these transformations into Eq. (105) results in a linear differential equation for g(u, t ) which contains derivatives of g with respect to t up to third order. The boundary conditions for this equation are (primes denote differentiation with respect to t ) g(., 0) = g’(u, 0) = g’(u, t.) = 0

(114)

C. FORBES DEWEY, JR.,

338

AND JOSEPH

F. GROSS

The differential equation for g(a,t ) is now integrated formally three times with respect to t , using the boundary conditions given by Eq. (114). The resulting integral equation is

g(., t ) = +t‘g’’(., 0) - J - ; J l M a ,t ) dt + A3 -A4

J-;j J- J-dt)g(a, t )dt + Y l ( t )

J-;J- Jo(t)g(a,4dt

(115)

T h e coefficients A3(a)and A4(a)are given by

A ~ ( u=)4 + 2E

+ 2/3,

A4(a)= 6

+ 4E + 2/3

(116)

and the terms J n ( t )are the nth integrals of the error function :

n=-1,0

J-,(X)

=

2

~

d;exp(-

x2),

Jo(x)= erf(x)

(1 17a)

n = 1,2,.., (117b) The term Y u , ( tis) defined by

where

A method of successive approximations is now applied to Eq. (115), substituting trial functions f ( a ,t ) for g(a,t ) in the three integrals and continuing until a trail function f is found that agrees satisfactorily with the function g computed from the integral equation. This scheme differs from Picard’s method in that the sequence of trial functionsf used in the integrals are suitably chosen integrable functions of t rather than the functions g(a,t ) obtained in the previous iteration step. Picard’s method converges absolutely, but in practice it usually cannot be continued analytically beyond one or two iterations. In the present scheme, convergence depends on the choice of trial functionsf but the integrals may be evaluated in closed form.

1. Evaluation of the Velocity Profile In this analysis based on the simplified momentum equation [Eq. (93)] we are most interested in the correction fl”(/3, 0) to the velocity gradient a t

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

339

the wall. It is therefore more convenient to work with the first derivative of Eq. (115), which is (after some rearrangement)

g’(.Pt)=

tg”(‘,o) +A4

1:1

- A 5 J l ( t ) g ( a , t ) + A6

Jo(t)g’(a,t)dt+ Y3(4 A5 = 3 +2E,

where

J:Jl(t)g’(a,t)dt

A6 =

(120)

2+2E

(121)

In following the technique of successive approximations, the terms g and g’ on the right-hand side of Eq. (120) must be replaced by the trial function

‘i’ I

g (a, m) = constant

FIG.3. Behavior of g andg’.

2 and its derivative 2’. The behavior of g and g‘ may be inferred from the boundary conditions g(u,0) = g’(u,0) = g’(u, m ) and is sketched in Fig. 3 . The behavior of Y3(t)and the two integrals appearing in Eq. (120) may be inferred from the behavior of g’ and the quantities Jn(t). Suppose we decompose the trial function f ( u , t ) into two parts, so that

f ( a , t ) = C(a).(t) (123) We then define ~ ( tto) be unity as t + 00, making f ( a , a)= C(a) # 0. The boundary conditions g(u, 0) = 0 and g’(u, 0) = 0 are automatically satisfied for all suitable trial functions f ( u , t ) ; the boundary condition g’(u, m) = 0 serves to evaluate both C(u)and g”(u, 0). The terms appearing on the righthand side of Eq. (120) have the following asymptotic forms :

340

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

The terms (yl, y z ,73) are numerical constants which are determined by the choice of the trial function ~ ( t ) . Substituting Eqs. (124)-(127) into Eq. (120) and applying the boundary condition g’(a, m) = 0, we obtain the following formulas for C(a)and g”(0):

g”(% 0) = c(a)[(I

+ As)- A4Y21 - 6(<2

- 1)

(128b)

) used Eqs. (128a) and (128b) to We have chosen several functions ~ ( tand determine C(a)and g”(a,0). The integrals appearing in Eq. (120) have been evaluated in closed form and the profiles g’(a, t ) have been determined for each trial function. Products and sums of the functions J n ( t )are useful in generating success) the integrals are easily ively more accurate trial functions ~ ( t because ) evaluated. Typical examples of ~ ( tare

.(t)

=

Jo(4 = JoZ(t)

and there are others. Weighted sums of the functions listed above were also used to effect a close fit to g’(a, t). The computed profiles g’(a, t ) differed in magnitude with , both the qualitative behavior and the the different functions ~ ( t )but quantity g”(u, 0) were relatively insensitive to the form of ~ ( t ) . For ,8 = E = 0, g’(u,t) has a maximum near t = 0.84 and decreases to about 12% of its maximum value at t = 2.0. The derivative g”(a,O) is approximately - 0.055. For ,B = 0, the term a@?)computed from Eq. (110) is 4,and

,8= E = 0,

fi”(0,O)a2X(a)g”(a, 0)

- 0.025

The Blasius solution to Eqs. (104) for ,8 = 0 is fo”(0,O)= 0.46960, and the final estimate of the correction to the skin-friction coefficient for ,B = E = 0 is

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS f”(0,O)=fo”(O,O) [l - 0.0536 +

* *

341

.]

The physical interpretation of this result agrees qualitatively with the sign of the correction. If E > 0, /3 is increasing and the local similarity value, fo”(/3, 0), would neglect “relaxation” effects and overestimate the shear at the wall. The correction reduces the skin-friction value by an amount proportional to E . The small magnitude (0.053) of the correction term is somewhat surprising, but this value is probably accurate to within about i20%.’ VI. Solutions for Large Values of the Pressure-Gradient Parameter p In using the concept of local similarity in highly accelerated flows, it is useful to have solutions of the laminar boundary-layer equations for large values of /3. The mathematical difficulties encountered in the limit /3+ 1may be illustrated by rewriting the streamwise momentum equation [Eq. (40)] in the form

B[(hf”)’+ff”] 1

-

(f’2 -

tw)9 -

Here X is redefined as the density-viscosity ratio at the wall (pp/pwpw). In the limit ,8 + a,Eq. (131) reduces from third to first order: 8+

m

The transverse-momentum equation and the energy equation remain of second order. As originally pointed out by Coles (52) and as later elaborated upon by Beckwith and Cohen ( 9 ) this leads to a singular perturbation problem in which the thickness of the velocity layer is of order /3-’/’ with respect to the total enthalpy layer and transverse velocity layers of order unity.2a The method of solution is similar to that employed in deriving a uniformly valid approximation to the Navier-Stokes equations in the limit of large Reynolds number [see Kaplun and Lagerstrom (54, Lagerstrom and Cole (55),and the recent book by Van Dyke (56)]. A. THEOUTERLIMITEQUATIONS Since the mathematical justification of this singular perturbation solution (more popularly called an inner- and outer-expansion procedure) has been 2 Bush estimated the correction term to be 0.204, but his result was obtained by asymptotically expanding an approximate solution rather than obtaining an approximate solution to the exact first-order equation. Zu Discussion of this problem also appears in an abbreviated version in Lagerstrom’s article (53).

342

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

discussed in detail by Coles (52),our purposes will be served by a merely cursory development of the governing equations. The present analysis extends the work of Beckwith and Cohen (9) to include a power-law temperature-viscosity relation, a constant but nonunit Prandtl number, and the mass transfer at the surface. Assume that appropriate outer representations of the dependent variables f , 8, and g are of the following forms:

f

=fo

+(wp)fl+

* * *

e = eo + (i/djj)el + -

(133) g =go + ( 1 / d p ) g l+ * * * Substituting these representations into the momentum equations and the and ~ m a l l e r the ,~ energy equations and dropping all terms of order F1l2 following “outer” equations are obtained : +.

1

1‘2

(134)

+fogol = 0

(136) The appropriate boundary conditions are found by (a) requiring that the outer equations satisfy the exact outer boundary conditions, and (b) by exact matching of the inner representations with the outer. The results may be written fo(0)=fw, eo(0) =go@) = 0, e o ( a ) =go(a) = 1 (137) Only one boundary condition onf o may be satisfied by the outer equations, because the outer limit equation forforeduces to first order. The condition3“ f’+ 1 as q + a is automatically satisfied by Eq. (134). The no-slip condition, f‘ + 0 as q + 0, must be satisfied by an inner solution which is valid in a region of extent j g - l i 2 with respect to the scale of the outer solution. in the In this approximation, the density-viscosity ratio X = (pp/prUpL,,) outer layer becomes (Ago’)’

3 T h e outer limit equations are properly obtained by applying the limit /3 + m to the full equations expressed in outer variables, with 7 held fixed. This gives identical results to those cited here. 3 O T h e exact boundary condition.

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

343

where

As v + 0, h + [(l - u2)/ts]w-1since fo’(0) = (tW/tJ1/* and fo‘(0)is nonzero in general.4 The energy and transverse momentum equations are coupled through the implicit appearance of both Bo andgo in Eqs. (135) and (136). If Pr = 1, 0, = g o , and the number of coupled ordinary differential equations is reduced from three to two. Setting the viscosity-temperature exponent w and the Prandtl number Pr equal to unity reproduces the equations of Beckwith and Cohen (9). If u2 = 0 (i.e., the local Mach number is zero), t, = 1, and the transverse momentum and energy equations are again uncoupled. One interesting case was pointed out by Coles and we extend his rzsult to generalized compressible flow. Let t, = t, = 1, A = 0 so that u2 = ( Um2/2He) ( U J U , ) ~ = ul.Then h = (1 - u ~ ) ~ andfa‘ -’ = 1 for all 7. In taking this limit with Pr # 1, the product of (&’) and (1 - tW)-’ approaches Oo’ so that the energy equation becomes

O,,” + Oo’(v+fw)Pr(l

-

0 ~ ) ’ - ~ [1 ul(l - Pr)]-l

with the boundary conditions

=0

(139)

eo+o

v+o,

eo+i

rl+co,

If we define the new variable x and the constants xo and W by

x = (v/v‘W)

-

xo

xo =- f w l r n W = [ l - u1 (1 - Pr)][Pr(l - ( T ~ ) ~ - ~ ] - - I then Eqs. (139) and (140) are satisfied by the solution

e = [erf ( x / d )+ erf (xo/z/Z)]/[l + erf (xo/zr2)1

(142)

For xo < 0, we note the identity erf

(-x) =

-

erf

(x)

(143)

Equations (134)-( 136) with the boundary conditions of Eq. (137) represent a two-point boundary-value problem for the complete solution of the outer limit equations. The derivatives Oo’(0) and go’(0) along with the 4 It should be noted that the inner and outer expansion procedure breaks down in the limit t , + 0, because the outer solution forfo satisfies the exact boundary conditions for the complete equations and the inner solution forf’ is simply zero.

344

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

integrals I t , 12,11(1),11(2),and 11(3)evaluated usingf,, do, andgo in place of

f,6, andg are given in Table VI. The integral IIis identically equal to zero. Beckwith and Cohen (9) calculated several of these quantities for the case of Pr = o = l,fw= 0, and nonunit values of t,.

B. THEINNER LIMITEQUATIONS Inasmuch as the order of the energy and transverse momentum equations is not reduced in taking the limit fl + co, the outer equations [Eqs. (135)(137)] represent complete solutions for the total enthalpy and transverse shear profiles for large /I. The termsfo and (fi2)’,which appear in the outer ’ in a region which equations, differ from the exact solutionsf and ( f 2 ) only smaller 2 in extent than the region of applicability of the outer equais F1j tions. Therefore, the outer equations asymptotically represent the complete solutions for 6 and g as /I + co, and in this limit the inner solutions for 6 and g are identically zero. The no-slip condition f‘(0)= 0 is satisfied by the inner limit equation forf; the inner equations for 6 andg, as noted previously, reduce to 6 = g = 0. T o examine the inner streamwise-momentum equation, it is necessary to introduce a new independent variable 75 and an inner representation for f: so that

q =4 3 %

+ (1/4F)f0(75) + .

f=fw

Jo’(?l)=fo‘(~)

*.

= U/%

(144) (145)

After rearrangement, the inner equation forfo’ becomes

and, dropping terms of order /Pand smaller, the result is where

(Aj?)’

- (&)2

+ tw/ts

=0

(147)

The boundary conditions onf;are found from the exact boundary conditions at the wall and the matching condition5 that the inner and outer representaa, with 75 large but fixed. The following tions off agree in the limit fl+ boundary conditions for the inner equation result:

Jo’(0) = 0,

o’(m) = ( t w / 4 ) 1 ’ 2

(149)

5 T h e matching condition applied here is elaborated by Van Dyke: inner representation of “outer representation” = outer representation of “inner representation.”

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

345

Since Eqs. (147) and (148) involve onlyfor and notfO, they represent a well-posed, second-order, two-point boundary-value problem. A more symmetrical form may be obtained by one final transformation : t = (tw/ts)”47j

Then the problem may be written (Aso’)’ - so2

+ 1 =0

with the boundary conditions s0(O) = 0,

and the definitions

h = [I

so( m)

=

1

(152)

- aso2]o-1

(153)

a = (I1 cos2(I/ts

(154) Values of sor(0) satisfying Eqs. (151)-(152) are given in Table VII for several values of w , (I, and For the special case of w = 1, Coles (52) pointed out that an analytic solution may be found in the form so = 1

3 sech2[t/dZ+ tanh-I

di]

(155) Using either the analytic solution for w = 1 or the numerical solutions for w # 1, the surface skin-friction derivativef,” is then found by reversing the previous transformations ; -

In general sor(0)depends on the three parameters 02,w, and t,; for w = 1, the value of so’(0) is $. VII. Discussion The large number of solutions listed in Tables I-VII suggests many possibilities for numerical correlations and comparisons with approximate results. Although an exhaustive discussion of these possibilities goes beyond our present purposes, we present selected examples in Figs. 4-14 of the use of the present solutions in understanding the influence of the similarity parameters on heat, mass, and momentum transfer. The local skin friction and heat transfer coefficients C,, and C, are defined by Eqs. (70) and (73). Following Dewey (ZO), they may be placed in a more symmetric form by using the Reynolds number

Re,, x

= Pe ue X I P e

(157)

Our thanks go to J. Avoesty of the RAND Corporation for suggesting a simple numerical quadrature technique for solving Eq. (1 51).

C. FORBES DEWEY, JR.,

346

AND JOSEPH

F. GROSS

and the dimensionless streamwise coordinate

With these definitions, the quantities CpLand C, become

+ (gw‘)2sin2A]

1/2

0.5

3.5

0.7

1 .o

1

tw = 0.6

3.0

2.5

w =

0.5 0.7

I .o

I

tw=0.15

2.0

1.5

I .o

0

0.5

1 .o

1.5

8

2.0

2.5

3.0

FIG.4. Variation of the skin friction coefficient with pressure gradient.

In Eqs. (159) and (160), only the terms appearing in the square brackets depend upon the similarity parameters ; the terms preceding the square brackets represent the external flow conditions and wall temperature. T h e Reynolds analogy function j = 2CdCf (161) is simply the ratio of the square brackets appearing in C,l and C, and consequently depends only upon the similarity parameters. In Fig. 4, the expected increase in the skin friction coefficient with increasing pressure gradient parameter /3 is shown. The relative increase in Cj/(Cj)ppois much greater for higher wall temperatures. This occurs

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

347

because the boundary layer is thicker and the response to increasing /3 is proportionately higher. Figure 5 shows similar behavior for the heat transfer coefficient. We have used the wall gradient parameter O,’[(l - tw)/(taw- t,)] in forming the heat transfer coefficient. As shown previously (4), 0,‘ varies greatly with /3 for

FIG.5. Variation of the heat transfer coefficient with pressure gradient.

t, > 0.6 because tawvaries with /3 if u # 0.6 The percentage increase here is less than for the skin friction but the trends with /3 and t, are the same. Lees (42) argued that the high density in the boundary layer near the wall for low wall temperatures “insulates” the wall from pressure gradient effects and the influence of /3 on C, is greatly reduced. This behavior is shown with decreasing values of t,. T h e next two figures deal with the Reynolds analogy function j = 2C,/Cf. Li and Gross (57)had earlier shown that large deviations from unity could occur in a hypersonic boundary layer even for t, e 1 and /3 = 0. Figure 6 demonstrates that j decreases with increasing /3, the decrease being faster with increasing wall temperature. This result can be easily explained by examining the limiting equations for /3 --f co. This heattransfer coefficient C, approaches an asymptote as /3 + co, whereas for large 6 It is particularly important to determine the proper adiabatic wall temperature in cases with mass injection because fa, is greatly decreased by injection (see Table 111). T h e assumption Pr = 1 predicts t., = 1 under all conditions, which is seriously in error for large injection and high local Mach numbers.

C. FORBES DEWEY,JR.,

348

AND JOSEPH

F. GROSS Pr = 0.7

f,=

0

t, = 1.0 QI

=o

t,=0.15

t W = 0.6

0.5

0

0.2

0.4

B

0.6

0.8

1 .o

FIG.6. Variation of the Reynolds analogy function with pressure gradient.

/l the skin-friction coefficient increases as /ll/*.The approach to this limiting

behavior is more rapid with larger wall temperatures. Similar behavior is shown in Fig. 7 for values of the hypersonic parameter u= 0, 0.5, 1.0.

0.15

0.60

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

349

Figure 8 shows the well known boundary-layer property that heat transfer is decreased by mass injection at the surface. The heat transfer reduction caused by a given value of fw decreases with increasing p. The reason will be explained below in the discussion on limiting solutions for large /3. Finally, the sweep angle parameter t, is varied in Fig. 9. When the sweep angle A is zero, there is skin friction only in the x direction [see Eq. (160)l. As the sweep angle increases, the span-wise term [g'(O) sin A] 1 .o

0.8

0.2

0

-0.2

-0.4

-0.6

-0.8

fw

FIG.8. Variation of the heat transfer coefficient with mass transfer.

contributes increasingly to the skin-friction coefficient. If the free-stream cos A], direction is sufficiently oblique, thex component of C,, [f"(0)(uc/um) no longer exerts an appreciable influence. The quantity [l +{g"(O) sin Alf"(0)cos

plotted in Fig. 9 is a relative measure of the contribution of the two skinfriction components, and approaches a limiting value as t, -to. Whalen(58) reported a similar result for displacement thickness for Pr = w = 1. I t is very difficult to obtain exact numerical solutions of the laminar boundary-layer equations for values of /3 greater than 2. The reason is simply that the singular behavior of f'(7) near the wall which exists for -+ m becomes dominant even for moderate values of 8. Conversely, the results obtained for p = m should be good representations of the behavior of the

C. FORBES DEWEY, JR.,

350

AND JOSEPH

F. GROSS

1 .o

-

0.9

E r4

6

-z

0.8

I

0

r, +

-"

-

0.7

C

0 m

+

c

0.6

u

1 .o

0.5

I

0

0.8

1

I

I

0.6

0.4

1

I

0.2

0

'I

FIG.9. Variation of skin friction coefficient with sweep angle. 1.25

Limit solutions

so

lo) = -3/4

1.20

1

1.151 1 .I5 0

I

I

I

I

II

I

I

0.5 a = -

u , cosl

.t

t S

FIG.10. Inner solutions for p

+ a.

I

'

I

'

1 .o

35 1

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

boundary layer for large but finite values of p. One of the important results that we wish to demonstrate is that by combining the exact numerical results for /3 < 5 given in Tables I-V and the limiting solutions for /3 -+ m given in Tables VI and VII, it is possible to estimate accurately the skin friction and heat transfer derivativesf,”, gWf,and 0,‘ for all positive values of /I. T h e skin friction results for i3 = m are displayed in Fig. 10. The influences of the two parameters w and a on the inner solutions for so’(0)are seen to be 2 .o

1.5

..-* C

0

L L

.C I

Y bl

1 .o

I

0

!

l

l

l

l

l

t

l

0.5 Limit parameter,

l

1 .o

l

l

l

l

1.5

-l/2

p

FIG.1 1 . Variation of skin friction with wall temperature for large 8.

relatively small. For 0.5 G w G 1.0 and 0 < a G 1.0, the value of so’(0) may be found by interpolation to better than 0.25%. The skin friction parameter fWff/3-’/* (t,/tJ-’/’’ approaches the limit of ~ ~ ’ as ( 06 )-+ m . T h e difficulties that were observed previously in calculating exact solutions for /?2 2 imply that this limit is approached very rapidly with increasing /I.This supposition is borne out by Figs. 11-13, where the skin friction parameter is shown as a function of p’/2. The limit parameter /3-1’z is suggested by the ordering procedure used to obtain the inner and outer equations. Solid lines indicate exact numerical solutions and dashed lines are extrapolations. Figure 11 illustrates the approach of the skin friction parameter to its asymptotic limit so’(0)for different wall temperatures, The limiting value is approached most rapidly for high wall temperatures. This result is to be expected from the behavior of the outer equations. Large values of t, increase the magnitude of the velocity difference across the inner layer,

352

C . FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

whereas for t, + 0, the distinction between the inner and outer layers breaks down and no proper limit is obtained. I t has also been found empirically that exact solutions are more difficult to obtain for large t, . The approach of the skin-friction parameter to its limiting value with increasing /?is illustrated in Fig. 12 for several values of the sweep parameter t, . From a numerical point of view, the accuracy of the present extrapolation procedure increases with increasing sweep. 2‘o

-

/

Limit parameter,

,3333

p-”’

FIG.12. Variation of skin friction with t , for large p.

I t is of interest to examine the behavior of the inner and outer equations with mass injection at the wall. Applications of these results include mass transfer cooling of rocket nozzles [Back and Witte, (59)]and blunt hypervelocity vehicles. The outer equations determine the heat transfer derivative 8,’ and they contain the boundary conditionfo(v -+ 0) =fw . Thus, surface mass transfer (f, < 0) acts to reduce 8,’ and consequently surface heat transfer even in the limit13 + a.On the contrary, the skin friction derivative fw” is found from the inner equation forfo’(0), Eq. (147), and the solution of this equation is independent of the value off,. In highly accelerated flows, therefore, the effects of blowing on skin friction become negligible in the limit + 00 with f, fixed. Figure 13 illustrates the behavior of the skin friction parameter as a function of the injection parameter 5”.The limit is approached smoothly for all J c , and accurate estimations of ffU” with decreasing values of /Pg-li2

353

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS 1.6-

l

I

I

-

4

1.4

-

/-. --------

I

s

IQ

:a L

1.2

-

1

1.0

.C

..d

-\

-

C

p--o:

1.1547 /

0.6’ 0

-

-

R = 0=0.7 t, = 0.6

-

a2=0

-

0.8 -

-

-

-0.4

-

c

t

Limit

/ -

t, = 1

’

I

’

I

’

0.5

I

I

I

’

1 .o

’

I

I

FIG.14. Variation of heat transfer with wall temperature for large B.

’

1.5

354

C. FORBES DEWEY, JR., AND JOSEPHF. GROSS

may be obtained for all ,6 by comparing the exact solutions for ,6 < 5 and the inner limit solutions of Fig. 10. The wall heat transfer is related by the modified Stewartson and HowarthDorodnitsyn transformations (Eqs. (10) and (11)) to 9,'. Inasmuch as go and go represent the complete solutions for 9 and g as ,6 + co, the values of %,'(O) and go'(0) represent the asymptotic limits of 6,' and g,' as ,6+ 00. I t should be emphasized that 9,' and g,' become independent of /3 as ,6 + co, demonstrating that the heat transfer predicted by a local similarity analysis in highly accelerated flows approaches a limiting value. Figure 14 shows the typical behavior of the heat-transfer derivative 6,' with increasing values of ,6. The approach of 9,' to its limiting value for ,6 + co, is seen to be smooth and (at least for the case of u1 = 0) monotonic. In an earlier paper (4,we demonstrated that the proper parameter to use in comparing different heat transfer calculations is %,'[(l- t,)/(t,, - t,)]. For u1 = 0 and t, = 1, the adiabatic wall temperature tawis unity for all Pr so that the heat transfer parameter reduces to 6,'. Although we have not been able to prove it analytically, it appears that tau = 1.0 for all values of Pr, t,, w , and u in the limit ,6 = co. This is a very surprising result and should be examined further. The boundary-layer displacement thickness 6" is defined by the relation

where the integrals I,and I 2 are given in the list of symbols. For ,6 = co, the integral I l is identically zero and the integral I z [Eq. (75)]

is recorded in Table VI. In the absence of sweep, the velocity profile is monotonic and 0 1 for some range of 7 and I 2becomes negative. With sweep, therefore, the displacement thickness may be either positive or negative, depending upon the particular parameters being considered. Numerical comparison between the calculated results for moderate /3 and the present results for ,B = a, suggests that 6" monotonically decreases with increasing 8. In the special case when ,6 = m and t, = t, = 1,f' = 1 and 6" is of the order of /3-'" times the scale of the boundary displacement layer thickness for ,6 = 0. The numerical results given in Table V display the variation of boundarylayer properties for various Prandtl numbers near unity. These results may be used to estimate recovery factors for Pr > 1 by combining the present

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

355

results with the asymptotic solution of Narasimha and Vasantha (60) for Pr s 1. Although these authors explicitly solved the problem of a flat plate (/I= 0) with o = 1 and low Mach numbers, there is no reason why a similar analsyis could not be conducted for /I > 0 and general compressible flow. As Narasimha and Vasantha demonstrate, interpolations accurate to better than 3% in the recovery factor can be made between first-order asymptotic solutions for Pr s 1 and exact numerical results for Pr of order unity. I n concluding this discussion, it is advisable to point out that the numerical values listed for the wall derivativesf,”, Ow’, and g,’ allow reconstruction of the complete velocity and total enthalpy profiles by standard numerical integration techniques. Whereas the boundary layer equations themselves, including the boundary conditions at 7 = 0 and 7 --f co, represent a twopoint boundary-value problem, one may solve the equations as an initialvalue problem if the wall derivativesf,”, Ow’, andg,‘ are known. Furthermore, by using the wall derivatives given here, multi-term expansions of the boundary layer equations may be constructed for small q.This property is useful in cases where additional “nonclassical” behavior occurs near the surface, for example, in MHD boundary layers where an electrostatic potential sheath exists for small 7.

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS KEY I SIMILAR SOLUTIONS FOR

w = Pr =

357

1 t

8

t

o

all

X

X

I'

X

X

I'

X

X

"

X

X

I'

X

X

"

X

X

'I

X

X

-0.8485

'I

X

X

-0.8755

I'

X

X

f"

0.0 (a)

0.05 0.1 0.2

0.0 -0.1414 -0.2828 -0.4243 -0.5657 -0.7071 -0.7782

0.0 0.0 0.0

0.25 0.2857

0.3 0.4

0.0 0.0 -0.2 -0.4 -0.6 0.0 0.0 -0.2 -0.4 -0.6

0.5

0.0

-0.5

0.75

0.0

("All

1.0 1 .o 0.1539 0.3333 0.6250 1.0 0.1000 1 .o 1 .o 1.0 1 .o 1 .o 1.0 1 .o

1 .o 1 .o

0.1000 0.1539 0.3333 0.6250 1 .o 0.1539 0.3333 0.6250 1 .o 0.1000 0.3333 1.0

0.15

0.2

0.b

0.5 0.6

0.8

1.0

X

X

X

X

X X

X

x x x

X

X

X

X

X

X

X

X

X

x

X

X X X

X X

X X

x x x x

x x x x

x x x x

x x x x

x x x

x x x

x x x x x

x x x x x

x x x x x

X

x

X

x x x x x x x x

x

X

x x x x

X

x

x

x

X

X

X

X

X

X

X

X

x

x

x

x

X

X

X

X

X

X

X

X

X

x x x

X

X

X

X

solutions for Pr =

x ~JJ E

x

1 and

B

x

= 0 are

2.0

x

linear in tW.

X

X

X

C. FORBES DEWEY, JR.,

358

AND JOSEPH

F. GROSS

KEY I (Continued) -

8

fw

1 .o

0.0

-0.5

-1.0

I *4

0 .o

1.5

0.0

1.8 2.0

2.4

2.8

3.4 4 .O 5.0

0.0 0.0

0.0

0.0

0.0

0.0 0.0

t

0.1000 0.1539 0.2500 0.3333 0.5000 0.6250 1 .o 0.1539 0.3333 0.6250 0.8333 1 .o 0.1539 0.3333 0.6250 1.0

o

t"

0.15

9.2

0.4

0.5

0.6

0.8

1.0 X

X

X

X

X

X

X

X

X X

X

X

X

X

x

X

X

X

x

x

x

x

x

2.0

x

x

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

x X

0.1000

0.1539 0.3333 0.6250 1.0 0.1000 0.1000 0.1539 0.3333 0.6250 1 .o 0.1539 0.3333 0.6250 1 .o 0.1000 0.1539 0.3333 0.6250 1 .o 1 .o 1.0 1.0

X

X

X

X

X

X

X

x

X

X

x

x

x

x

x

x

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

x X

x

x

x

x X

x

x

x X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X X

X

X

X

X

X

X

X X

X

X

X

X

X

X

X

X

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

359

KEY I1 SIMILAR SOLUTIONS FOR A POWER-LAW VISCOSITY RELATION, Pr = O , f , , = 0

0 .o

0.5

0.1

1 .o

0.1

0.2

all

all

all

0.0 0.5 1.0 0.0 0.5 1.0

...

... ... ... ... ... ... ... ... ...

x x x

x x x

x x x

x

x x x

x x x

x x x

x

x x x

x x x

x x x

x

x

x

x

0.5

1 .o

1.0

0.1

0.3333

0.5000 0.6250 1.0

1.0 1.0 1.0 1.0

1 .o

1.0

1.0

0.5

0.3333

1.0 1.0 1.0

1.0 1.0

1 .o

1.0 1.0 1.0 1 .o

1.0 1.0 1.0

...

x

0.3333

1.0

1.0 1.0

...

X

X

0.1

0.6250 1 .o 0.3333 0.5000

0.6250 1 .o

0.2857

0.0

0.5 1 .o

0.5

0.1

1 .o

1.0 1.0 1.0

...

... e . .

X X

X

X

X

X

X X X

X

X

X

X

X

X

X

x

x

x

x

x

x

X

X

X

X

X

x

X

X

x

X

x

X

X

X

X

X

X

X

X

x

X

x

X

X X

X

0.6250 1.0

1.0 1 .o

0.3333

1.0 1.0 0.0 0.5 1.0

1.0 1.0

... ... ...

X

1.0 1.0 1.0 1.0

1.0 1.0 1.0

X

X

0.5000 0.6250 1 .o

X

X

1 .o

1.0

0.6250 1 .o

0.3333

... ...

x

X

x

X X X

X

X

X

x

X

x

x

X

X

X

X

x

x

x

x

x

x

X

X X

C. FORBES DEWEY, JR.,

3 60

AND JOSEPH

F. GROSS

KEY I1 (Continued)

I .05 0 04

0.5

0.3333

0.6250 1.o

0.1

0.3333

1.0

1.0

0.0 0.5 1.0

.1

I

0.0

1 .o

0.3333 0.6250

1.0

0.5

0.5

0.3333 1 .o

0.15

1.o

-

1.1

taw

--

X

X

X

X

X

X

X

1.0

X

0.0 0.5 1.0

... ...

X

...

X

1.0

0.0

1.0

0.0 0.5

X

x

X

0.0 0.5 1.0

... ... ...

1.0 1.0

1.0 0.0 0.25 0.5

0.5

0.0 0.5 1.o

0.0 1.0

0.5

1.0

0.5

0.1

1 .o

0.0 0.5

... ... ... ... ... ... ...

X

x

x

X

X X

X

X

X

X

X

X X

X

X

x x

X

x

x

x

x

X X

X

X X

X X

X

x

x

x

x

x

x

x

X X

X X

x

X

X

X

x

X

X

x

X

x

x x x

x x x

x x x

x

x

x

x

x

x

x

x

X

x

x

x

X

x x

x x

x x

X X

X

X X

x

X

x X

x x

X

x

X X

X

x

X

X

X

... ... ... ... *.. ... ... ... ... ...

X

X

0.5

1.0 1.0 0.0 0.5 1 .o

1.o

1 .o

1.0

X

0.8

1.o

.6

X

0.6

0.1

.4

X

1.0 1.0

.15

0.5

1 .o

0.6250

-

tW

X X

x

X

x X

X

X

X

X

X

X

X

x X

X

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

361

KEY I1 (Continued)

I

t

1 .o

0.5

0.7

1.0

0.3333 0.6250 1 .o

0.3333 0.6250 1 .o

1.0 1.0 0.0 0.25 0.5 0.6 0.: 1 .o

0.0 0.0

1.0 1.0 0.0 0.5 0.9 1 .o

0.0 0.0

1.0

0.0

0.1539 0.3333 0.6250 1 .o

1.0 1.0 0.0

0.6

1.4

0.7

1.0

1.5

0.5

1.0

0.0 0.25

0.7

1.0

0.5 0.0

1.8

0.5

1.0

0.6

2.0

0.5

1.0

0.7

1.0

0.0 0.25 0.0

3.0

0.5

0.7

1.0

1.0

0.0 0.0

...

... ... ... ... ...

... ... ... ...

X

X

X

X

X

X

X

x

X

X

X

X

x x x x

X

X

X

X

X

X

X

x x x x

x x

x x

X X

x

x

x

x

X X

X

x

x x x x

x

x x x x

X

x

X

x X

X X

X

X

X

X

X

X

X

x

x

x

x

x

x

x

x

x x

X

x

X

X

X

x

X X

X

x X

X

x

...

x

x

x

.*.

X

x

x

x

x

X

X

X

x

x

x

x

x X

x

x x

x x

x x

x

X X

X

... ... ... ...

x

X

0.0 0.0

... ... ... ... ... ...

x

X

X

7

C. FORBES DEWEY, JR.,

362

AND

JOSEPH F. GROSS

KEY I11 S I M I L A R SOLUTIONS FOR

Pr= 0.7, f, # 0, ts = 1

tw

05 0. 15 0.0

0.5

0.0 -0.2 -0.4

-0.6 0.7

0.0

1 .o 1 .o

1 .o 1.o

-0.4

0.5

0.0

0.7

0.4

0.5.

0.0

X

0.5 1.o

X

X

X

X

0 .o 0.5 0.0 0.5

0.7

1 .o

0.0

1.o 1.o

-0.6 0.0

-0.2

-0.4

X

X

X

1.o 1.o 1.o 1.o

-0.4

X

X

0.0 -0.2 -0.4 -0.6 -0.2

X

X

-0.6

-0.4

X

X

X

1.o 1.o 1.o

-0.2

X

X

1.o

0.2

aw -

0.0

1.o

-0.6

I. 0

0.5

1.o

-0.2

0.4 0. 6

1.o

1.o

0.0 0.5 1.o

X X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

X

X

0.0 0.5 1.0

X

0.0

X

0.5 1 .o

X

X

X

X

X

X

X

X

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

363

KEY I11 (Continued)

P

UI

0.4

0.7

fw

01

-0.6

0.0 0.5

.G5

X

1 .o

0.5

0.7

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2

-1.4 0.75

0.7

0.0 -0.2

-0.4 -0.6 1.0

0.5

0.0

-0.2

-0.4 -0.6 0.7

1.0

0.0 -0.2

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5

0.0 0.0 0.0 0.0 0.0

-0.4 -0.6 -0.8 -1.2 -1.4 -1.6

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2

0.0 0.0 0.0 0.0 0.0 0.0 0.0

tw 0.15 9.4 @,h 1.0

X

X

x

x

x

x

X X

X

X

X

X

X X

'

X

x

x

x

x X

X X

X

x

x

x X

X

X

X

X

X

X

X

x

x

X

X

X

X

X

X

X

x

x

x

X

X

X

X

X X

X

x

x

x X X X X

X

X

X

X

X

X

X

X

X

X

X X

X

X

X

-1.4 -1.5

0.0 0.0

X

X

-1.6

0.0

X

X

364

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

KEY I11 (Continued)

P

W

2.0

0.5

fw

91

0.0

0.0 0.0

-0.4 0.7

-0.2

-0.4

5.0

0.5

0.0 0.0

-0.6

1 .o 0.0

-0.2

1.0

-0.6

1.0

X

X X

x x

x x

x

X X I

I

365

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

P 0.0

01 0.5

0.0 0.5 1.0

0.7

0.0 0.5 1.0

1.0

0.0 0.5 1.0

Pr 0.7 0.5 0.7 0.5 0.7 1.o

0.7 0.5 0.7 1 .o

-5

.6

- 8 1.0

1.1

2.0

taw

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

x

X

X

X

X

X

X

X

X

x

X

X

X

X

X

X

X

X

0.5

X

X

X

X

0.7

X

X

X

X

X

X

X

X

X

X

X

1 .o

0.1

0.15

0.1

0.0

0.7

0.2

0.5

1.0 1.0 0.0 1.0

0.7

X

0.0 0.5 1.0 1.0 0.0 1.0

0.7

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X X

X

X

0.7 0.7 0.7

X X

X

1.0 0.7

X

1.0

X

X

X

X

X

x X

x

*Notes: a) Solution 6 for Pr = 1 apply for all 01 b) For PI linear ii

X

X

0.7

1.0 0.7

X X

1.0

0.7

X

X

0.7

1.0

X

X

X

0.5 0.7 1 .o

1.0 1.0 0.0

X

X

0.7

0.5 0.1 1.0

0.0

.4

X

0.10

1.0

.2

X

1.0

0.3

.15

X

0.0

0.7 1.0

.1

X

1.0

0.2057 0.5

X

0.7 1.0

0.05

0.7 1.0

0 .05

X

X

X

X

x

x

X

X

x

x

x

X

X

X

X

x X

x

x

X

C. FORBES DEWEY, JR.,

366

AND JOSEPH

F. GROSS

KEY IV (Continued)

0 .05

0.4

0.5

0.1

0.0 0.5 1.0

0.0 0.5 1.0

1.0

1.0

0.5

0.0

0.5

0.1

0.25 0.5 0.6 0.8 1.0

0.1 1 .o 0.1 0.1 0.7 0.1 0.1

0.0

0.5

0.5 1.0

1.0

0.0

1.0

0.15

1.0

0.5 0.1

0.5

X

0.0

0.5

X

2.0

X

X X

x

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

x

x

X

aw

X

X

X X

X

x

x X

X

X

t

X

x

X

-

X X

X

x

x X

X

X

X

x

X

X

X

X

X

X

X X

X

X

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

x

X

x

X

x

x

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

X

X

x

x

X

X

X

x

X

X

X

X

X

X

X

X

X

X

x

x

x

x

x

x X

X X

X

X

X

x X

X

0.7

X

X

X

X

0.7

X

X

X

0.25

X

x

X

0.5 0.6

X

x

X

X

x

X

x

x

X

X

x

X

X

0.8 1.0

1.1

X X

X

0.1 1 .o

0.1 0.1 0.7

.8 1 . 0

X

0.1

0.5

.6

X

X

0.5 0.1

1.0

.5

X

0.1

0.7 0.7 1.0

.4

X

X

0.5

0.0 0.5 0.0

.2

X X

0.1 1 .o 0.1

0.5 0.1 1 .o

.15 X

0.1 0.7 0.1 0.1 1 .o 0.1 0.1

0.0

0.5 0.5

.0.7 0.1 0.1

.1

-

x X

X

X X X

X

X

X

X

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

367

KEY IV (Continued)

B 1.0

(u

0.7

1.0

1.4

1.5

1.8

2.0

b,

Pr 10 .o5

0.0

0.5 0.7 I 1.0 ~x

0.5 0.9 1.0

0.7 I 0.7 0.7

0.0

0.5 0.7 1.0

0.5 0.6 0.8 0.7 0.6

0.7 0.7

0.5

0.7

0.0

0.25

0.7 0.0 1.0 0.0

0.7 1.0

0.5

0.6 0.8

0.7

0.0

0.5 0.7

0.7

0.7

0.0

0.5

.4

.5

X X

-6

1.0

1.1

X

x

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

X

x

x

x x

.8

2.0

X

X

X

X

x x

X

X

X

x x

X

X

X X

x

x

X

X X X

X

X

X

X

X

X

X

X

x

X

X

X

X

X

x

X

x

0.7

0.25

.2

I

0.7

0.7

.15

X

0.7

0.5

0.5

.1

X

X

X X X

X

x

x

x

X

x

x

X

X

x

x

X

X

X X

X

X

X

X

X

X

X

X

X X

X

X

X

X

X

X

X

X

X

X

X

0.5

0.7 0.5

1.0

0.0

1.0

X

2.4

1.0

0.0

1.0

X

X

X

2.8

1.0 0.0

1.0

X

X

X

3.0

0.5 0.7

0.0 0.0

0.7 0.7

3.4

1.0

0.0

1.0

X

X

4.0

1.0

0.0

1.0

X

X

X

5.0

1.0

0.0

1.0

X

X

X

X

x

X

x

x

x

x

x

x

X

X

X

X

X

X

X

X X

X

C. FORBES DEWEY, JR.,

368

AND JOSEPH

F. GROSS

tW

B

0.0

S

0.01

fw

u1 0 . 0 5 0 . 1 5 0 . 2 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8

1.0

0.0

0.0

(h)

-0.2 0.03 0.05 0.1

0.3

-0.6 0.0

0.01 'Two

X

X

X

X

X

X

1.0

X

X

X

0.0

X

X

X

X

0.0

x

0.5 1.o

X

x

0) (1))

x

(b)

X X

X

X

X

-0.2 -0.6

0.0 0.0

X

X

X

X

X

X

0)

1.0

X

X

0.0 0.0 0.0 0.0 0.0 - 0 . 2 0.0 -0.6 0.0

X

X

X

(I>)

X

X

X

X

X

X

0.0 0.9 -0.2 1.0 0.0 0.0 0.0 0.0 0.0 0.0 - 0 . 5 0.0 0.0 0.0 0.9

0.0 0.0

taw

X

0.5

0.0 -0.2 -0.6 0.0 0.0

2.0

x

X

0.0

0.3

X

x

0.01

0.05 0.1 0.2

X

X

0.05

0.01

X

X

0.5

1.0

0.0 0.0

X

X

0.0 1.0 0.0 0.0

0.4

0.02 0.3

X

0.5 1 .o

1.1

(b )

X

(h)

0))

X X

X

X

X

X

X

X

X

X

X

X

(h) (1))

(b) X

X X

X

X

X

X X

x X

X

X X X

s e t s of answers are given f o r each c a s e .

The f i r s t s e t

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

369

KEY VI SOLUTION OF

THE

ic

fw

t

3

0.5

0.0

0.1000

0.9 1.0

0.3333

0.9 1.0

OUTERLIMITEQUATIONS FOR /3 +a,

(?I

2

1.0 0.0

0.5 1.0 0.0

0.5 1.o

0.9 1.0

1.0

0.9 1.0

1.0 0.0

0.5 -0.2

1.0 0.0

0.5 -0.4

1.0

0.9

1.0

0.7

0.0

-0.2

-0.4

1.0

0.0

1.0 0.0

Pr 0.7 0.7 0.7 0.7

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

0.5

0.7 0.7 0.7

1.0

0.7

1.0

0.c 0.5

0.7 0.7

0.3333

0.9 1.0

1.0 0.0 0.5

0.7 0.7 0.7

1.0

0.9

1.0 0.0

0.7

1.0

0.5

0.7

0.9 1.0

1.0 0.0

0.7

0.1000

1.0

1.0

0.1000 0.1536 0.3333 0.6250 1 .o

0.9

0.9

1.0

1.0 1.0

1.0 1.0

1.0

0.7

0.5

0.7 0.7

1.0

0.7

0.0

0.5

0.7 0.7

1.0

1.0

X

X

1.0

1.0

X

X

1.0

1.0 1.0 1.0

X

X

X

X

X

X

1.0 1.0

6)An analytic solution is available for the special case =; t

ts= 1

w

TABLE I

0

SIMILAR SOLUTIONS FOR w = Pr = 1

t

B 0

I

t

ts

W

fll(o)

e’(o)

= gl(o)

I~

I2

I

1

0 1 .o

-0.1414

1

- 0.2 82 8

0 1.0

1

0

-0.4243

1

-0.5657

1

0

, ~

0.4696 0.4696

0.4696 0.46%

0.4696 1.686

0.4696 0.4696

1.686 1.686

1.686 1.686

1.217 1.217

0.3700 0.3700

0.3700 0.3700

0.5114 1.896

0.5114 0.5114

1.896 1.896

1.896 1.896

1.385 1.385

0.2766 0.2766

0.2766 0.2766

0.5594 2.163

0 -5594

0.5594

2.163 2.163

2.163 2.163

1.603 1.603

0 1 .o

0.1907 0.1907

0.1907 0.1907

0-6150 2.516

0.6150 0.6150

2.516 2.516

2.516

2.516

1.901 1.901

0 1 .o

0.1143 0.1143

0.1143 0.1143

0.6800 3 -021

0.6800 0.6800

3.021 3.021

3.021 3.021

2.341 2.341

1 .o

‘ 1

-0.7071

1

0 1.0

0.0502 0.0502

0.0502 0.0502

0.7573 3.862

0.7573 0.7573

2.862 3.862

3.862 3.862

3.105 3.105

-0.7782

1

0 1 .o

0.0243 0.0243

0.0243 0.0243

0.8025 4.628

0.8025 0.8025

4.628 4.628

4.628 4.628

3.825 3.825

-0.8485

1

0 1 .o

0.0048 0 -0048

0.0048 0.0048

0.8533 6.404

0.8533 0.8533

6-403 6.403

6.403 6.403

5.550 5.550

-0.8755

1

0 1 .o

0.0003 0 0003

0 -0003 0.0003

0.8621 9.591

0.8621 0.8621

9.591 9.591

9.591 9.591

8.729 8.729

-

0

?i

n

:Tf 0

arn P

TABLE I (Continued)

B

fw

tw

ts

0.05

0

1

0 0.5 1 .o

0.4848 0.5082 0.5311

0.4733 0.4773 0.4812

0.4453 1.023 1.593

0 -4626 0.4570 0.4514

1.675 1.663 1.652

1.654 1.623 1.593

1.208 1.199 1.191

0.1

0

1

0.15 0.4 0.6 1 .o

0.5123 0.5347 0.5523 0.5870

0.4788 0.4825 0.4854 0.4909

0.5910 0.8668 1.085 1.516

0.4532 0.4479 0.4439 0.4354

1.658 1.647 1.639 1.624

1.607 1.579 1.558 1.516

1.196 1.188 1.182 1.176

0.2

0

0.1539

0

0.5 1 .o

0.7839 1.222 1.611

0.5349 0.5845 0.6221

0.3600 0.7895 1.180

0.3158 0.1958 0.0777

1.503 1.393 1.320

1.101 0.7174 0.4077

1.081 0.9999 0.9456

0.3333

0 0.5 1 .o

0.6238 0.8506 1.058

0.5039 0.5344 0.5593

0.3761 0.8479 1.287

0.3981 0.3418 0.2855

1 585 1.508 1.450

1.383 1.154 0.9607

1.142 1.084 1.042

0.625

0 0.5 1.0

0.5543 0.6836 0.8047

0.4891 0.5082 0.5248

0.3842 0.8817 1.355

0.4314 0.4014 0.3712

1.627 1.576 1.533

1.515 1.373 1.247

1.173 1.134 1.103

1

0 0.5 1 .o

0.5233 0.6070 0.6867

0.4821 0.4951 0.5069

0.3881 0.8995 1.392

0.4457 0.4271 0.4082

1.648 1.612 1.580

1.576 1.480 1.392

1.188 1.161 1.138

0.1

0

1.017 1.756 2.401

0.5737 0.6425 0.6921

0.3304 0.7123 1.054

- 0.0248

0.1905

1.412 1.280 1.199

0.7389 0.1840 -0.2584

1.014 0.9169 0.8573

0.25

0

0.5 1 .o

-0.2332

-

w -4 to

TABLE 1 (Continued)

B 0.2857

t f "

tw

0

1

0 0.2 0.4 0.6 (L. 8 1 .o

0.5419 0.5883 0.6334 0.6774 0.7205 0.7627

0.4862 0.4932 0.4998 0.5061 0.5121 0.5179

0.3629 0.5593 0.7517 0.9406 1.126 1.309

0.4382 0.4285 0.4186 0.4087 0.3987 0.3887

1.636 1.617 1 * 599 1.582 1.566 1.551

1.542 1.491 1.443 1.396 1.352 1.309

1.179 1.165 1.151 1.139 1.127 1.116

-0.2

1

0 0.2 0.4 0.6 0.8 1 .o

0.4032 0.4547 0 5042 0.5520 0.5983 0.6434

0.3511 0.3604 0.3690 0.3769 0.3843 0.3913

0.3929 0.6183 0.8367 1.049 1.257 1.460

0.4909 0.4781 0.4651 0.4522 0.4392 0.4262

1.910 1.875 1.843 1.815 1.789 1.766

1.793 1.716 1.646 1.580 1.519 1.460

1.400 1.373 1.348 1.327 1.307 1.289

-0 -4

1

0 0.2 0.4 0.6 0.8 1.0

0.2758 0.3335 0.3876 0.4392 0.4886 0.5362

0.2301 0.2425 0.2534 0.2631 0.2720 0.2802

0.4252 0.6877 0.9379 1.178 1.411 1.637

0.5543 0.5370 0.5197 0.5025 0.4854 0.4684

2.279 2.209 2.151 2.102 2.059 2.021

2.133 2.009 1.901 1.805 1.717 f .637

1.707 1.651 1.605 1.566 1.532 1.501

-0.6

1

0

0.1629 0.2279 0.2867 0.3414 0.3931 0.4426

0.1273 0.1436 0.1568 0.1682 0.1782 0.1873

0.4591 0.7712 1.060 1.332 1.593 1.843

0.6318 0.6076 0.5839 0.5608 0.5381 0.5159

2.810 2 -656 2.545 2.457 2.385 2.324

2.628 2.402 2.227 2.081 1.955 1.843

2.169 2.038 1.945 1.872 1.812 1.762

0.2 0.4 0.6 0.8 1 .o

-

TABLE I (Continued)

B 0.3

0.4

t

fW

0

W

1

0 0.2 0.4 0.6 0.8 1 .o 2.0

0.5448 0.5931 0.6402 0.6860 0.7309 0.7748 0.9829

0.4868 0.4941 0.5009 0.5074 0.5136 0.5195 0.5457

0.3591 0.5539 0.7446 0.9318 1.116 1.297 2.165

0.4371 0.4270 0.4168 0.4065 0.3961 0.3857 0.3334

1.634 1.614 1.596 1.578 1.562 1 547 1.484

1.537 1.484 1.434 1.386 1.341 1.297 1.099

1.178 1.163 1.149 1.136 1.124 1.113 1.066

-

0

1

0 0.2 0.4 0.6 0.8 1 .o

0.5639 0.6254 0.6850 0.7429 0.7993 0.8544

0.4908 0.4997 0.5079 0.5157 0.5231 0 5300

0.3350 0.5198 0.7001 0.8765 1.049 1.219

0.4299 0.4175 0.4050 0.3923 0.3795 0.3667

1.623 1.599 1.577 1.557 1.538 1.521

1.505 1.441 1.381 1.325 1.271 1.219

1.170 1.152 1.135 1.120 1.107 1.094

-0.2

1

0 0.2 0.4 0.6 0.8 1 .o

0.4246 0.4924 0.5571 0.6194 0.6795 0.7378

0.3565 0.3681 0.3785 0.3881 0.3970 0.4052

0.3591 0.5689 0.7716 0.9680 1.159 1.346

0.4809 0.4648 0 4485 0.4322 0.4157 0.3993

1.889 1.846 J.809 1.776 1.747 1.720

1.743 1.649 1.565 1.487 1.414 1.346

1.384 1.351 1.322 1.297 1.275 1.254

0 0.4 0.6

0.2959 0.4413 0.5077 0.5711 0.6321

0.2363 0 2643 0.2757 0.2861 0.2956

0.3842 0.8547 1.074 1.286 1.491

0.5423 0.4994 0.4780 0.4567 0.4357

2.242 2.096 2.042 1.995 1.954

2.062 1.791 1.681 1.582 1.491

1.678 1.561 1.518 1.480 1.448

-0.4

1

0.8

1 .o

-

-

w

TABLE I (Continued)

P

0.4

0.5

-0.6

0

1

0.2 0 .& 0.6 0.8 1 .o

0.1809 0.2648 0.3400 0.4097 0.4756 0.5384

0.1340 0.1532 0.1686 0.1816 0.1931 0.2034

0.4094 0.6923 0.9524 1.197 1.431 1.656

0.1

0 0.5 1 .o

1.365 2.600 3.661

0.6211 0.7113 0.7739

0.1539

0 0.5 1 .o

1.089 1 966 2.728

-

0.3333

0 0.5 1 .o

-

0.625

1

0.6171 0.5878 0.5590 0.5307 0.5031 0.4760

2.741 2.572 2.453 2.362 2.288 2.225

2.520 2.267 2.074 1.915 1.778 1.656

0.2704 0 5882 0.8688

0.0133 -0.3417 -0.6827

1.315 1.169 1.085

0.2987 -0.4635 -1.075

0.9427 0.8355 0.7738

0.5828 0.6577 0.7111

0.2827 0.6257 0.9314

0.1703 -0.0671 -0.2988

1.393 1.256 1.172

0.6761 0.0812 -0.3942

1.000 0.8985 0.8374

0.7812 1 248 1.662

0.5328 0.5833 0.6215

0.3000 0.6856 1.037

0.3311 0.2195 0.1073

1.509 1.398 1.324

1.138 0.7649 0.4623

1.086 1.003 0.9480

0 0.5 1.0

0.6438 0.9167 1.165

0.5070 0 * 5410 0.5684

0.3097 0.7244 1.110

0.3961 0.3371 0.2769

1.577 1.494 1.432

1.366 1.122 0.9165

1.136 1.074 1.028

0 0.15 0.2 0.4 005 0.6 0.8 1 .o 2.0

0.5811 0.6368 0.6550 0.7262 0.7609 0.7952 0.8623 0.9277 1.235

0.4942 0.5020 0.5045 0.5140 0.5185 0.5228 0.5311 0.5390 0.5729

0.3146 0.4474 0.4909 0.6625 0.7469 0.8301 0 9941 1.155 1.920

0.4238 0.4131 0.4095 0.3949 0.3876 0.3802 0.3653 0.3503 0.2746

1.613 1.592 1.586 1.561 1 * 550 1.539 1.519 1.500 1.424

1.477 1.422 1.404 1.337 1.304 1.273 1.212 1.155 0.8988

1.162 1.147 1.142 1.124 1.115 1.107 1.092 1.078 1.022

0

1

-

-

2.111 1.968 1.869 1.793 1.732 1.681

r

0

z m

8

r

TABLE I (Continued)

0

fW

0.5

-0.5

t

tW

0.1539

0

0.6921 1.651

0.2956 0.3896

0.3132 0.7224

0.1742 -0.1967

1.846 1.550

0.3333

0 0.5

0.4265 0.9556 1.395

0.2377 0.3081 0.3540

0.3394 0.8202 1.233

0.4173 0.2252 0.0458

2.117 1.814 1.660

1.548 0-8506 0.3782

1.588 1.346 1.225

0.625

0 0.5 1 .o

0.3067 0.6290 0.9003

0.2049 0.2581 0.2941

0.3560 0.8954 1.365

0.5219 0.4127 0.3081

2.312 2.028 1.874

1.974 1.428 1.060

1.744 1.515 1.392

1

0 0.5 1 .o

0.2512 0.7414 0.6594

0.1873 0.2290 0.2583

0.3654 0.9464 1 -460

0.5685 0.4981 0.4294

2 -433 2.179 2.031

2.209 1.764 1.460

1.843 1.635 1.517

0.1

0 0.5 1 .o

0.6515 0.7549 0.8254

1.260 1.109 1.024

0.0149 -0.8868 -1.617

0.9023 0.7914 0.7294

0 0.5 1.0

0.5485 0.6089 0.6534

0.2329 0.5136 0 * 7599 0.2605 0.6046 0.9152

-0.1147 -0.5742 -1.015

0.3333

1.632 3.278 4.685 0.8796 1.509 2.062

0.2935 0.1487 0.0030

1.472 1.347 1.268

1.011 0.5673 0.2101

1.058 0.9655 0.9067

1

0 0.15 0.4 0.5 0.6 1.0

0.6181 0 6948 0.8173 0 * 8648 0.9112 1.090

0.5012 0.5112 0.5262 0.5318 0.5371 0.5568

0.2748 0.3953 0.5896 0.6654 0.7402 1.031

0.4120 0.3984 0.3751 0.3656 0.3560 0.3174

1.594 1.568 1.531 1.517 1.504 1.460

1.423 1.355 1.250 1.211 1.173 1.031

1.148 1.129 1.101 1.091 1.081 1.048

0.5

1 .o

0.75

0

-

0.8092 -0.1245

1.373 1.140

w

cn

w

TABLE 1 (Continued)

. I

o\

B 1

t f "

0

t

s

W

0.1

0 0.5 1.0

1.850 3.859 5.567

0.6735 0.7865 0.8625

0.2063 0.4615 0.6842

-0.2130 -0.7556 -1.275

1.223 1.069 0.9846

-0.1898 -1.197 -2.019

0.8754 0.7624 0.7007

0.1539

0 0.5 1 .o 2.0

1.431 2.866 4 094 6.254

0.6253 0.7212 0.7870 0.8817

0 2166

0.4934 0.7370 1.173

0.0235 -0.3404 -0.6945 -1.370

1.309 1.158 1.072 0.9673

0.3071 -0.4747 - 1 * 105 -2.164

0.9382 0.8270 0.7640 0.6879

0.25

0

1.102 2.079 2.923

0.5815 0.6593 0.7143

0.2267 0.5277 0.7955

0.1957 -0.0320 -0.2587

1.398 1.256 1.172

0.7265 0.1385 -0.3327

1.003 0.8987 0.8363

0.5 1.0

0.9607 1 734 2.409

0 * 5603 0.6280 0.6770

0.2318 0.5466 0.8289

0.2652 0.0942 -0.0781

1.445 1.312 1.230

0.9194 0.4243 0.0271

1.038 0.9396 0.8787

0 0.5 1 .o 2 .o

0.8105 1.364 1.853 2.726

0.5358 0.5904 0.6313 0.6931

0.2379 0.5711 0.8735 1.425

0.3346 0.2218 0.1061 -0.1244

- 1.386

1.503

1.309 1.206

1.135 0.7498 0.4382 -0.0782

1.081 0.9938 0.9367 0.8613

0

0.5 1 .o

0.7475 1.206 1.615

0.5248 0.5728 0.6094

0 * 2408 0.5833 0.8965

0.3622 0.2730 0.1805

1.531 1.424 1.350

1.229 0.8962 0.6242

1.102 1.021 0.9671

0 0.5 1 .o

0.6823 1.041 1.364

0.5130 0.5530 0.5845

0.2439 0.5975 0.9240

0.3897 0.3239 0.2551

1.562 1.468 1.401

1.330 1.056 0.8287

1.124 1.054 1.004

0.5 1 .o 0.3333

0.5

0.625

0.8333

0

-

-

TABLE I (Continued) ~

B 1

t

fW

0

-0.5

f"(0)

O'(0) = g'(0)

I1

~~

~ ~ ( i )1 ~ ( 2 )

1~(3)

0.15 0.2 0.4 0.5 0.6 0.8 1.0 2.0

0.6489 0.7445 0.7755 0.8963 0.9548 1.012 1.124 1.233 1.737

0.5067 0.5183 0.5219 0.5357 0.5421 0.5482 0.5597 0.5705 0.6156

0.2456 0.3570 0.3934 0.5360

0.4033 0.3875 0.3821 0.3603

1.579 1.550 1.541

1.508

1.383 1.305 1.280 1.186

1.137 1.115 1.109 1.084

0.6056 0.6743 0.8089 0.9402 1.560

0.3491 0.3380 0.3153 0.2923 0.1760

1.493 1.479 1.454 1.430 1.340

1.142 1.099 1.018 0.9402 0.6010

1.073 1.063 1.043 1.026 0.9595

0.1539

0 0.5 1.0

0.9717 2.512 3.774

0.3391 0.4536 0.5249

0.2288 0.5475 0.8164

-0.0152 -0.5454 -1.030

1.690 1.399 1.262

0.7219 -0.8079 -1.635

1.250 1.023 0.9188

2

0.3333

0 0.5 1.0

0.5719 1.426 2.129

0.2678 0.3558 0.4115

0.2480 0.6236 0.9434

0.3277 0.0549 -0.2016

1.970 1.657 1.505

1.215 0.3895 -0.1807

1.471 1.222 1.104

0.625

0 0.5 1.0

0.3910 0.9146 1.349

0.2261 0 2945 0.3390

0.2603 0.6837 1.048

0.4744 0.3208 0.1719

2.186 1.875 1.717

1.732 1.083 0.6471

1.642 1.392 1.268

r2

0

0.5 1.0

0.3067 0.6669 0.9692

0.2031 0.2580 0.2950

0.2675 0.7255 1.125

0.5392 0.4414 0.34Ltl

2.327 2.035 1.878

2.024 1.485 1.125

1.756 1.519 1.394

0 0.5 1.0

0.5068 2.144 3.416

0.1298 0.2520 0.3212

0.2363 0.6063 0.9018

-0.0286 -0.7956 -1.444

2.277 1.685 1.478

0.4951 -1.199 -2.267

1.766 1.270 1.103

1

1

-1 .o

tW

0.1539

0

-

tn

P

2

0

5

%

P

7 P

M

0 C

*2 0

5

TABLE I (Continued)

B

-1 .o

1

1.4

0

0.3333

0 0.5 1 .o

0.2159 1.139 1.847

0.0699 0.1652 0.2184

0.2570 0.7202 1.071

0.4449 0.0174 -0.3663

2.918 2 -091 1.830

1.892 0.3794 -0.4481

2.319 1.601* 1.385

0.625

0 0.5 1 .o

0.0970 0.6740 1.112

0.0369 0.1126 0.1546

0.2701 0.8053 1.225

0.6648 0.3855 0.1513

3.604 2.484 2.170

2.963 1.343 0.6582

2.934 1.931 1.665

1

0 0.5 1 .o

0.0481 0.4519 0.7566

0.0202 0.0822 0.1168

0.2780 0.8805 1.351

0.7624 0.5713 0.4058

4.234 2.817 2.459

3.801 1.989 1.351

3.523 2.218 1.908

0

2.135 4.657 6.790

0.6990 0.8234 0.9059

0.1762 0.4025 0.5991

-0.3318 -0.9785 -1.596

1.183 1.027 0.9429

-0.4251 -1.561 -2.495

0.8463 0.7316 0.6703+

0.5 1 .o

1.677 3.566 5.172

0.6507 0.7591 0.8321

0.1791 0.4190 0.6286

-0.0687 -0.5174 -0.9538

1.264 1.108 1.021

0.0950 -0.8020 -1.531

0.9053 0.7901 0.7268

0.3333

0 0.5 1 .o

1.091 2.115 3 .OOO

0.5772 0.6555 0.7107

0.1924 0.4668 0.7107

0.2249 0.0144 -0.1984

1.409 1.266 1.180

0.7930 0.2264 -0.2279

1.011 0.9051 0.8419*

0.625

0

0.8233 1.435 1.974

0.5362 0.5928 0.6350

0.2004 0.5008 0.7728

0.3422 0.2330 0.1190

1.504 1.383 1.305

1.148 0.7644 0.4535

1.081 0.9915 0.9334

0.1

1 .o

0

0.1539

0

0.5 1 .o

*

Il

f"(0)

W

0.5

1.5

= g'(0)

t

ts

fW

-4

Convergence to 10

O'(0)

TABLE I (Continued)

B

t W

fii(o)

0 0.15 0.2 0.4 0.5 0.6 0.8 1 .o 2.0

0.6987 0.8278 0.8695 1.031 1.109 1.185 1.334 1.477 2.140

0 0.5

0.1

ei@)

I1

I2

Il(U

11(2)

5(3)

0.5147 0.5289 0.5333 0.5498 0.5574 0.5646 J . 5781 0.5906 0.6423

0.2049 0.3035 0.3356 0.4610 0.5220 0.5821 0.6996 0.8141 1.352

0.3914 0.3725 0.3661 0.3395 0.3259 0.3122 0.2844 0.2562 0.1128

1.558 1.524 1.514 1.476 1.460 1.444 1.416 1.390 1.294

1.326 1.235

1.122 1.096 1.088 1.060 1.048 1.036 1.015 0.9963 0.9251*

1 .o

2.368 5.345 7.854

0.7176 0.8504 0.9378

0.1549 0.3623 0.5387

-0.4203 -1.140 -1.842

1.156 0.9987 0.9148

-1.811 -2.847

0.8266 0.7107 0.649P

0 0.5 1 .o

2.471 5.660 8.342

0.7252 0.8616 0.9509

0.1464 0.3439 0.5144

-0-4568 -1.218 -1.945

1.146 0.9875 0.9039

-0.6615 -1.936 -2.991

0.8188 0.7026 0.6418*

0.1539

0 0.5 1.0

1.871 4.156 6.088

0.6681 0.7055 0.8635

0.1542 0.3694 0.5563

-0.1330 -0.6442 - 1.141

'1.236 1.076 0.9893

-0.0466 -1.025 - 1.824

0.8842 0.7666 0.7034

0.3333

0 0.5 1 .o

1.194 2.437 3.504

0.5891 0.6749 0.7345

0.1661 0.4132 0.6315

0.1072 -0.0423 -0.2848

1.385 1.235 1.148

0.7086 0.0925 -0.4016

0.9932 0.8825 0.8183*

0.625

0 0.5 1 .o

0.8837 1.629 2.281

0.5444 0.6073 0.6533

0.1734 0.4505 0.6892

0.3207 0.2095 0.0752

1.485 1.356 1.275

1.094 0.6839 0-3377

1.067 0.9713* 0.9111

fw

tS

= g'(0)

~~

1.5

1 .8

2

0

0

0

*Convergence

1

0.1

-4 to 10

.

1.206

1.097 1.046 0.9966 0.9026 0.8141 0.4268 -0.5935

TABLE I (Continued)

6 2

2 -4

fw 0

0

*Convergence

t"

tS

0 0.15 0.2 0.4 0.5 0.6 0.8 1 .o 2 .o

0.7386 0.8972 0 * 9483 1.146 1.241 1.333 1.513 1.687 2 -488

0.5206 0.5367 0.5417 0.5601 0.5686 0.5766 0.5915 0.6052 0.6615

0.1775 0.2673 0 * 2965 0.4101 0.4653 0.5194 0.6254 0.7284 1.193

0.3837 0.3626 0.3553 0.3254 0.3101 0.2945 0.2629 0.2309 0.0587

0.1539

0 0.5 1.o

2.001 4.574 6.741

0.6789 0.8018 0.8830

0.1394 0.3462 0.5131

-0.1722 -0.7022 -1.258

0.3333

0 0.5 1.o

1.264 2.666 3 864

0.5965 0.6871 0.7493

0.1504 0.3906 0.5838

0.625

0 0.5 1.o

0 9248 1.767 2.501

0.5495 0.6164 0 * 6648

1

0

0.7659 1.335 1* 838

0.5244 0.5757 0.6145

1

0.5 1.o

-4

t o 10 -3 ClCConvergence to 10

.

.

-

1.544 1.506 1.495 1.454 1.436 1.420 1.390 1.363 1.263

1.288 1.187 1.156 1.036 0.9804 0.9266 0.8245 0.7284 0.2908

1.111 1.082 1.074 1.044 1.030 1.018 0 9957* 0.9760* 0.9022*

1.219 1.058 0.9708

-0.1309 -1.119 -2.003

0.8718 0.7529* 0 6 898"

0.1805 -0.0631 -0.3389

1.370 1.217 1.129

0.6582 0 0406 -0.5072

0.9826 0.8691* 0.8044*

0.1573 0.4113 0.6387

0.3208 0.1881 0.0481

1.473 1.340 1.257

1.061 0.6213 0.2678

1.059 0.9593 0.8978*

0.1611 0.7309 0.6762

0.3791 0.3005 0.2151

1.535 1.422 1.347

1.265 0.9408 0.6762

1 .lo4 1.020* 0.9637

-

-

B 2.8

t

fW

0

t

W

1.0

2 * 816 6.774 0.083

0.7484 0.8961 0.9961

0.1209 0.2933 0.4412

-0.5687 -1.439 -2.268

1.115 0.9550 0.8721

0 1 .o

2.117 7.342

0.6877 0.8991

0.1443 0.4784

-0.1507 -1.355

0.3333

0 1.0

1.325 4.196

0 6026 0.7617

0.1379 0.5460

0.625

0

0.5 1 .o

0.9612 1.895 2 704

0.5538 0.6241 0.6744

1

0 0.5 1 .o

0.7901 1.421 1.978

1

0.5 1.0

0.1

0

0.1539

0.5

-0.8655 -2.268 -3.436

0.7961 0.6787 0.6186*

1.206 0 9562

- 0.0900 -2.148

0.8619* 0.6791*

0.1671 -0.3826

1.359 1.114

0.6179 -0 5904

0.9739 0.7933*

0.1444 0.3874 0.5979

0.3144 0.1741 0.0256

1.464 1.327 1.243

1.035 0.5778 0.2111

1.052 0.9494 0.8870*

0.5275 0.5817 0.6223

0.1480 0.4033 0.6341

0.3756 0.2928 0.2023

1.527 1.411 1.333

1.246 0.9088 0.6341

1.098 1.011* 0.9537

1.541 2.170

0.5893 0.6320

0.3697 0.5929

0.2835 0.1919

1.396 1.317

0.8699 0.5938

0.9416*

0.1205 0.3434 0.5447

0.3684 0.2762 0.1751

1.511 1.385 1.305

1.207 0.8393 0.5447

1.086 0.9918 0.9320**

0.1052 0.3096 0.4923

0.3647 0.2668 0.1586

1.502 1.368 1.288

1.184 0.7994 0.4923

1.079 0.9796 0.9196-

-

-

3-4

0

4

0

1

0 0.5 1 .o

0.8502 1.650 2.347

0.5346 0.5955 0.6401

5

0

1

0 0.5 1.0

0 * 8907 1.815 2.616

0 6042

*Convergence

. ,,Convergence to 10- 3 . to

-4 10

-

0.5389

0.6509

-

-

1 .ooo

SIMILAR SOLUTIONS FOR

A

w

TABLE I1 POWER-LAW VISCOSITY RELATION,Pr = O , f w = 0

fl'(0)

e'(o)

g'(0)

0.3490 0.4143 0.4399 0.4696

0.3034 0.3625 0.3861 0.4139

0.3734 0.4394 0.4642 0.4876

00

N

I1

I2

11(1)

11(2)

11(3)

0.3490 0.4143 0.4399 0.4696

0.4037 0.7886 1.089 1.686

0.3490 0.4144 0.4399 0.4696

1.411 1.571 1.626 1.686

1.411 1.571 1.626 1.686

1.185 1.303 1.343 1.385

0.2963 0.3341 0.3246 0.0

0.3734 0.4394 0.4642 0.4876

0.4608 0.8600 1.167 1.645

0.3735 0.4394 0.4642 0.4876

1.452 1.610 1.663 1.710

1.452 1.610 1.663 1.710

1.166 1.250 1.240 0.7665

0.3507 0.4328 0.4987 0.5208 0.5343

0.2527 0.3026 0.3045 0.2392 0.0

0.3507 0.4328 0.4987 0.5208 0.5343

0.3514 0.5519 0.9513

1.304 1.523 1.672 1.717 1.742

1.304 1.523 1.672 1.717 1.742

1.003 1.142 1.201

1.546

0.3494 0.4303 0.4924 0.5124 0.5239

0.9846

1.1

0.5491

1.521

0.5491

1.953

0.5301

1.753

1.753

L-000

0.15 0.4 0.6 1 .o 0.15 0.4 0.6 0.9162

0.3920 0.4353 0.4514 0.4696 0.4089 0.4512 0.4663 0 -4802

0.3428 0.3820 0.3969 0.4139 0.3250 0.3431 0.3263 0.0

0.3920 0.4353 0.4514 0 -4696

0.4414 0.8137 1.106 1.686

0.3921 0.4353 0.4514 0.4696

1.508 1.614 1.649 1.686

1.508 1.614 1.649 1.686

1.255 1.334 1.359 1.385

0.4089 0.4512 0.4663 0.4802

0.4970 0.8778 1.173 1.636

0.4089 0.4512 0.4663 0.4802

1.538 1.640 1.672 1.700

1.537 1.640 1.672 1.700

1.224 1.270 1.248 0.7691

? 0

0.5

0.7

all

all

0

0.5

-

1

-

0

-

0.5

-

0.15 0.4 0-6 1 .o 0.15 0.4 0.6 0.9151 0.05 0.15 0.4 0.6 0.8009

1.251

1.163

U M

"g

TABLE I1 (Continued) n

0

0.7

all

all

1

0.1

0.5

1

1

-

-

0.05 0.15 0.4 0.6 0.8187

0.3924 0.4451 0.4835 0 4960 0.SO34

0.2843 0.3136 0.3025 0.2420 0.0

0.3924 0.4451 0.4835 0.4960 0 5034

-

0.3963 0.5766 0.9637 1.253 1.580

0.3923 0.4448 0.4832 0.4947 0.5029

1.442 1.586 1.681 1.707 1,725

1.442 1.586 1.681 1.707 1.725

1.100 1.188 1.195 1.134 0.8037

1.1

0.5097

1.291

0.5097

1.982

0.5085

1.737

1.737

2.444

0.15 0.4 0.6 1 .o

0.4696 0.4696 0.4696 0.4696

0.4139 0.4139 0.4139 0.4139

0.4696 0.4696 0.4696 0.4696

0.5091 0.8554 1.132 1.686

0.4696 0.4696 0.4696 0.4696

1.686 1.686 1.686 1.686

1.686 1.686 1.686 1.686

1.385 1.385 1.385 1.385

0.15 0.4 0.6 0.9179

0.4696 0.4696 0.4696 0.4696

0.3739 0.3572 0.3281 0.0

0.4696 0.4696 0.4696 0.4696

0.5595 0.9057 1.183 1.623

0.4696 0.4696 0.4696 0.4696

1.686 1.686 1.686 1.686

1.686 1.686 1.686 1.686

1.326 1.301 1-259 0.7732

1

0.15 0.4 0.6 0.8357

0.4696 0.4696 0.4696 0.4696

0.3339 0.3006 0.2439 0.0

0.4696 0.4696 0.4696 0.4696

0.6097 0.9560 1.233 1.559

0.4696 0.4696 0.4696 0.4696

1.686 1.686 1.686 1.686

1.686 1.686 1.686 1.686

1.267 1.217 1.134 0.7729

1

0.05 0.15 0.4 0.6 0.7948

0.3780 0.4733 0.5655 0.6066 0.6376

0.2597 0.3112 0.3134 0.2435 0.0

0.3623 0.4476 0.5194 0.5450 0.5613

0.3187 0.5088 0.8753 1.149 1.409

0.3414 0.4197 0.4717 0.4840 0.4882

1.291 1.513 1.652 1.690 1.709

1.260 1.464 1.575 1.591 1.590

0.9912 1.124 1.166 1.106 0.8825

1.1

0.6789

1.644

0.5909

1.785

0.4852

1.709

1.566

2.185

0

-

0.5

-

TABLE I1 (Continued)

f"(0)

0.1

0.7

I2

11(3)

0.3333

1

1

0.15 0.8151

0.5523 0.7916

0.3304 0.0

0.4188 0.5580

0.5197 1.375

0.3964 0.3793

1.531 1.602

1.387 1.265

1.134 0.6223

0.5

1

1

0.15 0.814(

0.5191 0.7020

0.3257 0.0

0.4643 0.5435

0.5246 1.399

0.4142 0.4222

1.550 1.639

1.452 1.407

1.149 0.6643

0.15 0.813

0.5056 0.6651

0.0

0.4615 0.5373

0.5270 1.411

0.4212 0.4394

1.559 1.656

1.479 1.467

1.155 0.6794

0.2900 0.3206 0.3096 0-2463 0.0

0.4021 0.4573 0.5008 0.5164 0.5274

0.3614 0.5303 0.8823 1.150 1.428

0.3834 0.4320 0.4608 0.4655 0.4653

1.430 1.571 1.654 1.674 1.682

1.394 1.521 1.576 1.575 1.561

1.087 1.166 1.157 1.063 0.7109

0.625

0.2

I1

1

1

0.3237

1

1

-

0.05 0.15 0.4 0.6 0.8121

0.4202 0.4850 0.5492 0.5808 0.6083

1.1

0.6420

1.492

0.5447

1.659

0.4751

1.702

1.569

0.8969

1

1

1

-

0.15 0.4 0.6 0.8291

0.5085 0.5331 0.5525 0.5744

0.3381 0.3051 0.2459 0.0

0.4781 0.4824 0.4857 0.4894

0.5563 0.8715 1.121 1.403

0.4529 0.4460 0.4404 0.4341

1.660 1.648 1.638 1.627

1.610 1.576 1.549 1.520

1.240 1.174 1.071 0.6798

0.5

0.3333

1

1

0.15 0.790

0.6229 1.033

0.3336 0.0

0.4808 0.6313

0.4625 1.214

0.3458 0.3112

1.451 1.562

1.198 1.023

1.064 0.8013

0.625

1

1

0.5441 0.8240

0.3235 0.0

0.4668 0.5986

0.4713 1.286

0.3918 0.4105

1.489 1.653

1.349 1.319

1.095 0.7607

1

1

-

0.15 0.792 0.05 0.15 0.4

0.4015 0.5091 0.6243

0.2657 0.3188 0.3209

0.3718 0.4604 0.5370

0.2977 0.4752 0.8161

0.3366 0.4114 0.4550

1.285 1.507 1.640

1.230 1.419 1.499

0.9808 1.110 1.139

TABLE I1 (Continued)

0.2

0.5

0.7

1

0.3333

1

0.6818 0.7268

1.1

0.7913

1.748

0.6069

1.659

0.15 0.6 0.8123

0.6428 0.9193 1.024

0.3431 0.2732 0.0

0.4908 0.5747 0.5951

0.4769

0.2467 0.0

0.5655 0.5838

1.069 1.303

0.4608 0.4589

1.673 1.686

1.494 1.474

1.062 0.8130

0.4479

1.680

1.425

2.337

1.011 1.243

0.3568 0.3134 0.2787

,1.495 1.546 1.535

1.240 1.062 0.9563

1.098 0.9341 0.5252

0.5

1

0.15 0.8095

0.5828 0.8663

0.3352 0.0

0.4798 0.5727

0.4843 1.275

0.3892 0.3566

1.526 1.587

1.350 1.184

1.122 0.5789

0.625

1

0.15 0.6 0.8085

0.5581 0.7377 0.8004

0.3318 0.2567 0.0

0.4752 0.5471 0.5627

0.4875 1.047 1.290

0.4021 0.4028 0.3878

1.539 1.614 1.612

1.397 1.340 1.282

1.133 0.9888 0.6020

0.05 0.15 0.4 0.6 0.8064

0.4445 0.5202 0.6068 0.6548 0.6979

0.2950 0.3265 0.3147 0.2470 0.0

0.4103 0.4678 0.5155 0.5338 0.5465

0.3356 0.4925 0.8141 1.059 1.314

0.3766 0.4216 0.4426 0-4415 0.4350

1.421 1.560 1.633 1.646 1.653

1.358 1.470 1.495 1.473 1.439

1.076 1.150 1.136 1.036 0.6471

1.1

1

1

0.6 0.7897

0.7539

1.489

0.5607

1.657

0.4223

1.647

1.386

2.709

0.3333

1

0.15 0.8297

0.6768 1.001

0.3604 0.0

0.5112 0.5550

0.4959 1.207

0.3793 0.2769

1.565 1.458

1.316 0.9579

1.160 0.4980

0.625

1

0.15 0.8257

0.5840 0.7700

0.3475 0.0

0.4935 0.5215

O.SO82 1.260

0.4214 0.3670

1.615 1.540

1.475 1.251

1.199 0.5734

w

TABLE I1 (Continued)

0.2

1

1

1

0.2857

0.5

0.3333

1

1

0.625

1

1

1

0

-

0.5

-

1

0.7

-

0.3333

1

1

0.5

1

1

m

0.5424 0.5883 0.6633

0.3414 0.3085 0.0

0.4851 0.4926 0.5044

0.5143 0.8065 1.288

0.4395 0.4270 0.4057

1.639 1.618 1.586

1.550 1.491 1.397

1.219 1.141 0.6165

0.15 0 7933

0.6898 1.209

0.3436 0.0

0.4965 0.6565

0-4359 1.168

0.3175 0.2198

1.434 1.553

1.101 0.7817

1.044 0.6157

0.15 0.7887

0.5843 0.9251

0.3307 0.0

0.4787 0.6191

0.4461 1.210

0.3792 0.3727

1.481 1.632

1.296 1.196

1.081 0.7071

0.15 1 .o

0.4283 0-7627

0.3157 0.4534

0.3643 0.5179

0.3292 1.309

0.3342 0.3887

1.360 1.551

1.302 1.309

1.144 1.280

0.15 0.9071

0.4605 0.7683

0.3101 0.0

0.3934 0.5397

0.3743 1.282

0.3537 0.4019

1.406 1.595

1.328 1.335

0.5667

0.05 0.15 0.4 0.6 0.7849

0.4209 0.5371 0-6701 0.7401 0.7952

0.2710 0.3246 0.3266 0.2490 0.0

0.3808 0.4704 0.5505 0.5810 0.6012

0.2779 0.4503 0.7737 1.013 1.223

0.3299 0.4053 0.4428 0.4439 0.4374

1.268 1.502 1.631 1.661 1.667

1.199 1.385 1.446 1.426 1.393

0.9696 1.100 1.120 1.032 0.7890

1.1

0.8787

1.862

0.6294

1.548

0.4195

1.640

1.325

2.232

0.15 0.8093

0.7114 1.197

0.3521 0.0

0.5050 0.6200

0.4478 1.159

0.3278 0.2053

1.473 1.496

1.139 0.7527

1.075 0.4696

0.6315 10.9897

0.3421 0.0

0.4913 0.5928

0.4561 1.193

0.3710 0.3089

1.510 1.556

1.280 1.034

1.104 0.5290

0.15 0.4 0.8235

-

-

00

::::62

1.123

TABLE I1 (Continued)

0.2857

0.4

0.7

0.625

1

1

0.15 0.8049

0.5984 0.9027

0.3378 0.0

0.4853 0.5806

0.4598 1.209

0.3883 0.3504

1.526 1.585

1.339 1.157

1.117 0.5540

1

1

-

0.05 0.15 0.4 0.6 0.8022

0.4636 0.5475 0.6512 0.7118 0.7664

0.2988 0.3310 0.3194 0.2499 0.0

0.4171 0.4758 0.5259 0.5459 0.5603

0.3143 0.4657 0.7744 1.007 1.236

0.3707 0.4143 0.4298 0.4241 0.4133

1.411 1.553 1.624 1.635 1.635

1.329 1.433 1.439 1.402 1.356

1 A068 1.138 1.108 0 9885 0.6055

-

1.1

0.8401

1.565

0.5771

1.557

0.3950

1.621

1.286

2.709

1

1

1

-

0.15 0.4 0.6 0.8194

0.5684 0.6308 0.6793 0.7310

0.3438 0.3108 0.2473 0.0

0.4901 0.4999 0.5072 0.5148

0.4846 0.7611 0.9773 1.210

0.4299 0.4134 0.4001 0.3854

1.625 1.598 1.578 1.558

1 508 1.432 1.374 1.314

1 204 1.117 0.9918 0.5758

0.5

0.3333

1

0

0.15 0.8938

0.6910 1.434

0.3363 0.0

0.4393 0.6268

0.3396 1.081

0.2834 0.1360

1.311 1.410

1.024 0.5590

1.030 0.4240

0.5

0.15 0.8498

0.7224 1.431

0.3403 0.0

0.4643 0.6476

0.3656 1.087

0.2837 0.1264

1.346 1.452

1.012 0.5407

1.022 0.4055

1

0.15 0.7898

0.7703 1.412

0.3550 0.0

0.5146 0.6872

0.4069 1.084

0.2845 0.1311

1.418 1.523

0.9944 0.5552

1.023 0.5541

0

0.15 0.9407

0.5342 1 *044

0.3219 0.0

0.3972 0.5669

0.3279 1.156

0.3244 0.3039

1.342 1.484

1.204 1.009

1.092 0.5241

0.5

0.15 0.8738

0.5674 1.044

0.3221 0.0

0.4255 0.5893

0.3616 1.149

0.3359 0.3085

1.383 1 530

1.213 1.019

1.077 0.4872

0.625

1

-

-

TABLE I1 (Continued)

0.4

0.5

0.7

0.15 0.7829

0.6332 1.045

0.3393 0.0

0.4928 0.6439

0.4182 1.117

0.3648 0.3334

1.473 1.599

1.236 1.070

1.067 0.6992

0.15 1 .o

0.4540 0.8544

0.3192 0.4632

0.3686 0.5300

0.3094 1.219

0.3303 0.3667

1.347 1.521

1.273 1.219

1.133 1.257

0.15 0.9048

0.4887 0.8564

0.3141 0.0

0.3992 0.5533

0.3511 1.194

0.3483 0.3786

1.394 1.570

1.296 1.244

1.111 0.5232

0.05 0.15 0.4 0.6 0.7815

0.4416 0.5714 0.7260 0.8112 0.8782

0.2757 0.3318 0.3334 0.2515 0.0

0.3870 0.4824 0.5667 0.5995 0.6203

0.2694 0.4229 0.7270 0.9516 1.149

0.3321 0.3987 0.4291 0.4246 0.4135

1.289 1.499 1.624 1.651 1.659

1.188 1.349 1.386 1.351 1.306

0.9672 1.090 1.099 0.9983 0.7155

1.1

0.9837

1.924

0.6489

1.474

0.3884

1.644

1.218

2.562

0

0.15 0.8964

0.7382 1.428

0.3634 0.0

0.4742 0.6161

0.3620 1.072

0.3039 0.1408

1.385 1.390

1.071 0.5692

1.081 0.4242

0.5

0.15 0.8545

0.7613 1.418

0.3602 0.0

0.4911 0.6275

0.3841 1.070

0.3005 0.1346

1.406 1.414

1.055 0.5585

1.065 0.4054

1

0.15 0.8052

0.7938 1.403

0.3622 0.0

0.5212 0.6471

0.4162 1.068

0.2945 0.1234

1.450 1.458

1.029 0.5395

1.052 0.4289

0

0.15 0.9414

0.5807 1.040

0.3550 0.0

0.4369 0.5619

0.3527 1.152

0.3550 0.3027

1.429 1.475

1.274 1.007

1.152 0.5242

0.5

0.15 0.8763

0.6049 1.032

0.3457 0.0

0.4564 0.5743

0.3834 1.136

0.3594 0.3044

1.456 1.502

1.272 1.013

1.126 0.4875

0.625

1

1

0

-

0.5

-

1

-

0.3333

0.625

1

1

1

TABLE I1 (Continued)

0.4

0.7

0.625

1

1

0

0.5

0.4292 1.122

0.3726 0.3077

1.513 1.559

1.275 1.021

1 .I00 0.5053

0.3347 1.219

0.3687 0.3667

1.443 1.521

1.358 1.219

1.204 1.257

0.4344 0.4909 0.5159 0.5430

0.3749 0.6569 0.8701 1.185

0.3781 0.3955 0.3906 0.3728

1.475 1.544 1.554 1.551

1.366 1.362 1.318 1.235

1.166 1.176 1.119 0.5240

0.4 0.6 0.7974 1.1

0.4864 0.5007 0.7053 0.7811 0.8490 0.9451

0.3034 0.4246 0.3363 0.4853 0.3239 0.5389 0.2500 0.5609 0.5763 0.0 1.636 0.5953

0.2943 0.4358 0.7220 0.9380 1.151 1.460

0.3654 0.4063 0.4154 0.4050 0.3887 0.3616

1.404 1.546 1.608 1.614 1.617 1.603

1.300 1.393 1.377 1.326 1.265 1.172

1.058 1.126 1.092 0.9700 0.5624 2.887

0.15 0.8005

0.6473 1.025

0.3448 0.0

0.4971 0.6007

-

0.15 1.0

0.5027 0.8544

0.3597 0.4632

0.4129 0.5300

-

0.15

0.5281 0.6583 0.7386 0.8470

0.3422 0.3671 0.3500 0.0

1

0.4 0.6 0.905f 1

-

0.05 0.15

1

'

0.3333

1

1

0.15 0.8231

0.8319 1.383

0.3759 0.0

0.5366 0.5993

0.4289 1.033

0.3173 0.1439

1.500 1.363

1.098 0.5815

1.103 0.3952

0.625

1

1

0.15 0.817s

0.6727 0.9951

0.3563 0.0

0.5094 0.5520

0.4438 1.091

0.3887 0.2967

1.571 1.467

1.342 1.006

1.158 0.4799

1

0

0.15 1 .O

0.5901 0.8544

0.4327 0.4632

0.4926 0.5300

0.3806 1.219

0.4380 0.3667

1.619 1.521

1.514 1.219

1.333

0.15 0.9071

0.5948 0.8337

0.3901 0 .o

0.4942 0.5283

0.4156 1.173

0.4286 0.3646

1.614 1.525

1.487 1.221

1.261 0.5248

0.5

-

-

1.257

w

TABLE I1 (Continued)

Q 0

0.4

1

1

1

-

0.15 0.4 0.6 0.8147

0.5998 0.6824 0.7461 0.8125

0.3465 0.3134 0.2474 0.0

0.4959 0.5082 0.5173 0.5264

0.4515 0.7109 0.9129

0 -4192 0 3980 0.3809 0.3624

1.608 1.575 1.552 1.529

1-460 1.366 L.295 1.224

1.187 1.092 0.9564 0.5334

0.5

0.5

0.3333

1

0.5

0.3469 0.0

0.9453 0.3868

1.006 0.3726

-

0.2520 0.3218 0.3940 0.4266 0.4705 0.4791

1.155 1.250

0 2644 0.0638 0 2666 0.3274 0.3658 0.3685 0.3503 0.3431

1.328 1.424

0

0.4739 0.6662 0.2923 0.3720 0.4533 0.4899 0.5390 0.5486

0.3450 1.022

1

0.7824 0.15 0.8488 1.599 0.3518 0.05 0.4747 0.15 0.6472 0.4 0.7511 0.6 0.9277 1.0 1.1 0.9681

1.153 1.336 1.455 1.484 1.500 1.500

1.103 1.251 1.290 1.258 1.155 1.126

0.9820 1.125 1.212 1.232 I. 240 1.240

0.05 0.15 0.9523

0.3653 0.4909 0.9260

0.2517 0.3187 0.0

0.3038 0.3857 0.5494

0.1840 0.3127 1.142

0.2733 0.3245 0.3549

1.171 1.358 1.522

1.112 1.259 1.165

0.05 0.15 0.4 0.6 0.903 1.1

0.3827 0.5114 0.6867 0.7908 0.9266 1.007

0.2532 0.3171 0.3637 0.3539

1 -024

0.3189 0.4037 0.4878 0.5247 0.5635 0.5830

0.2001 0.3339 0.6147 0.8245 1.131 0.1325

0.2826 0.3444 0.3794 0.3786 0.3612 0 3446

1.194 1.385 1.507 1.537 1.552 1.553

1.124 1.271 1.303 1.264 1.179 1.117

0.05 0.15 0.8826

0.3913 0.5215 0.9280

0.2546 0.3173 0.0

0.3265 0.4128 0.5708

0.2077 0.3438 1.127

0.2874 0.3496 0 3645

1.206 1.399 1.566

1.130 1.277 1 * 185

0.9772 1.114 0.4897 0.9728 1.103 1.147 1.098 0.4935 2.080 0.9712 1.098 0.4953

0.25

-

0.5

-

0.6

-

0.0

1.125

0.1704 0.2946 0.5628 0.7652

-

-

-

-

-

TABLE I1 (Continued)

0.5

0.5

1

0.8

-

f"(0)

e'(o)

g'(0)

0.4138 0.4928 0.5475 0.9335

0.2603 0.2989 0.3204 0.0 0.2815 0.3387 0.3385 0.2521

1.1

0.4614 0 * 6009 0.7723 0.8688 0.9439 1.069 0.4252 0.5243 0.6713 0.7643 0.9277 0.9657

0.05 0.15 0.4 0.6 0.9040 1.1

0 4500 0.5513 0.6986 0.7913 0.9169 0 9920

0.05 0.15 0.4

0.5048 0.6073 0.7489

0.05 0.1 0.15

0.8393 1

0.7

1

0

-

-

0 -05 0.15 0.4 0.6 0.7762 1.1 0.05 0.15

0.4 0.6 1 .o

0.5

-

1

-

-

Il(3)

I1

I2

11(1)

0.3471 0.4028 0.4371 0 5906 0.3970 0.4957 0.5831 0.6156 0.6374 0.6666

0.2261 0.3017 0.3675 1.121

0.3007 0.3419 0.3638 0.3731

1.236 1.366 1.435 1.605

1.146 1.249 1.294 1.201

0 2496 1.386 0.6808 0.8997 1.080 1 404

0.3233 0.3871 0 -4149 0 * 4094 0.3958 0.3640

1.258 1.460 1.586 1.629 1 a634 1.632

1.155 1 -306 1.334 1.294 1.242 1.138

1.074 1.089 0.9863 0.7265 2.655

0 2008 0.3182 0.5779 0.7751 1.155 1.247

0.3248 0.3652 0.3823 0.3768 0.3503 0.3420

1.307 1.433 1.499 1.508

1.244 1.334 1.324 1.275 1.155 1.124

1.098 1.196 1.244 1 * 249 1.240 1.237

0.3013 0.3452 0.3710 0.3535 0.0 0.9870

0.3578 0.4164 0.4750 0.5018 0.5390 0 5464 0.3784 0.4388 0.4975 0.5238 0.5526 0.5674

1.253 1.340 1.321 1.266 1.170 1.105

1.075 1.157 1.162 I. 101 0.4941 2.030

0.430'5 0.4928 0.5488

0.3339 0.3733 0.3867 0.3784 0.3557 0.3379 0.3616 0 * 4004 0.4047

1.339

0.3071 0 3405 0.3277

0.2322 0.3560 0.6238 0.8258 1.122 1.308 0.2794 0.4134 0.6862

1.278 1.363

1.051 1.118

0.0

2.001

0.3109 0.3625 0 4142 0.4378 0.4705 0.4770

-

-

-

-

1.500

1.496 1.465 1.530 1.538 1.532 1.524

-

1.400 1 540 1 * 599

1.332

0 * 9680

1.052 1.090 0 -4989

-

0 9530

1.076

w

TABLE I1 (Continued)

0.5

0.7

1

0.75

1

1

1

-

0.8370 0.9146 1.029

0.0

1.700

0.5719 0.5886 0.6095

0.8938 1.087 1.386

0.3900 0.3711 0.3372

0.15 0.4 0.6 1.0

0.6136 0.7105 0.7850 0.9277

0.4358 0.4471 0.4555 0.5390

0.4964 0.5103 0.5205 0.5390

0 3609 0.6026 0 7906 1.155

0.15 0 -4 0.6 0.8111 1.1

0.6249 0.7237 0.7997 0.8773 0.9800

0.3485 0.3152 0.2474 0.0, 1.409

0 5003 0.5145 0.5248 0.5349 0.5478

0.4272 0.6743 0.8662 1.064 1.328

0.05 0.15 0.4 0.6 0.8994 1.1

0.4131 0.5618 0.7768 0.9092 1.083 1.189

0.3251 0.4129 0.5020 0.5419 0.5840 0.6061

0.05 0.15 0.4 0.6 1 .o

0.4557 0.5726 0.7568 0.8770 1.090 1.140

0.2576 0.3234 0.3721 0.3616 0.0 1.085 0.3150 0.3683 0.4232 0.4489 0.4849 0.4922

0.3629 0.4237 0.4862 0.5155 0.5568 0.5652

0.6 0.7934

0

-

1

-

0.5

1

0.5

-

0.7

1

0

-

9 N

1.1

1.1

0.2518

-

1.268 1.200 1.090

0.9366 0.5470 2.953

0.4331 0.4092 0.3897 0.3503

1.607 1.603 1.588 1.609 1.572 1.545 1.500

1.487 1.380 1.301 1.155

1.325 1.296 1.276 1.240

0.4113 0.3866 0.3667 0.3454 0.3161

1.596 1.559 1.533 1.508 1.478

1.425 1.318 1.238 1.159 1.058

1.174 1.073 0.9303 0.5038 2.702

0.1796 0 3000 0.5518 0.7391 1.008 1.183

0.2784 0.3367 0.3630 0.3549 0.3264 0.3017

1.092 1.223 1.224 1.165 1.052 0.9717

0.9608 1.086 1.121 1.064 0.4382 2.118

0.1800 0.2857 0.5183 0.6939 1.031 1.113

0.3207 0.3582 0.3682 0.3565 0.3174 0.3059

1.182 1.368 1.483 1.508 1.518 1.517 1.292 1.412 1.470 1.476 1.460 1.454

1.211

1.085 1.179 1.221 1.223 1.209 1.204

-

-

1.288 1.251 1.183 1.031 0.9922

I'ABLE I1 (Continued)

I2 0.75

1

0.7

0.5

1

0.5

-

0.05 0.15 0.4 0.6 0.9004 1.1

0.4833 0.6029 0.7885 0.9089 1.072 1.173

0.3059 0.3512 0-3787 0.3605 0.0

0.3850 0.4478 0.5108 0 5400 0.5719 0.5890

-

0.3333

1

0

0.05 0.15 0.8980

0.6684 0.9742 2.298

0.625

1

0

0.05 0.15 0.9429

0.4913 0.6981 1.580

1

0

-

0.05 0.15 0 -4 0.6 1 .o 1.1

0.4O25 0.5612 0.8060 0.9615 1.233 1.295

1.045 0.2862 0.3695 0.3626 0.4740 B.0 0.6980 0.2665 0.3280 0 * 3394 0.4198 0.0 0.6190 0.2584 0* 3004 0.3317 0.3844 0.4097 0.4730 0 4460 0 5140 0.4959 0.5705 0.5058 0.5817

0.05 0.15 0.9490

0.4187 0.5808 1.224

0.2588 0.3291 0.0

0.3132 0.3998 0.5829

0.05 0.15 0.4

0.4395 0.6057 0.8558

0.2612 0.3285 0.3787

0.3302 0 4204 0.5132

0.25

-

0.5

-

-

-

-

0.2072 0.3187 0.5591 0.7396 1 .oooo

1.169

0.3279 0.3690 0-3692 0.3542 0.3215 0.2963

1.325 1 -446 1 503 1.507 1 *496 1.485

-

1.215 1.287 1.240 1.166 1.044 0.9624

1.061 1.139 1.135 1.065 0.4385 2.062

-

0.1550 0.2546 0.8082

0.2033 0.2097 -0.1293

1.075 1.228 1.287

0.1496 0.2520 0.8788

0.2520 0.2948 0.1736

1.115 0.1280 1.378

0.9741 1.052 0.6162

0.9235 1.042 0.4018

0.1408 0.2437 0.4631 0.6272 0 9402 1.016

0.2617 0.31% 0 3430 0.3344 0.2923 0.2791

1.127 1.300 1.404 1 -426 1.430 1.429

1.053 1.175 1.167 1.102 0.9402 0.8978

0.9605 1 .O% 1.172 1.186 1.186 1.184

0.1514 0.2578 0 * 9289 0.1641 0.2746 0.5051

0* 2673 1.146 0.3230 1.323 0.2955 1.458 0.2755 1.172 0.3311 1.355 0.3507 1.465

1.059 1.180 0.9490

0.9556 1.084 0.3943

1.068 1.187 1.166

0.9516 1.074 1.102

-

0.7781 0 8644 0.7633 0.9629 - 0 -0618 0.2881

w

TABLE I1 (Continued)

0.5

-

0.6

-

0.8

-

~

1

0.5

1

1

~

-

0.6 0.8965 1.1

1.013 1.219 1.349

0.3675 0.0 1.135

0 * 5554 0.6000 0.6241

0.6760 0.9187 1 .080

0.3369 1.488 0 * 3000 1.495 0.2689 1.492

1.092 0 9601 0.8652

1.039 0.3992 2.149

0.05 0.1 0.15 0.8748

0.4499 0.5467 0.6181 1.218

0.2632 0.3047 0.3292 0.0

0.3390 0.3954 0.4310 0.6091

0.1700 0.2298 0.2825 0.9152

0.2799 0.3169 0.3356 0.3025

1.185 1.307 1.371 1.513

1.073 1.159 1.192 0.9655

0.9504 1.033 1.070 0.4013

0.05 0.1 0.15 0.8290

0.4774 0.5774 0.6501 1.218

0.2708 0.3115 0.3344 0.0

0.3632 0.4227 0.4599 0.6344

0.1848 0.2472 0 3016 0,9094

0.9490 1.029 1.063 0.4061

0.6 0.7640 1.1

0.5381 0.7166 0.9653 1.110 1.220 1.423

0 * 3014 0 3623 0.3634 0.2598

0.4277 0 * 5342 0.6465 0.6760 0.7006 0.7483

0.2068 0.3297 0.5532 0.7445

1.221 1.348 1.415 1 * 565 1.244 1.473 1.488 1.603 1.606 1.517

1.086 1.173 1.205 0.9788

0.05 0.15

0.2926 0.3302 0.3486 0 * 3090 0.3133 0.3745 0.3694 0.3577 0.3329 0.2860

0.4338 0.5111 0.6871

0.1784 0.2328 0.2701 0 2224 0 * 8031 -0.1164

1.208 1.299 1.265

0.3929 0.4609 0.6140

0.1735 0.2692 0.8765

0.2985 0.3208 0.1743

1.260 1 a366 1.368

1.084 1.109 0.6178

1.028 1.102 0.4019

0.3670 0.4295

0.1643 0.2614

0.3177 0.3532

1.280 1.396

1.186 1.253

1.076 1.167

0.4

0.7

9

P

-

0.0

2.402

0.3333

1

0

0.05 0.15 0.9006

0.7685 1.034 2.295

0.625

1

0

0.05 0.15

0.5761 0.7516 1.577

0.3367 0.3912 0.0 0.3206 0.3736 0.0

1

0

-

0.05

0.4821 0.6146

0.3183 0.3729

0.9435 0.15

-

0.8838

1.136

-

-

1 a087 1.221 1.163 1.104 1.027 0.8877

0.9266 1.049 1.016 0.8990 0.6067 2.484 3.8477 0.9573 '0.7915 1.012 -0.0368 0.2890

c)

*

ra

TABLE I1 (Continued) I

4

1

0.7

1

0

0.5

0.9

1

1

-

-

-

-

Wo)

ei(o)

g'(0)

0.4 0.6 1 .o 1.1

0.8317 0.9756 1.293

0.4302 0.4574 0.4959 0.5037

0.05 0.1s 0.4 0.6 0.8975 1.1

0.5120 0.6478 0.8671 1.012 1.207 1.331

0.15 0.4 0.6 0.8093

I1

I2

11(1)

I p )

11(3)

0.4950 0.5262 0.5705 0.5795

0.4741

0.6343 0.9402 1.014

0.3576 0.3413 0.2923 0.2783

1.669 1.450 1.430 1.424

1.197 1.116 0.9402 0.8964

1.205 1.203 1.186 1.181

0.3095 0.3560 0.3848 0.3658 0.0 1.091

0.3902 0.4550 0.5214 0.5526 0.5869 0.6057

0.1884 0.2908 0.5110 0.6759 0.9113 1.067

0.3235 0.3570 0.3560 0.3359 0.2955 0.2644

1.314 1.432 1.4W 1.470 1.457

1.187 1.248 1.180 1.092 0.9523 0.8578

1.051 1.126 1.115 1.039 0.3995 2.089

0.6940 0.9117 1.055 1.192

0.3506 0.3494 0.2822 0.0

0.4955 0.5617 0.5919 0.6160

0.3239 0.5505 0.7195 0.8902

0.3702 0.3612 0.3019

1.491 1.544 1.543 1.534

1.254 1.173 1.075 0.9683

1.095 1.037 0.8892 0.4096

0.05 0.5809 0.15 0.7188 0.4 0.9317 11.071 0.6 0.7779 1.185 1.1 1.380

0.3222 0.3576 0.3426 0.2551 0.0 1.970

0 ;548 0.~230 0.5891 0.6162 0.6372 0.6650

0.2275 0.3372 0.5616 0.7336 0.8753 1.130

0.3488 0.3810 0.3668 0.3379 0.3108 0.2547

1.387 1.528 1.562 1.579 1.549 1.527

1.202 1.262 1.175 1.075 0.988) 0.8266

1.023 0.8527 0.5073 3.035

1.272 1.077

0.1107 -1.207

0.9563 0.1603

1.233

0.3351

0.1539

1

0

0.15 0.8838

1.764 3.839

0.4779 0.0

0.6465 0.7797

0.2809 0.0612 0-7071 -0.7567

0.3333

1

0

0.1s 0.904s

1.132 2.290

0.4391

0.5730 0.6715

0.2962 0.7960

0.0

1.484

0.2436 1.420 -0.0978 1.236

1.026

1.08R

0.8397 1.094 -0.0008 0.2903

CA C

2

5

$

F U

>

F 2z

M

0

>

2 v,

w

s

TABLE I1 (Continued)

-~

1

1.4

1.5

1

0.7

0.5

0.625

1

1

0

-

1

0.6

-

1

0

0

-

0.25

-

0.5

-

0.15 0.9444

0.8444 1.572

0.4329 0.0

0.5321 0.6067

0.15 0 -4 1 .o

0.7104 0.8735 1.233

0.4473 0.4637 0.4959

0.5107 0.5308 0.5705

0.15 0.4 0.6 0.8721

0.3596 0.3830 0.3516

1.1

0.7200 0.9874 1.167 1.389 1.564

0.05 0.15 0.4 0.6 1 .o

0.4427 0.6304 0.9337 1.131 1.477

0.2628 0.3383 0.4200 0.4585 0.5119

0.05 0.15 0.9469

0.4608 0.6526 1.464

0.05

0.4842 0.6810 0.9917 1.192 1.453

0.2636 0.3361 0.0 0.2667 0.3362 0.3888 0.3763 0 .o

0.4719 0.5427 0.5765 0.6109 0.6343 0 3060 0.3928 0.4860 0.5298 0.5906 0.3197 0.4095 0.6045 0.3381 0.4319 0.5304 0.5760 0.6241

0.15 0.4 0.6 0.8922

0.0

1.295

-

0.2992 0.8733 0.2934 0.4921 0 9402

0.3657 0.1753

1.516 1.354

1.210 0.6204

1.207 0 4024

0.4171 0.3814 0.2923

1.572 1.521 1.430

1.395 1.246 0.9402

1.296 1.257 1.186

0.2640 0.4622 0.6100 0.8028 0.9587

0.3505 0.3403 0.3125 0 * 2653 0.2214

1.426 1.472 1.469 1.453 1.437

1.200 1.105 0.9983 0.8486 0.7246

1.101 1.071 0.9707 0.3577 2.341

0.1225 0.2123 0.4030 0 5448 0.8141

0.2590 0.3119 0.3291 0.3134 0.2562

1.110 1.277 1.374 1.391 1.390

1.022 1.128 1.092 1.008 0.8141

0.9465 1.078 1.148 1.159 1.155

0.1312 0.2242 0.8034

0.2640 0.3163 0.2585 0.2715 0.3234 0.3332 0.3111 0.2620

1.130 1.302 1.421

1.026 1.130 0.8216

0.9418 1 -066 0.3415

1.158 1.336 1.440 1.459 1.463

1.033 1.136

0.9384 1 SO56 1.075 1.004 0.3466

-

0.1419 0.2383 0.4389 0.5871 0.7937

1 .ON

0.9885 0.8311

-

TABLE I1 (Continued)

f"(0)

0.3138 0.3196 0.2561

1.262 1.415 1.390

1.150 1.021 0.8140

1.061 1.176 1.155

0 3083

0.3273 0.2993 0.1938

1.286 1.347 1.451 1.469 1.467

1.095 1.116 1.047 0.9388 0.6373

1.010 1 -044 1.048 0.9526 2.484

0.1419 0.5504 0.8140

0.1 0.15 0.4 0.6 1.1

0.6421 0.7354 1.080 1.303 1 788

0.3155 0.3414 0.3877 0.3616 1.425

0.4115 0.4496 0.5524 0.6000 0.6786

0.1856 0.2285 0.4182 0.5578 0.8857

0.05 0.15 0.4 0.6 1.0

0.4766 0.6893 1.043 1.276 1.687

0.2661 0.3432 0.4276 0.4677 0.5235

0.3103 0.3991 0.4956 0.5414 0.6052

0.1097 0.1906 0.3617 0.4884 0.7282

0.2573 0.3082 0.3196 0.2988 0.2308

1.097 1.261 1.352 1.367 1.363

0.9992 1.096 1.040 0.9445 0.7282

0.9364 1.065 1.131 1.140 1.134

0.25 -

0.05 0.15 0.9452

0.4962 0.7137 1.669

0.2672 0.3414 0 -0

0.3246 0.4167 0 6204

0.1172 0.2009 0.7180

0.2619 0.3119 0.2326

1.119 1.287 1.396

1.003 1.096 0.7348

0.9319 1.053 0.3063

0

0.05 0.15 0.4 0.6 1 .o

0.5640 0.7473 1.071 1.291 1.687

0.3270 0.3851 0.4484 0.4793 0.5235

0.3781 0.4449 0.5179 0.5537 0.6052

0.1264 0.2028 0.3690 0.4930 0.7282

0.3112 0.3417 0.3326 0.3046 0.2308

1.249 1.357 1.396 1.391 1.363

1.125 1.168 1.067 0.9561 0.7282

1.051 1 136 1.164 1.158 1.134

-

1.8

0.5

1

0.6

-

1

I1 (3)

0.3733 0.5420 0.5906

0

0.7

I1(2)

0.3233 0.4700 0.5119

1

1

I1(1)

0.5266 1.145 1.477

0.7

0.5

I2

0.05 0.6 1 .o

1.5

2

g'(0)

0

-

-

-

-

-

0.3241

-

w

TABLE I1 (Continued) I

n

3

0.5

0.7

Q

1

1

*Convergence

0

to

-

-

\o 00

0.15 0.4 0.6 1.0 0.05

0.15 0.4 0.6 1 .o

0* 7883 1.228 1.522 2.044 0.6255 0.8493 1.257 1.537 2.044

0.3503 0.4383 0.4805 0.5397

0.4082 0-5093 0.5578 0.6258

0.1616 0.3072 0.4142 0.6158

0.3036 0.3074 0.2796 0.1967

1.238 1.323 1.335 1.328

1.051 0.9723 0.8603 0.6158

1.047 1.109* 1.115*

0.3324 0.3926 0.4593 0.4923 0.5397

0.3849 0.4545 0.5319 0.5702 0.6258

0 1057 0.1711 0.3125 0.4176 0.6158

0.3083 0.3360 0.3194 0.2847 0.1967

0.1231 1.334 1.367 1.359 1.328

1.091 1.121 0.9973 0.8708 0.6158

1.037 1.118 1.141 1.133 1.107

-

1.107

P

hj

I2

L?

5e

TABLE 111

SIMILAR SOLUTIONS FOR Pr = O.7,fM,# 0, t, ~~

B 0

~

UI

0.5

0.7

-

-

t

I1

I2

11(1)

11(2)

11(3)

0.4328 0.5343

0.5519 1.546

0.4303 0.5239

1.523 1.742

1.523 1.742

1.142 0.9846

0.2093 0.0

0.2920 0.3897

0.6331 1.697

0.4888 0.5774

1.836 2.004

1.836 2.004

1.415 1.368

0.1262

0.1663 0.2563

0.7514 0.5620 1.898 0.6411

2.318

0.0

2.366

2.318 2.366

1.843 1.840

f"(0)

9'(0)

g'(0)

0.15 0.8009

0.4328 0.5343

0.3026 0.0

1

0.15 0.7755

0.2920 0.3897

1

0.15 0.7455

0.1663 0.2563

=1

0

1

-0.2 -0.4

1

-~

~

f"

=

W

-0.6

1

0.15 0.4 0.6

0.0629 0.1135 0.1315

0.0556 0.0720 0.0435

0.0629 0.1135 0.1315

0.9530 1.517 1.956

0.6548 0.6980 0.7123

3.224 2.981 2.934

3.224 2.981 2.934

2.672 2.440 2.444

0

0

0.15 1 .o

0.3920 0.4696

0.3428 0.4139

0.3920 0.4696

0.4414 1.686

0.3921 0.4696

1.508 1.686

1.508 1.686

1.255 1.385

0.5

0.15 0.9162

0.4089 0.4802

0.3250

0.4089 0.4802

0.4970 1.636

0.4089 0.4802

1.538 1.700

1.537 1.700

1.224 0.7691

0.15 0.8187

0-4451 0.5034

0.3136 0.0

0.4451 0.5034

0.5766 1.580

0.4448 0.5029

1.586 1.725

1.586 1.725

1.188 0.8037

0

0.15 1.0

0.2559 0.3305

0.2422 0.3108

0.2559 0.3305

0.5611 1.999

0.4559 0.3305

1.869 1.999

1.869 1.999

1.539 1.623

0.5

0.15

0.2718 0.3400

0.2281 0.0

0.2718 0.3400

0.6022 1.887

0.4718 0.5400

1.884 2.004

1.884 2.004

1.508 1.210

1

-0.2

0.9028

0.0

TABLE 111 (Continued)

P 0

=1

UJ

f"

0.7

-0.2

1

0.15 0.7925

0.3050 0.3612

0.2205 0.0

0.3050 0.3612

0.6625 1.757

0.5048 0.5605

1.901 2.011

1.901 2.011

1.458 1.228

-0.4

0

0.15 1 .o

0.1371 0.2049

0.1481 0.2126

0.1371 0.2049

0.7445 2.447

0.5371 0.6049

2.443 2.447

2.443 2.447

1.998 1.968

0.5

0.15 0.8868

0.1509 0.2128

0.1398 0.0

0.1509 0.2128

0.7580 2.239

0.5509 2 -422 0.6128' 2.438

2.422 2.438

1.957 1.763

1

0.15 0.7616

0.1795 0.2311

0.1369 0.0

0.1795 0.2311

0.7824 1.996

0.5792 0.6301

2.373 2.415

2.373 2.415

1.872 1.754

0

0.15 0.4 0.6

0.0433 0.0728 0.0843 0.0975

0.0620 0.0435 0.0941 0.0728 0.1063 0.0843 0.1201 0.0975

1.081 1.705 2.200 3.181

0.6433 0.6728 0.6843 0.6975

3.613 3.321 3.249 3.181

3.613 3.321 3.249 3.181

2.979 2.695 2.622 2.552

0.5

0.15

0.8682

0.0533 0.1033

0.0615 0.0

0.0533 0.1033

1.032 2.805

0.6533 0.7034

3.460 3.143

3.460 3.143

2.856 2.563

0.15 0.4 0.6 0.7248

0.0745 0.1041 0.1138 0.1177

0.0651 0.0693 0.0432 0.0

0.0745 0.1041 0.1138 0.1177

0.9795 1.591 2.068 2.364

0.6737 0.7026 0.7119 0.71 57

3.214 3.091 3.063 3.052

3.214 3.091 3.063 3.052

0.15 0.7897

0.5091 0.7268

0 .o

0.3188

0.4604 0.5m

0.4752 1.303

0.4114 0.4589

1.507 1.686

1.419 I .474

1.110 0.8130

0.15

0.3733 0.5874

0.2286 0.0

0.3250 0.4463

0.5289 1.390

0.4638 0.5021

1.779 1.901

1.670 1.653

1.343 1.107

-0.6

W

1.o

1

0.2

0.5

0 -0.2

1 1

0.7625

2.629 2.501

2.486 2.502

TABLE I11 (Continued)

B

UJ

f"

0.2

0.5

-0.4

0.5

W

3.2525 3.4579

0.1494 0.0

0.2056 0.3216

0.5983 1.476

0.5516

0.15 0.4 0.6 0.6882

0.1517 3.2611 0.3192 0.3406

0.0839 0.0975 0.0498 0.0

0.1080 0.1760 0.2039 0.2114

0.6917

0.6061

1b.111

0.15 0.8064

D.5202 0.6979

0.3265 0.0

0.4678 0.5465

-0.2

0.15 0.7765

0.3846 0.5614

0.2361 0.0

-0.4

0.15

0.2633 0.4352

0.1562

0.7400

0.15 0.4 0.6 0.6950

0

-0.6

0.4

t

0.15 0.7282

-0.6

0.7

01

0 -0.2 -0.4

0.5277

2.154 2.158

2.017 1.875

1.670 1.466

0.6180 0.6110 0.6115

2.701 2.541 2.485 2.498

2.526 2.302 2.193 2.165

2.158 1.984 1.916 1.855

0.4925 1.314

0.4216 0.4350

1.560 1.653

1.470 1.439

0.6471

0.3329 0.4122

0.5479 1.408

0.4749 0.4797

1.836 1.880

1.724 1.628

1.383 0.9855

0.2135 0.2910

0.6179

0.0

0.5395 0.5328

2.212 2.164

2.070 1.866

1.708 1.367

0.2487 0.3000 0.3219

0.1610

0.0896 0.0914 0.0484 0.0

0.1598 0.1793 0.1865

0-1150 0.7106 1.145 1.470 1.625

0.6184 0.6178 0.6041 0.5963

2.752 2.617 2.551 2.529

2.568 2.356 2.230 2.177

2-186 2.018 1.902 1.807

0.15 0.7815

0.5714 0.8782

0.3318 0.0

0.4824 0.6203

0.4229 1.149

0.3987 0.4135

1.499 1.659

1.349 1.306

1.090 0.7155

0.15

0.7522

0.4377 0.7383

0.2432

0.3499 0.4865

0.4634 1.202

0.4479 0.4509

1.749 1.847

1.568 1.448

1.300 0.9953

0.15 0.7172

0.3179 0.6060

0.1653

0.2330 0.3638

0.5129 1.257

0.5069 0.4946

2.078 2.072

1.857 1-623

1.582 1.296

0.0 0-0

1.427 1.586

1.511

1.150

B 0.4

t

f"

U

0.5

-0.G

1

0.15 0.6749

0.2160 0.4828

0 .o

0.1006

0 * 1363 0.2548

0.5746 1.313

0.5780 0.5464

2.521 2.347

2.250 1.842

1.971 1.628

0.7

0

0

0.15 1.0

0.5027 0.8544

0.3597 0.4632

0.4129 0.5300

0.3347 1.219

0.3687 0.3667

1.443 1.521

1.358 1.219

1.204 1.257

0.5

0.15 0.9058

0.5281 0 8470

0.3422 0.0

0.4344 0.5430

0.3749 1.185

0.3781 0.3728

1.475 1.551

1.366 1.235

1.166 0.5240

1

0.15 0.7974

0.5807 0.8490

0.3363 0.0

0.4853 0.5763

0.4358 1.151

0.4063 0.3887

1.546 1.617

1.393 1.265

1.126 0.5624

0

0.15 1 .o

0.3786 0.7377

0.2662 0.3727

0.2851 0.4052

0.3934 1.346

0.4211 0.3993

1.730 1.720

1.608 1.346

1.429 1.408

0-5

0.15 0.8880

0.4002 0.7202

0.2511 0-0

0.3054 0.4156

0.4259 1.280

0.4298 0.4083

1.753 1.754

1.608 1.373

1.391 0.8373

1

0-4468 0.15 0-7647 0.7109

0.2472 0.0

0.3531 0.4452

0.4760 1.207

0.4559 0.4281

1.800 1.816

1.615 1.416

1.340 0.8903

0

0.15 1 .o

0.2707 0.6321

0.1829 0.2903

0.1768 0.2956

0.4657 1.491

0.4845 0.4357

2.113 1.954

1.940 1.491

1.735 1.588

0.5

0.15 0.8656

0.1706 0.0

0-1944 0.4882 0.3022 1.383

0.4923 0.4495

2.122 1.996

1.930 1.540

1.696 1.171

1

0.15 0.7256

0.2875 0.6025 0.3263 0.5810

0.1687 0.0

0.2362 0.3264

0.5251 1.266

0.5155 0.4141

2.133 2.060

1.906 1.602

1.624 1.223

0

0.15 1 .o

0.1826 0.5384

0.1124 0.2175

0.0933 0.2034

0.5537 1.656

0.5611 0.4760

2.632 2.225

2.391 1.656

2.162

-0.2

-0.4

-0.6

1

W

-

1.800

TABLE 111 (Continued)

B 0.4

0.5

fw

0.7

0.1

f**(O)

e*(o)

g*(o)

0.1580 0.1830 0.2054

I~

I2

I10)

11(2)

11(3)

0.9262 1.193 1.493

0.5546 0.5308 0.4978

2.449 2.364 2.286

2.081 1.913 1.743

1.925 1.801 1-531

0.5872 0.5692 0.5415 0.5298

2.579 2.454 2.384 2.360

2.299 2.052 1.894 1.838

2.015 1.834 1.680 1.596

0.5

0.4 0.6 0.8367

0.3251 0.4078 0.4950

0.1308 0.1207 0.0

1

0.15 0.4 0.6 0.6772

0.2228 0.3517 0.4316 0.4601

0.1031 0.1391 0.5857 0.1018 0-1914 0-9521 0.0472 0.2151 1.222 0.0 0.2226 1.323

0

0.15 1 .o

0.5243 0.9277

0-3625 0.4164 0.4705 0.5390

0-3182 0.3652 0.3503 1.155

1.433 1.500

1.334 1.155

1.196 1.240

-0.2

0

0.15 1.0

0.4014 0.8126

0.2699 0.3810

0.2895 0.4153

0.3704 1.266

0.4163 0.3796

0.1689

1.711

1.572 1.013

1.414 1.014

-0.4

0

0.15 1.0

0.2942 0.7077

0.1874 0.2995

0.1822 0.3065

0.4331 1.391

0.4777 2.077 0.4121 -1.909

1.883 1.391

1.706 1.553

-0.6

0

0.15 0.4 0.6 1 .o

0.2057 0.3534 0.4482 0.6138

0.1178 0.1682 0.1927 0.2274

0.0990 0.1511 0.1771 0.2146

0.5079 0.8354 1.077 1.532

0.5518 0.5357 0.5096 0.4477

2.562 2.379 2.289 2.162

2.297 1.997 1.820 1.532

2.105 1.936 1.857 1.750

-0.8

0

0.5 1.0

0.3255 0.5314

0.1216 0.1656

0.0968 0.1410

1.081 1.689

0.5853 0.4866

2.718 2.447

2.186 1.689

1.979

-1.0

0

0.15 1 .o

0.0939 0.4604

0.0284 0.1148

0.0140 0.6933 0-0859 1.863

0.7472 0.5489

4-045 2.763

3.601 1-863

3.421 2.240

-1.2

0

0.15 0.4

0.0683 0.1774

0.0099 0.0341

0.0031 0.0169

0.7968 1.218

0.8700 0.7683

5.060 3.872

4.531 3.160

4.393 3.236

-0.6

0

2.211

+ 0

TABLE I11 (Continued)

P

B

fw

=1

-1.2

0

t

W ~~

0.5

0.7

0.75

0.7

0.5

1 .o

0.2563 0.4006

0.0499 0.0753

0.0281 0.0481

1. 513 2.052

0.6998 0.5745

3.506 3.105

2.672 2.052

2.896 2.536

0

1 .o

'3.3510

0.0464

0.0244

2.256

0.6229

3.467

2.256

2.862

0.5

0.15 0.9004

0.6029 1.072

0.3512 0.0

0.4478 0.5719

0.3187 1.0000

0.3690 0.3215

1.446 1.496

1.287 1.044

1.139 '3.4385

-0.2

0.5

0.15 0.8808

0.4771 0.9440

0.2618 0.3213 0.0 0.4471

0.3535 1 059

0.4119 0.3495

1.701 1.674

1.496 1.146

1.344 0.7242

-0.4

0.5

0.15 0.8560

0.3651 0.8222

0.1830 0.0

0.2127 0.3941 0.3354 1.120

0.4696 0.3825

2.027 1.882

1.764 1.267

1.612

-0.6

0.5

0.15 0.4 0.6 0.8240

0.2697 0.4588 0.5817 0.7064

0.1171 0.1455 0.1311 0.0

0.1256 0.1844 0.2134 0.2387

0.4411 0.7349 0.9500 1.179

0.5389 0 5076 0.4693 0.4221

2.446 2.284 2.200 2.126

2.111 1.798 1.604 1.415

1.965 I . 772 1.635 1.338

0

0

0.15 1.0

0.5612 1.233

0.3317 0.4959

0.3844 0.5705

0.2437 0.9402

0.3176 0.2923

1.300 1.430

1.175 0.9402

1.096 1.186

-0.2

0

0.15 1 .o

0.4461 1.121

0.2441 0.4090

0.2633 0.4497

0.2787 1.009

0.3674 0.3119

1.570 1.593

1.394 1.009

1.312 1.310

-0.4

0

0.15 1 .o

0.3469 1.018

0.1684 0.3298

0.1641 0.3428

0.3189 1.085

0.4279 0.3330

1.917 1.778

1.675 1.085

1.595 1.452

-0.6

0

0.4 0.6

0.5028 0.6559 0.9226

0.1763 0.2106 0.2591

0.1621 0.1987 0.2512

0.6238 0.8128 1.167

0.4791 0.4431 0.3556

2.174 2.094 1.984

1.693 1.496 1.167

1.783 1.709 1.612

-1.4

1

0.6

0

1 .o

1.022

. c -

TABLE 111 (Continued)

1

0.7

1

Q1

W

0

0

0.15 1 .o

0.6146 1.233

0.3729 0.4959

0.4295 0.5705

0.2614 0.9402

0.3532 0.2923

1.396 1.430

1.253 0.9402

1.167 1.186

-0.2

0

0.15 1.0

0.4947 1.121

0.2828 0.4090

0.3053 0.4497

0.2944 1.009

0.4004 0.3119

1.647 1.593

1.454 1.009

1.364 1.310

-0.4

0

0.15 1.0

0.3886 1.018

0.2028 0.3298

0.2002 0.3320 0.3428 1.085

0.4566 0.3330

1.965 1.778

1.707 1.085

1.617 1.452

-0.6

0

0.15 0.4 0.6 1.0

0.2983 0.5189 0.6647 0.9226

0.1350 0.1912 0.2191 0.2591

0.1175 0.1770 0:2072 0.2512

0.3744 0.6296 0.8166 1.167

0.5239 0.4894 0.4481 0.3556

2.366 2.197 2.109 1.984

2.028 1.704 1.503 1.167

1.945 1.791 1.716 1.612

-0.8

0

1 .o

0.8354 0.1976

0.1758

1.255

0.3799

2.212

1.255

1.793

-1.2

0

1.0

0.6858

0.1034

0.0731

1.452

0.4330

2.723

1.452

2.218*

-1.4

0

1.0

0.6229

0.0704

0.0429

1.561

0.4620

3.000

1.561

2.460*

-1.6

0

1 .o

0.5674 0.0457

0.0234

1.675

0.4924

3.287

1.675

2.721*

0

0

0.15 1 .o

0.7104 1.233

0.4473 0 -4959

0.5107 0.5705

0.2934 0.9402

0.4171 0.2923

1.572 1.430

1.395 0.9402

1.296 1.186

-0.2

0

0.15 1 .o

0.5842 1.121

0.3536 0.4090

0.3823 0.4497

0.3236 1.009

0.4607 0.3119

1.798 1.593

1.572 1.009

1.469 1.310

-0.4

0

0.15 1 .o

0.4688 1.018

0.2677 0.3298

0.2691 0.3428

0.3575 1.085

0.5112 0.3330

2.076 1.778

1.788 1.085

1.682 1.452

*convergencet o

10-4.

TABLE I I I (Continued)

1

2

1

0.5

0.7

-0.6

0

0.15 1 .o

0.3658 0.9226

0.1912 0.2591

0.1740 0.2512

0.3957 1.167

-0.8

0

0.15 1.0

0.2774 0.8354

0 e 1260 0.1976

0.0999 0.1758

0.4383 1.255

0.5702 0.3556 0.6392 0.3799

- 1 .o

0

0.15

0.2057

0.0742

0.0485

0.4856

-1.2

0

0.15 1.0

0.0374 0.1034

0.0187 0.0731

-1.4

0

0.15

0.1524 0.6858 0.1171 0.6229

0.0155 0.0704

-1.6

0

0.15

0.0960 0.5674

0

0

0.4 0.6 1 .o

1.043 1.276 1 a687 1.065 1.475

1 .o 1 .o

2.419 1.984

2.054 1.167

1.951 1.612

2.842 2.212

2.388 1.255

2.294 1 793

0.7201

3.366

2.813

2.738

0.5372 1.452

0.8151 0.4330

4.007 2.723

3.351 1.452

3.311 2.218*

0.0054 0.0429

0.5923 1.561

0.9248 0.4620

4.764 3.000

4.012 1.561

4.023 2.460*

0.0052 0.0457

0.0011 0.0234

0.6497 1.675

1.046 0.4924

5.602 3.287

4.764 1.675

4.840 2.721*

0.4276 0.4677 0.5235

0.4956 0.5414 0.6052

0.3617 0.4884 0.7282

0.3196 0.2988 0.2308

1.352 1.367 1.363

1.131 1 140

0.3048 0.3605

0.3174 0.3802

0.5552 0.8100

0.3544 0.2549

1.727 1.664

1.040 0.9445 0.7282 1.123 0.8100

-

-

1.134

-0.4

0

0.6 1 .o

-0.2

0

0.6

1 .o

1.180 1.578

0.3930 0.4384

0.4336 0.4863

0.5248 0.7678

0.3304 0.2425

1.557 1.506

1.039 0.7678

1.286* 1 243

-0.4

0

0.4 0.6

0.9597 1.181

0.2038

0.4094 0.4464

0.4792 0.6222

0.3915 0.3327

1.889 1.848

1.226 1.036

1.245 1.035

-0.6

0

0.15 0.4

0.4291 0.7594

0.1520 0.2140

0.1364 0.2032

0.2640 0.4556

0.5010 0.4481

2.206 2.047

1.807 1.459

1.815 1.673

'convergence

to

1.420* 1.365

-

PI

d

(v

0

. ?

PI

rl

4

v1

:

0.

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

0

m

00

4 4

d F - 4

n

m I

0

N h PI

h

N03

..

4 0

w m e mln

? ? '9

I4

U

h

4

U N

Y h

0

h

d

. . ? ?

A

N d d

y m

v1

m m dpl m w

d

Y 4 d

H

m N 0

w m

*

d

0

?

m

0

0

w

4

. . ;=. 0

w

06 d d 0

m

00

YN

m a m

h N N h Q v1

00

2.!:

*cow*

00

..

N +I

d

I4

n

0

-

U

v1

m o -3

0

N

N

m

0 03

h

N d

:?

WPI

47 04

F - 4 h

v)

00

-3m N N

. . f;. .

N-3

n

a l

-0, h

-

3

Y 0

ry

u

b"

2 3

m.

407

TABLE IV SIMILAR SOLUTIOXS FOR t,

0

0.5

0

0.5

l,f,, = 0

f"(0)

e"o)

g'(0)

0.15 0.4 0.6 1 .o

0.3490 0.4143 0.4399 0.4696

0.3034 0.3625 0.3861 0.4139

0 3490

0.5

0.15 0.4 0.6 0 8482

0.3816 0.4437 0.4662 0.4834

0.7

0.15 0.4 0.6 0.9151 0.15 0.4 0.6

0.7

I1

I2

I p )

I1(2)

0.4143 0.4399 0.4696

0.4037 0-7886 1.089 1.686

0.3490 0.4144 0.4399 0.4696

1.411 1.571 1.626 1.686

1.411 1.571 1.626 1.686

1.185 1.303 1.343 1.385

0.2443 0.2608 0.2287 0.0

0.3816 0.4437 0.4662 0.4834

0.3951 0.8234 1.163 1.584

0.3816 0.4437 0.4662 0.4834

1.478 1.623 1.670 1.705

1.478 1.623 1.670 1.705

1.274 1.332 1.268 0.7940

0.3734 0.4394 0.4642 3.4876

0.2963 0.3341 0.3246 0.0

0.3734 0.4394 0.4642 0.4876

0.4608 0.8600 1.167 1.645

0.3735 0.4394 0.4642 0.4876

1.452 1.610 1.663 1.710

1.452 1.610 1.663 1.710

1.166 1.250 1.240 0.7665

0.4532 0.5043 0.5232

0.2166 0.1716 0.0450

0.4532 0.5043 0.5232

0.4784 0.9359 1.251

0.4315 '1.532 0.4949 1.687 0.5106 1.721

1.532 1.687 1.721

1.240 1.252 1.174

0.05 0.15 0.4 0.6 0.8009

0.3507 0.4328 0.4987 0.5208 0.5343

0.2527 0.3026 0 3045 0.2392 0.0

0.3507 0.4328 0.4987 0 5208 0.5343

0.3514 0.5519 0.9513 1.251 1.546

0.3494 0.4303 0.4924 0.5124 0.5239

1.304 1.523 1.672 1.717 1.742

1.304 1.523 1.672 1.717 1.742

1.003 1.142 1.201 1.163 0.9846

1.1

0.5491

1.521

0.5491

1.953

0.5301

1.753

1.753

2 .ooo

-

1

=

0.5

0.7

-

-

-

TABLE IV (Continued) ~

0

0.5

all

1

f"(0)

e'(0)

g'(0)

1 .o 1.1

0.4223 0.4988 0.5264 0.5541 0.5550

0.4223 0.4988 0.5264 0.5541 0.5550

0.4223 0.5723 0.4988 0.9284 0.5264 1.192 0.5541 1.698 0.5550 1.871

0.15

0.4 0.6

0.7

I1

~~

~

I#)

I1(2)

I1(3)

0.4146 0.4734 0.4939 0.5179 0.5345

1.466 1.611 1.657 1.709 1.749

1.466 1.611 1.657 1.709 1.749

1.051 1.137 1.163 1.192 1.215

l2

0

0.7

0.15 0.4 0.6 1 .o

0.3920 0.4353 0.4514 0.4696

0.3428 0.3820 0.3969 0.4139

0.3920 0.4353 0.4514 0.4696

0.4414 0.8137 1.106 1.686

0.3921 0.4353 0.4514 0.4696

1.508 1.614 1.649 1.686

1.508 1.614 1.649 1.686

1.255 1.334 1.359 1.385

all

1

0.6 1. -0

0.4498 0.4696

0.4498 0.4696

0.4498 0.4696

1.167 1.686

0-4498 1.645 0.4696 1.686

1.645 1.686

1.196 1.217

0.5

0.7

0.15 0.4 0.6 0.9162

0.4089 0.4512 0.4663 0.4802

0.3250 0.3431 0.3263 0.0

0.4089 0.4512 0.4663 0.4802

0.4970 0.8778 1.173 1.636

0.4089 0.4512 0.4663 0.4802

1.538 1-640 1.672 1.700

1.538 1.640 1.672 1.700

1.224 1.270 1.248 0.7691

1

0.5

0.15 0.4 0.6 0.6729

0.4517 0.4854 0.4954 0.4978

0.2254 0.1793 0.0733 0.0

0.4517 0.4854 0.4954 0.4978

0.5519 0.9694 1.294 1.412

0.4512 0.4844 0.4941 0.4965

1.610 1.690 1.712 1.717

1.610

1.690 1.712 1.717

1.245 1.201 1.044 0.9324

0.05 0.15 0.4 0.6 0.8187

0.3924 0.4451 0.4835 0.4960 0.5034

0.2843 0.3136 0.3025 0.2420 0.c

0.3924 0.4451 0.4835 0.4960 0.5034

0.3963 0.3923 0.5766 0.4448 0.9637 . 0.4832 0.4947 1.253 0.5029 1.580

1.442 1.586 1.681 1.707 1.725

1.442 1 * 586 1.681 1.707 1.725

1.100 1.188 1.195 1.134 0.8037

1.1

0.5097

1.291

0.5097

1.982

1.737

1.737

2.444

0.7

0.5085

TABLE IV (Continued) ~

B 0

w

~

f"(0)

I1

I2

Il(1)

Il (2)

I1(3)

0.4386

0.6046

0.4369

1.555

1.555

1.118

0.4696 0.4696

0.4696 1.686

0.4696 0.4696

1 *686 1.686

1.686 1.686

1.217 1.217

0.4139 0.4139 0.4139 0.4139

0.4696 0.4696 0.46% 0.4696

0.5091 0.8554 1.132 1.686

0-4696 0.4696 0.46% 0.4696

1.686 1.686 1.686 1.686

1.686 1.686 1.686 1.686

1.385 1.385 1.385 1.385

0.4696 0.4696 0.4696 0.4696

0.3029 0.2763 0.2311 0.0

0.4696 0.4696 0.4696 0.4696

0.4671 0.8593 1.173 1.569

0.4696 0.4696 0.4696 0.4696

1.686 1.686 1.686 1.686

1.686 1.686 1.686 1.686

1.434 1.378 I . 283 0.7952

0.15 0.4 0.6 0.9179

0.4696 0.4696 0.4699 0.4696

0.3739 0.3572 0.3281 0.0

0.4696 0.4696 0.4699 0.4696

0.5595 0.9057 1.177 1.623

0.4696 0.4696 0.4684 0.4696

1.686 1.686 1.683 1.686

1.686 1.686 1.683 1.686

1.326 1.301 1.266 0.7732

0-5

0.15 0.4 0.6

0.4696 0.4696 0.4696

0.2301 0.1860 0.0955

0.4696 0.4696 0.4696

0.5812 0.9738 1.287

0.4696 0.4696 0.4696

1.686 1.686 1.686

1.686 1.686 1.686

1.300 1.188 0.9983

0.7

0.15 0.4 0.6 0.8357

0 * 4696 0.4696 0.4696 0.4696

0.3339 0.3006 0.2439 0 .o

0.4696 0.4696 0.4696 0.4696

0.6097 0.9560 1.233 1.559

0.4696 0.4696 0.4696 0.4696

1.686 1.686 1.686 1.686

I .686

1.686 1.686 1.686

1.267 1.217 1.134 0.7729

0.15

0.4696

0.4696

0.4696

0.6521

0.4696

1.686

1.686

1.217

g'(0)

Q1

Pr

0.7

all

1

0.15

0.4386

0.4386

1

all

1

0

1 .o

0.4696 0.4696

0.4696 0.4696

0

0.7

0.15 0.4 0.6 1 .o

0.4696 0.4696 0.4696 0.4696

0.5

0.5

0.15 0.4 0.6 0.8522

0.7

1

all

1

TABLE IV (Continued)

0.05

1

all

1

0 0.5 1 .o

0.4848 0.5082 0.5311

0.4733 0.4773 0.4812

0.4733 0.4773 0.4812

0.4453 1.023 1.593

0.4626 0.4570 0.4514

1.675 1.663 1.652

1.654 1.623 1.593

1.208 1.199 1.191

0.1

0.5

1

0.7

0.05 0.15 0.4 0.6 0.7948

0.3780 0.4733 0.5655 0.6066 0.6376

0.2597 0.3112 0.3134 0.2435 0.0

0.3623 0.4476 0.5194 0.5450 0.5613

0.3187 0.5088 0.8753 1.149 1.409

0.3414 0.4197 0.4717 0.4840 0.4882

1.291 1.513 1.652 1.690 1.709

1.260 1.464 1.575 1.591 1.590

0.9912 1.124 1.166 1.106 0.8825

0.7

1

0.7

0.05

0.4202 0.4850 0.5492 0.5808 0.6086

0.2900 0.3206 0.3096 0.2463 0.5096

0.4021 0.4573 0.5008 0.5164 0.5274

0.3614 0.5303 0.8823 1.150 1.428

0.3834 0.4320 0.4608 0.4655 0.4653

1.430 1.571 1.654 1.674 1.682

1.394 1.521 1.576 1.575 1.561

1.087 1.166 1.157 1.063 0.7109

0.15

0.4 0.6 0.8120

1

all

1

1

0.7

1.1

0.6420

1.492

0.5447

1.659

0.4751

1.702

1.569

0.8969

0.15

0.5123

0.4 0.6 1 .o

0.5347 0.5523 0.5870

0.4788 0.4825 0.4054 0.4909

0.4788 0.4825 0.4854 0.4909

0.5909 0.8668 1.085 1.516

0.4531 0.4480 0.4438 0.4355

1.658 1.647 1.639 1.624

1.607 1.595 1.558 1.516

1.196 1.188 1.182 1.170

0.15 0.4 0.6 0.8290

0.5085 0.5331 0.5525 0.5744

0.3381 0.3051 0.2459 0.0

0.4781 0.4824 0.4857 0.4894

0.5563 0.8715 1.121 1.403

0.4529 0.4460 0.4404 0.4341

1.660 1.648 1.638 1.627

1.610 1.576 1.549 1.520

1.240 1.174 1.071 0.6798

N

B

u

r

u1

Pr

tw

f"(0)

e'(0)

g'(0)

I1

12

Il(1)

Il(2)

I1(3)

0.15

0.7

0

0.7

0.6

1.145

0.4700

0.5420

0.5504

0.3196

1.415

1.021

1.176

0.2

0.5

1

0.7

0.05 0.15 0.4 0.6 0.7897

0.4015 0.5091 0.6243 0.6818 0.7268

0.2657 0.3188 0.3209 0.2467 0.0

0.3718 0.4604 0.5370 0.5655 0.5838

0.2977 0.4752 0.8161 1.069 1.303

0.3366 0.4114 0.4550 0.4608 0.4589

1.285 1.507 1.640 1.673 1.686

1.230 1.419 1.499 1.494 1.474

0.9808 1.110 1.139 1.062 0.8130

1.1

0.7913

1.748

0.6069

1.659

0.4479

1.680

1.425

2.337

0.05 0.15 0.4 0.6 0.8064

0.4445 0.5202 0.6068 0.6548 0.6979

0.2950 0.3265 0.3147 0.2470 0.0

0.4103 0.4678 0.5155 0.5338 0.5465

0.3357 0.4925 0.8141 1.059 1.314

0.3766 0.4216 0.4426 0.4415 0.4350

1.422 1.560 1.633 1.646 1.653

1.358 1.470 1.495 1.473 1.439

1.076 1.150 1.136 1.036 0.6471

1.1

0.7539

1.489

0.5607

1.657

0.4223

1.647

1.386

2.709

0 0.5

1.0

0.5233 0.6070 0.6867

0.4821 0.4951 0.5069

0.4821 0.4951 0.5069

0.3881 0.8995 1.392

0.4457 0.4271 0.4082

1.648 1.612 1.580

1.576 1.480 1.392

1.188 1.161 1.138

0.15 0.4 0-8235

0.5424 0.5883 0.6633

0.3414 0.3085 0.0

0.4851 0.4926 0.5044

0.5143 0.8065 1.288

0.4395 0.4270 0.4057

1.639 1.618 1.586

1.550 1.491 1.397

1.219 1.141 0.6165

0.15 1.0

0.4283 0.7627

0.3157 0.4534

0.3643 0.5179

0.3292 1.309

0.3342 0-3887

1.360 1-551

1.302 1.309

1.144 1.280

0.7

1

1

all

1

0.2857

0.5

0

0.7

1

0.7

0.7

9 r

0

z

m

G3 U

j3 L(

F!

* 3

L(

g m

zr

Fg (r,

TABLE IV (Continued)

0.2857

0.5

0.7

1

1

0.5

0.7

1

0.7

1

all

1

0.7

1

0.7

f"(0)

9"O)

g'(0)

0.15 0.9071

0.4605 0.7683

0.3101 0.0

0.05 0.15 0.4 0.6 0.7849

0.4209 0.5371 0.6701 0-7401 0.7952

1.1

I1

I2

0.3934 0.5397

0.3742 1-282

0.3537 0.4019

0.2710 0-3246 0.3266 0.2490 0.0

0.3808 0-4704 0.5505 0.5810 0.6012

0.2779 0.4503 0.7737 1.013 1.223

0.8787

1.862

0.6294

0.05 0.15 0.4 0.6 0.8022

0.4636 0.5475 0.6512 0.7118 0.7664

0.2988 0.3310 0.3194 0.2499 0.0

1.1

0.8401

0 0.2 0.4 0.6 0.8

I

p

11(2)

11(3)

1.406 1-595

1.328 1.335

1.123 0.5663

0.3299 0.4053 0.4428 0.4439 0.4374

1.268 1.502 1.631 1.661 1.667

1.199 1-385 1.446 1.426 1.393

0.9696 1-100 1.120 1.032 0.7890

1.548

0.4195

1.640

1.325

2.232

0.4171 0.4758 0.5259 0.5459 0.5603

0.3143 0.4657 0.7744 1.007 1.236

0.3707 0.4143 0.4298 0.4241 0.4133

1.411 1.553 1.624 1.635 1.635

1.329 1.433 1.439 1.402 1.356

1.068 1.138 1.108 0.9885 0.6055

1.565

0.5771

1.557

0.3950

1.621

1.286

2.709

1.o

0.5419 0.5883 0.6334 0.6774 0.7205 0.7627

0.4862 0.4932 0.4998 0.5061 0.5121 0.5179

0.4862 0.4932 0.4998 0.5061 0.5121 0.5179

0.3629 0.5593 0.7517 0.9406 1.126 1.309

0.4382 0.4285 0.4186 0.4087 0.3987 0.3887

1.636 1.617 1.599 1.582 1.566 1.551

1.542 1.491 1.443 1.396 1.352 1.309

1.179 1.165 1.151 1.139 1.127 1.116

0.15 0.4

0.5684 0.6308

0.3438 0.3108

0.4901 0.4999

0.4846 0.7611

0.4299 0.4134

1.625 1.598

1.508 1.432

1.204 1.117

TABLE IV (Continued)

fii(o)

ei(o)

gi(o)

I2

11(1)

11(3)

0.2857

1

1

0.7

0.6 0.8194

0-6793 0.7310

0.2473 0.0

0.5072 0.5148

0.9773 1.210

0.4001 0.3854

1.578 1.558

1.374 1.314

0.9918 0.5758

0.3

1

all

1

0 0.2 0.4 0.6 0.8 1 .o 2 .o

0.5448 0.5931 0.6402 0.6860 0.7309 0.7748 0.9829

0.4868 0.4941 0.5009 0.5074 0.5136 0.5195 0.5457

0.4868 0-4941 0.5009 0.5074 0.5136 0.5195 0.5457

0.3591 0.5539 0.7446 0.9318 1.116 1.297 2.165

0.4371 0.4270 0.4168 0.4065 0.3961 0.3857 0.3334

1.634 1.614 1.596 1.578 1.562 1.547 1.484

1.537 1.484 1.434 1.386 1.341 1.297 1.099

1.178 1.163 1.149 1.136 1.124 1.113 1.066

0.4

0.5

0

0.7

0.15 1.0

0.4540 0.8544

0.3192 0.4632

0.3686 0.5300

0.6094 1.219

0.3303 0.3667

1.347 1.521

1.273 1.219

1.133 1.257

0.5

0.7

0.15 0.9048

0.4887 0.8564

0.3141 0.0

0.3992 0.5533

0.3511 1.194

0.3483 0.3786

1.394 1.570

1.296 1.244

1.111 0.5232

1

0.7

0.05 0.15 0.4 0.6 0.7815

0.4416 0.5714 0.7260 0.8112 0.8782

0.2757 0.3318 0.3334 0.2515 0.0

0.3870 0.4824 0.5667 0.5995 0.6203

0.2694 0.4229 0.7270 0.9516 1.149

0.3321. 0.3987 0.4291 0.4246 0.4135

1.289 1.499 1.624 1.651 1.659

1.188 1.349 1.386 1.351 1.306

0.9672 1.090 1.099 0.9983 0.7155

1.1

0.9837

1.924

0.6189

1.474

0.3884

1.644

1.218

2.562

0

0.7

0.15 1 .o

0.5027 0.8544

0.3597 0.4632

0.4129 0.5300

0.3347 1.219

0.3687 0.3667

1.443 1.521

1.358 1.219

1.204 1.257

0.5

0.7

0.15 0.4

0.5281 0.6583

0.3422 0.3671

0.4344 0.4909

0.3749 0.6569

0.3781 0.3955

1.475 1.544

1.366 1.362

1.166 1.176

0.7

TABLE IV (Continued)

0.4

0.7

1

0.5

0.7

0.6 0.9058

0.7386 0.8470

0.3500 0.0

0.5159 0.5430

0.8701 1.185

0.3906 0.3728

1.554 1.551

1.318 1.235

1.119 0.5240

1

0.7

0.05 0.15 0.4 0.6 0.7974

0.4864 0.5807 0.7053 0.7811 0.8490

0.3034 0.3363 0.3239 0.2500 0.0

0.4246 0.4853 0.5389 0.5609 0.5763

0.2943 0.4358 0.7220 0.9380 1.151

0.3654 0.4063 0.4154 0.4050 0.3887

1.404 1.546 1.608 1.614 1.617

1.300 1.393 1.377 1.326 1.265

1.058 1.126 1.092 0.9700 0.5624

1.460

all

0

1.1

0.9451

1.636

0.5953

0.3616

1.603

1.172

2.887

1

0 0.2 0.4 0.6 0.8 1 .o

0.5639 0.6254 0.6850 0.7429 0.7993 0.8544

0.4908 0.4997 0.5079 0.5157 0.5231 0.5300

0 -4908 0.3350

0.4997 0.5079 0.5157 0.5231 0.5300

0.5198 0.7001 0.8765 1.049 1.219

0.4299 0.4175 0.4050 0.3923 0.3795 0.3667

1.623 1.599 1.577 1.557 1.538 1.521

1.505 1.441 1.381 1.325 1.271 1.219

1.170 1.152 1.135 1.120 1.107 1.094

0.7

0.15 1.0

0.5901 0.8544

0.4327 0.4632

0.4926 0.5300

0.3806 1.219

0.4380 0.3667

1.619 1.521

1.514 1.219

1.333 1.257

0.5

0.7

0.15 0.9074

0.5948 0.8337

0.3901 0.0

0.4942 0.5283

0.4156 1.173

0 * 4286 0.3646

1.614 1.525

1.487 1.221

1.261 0.5248

1

0.7

0.15 0.4 0.6 0.8147

0.5998 0.6824 0.7461 0.8125

0.3465 0.3134 0.2474 0.0

0.4959 0.5082 0.5173 0.5264

0.4515 0.7109 0.9129 1.125

0.4192 0.3980 0.3809 0.3624

1.608 1.575 1.552 1.529

1.460 1.366 1.295 1.224

1.187 1.092 0.9564 0.5334

"1

0.5

0.5

Pr

tw

0.5

0.15 0.4 0.6

0.4646 0.6386 0.7451

0.2823 0.3456 0.3742

0.3746 0.4546 0.4905

0.2303 0.5099 0.7282

0.3495 0.3837 0.3809

1.372 1.476 1.496

1.324 1.342 1.292

1.286 1.387 1.410

0.7

0.05 0.15 0.4 0.6 1 .o 1.1

0.3518 0.4747 0.6472 0.7511 0.9277 0.9681

0.2520 0.3218 0.3940 0.4266 0.4705 0.4791

0.2923 0.3720 0.4533 0.4900 0.5390 0.5486

0.1704 0.2946 0.5628 0.7652 1.155 1.250

0.2666 0.3274 0.3658 0.3685 0.3503 0.3431

1.153 1.336 1.455 1.484 1.500 1.500

1.103 1.251 1.290 1.258 1.155 1.126

0.9820 1.125 1.212 1.232 1.240 1.240

all

1

0.15 0.4 0.6

0.4846 0.6558 0.7572

0.3689 0.4517 0.4890

0.3689 0.4517 0.4890

0.3530 0.6109 0.7989

0.3081 0.3503 0.3578

1.301 1.433 1.471

1.185 1.243 1.227

0.9788 1.054 1.071

0.25

0.7

0.05 0.15 0.952:

0.3653 0.4909 0.9260

0.2517 0.3187 0.0

0.3038 0.3857 0.5494

0.1840 0.3127 1.142

0.2733 0.3345 0.3549

1.171 1.358 1.522

1.112 1.259 1.165

0.9772 1.114 0.4897

0.5

0.7

0.05 0.15 0.4 0.6 0.903C 1.1

0.3827 0.5114 0.6867 0.7908 0.9266 1.007

0.2532 0.3171 0.3637 0.3539 0.0 1.024

0.3189 0.4037 0.4878 0.5247 0.5635 0.5830

0.2001 0.3339 0.6147 0.8245 1.131 1.325

0.2826 0.3444 0.3794 0.3786 0.3612 0.3446

1.194 1.385 1.507 1.537 1.552 1.553

1.124 1.271 1.303 1.264 1.179 1.117

0.9728 1.103 1.147 1.098 0.4935 2.080

0.6

0.7

0.05 0.15 0.882t

0.3913 0.5215

0.2546 0.3173 0.0

0.3265 0.4128 0.5708

0.2077 0.3438 1.127

0.2874 1.206 0.3496 1.399 0.3645 1.566

L .277

1.130

0.9712 1.098 0.4953

0

0.9280

1.185

TABLE IV (Continued)

0.5

0.5

0.05 0.1 0.15 0.8393

0.4138 0.4928 0.5475 0.9335

0.2603 0.2989 0.3204

0.3471 0.4028 0.4371 0.5906

0.2261 0.3017 0.3675 1.121

0.3007 0.3419 0.3638 0.3731

1.236 1.366 1.435 1.605

1.146 1.249 1.294 1.201

0.9680 1.052 1.090 0.4989

0.05 0.15 0.4 0.6 0 - 7762

0.2815 0.3380 0.3385 0.2521

1.1

0.4614 0.5998 0.7723 0.8688 0.9439 1.069

2.001

0.3970 0.4938 0.5831 0.6156 0.6374 0.6666

0.2496 0.3980 0.6808 0.8997 1.080 1.404

0.3233 0.3908 0.4149 0.4094 0.3958 0.3640

1.258 1.479 1.586 1.629 1.634 1.632

1.155 1.315 1.334 1.294 1.242 1.138

0.9530 1.079 1.089 0.9863 0.7265 2.655

0.5

0.15

0.5099

0.3173

0.4161

0.2437

0.3881

1.462

1.408

1.370

0.7

0.05 0.15 0.4 0.6 1 .o 1.1

0.4252 0.5243 0.6713 0.7643 0.9277 0.9657

0.3109 0.3625 0.4142 0.4378 0.4705 0.4770

0.3578 0.4164 0.4750 0.5018 0.5390 0.5464

0.2008 0.3182 0.5779 0.7751 1.155 1.247

0.3248 0.3652 0.3823 0.3768 0.3503 0.3420

1.307 1.433 1.499 1.508 1.500 1.496

1.244 1.334 1.324 1.275 1.155 1.124

1.098 1.196 1.244 1.249 1.240 1.237

all

1

0.4 0.6 1.0

0 6825 0.7720 0.9277

0.4754 0.5022 0 5390

0.4754 0.5022 0.5390

0.6305 0.8110 1.155

0.3673 0.3665 0.3503

1.482 1.497 1.500

1.279 1.245 1.155

1.080 1.085 1.078

0.5

0.7

0.05 0.15 0.4 0.6 0.9040 1.1

0.4500 0.5513 0.6986 0.7913 0.9169 0.9920

0.3013 0.3452 0.3710 0.3535

0.3784 0.4388 0.4975 0.5238 0.5526 0.5674

0.2322 0.3560 0.6238 0.8258 1.122 1.308

0.3339 0.3733 0.3867 0.3784 0.3557 0.3379

1.339 1.465 1.530 1.538 1.532 1.524

1.253 1.340 1.321 1.266 1.170 1.105

1.075 1.157 1.162 1.101 0.4941 2.030

0.8

1

0.7

0

0.7

0.7

-

0.0

0.0

-

0.0

0.9870

P, 00

t

0.5

3

w

0.7

1

Ul

P=

1

0.5

0.7

all

f"(O)

e*(o)

g'(0)

0.15 0.4 0.6

0.6047 0.7484 0.8397

0.2377 0.1805 0.0407

0.05 0.15 0.4 0.6 0.7934 1.1

0.5048 0.6073 0.7489 0.8370 0.9146 1.029

0 0.1 0.15 0.2 0.4

I1

I2

11(1)

I I W

11(3)

0.4964 0.5481 0.5706

0.4006 0.7066 0.9314

0.4034 0.3951 0.3745

1.558 1.600 1.591

1.384 1.316 1.235

1.157 1.015 0.7591

0.3071 0.3405 0.3277 0.2518 0.0 1.700

0.4305 0.4928 0.5488 0.5719 0.5886 0.6095

0.2794 0.4134 0.6862 0.8938 1.087 1.386

0.3616 0.4004 0.4047 0.3900 0.3711 0.3372

1.400 1.540 1.599 1.607 1.603 1.588

1.278 1.363 1.332 1.268 1.200 1.090

1.051 1.118 1.076 0.9366 0.5470 2.953

2.0

0.5811 0.6184 0.6368 0.6550 0.7262 0.7609 0.7952 0.8623 0.9277 1.235

0.4942 0.4995 0.5020 0.5045 0.5140 0.5185 0.5228 0.5311 0.5390 0.5729

0.4942 0.4995 0.5020 0.5045 0.5140 0.5185 0.5228 0.5311 0.5390 0.5729

0.3146 0.4034 0.4474 0.4909 0.6625 0.7468 0.8301 0.9941 1.155 1.920

0.4238 0.4167 0.4131 0.4095 0.3949 0.3876 0.3802 0.3653 0.3503 0.2746

1.613 1.599 1.592 1.586 1.561 1.550 1.539 1.519 1.500 1.424

1.477 1.440 1.422 1.404 1.337 1.304 1.273 1.212 1.155 0.8988

1.162 1.152 1.147 1.142 1.124 1.115 1.107 1.092 1.078 1.022

0.5

0.15 0.4 0.6 1.0

0.5913 0.6955 0.7754 0.9277

0.3806 0.3913 0.3990 0.4130

0.4907 0.5067 0.5183 0.5390

0.2686 0.5385 0.7484 1.155

0.2570 0.4262 0.4011 0.3503

1.626 1.582 1.552 1.500

1.561 1.430 1.333 1.155

1.521 1.486 1.461 1.419

0.7

0.15 0.4 0.6 1.0

0.6136 0.7105 0.7850 0.9277

0.4358 0.4471 0.4555 0.4705

0.4964 0.5103 0.5205 0.5390

0.3609 0.6026 0.7906 1.155

0.4331 0.4092 0.3897 0.3503

1.609 1.572 1.545 1.500

1.487 1.380 1.301 1.155

1.325 1.296 1.276 1.240

1

tw

0.5 0.6 0.8 1.0 0

r U m

-2

TABLE IV (Continued)

B

0.5

0.75

W

1

u1

Pr

1

p Ip)

11(3)

0.4086 0.3760 0.3496 0.3409

1.599 1.556 1.525 1.516

1.425 1.293 1.195 1.165

1.189 0.9936 0.7003 0.5362

0.4272 0.6743 0.8662 1.064 1.328

0.4113 0.3866 0.3667 0.3454 0.3161

1.596 1.559 1.533 1.508 1.478

1.425 1.318 1.238 1.159 1.058

1.174 1.073 0.9303 0.5038 2.702

t"

fil(0)

e'(0)

g'(0)

0.5

0.15 0.4 0.6 0.6653

0.6160 0.7243 0.8071 0.8335

0.2440 0.1831 0.0693 0.0

0.4990 0.5154 0.5273 0.5309

0.4148 0.6966 0.9150 0.9852

0.7

0.15 0.4 0.6 0.8111 1.1

0.6249 0.7237 0.7997 0.8773 0.9800

0.3485 0.3152 0.2474 0.0 1.409

0.5003 0.5145 0.5248 0.5349 0.5478

I2

0.5

0.5

0.7

0.05 0.15 0.4 0.6 0.8994 1.1

0.4131 0.5618 0.7768 0.9092 1.083 1.189

0.2576 0.3234 0.3721 0.3616 0.0 1.085

0.3251 0.4129 0.5020 0.5419 0.5840 0.6061

0.1796 0.3000 0.5518 0.7391 1.008 1.183

0.2784 0.3367 0.3630 0.3549 0.3264 0.3017

1.182 1.368 1.483 1.508 1.518 1.517

1.092 1.223 1.224 1.165 1.052 0.9717

0.9608 1.086 1-121 1.064 0.4382 2.118

0.7

0

0.7

0.05 0.15 0.4 0.6 1 .o 1.1

0.4557 0.5726 0.7568 0.8770 1.090 1.140

0.3150 0.3683 0.4232 0.4489 0.4849 0.4922

0.3629 0.4237 0.4862 0.5155 0.5568 0.5652

0.1800 0.2857 0.5183 0.6939 1.031 1.113

0.3207 0.3582 0.3682 0.3565 0.3174 0.3059

1.292 1.412 1.470 1.474 1.460 1.454

1.211 1.288 1.251 1.183 1.031 0.9922

1.085 1.179 1.221 1.223 1.209 1.204

P N

TABLE IV (Continrred)

~

0.75

0.7

1

1

0.5

0

~~

0.5

all

0

a11

0.7

0.05 0.15 0.4 0.6 0.9044 1.1

0.4833 0.6029 0.7885 0.9089 1.072 1.173

0.3059 0.3512 0.3787 0.3605 0.0' 1.045

0.3850 0.4478 0.5108 0.5400 0.5719 0.5890

0.2072 0.3187 0.5591 0.7396

0

0.15 0.4 0.5 0 -6 1 .o

0.6181 0.6948 0.8173 0.8646 0.9112 1.090

0.5012 0.5112 0.5262 0.5318 0.5371 0.5568

0.5

0.15 0.4 0.6 1 .o

0.5440 0.7923 0.9521 1.233

0.7

0.05 0.15 0.4 0.6 1 .o 1.1

1

0.15 0.4 0.6 1 .o

1

1.169

0.3279 0.3690 0.3692 0.3542 0.3215 0.2963

1.325 1.446 1.503 1.507 1.496 1.485

1.215 1.28-7 1.240 1.166 1.044 0.9624

1.061 1.139 1.135 1.065 0.4385 2.062

0.5012 0.5112 0.5262 0.5318 0.5371 0.5568

0.2748 0.3953 0.5896 0.6654 0.7402 1.031

0.4120 0.3984 0.3751 0.3656 0.3560 0.3174

1.594 1.568 1.531 1.517 1.504 1.460

1.423 1.355 1.250 1.211 1.173 1.031

1.148 1.129 1.101 1.091 1.081 1.048

0.2891 0.3575 0.3894 0.4334

0.3857 0.4734 0.5142 0.5705

0.1968 0.4248 0.6005 0.9402

0.3473 0.3674 0.3516 0.2923

1.340 1.428 1.439 1.430

1.269 1.234 1.146 0.9402

1.261 1.349 1.364 1.364

0.4025 0.5612 0.8060 0.9615 1.233 1.295

0.2584 0.3317 0.4097 0.4460 0.4959 0.5058

0.3004 0.3844 0.4730 0.5140 0.5705 0.5817

0.1408 0.2437 0.4631 0.6272 0.9402 1.016

0.2617 0.3476 0.3430 0.3344 0.2923 0.2791

1.127 1.300 1.404 1.426 1.430 1.429

1.053 1.175 1.167 1.102 0.9402 0.8978

0.9605 1.096 1.172 1.186 1.186 1.184

0.5789 0.8204 0.9714 1.233

0.3826 0.4722 0.5137 0.5705

0.3826 0.4722 0.5137 0.5705

0.2868 0.4985 0.6518 0.9402

0.2921 0.3219 0.3196 0.2923

1.261 1.380 1.411 1.430

1.092 1.107 1.062 0.9402

0.9475 1.014 1.026 1.026

1.0000

TABLE IV (Continued)

1

0.5

0.25

0.7

0.05 0.15 0.9490

0.4187 0.5808 1.224

0.2588 0.3291 0.0

0.3132 0.3998 0.5829

0.1514 0.2578 0.9289

0.2673 0.3230 0.2955

1.146 1.323 1.458

1.059 1.180 0.9490

0.9556 1.084 0.3943

0.5

0.7

0-05 0.15 0.4 0.6 0.8965 1.1

0.4395 0.6057 0.8558 1.013 1.219 1.349

0.2612 0.3285 0.3787 0.3675 0.0 1.135

0.3302 0.4204 0.5132 0.5554 0.6000 0.6241

0.1641 0.2746 0.5051 0.6760 0.9187 1.080

0.2755 0.3311 0.3507 0.3369 0.3000 0.2689

1.172 1.355 1.465 1.488 1.495 1.492

1.068 1.187 1.166 1.092 0.9601 0.8652

0.9516 1.074 1. l o 2 1.039 0.3992 2.149

0.6

0.7

0.05

0.4499 0.5467 0.6181 1.218

0.2632 0.3047 0.3292

0.3390 0.3954 0.4310 0.6091

0.1700 0.2298 0.2825 0.9152

0.2799 0.3169 0.3356 0.3025

1.185 1.307 1.371 1.513

1.073 1.159 1.192 0.9655

0.9504 1.033 1.070 0.4013

0.2708 0.3115 0.3344 0.0

0.3632 0.4227 0.4599 0.6344

0.1848 0.2472 0.3016 0.9094

0.2926 0.3302 0.3486 0.3090

1.221 1.348 1.415 1.565

1.O% 1.173 1.205 0.9788

0.9490 1.029 1.063 0.4061

0.1

0.15 0.8748

0.7

0.0

0.8

0.7

0-05 0.1 0.15

0.8290

0.4774 0.5774 0.6501 1.218

1

0.7

0.05 0.15 0.4 0.6 0.7640 1.1

0.5381 0.7166 0.9653 1.110 1.220 1.423

0.3014 0.3623 0.3634 0.2598 0.0 2.402

0.4277 0.5342 0.6465 0.6760 0 7006 0.7483

0.2068 0.3297 0.5532 0.7445 0.8838 1.136

0.3133 0.3745 0.3694 0.3577 0.3329 0.2860

1.244 1.473 1.&88 1.603 1.606 1.517

1 .087 1.221 1.163 1.104 1.027 0.8877

0.9266 1.049 1.016 0.8990 0.6067 2.484

0.1 0.4 0.6

0.5341 8.151 0.9645

0.3060 0.3750 0.3992

0.4036 0.4934 0.5251

0. l G O l 0.4327 0.6061

0.3739 0.3824 0.3585

1.396 1.468 1.461

1.341 1.265 1.160

1.312 1.387 1.385

0

0.5

-

8 N

TABLE IV (Continued)

~

1

0.7

~~

-

~~~

_

_

0

0.7

0.5 0.15 0- 4 0.6 1 .o 1.1

0-4821 0.6146 0.8317 0.9756 1.233 1.293

0.3183 0.3729 0.4302 0.4574 0.4959 0.5037

0.3670 0.1643 0.4295 0.2614 0.4950 0.4741 0.5262 0.6343 0.5705 0.9402 0.5795 1.014

a1

1

0.15 0.6 1 .o

0.6381 0.9873 1.233

0.4310 0.5271 0.5705

0.4310 0.5271 0.5705

0.5

0.7

0.05 0.15 0.4 0.6 0.8975 1.1

0.5120 0.6478 1 505 1.012 1.207 1.331

0.3095 0.3560 1.024 0.3658 0.0 1.091

0.3902 0.1884 0.3235 0.4550 0.2908 0.3573 1.366 0 2288 0.1155 0.5526 0.6759 0.3359 0.5869 0.9113 0.2955 0.6057 1.067 0.2644

1.187 1.314 1.051 1.432 1.126 1.248 0 * 4640 0.4537 0.3749 1.039 1.484 1.092 0.9523 0.3995 1.470 0.8578 2.089 1.457

0.3118 0.6605 0.9402

0.3177 0.3532 0.3576 0.3413 0.2923 0.2783

1.280 1.396 1.449 1.450 1.430 1.424

1.186 1.253 1.197 1.116 0.9402 0.8964

1.076 1.168 1.205 1.203 1.186 1.181

0.3262 1.363 0.3268 1.438 0.2923 1.430

1.167 1.076 0.9402

1.007 1.040 1.026

-

0.9

0.7

0.15 0.4 0.6 0.809:

0.6940 0.9117 1.055 1.192

0.3506 0.3494 0.2822 0.0

0.4955 0.5617 0.5919 0.6160

0.3239 0.5505 0.7195 0.8902

0.3702 0.3612 0.3351 0.3019

1.491 1.544 1.543 1.534

1.254 1.173 1.075 0.9683

1.095 1.037 0.8892 0.4096

1

0.7

0.05 0.15 0.4 0.6 0.7775 1.1

0-5809 0.7188 0.9317 1.071 1.185 1.380

0.3222 0.3576 0.3426 0.2551 0.0 1.970

0.4548 0.5230 0.5891 0.6162 0.6372 0.6650

0.2275 0.3372 0.5616 0.7336 0.8753 1.130

0.3488 0.3810 0.3668 0.3379 0.3108 0.2547

1.387 1.528 1.562 1.579 1.549 1.527

1.202 1.262 1.175 1.075 0.9880 0.8266

1.026 1.088 1.023 0.8527 0.5073 3.035

TABLE I\' (Continued)

1

1

I

Pr

UI

0

all

f '1 (0)

I2 ~~

0.5

0.4 0.6 1.o

0.8524 0.9840 1.233

0.4037 0.4145 0.4334

0.5261 0.5422 0.5705

0.4458 0.6149 0.9402

0.4066 0.3691 0.2923

1* 535

1.495 1.430

1.315 1.182 0.9402

1.449 1.417 1.364

0.7

0 0.15 0.4 0.5 0 06 1.0

0.6071 0.7104 0.8735 0.9362 0.9976 1.233

0.4362 0.4473 0.4637 0.4696 0.4753 0.4959

0.4970 0.5107 0.5308 0.5382 0.5452 0.5705

0 1695 0.4376 0.2934 0.4171 0.4921 0.3814 0 5692 0.3669 0.6454 0.3522 0.9402 0.2923

1.607 1.572 1.521 1 SO4 1.487 1.430

1.494 1.396 1.246 1.191 1.138 0.9402

1.324 1.296 1.257 1.244 1.231 1.186

0

0.6489 0.7445 0.7755 0.8963 0.9548 1.012 1.124 1.233 1.737

0.5067 0.5183 0 * 5219 0.5357 0.5621 0.5482 0.5597 0.5705 0.6156

0.5067 0.5183 0.5219 0.5357 0.5421 0.5482 0.5597 0.5705 0.6156

0.2456 0.3570 0.3934 0.5360 0.6056 0.6743 0.8089 0.9402 1.561

1.579 1.550 1.541 1.508 1.493 1.479 1.454 1.430 1.340

1.383 1.305 1.280 1.186 1.142

1.137 1.115 1.109 1.084 1.073

1.099 1.018 0.9402 0.6010

1.063 1.043 1.026 0.9595

1

0.15 0.2 0.4 0.5 0.6 0.8 1.o 2 .o

-

-

0.4033 0.3875 0.3821 0.3603 0.3491 0.3380 0.3153 0.2923 0.1761

-

m

2 0

5

%

kc?: 1: U L-

?

r 8

M

0

5

2 0

5

TABLE IV (Continued)

1.4

1.5

0.6

0.7

0.1

0.5977

0.3107

0.4043

0.2042

0.3118

1.295

1.122

1.020

0.8

0.7

0.1

0.6320

0.3189

0.4343

0.2195

0.3248

1.341

1.135

1.018

0.7

0.6

0.7

0.15 0.4 0.6 0.8721 1.1

0.7200 0.9874 1.167 1.389 1.564

0.3596 0.3830 0.3516 0.01.295

0-4719 0.5427 0.5765 0.6109 0.6343

0.2640 0.4622 0.6100 0.8028 0.9587

0.3505 0.3404 0.3125 0.2653 0.2214

1.426 1.472 1.469 1.453 1.437

1.200 1.105 0.9983 0.8486 0.7246

1.101 1.071 0.9707 0.3577 2.341

0.5

0

0.7

0.05 0.15 0.4 0.6 1 .o

0.4427 0.6304 0.9337 1.132 1.477

0.2628 0.3383 0.4200 0.4585 0.5119

0.3060 0.3928 0.4860 0.5298 0.5906

0.1225 0.2123 0.4030 0.5448 0.8141

0.2590 0.3119 0.3291 0.3134 0.2562

1.110 1.277 1.374 1.391 1.390

1.022 1.128 1.092 1.008 0.8141

0.9465 1.078 1.148 1.159 1.155

0.25

0.7

0.05 0.15 0.9469

0.4608 0.6526 1.464

0.2636 0.3361 0.0

0.3197 0.4095 0.6045

0.1312 0.2242 0.8034

0 2640 0.3163 0.2585

1.130 1.302 1.421

1.026 1.130 0.8216

0.9418 1.066 0.3415

0.5

0.7

0.05 0.15 0.4 0.6 0.8922

0.4842 0.6810 0.9917 1.192 1.453

0.2667 0.3362 0.3888 0.3763 0.0

0.3381 0.4319 0.5304 0.5760 0.6241

0.1419 0.2383 0.4389 0.5871 0.7937

0.2715 0.3234 0.3332 0.2620

1.158 1.336 1.440 1.459 1.463

1.033 1.136 1.084 0.9885 0.8311

0.9384 1 .056 1.075 1.004 0.3466

0.05 0.15 0.4

0.5266 0.6863 0.9604 1.477

0.3233 0.3799 0.4407 0.5119

0.3733 0.4383 0.5082 0.5906

0.1419 0.2267 0.4117 0.8140

0.3138 0.3462 0.3427 0.2561

1.262 1.373 1.418 1.390

1.150 1.203 1.120 0.8140

1.061 1.149 1.181 1.155

0.5

0.7

0

0.7

1 .o

-

0.3211

T A B L E I V (Continued)

1.5

1.8

1

1

0.

0.6

0.7

0 0.15 0.2 0.4 0.5 0.6 0.8 1.0 2.0

0.6987 0.8278 0.8695 1.031 1.109 1.185 1.334 1.477 2.140

0.5147 0.5289 0.5333 0.5498 0.5574 0.5646 0.5781 0.5906 0.6423

0.1 0.15

0.6421 0.7354 1.080 1.303 1.788

0.3155 0.3414 0.3877 0.3616 1.425

0.4 0.6 1.1 2

0.5

0.5498 0.5574 0.5646 0.578i 0.5906 0.6423

0.2049 0.3035 0.3356 0.4610 0.5220 0.5821 0.6996 0.8141 1.352

0.3914 0.3725 0.3661 0.3395 0.3259 0.3122 0.2844 0.2562 0.1128

1.558 1.524 1 514 1.476 1.460 1.444 1.416 1.390 1.294

1.326 1.235 1.206 1.097 1.046 0.9966 0.9026 0.8141 0.4268

1.122 1.096 1.088 1.060 1.048 1.036 1.015 0.9963 0.9251

0.4115 0.4496 0.5524 0.6000 0.6786

0.1856 0.2285 0.4182 0.5578 0.8857

0.3083 0.3241 0.3273 0 2993 0.1938

1.286 1.347 1.451 1.469 1.467

1.095 1.116 1.047 0.9388 0.6373

1.010 1.044 1.048 0.9526 2.484

0.5147 0.5289 0.5333

-

-

0.8

0-7

0.1

0.6795

0.3249

0.4436

0.1991

0-3212 1.335

1.108

1.010

0

0.5

0.15 0-4 0.6

0.6650 1.024 1.263

0.2975 0.3712 0.4063

0.3994 0.4955 0.5412

0.1598 0.3368 0.4711

0.3454 0 3508 0.3209

1.304 1.377 1.381

1.207 1.122 0.9984

1.232 1.308 1.318

0.7

0.05 0.15 0.4 0.6 1 .o

0.4766 0.6893 1.043 1.276 1.687

0.2661 0.3432 0.4276 0.4677 0.5235

0.3103 0.3991 0.4956 0.5414 0.6052

0.1097 0.1906 0.3617 0 -4884 0.7282

0.2573 0.3082 0.3196 0.2988 0.2308

1.097 1.261 1.352 1.367 1.363

0.9992 1.096 1.040 0.9445 0.7282

0.9364 1.065 1.131 1.140 1.134

*Convergence

.

to 10- 4

-

*

-

~

0 2

-

~

~

o1

Pr

0.5

0.25

0.7

0.05 0.15 0.945

0.4962 0.7137 1.669

0.2672 0.3414

0.0

0.3246 0.4167 0.6204

0.1172 0.2009 0.7180

0.2619 0.3119 0.2326

1.119 1.287 1.396

1.003 1.096 0.7348

0.9319 1.053 0.3063

0.7

0

0.5

0.15 0 -4 0.6 1.o

0.7168 1.049 1.276 1.687

0.3333 0.3890 0.4163 0.4555

0.4419 0.5157 0.5523 0.6052

0.1673 0.3421 0.4748 0.7282

0.3821 0.3646 0.3268 0.2308

1.394 1.418 1.403 1.363

1.286 1.150 1.010 0.7282

1.316 1.347* 1.339* 1-310

0.7

0.05 0.15 0-4 0.6 1.o

0.5640 0.7473 1.071 1.291 1.687

0.3270 0.3851 0.4484 0.4793 0.5235

0.3781 0.4449 0.5179 0.5537 0.6052

0.1264 0.2028 0.3690 0.4930 0.7282

0.3112 0.3417 0.3326 0.3046 0.2308

1.249 1.357 1.396 1.391 1.363

1.125 1.168 1.067 0.9561 0.7282

1SO51 1.136 1.164 1.158 1.134

0.

0.5

0.15 0.4 0.6

0.7679 1. l o 7 1.345

0.2937 0.2994 0.2449

0.4734 0 5492 0.5865

0.2029 0.3899 0.5308

0.3621 0.3308 0.2838

1.421 1.442 1.427

1.217 1.053 0.8976

1.193 1.105 0.9170

all

1

0 0.15 0.2 0.4 0.5 0.6 0.8 1.0 2.0

0.7386 0.8972 0.9483 1.146 1.241 1.333 1.513 1.687 2.488

0 * 5206 0.5367 0.5417 0.5601 0.5686 0.5766 0.5915 0.6052 0.6615

0.5206 0.1775 0.5367 0.2673 0.5417 0 * 2965 0 * 5601 0.4101 0.5686 0.4653 0.5766 0.5194 0.5915 0.6254 0.6052 0.7284 0.6615 0.1211

0.3837 0.3626 0.3553 0.3254 0.3101 0.2945 0.2629 0.2309 0.0676

1.544 1.506 1.495 1.454 1.436 1.420 1.390 1.363 1.263

1.288 1.187 1.156 1SO36 0.9804 0.9267 0.8245 0.7284 0.3086

1.111 1.082 1.074 1.044 1.030 1.018 0.9957* 0.9760* 0.9023*

u)

1

*cmvergence t o 10-4.

tw

TABLE IV (Continued)

fii(o) 2.4

1

all

1

e@(o)

g@(o)

x1

I2

I I W

11(3)

0.5 1 .o

0.7659 1.335 1.838

0.5244 0.5757 0.6145

0.5244 0.5757 0.6145

0.1611 0.4309 0.6762

0.3791 0.3005 0.2151

1-535 1.422 1.347

1.265 0.9408 0.6762

1.104 1.020 0.9637

0

*

2.8

1

all

1

0 0.5 1.0

0.7901 1.421 1.978

0.5275 0.5817 0.6223

0.5275 0.5817 0.6223

0.1480 0.4033 0.6341

0.3756 0.2928 0.2023

1.527 1.411 1.333

1.246 0.9088 0.6341

1.098 1.011 * 0.9537

3

0.5

0

0.7

0.15 0.4 0.6 1 .o

0.7883 1.228 1.522 2.044

0.3503 0.4383 0.4805 0.5397

0.4082 0.5093 0.5578 0.6258

0.1616 0.3036 0.3072 0.3074 0.4142 0.2796 0.6158 0.1967

1.238 1.323 1.335 1.328

1.051 0.9723 0.8603 0.6158

1.047 1.109* 1.115* 1.107

0.7

0

0.7

0.05 0.15 0.4 0.6

0.6255 0.8493 1.257 1.537 2.044

0.3324 0.3926 0.4593 0.4923 0.5397

0.3849 0.4545 0.5319 0.5702 0.6258

0.1057 0.1711 0.3125 0.4176 0-6158

0.3083 0-3360 0.3194 0.2847 0.1967

0.1231 1.334 1.367 1.359 1.328

1.091 1-121 0.9973 0.8708 0-6158

1.037 1.118 1.141 1.133 1-10)

all

1

0 0.5

0.8221 .1.541 2.170

0.5314 0.5893 0.6320

0.5314 0.5893 0.6320

0.1325 0.3697 0.5929

0.3715 0.2835 0.1919

1.518 1.396 1.317

1.224 0.8699 0.5938

1.092 1.000 0.9416*

0

0.8502 1.650 2.347

0.5346 0.5955 0.6401

0.5346 0.1205 0.3684 0.5955 0.3434 0.2762 0.6401 0.5447 0.1751

1.511 1.385 1.305

1.207 0.8393 0.5447

1.086 0.9918 0.9320*

0.8907 1.815 2.616

0.5389 0.6042 0.6509

0.5389 0.6042 0.6509

1.502 1.368 1.288

1.184 0.7994 0.4992

1.079 0.9796 0.9196*

1 .o

3.4

1

4

1

1.o

all

1

0.5 1.0 5

1

*Convergence

all

1

0

0.5 1.0 to 10-4.

0.1052 0.3096 0.4923

0.3647 0.2668 0.1586

TABLE V SIMILAR SOLYTIONS FOR

0

0.01

0

0

0.15 0.4 0.6

Lo(=) 0.5

0.15 0.6 0.9152

1

0.15 0.6 0.7

A

SVTHERLAND VISCOSITY-TEMPERATURE RELATION, Pr = 0.7,t,

=

1

0.3564 0.3554 0.4162 0.4161 0.4407 0.4407 0.4696 0.4696

0.3100 0.3093 0.3643 0.3642 0.3868 0.3868 0.4139 0.4139

0.3564 0.3554 0.4162 0.4161 0.4407 0.4407 0.4696 0.4696

0.4105 0.4093 0.7909 0.7906 1.090 1.090 1.686 1.686

0.3563 0.3554 0.4162 0.4161 0.4406 0.4406 0.4696 0.4696

1.430 1.425 1.575 1.574 1.628 1.628 1.686 1.686

1.430 1.425 1.575 1.574 1.628 1.628 1.686 1.686

1.200 1.195 1.306 1.306 1.344 1.344 1.385 1-385

0.3804 0.3797 0.4644 0.4644 0.4871 0.4871

0.3019 0.3014 0.3247 0.3247 0.0 0.0

0.3804 0.3797 0.4644 0.4644 0.4871 0.4871

0.4681 0.4672 1.167 1.167 1.645 1.645

0.3804 0.3798 0.4643 0.4643 0.4871 0.4871

1.470 1.467 1.664 1.664 1.710 1.710

1.470 1.467 1.664 1.664 1.710 1.710

1.179 1.176 1 240 1.240 0.7667 0.7666

0.4327 0.4333 0.5125 0.5080 0.5197 0.5141

1.540 1.537 1.726 1.709 1.742 1 722

1.540 1.537 1.726 1.709 1.742 1.722

1 149 1.154 1.102 1.175 1.011 1.127

0.4326 0.4357 0.5126 0.5191 0.5198 0.5266

a S o l u t i o n independent of s .

0.3047 0.3050 0.2467 0.2378 0.1765 0.1609

0.4326 0.4357 0.5126 0.5191 0.5198 0.5266

0.5637 0.5558 1.285 1.239 1.438 1.384

-

-

-

0.5317 0.5174 0.5136

-0.5368 0.5159 0.5121 0.5486 0.5205 0.5184

B

S

fw

bl

tw

0

0.01

-0.2

0

0.15 0.4 0.6 l.O(a)

-0.6

0

0.15 0.4 0.6 l.O(a)

f"(0)

e'(0)

g'(0)

0.2221 0.2211 0.2794 0.2792 0.3028 0.3028 0.3305 0.3305

0.2109 0.2101 0.2632 0.2631 0.2848 0.2848 0.3108 0.3108

0.0229 0.0222 0.0598 0.0597 0.0767 0.0767 0.0975 0.0975

Il

I2

0.2221 0.2211 0.2794 0.2792 0.3028 0.3028 0.3305 0.3305

0.5369 0.5355 0.9754 0.9751 1.318 1.318 1.999 1.999

0.0376 0.0368 0.0802 0.0800 0.0982 0.0982 0.1201 0.1201

0.0229 0.0222 0.0598 0.0597 0.0767 0.0767 0.0975 0.0975

I

p

11(2)

5(3)

0.4221 0.4211 0.4794 0.4792 0.5028 0.5027 0.5305 0.5305

1.824 1.819 1.922 1.921 1.958 1.958 1.999 1.999

1.824 1.819 1.922 1.921 1.958 1.958 1.999 1.999

1.514 1.510 1.577 1.576 1.599 1.599 1.623 1.623

1.176 1.180 1.752 1.752 2.230 2.230 3.181 3.181

0.6228 0.6221 0.6598 0.6597 0.6768 0.6767 0.6975 0.6975

4.137 4.154 3.437 3.438 3,299 3.299 3.181 3.181

4.137 4.154 3.437 3.438 3.299 3.299 3.181 3.181

3.483 3.499 2.809 2.810 2.672 2.672 2.552 2.552

0.03

0

1

0.4

0.48GO 0.4939

0.3052 0.3025

0.4860 0.4935

0.9686 0.9398

0.4863 0.4854

1.677 1.667

1.677 1.667

1.183 1.211

0.05

0

1

0.15

0.4421 0.4452 0 -4823 0.4890 0.5007 0.5074

0.3134 0.3139 0.3045 0.3028 0.2492 0.2418

0.4421 0.4452 0.4823 0.4890 0.5007 0.5074

0.5759 0.5782 0.9640 0.9566 1.271 1.262

0.4422 0.4451 0.4823 0.4865 0.5007 0.5052

1.587 1.587 1.677 1.677 1.716 1.716

1.587 1.587 1.677 1.677 1.716 1.716

1.190 1.187 1.188 1.200 1.112 1.134

0.4074 0.4011 0.4309 0.4299

0.3559 0.3511 0.3778 0.3770

0.4074 0.4011 0.4309 0.4299

0.4577 0.4491 0.8095 0.8073

0.4074 0.4011 0.4309 0.4299

1.561 1.529 1.608 1.603

1.561 1.529 1.608 0.603

1.298 1.270 1.330 1.326

0.4 0.6 0.1

0

0

0.15 0.4

aSolution independent of s .

Lur 0.5317 0.5174 0.5136

-0.5317 0.5174 0.5136

-0.5761 0.7027 0.6208 0.5944 0.7384 0.6497

P N \o

L 2 0

'I'ABLE \' (Continued)

0

0.1

0

0.6

0

1 0.5

0.15 0.4

0.3

0

0.15

0

0.4 0.6 0.5

0.15 0.4 0.6 0.9151 0.15

1

0.6 ~~

-

0.4471 0.3928 0.4468 0.3926 0.4696 0.4139 0.4696 0.4139

0.4471 0.4468 0.4696 0.4696

1.100 1.099 1.686 1.686

0.4471 0.4468 0 -4696 0.4696

1.641 1.640 1.686 1.686

1.641 1.640 1.686 1.686

1.354 1.353 1.385 1.385

0.6210

0.4269 0.4224 0.4499 0.4495

0.3393 0.3358 0.3422 0.3419

0.4269 0.4224 0.4499 0.4495

0.5168 0.5108 0.8761 0.8753

1.264 1.246 1.268 1.267

0.7708

0.4140 0.4065 0.3983 0.3969 O.GO24 0.4020

0.4713 0.4615 0.4532 0.4513 0.4574 0.4569

0.5150 0.5021 0.8373 0.8332 1.115 1.114

1.591 1.570 1.637 1.635 1.715 1.668 1.656 1.648 1.663 1.660

1.591 1.570 1.637 1.635

0.4713 0.4615 0.4532 0.4513 0.4575 0.4569

0.4269 0.4223 0.4499 0.4495 0.4713 0.4615 0.4532 0.4513 0.4573 0.4569

1.715 1.668 1.656 1.648 1.663 1.660

1.412 1.371 1.365 1.358 1.367 1.367

0.4803 0.4738 0.4631 0.4624 0.4676 0.4677 0.4876 0.4876

0.3823 0.3773 0.3522 0.3517 0.3274 0.1274 0.0 0.0

0.4803 0.4738 0.4631 0.4624 0.4676 0.4677 0.4876 0.4876

0.5722 0.5639 0.8960 0.8947 1.177 1.177 1.645 1.645

0.4803 0.4738 0.4623 0.4630 0.4676 0.4677 0.4876 0.4876

1.726 1.697 1.671 1.668 1.678 1.678 1.710 1.710

1.725 1.697 1.670 1.668 1.678 1.678 1.710 1.710

1.356 1.333 1.291 1.289 1.252 1.252 0.7664 0.7664

0.4779 0.4792 0.4723 0.4788

0.3416 0.3416 0.2493 0.2430

0.4779 0.6203 0.4792 0.6216 0.4735 1.238 0.4788 1.240

0.4779 0.4792 0.4735 0.4785

1.725 1.723 1.692 1.693

1.725 1.723 1.692 1.693

1.300 1.296 1.135 1.134

aSoIution independent of s .

--

?

r

z

0

m

(I)

0.6722 0.9717 0.8437 0.7918 1.019 0.8837

r

0.8266

z

0.5000 1.101 0.8838

0

$

(I)

TABLE V (Continued)

B 0

S

fw

01

tw

0.3

-0.2

0

0.15 0.4

0.6 l.O(a)

-0.6

0

0.15 0.4 0.6 l.O(a)

0.4

0

0.05

1

0.15 0.4

1.1 _

_

~

~~~

f"(0)

e'(0)

g'(0)

0.3328 0.3227 0.3148 0.3129 0.3188 0.3183 0.3305 0.3305

0.3116 0.3036 0.2959 0.2944 0.2998 0.2994 0.3108 0.3108

0.1005 0.0914 0.0858 0.0841 0.0886 0.0882 0.0975 0.0975 0.5769 0.5808 0.7056 0.7130 0.9696 0.9706

~

a S o l u t i o n independent of s .

Il

I2

Il(U

11(2)

11(3)

0.3328 0.3227 0.3148 0.3129 0.3188 0.3183 0.3305 0.3305

0.6259 0.6131 1.012 1.007 1.337 1.336 1.999 1.999

0.5328 0.5227 0.5149 0.5129 0.5188 0.5183 0.5305 0.5305

2.033 1.984 1.979 1.971 1.983 1.980 1.999 1.999

2.033 1.984 1.979 1.971 1.983 1.980 1.999 1.999

1.656 1.613 1.613 1.605 1.614 1.612 1.623 1.623

0.1226 0.1138 0.1077 0.1061 0.1108 0.1104 0.1201 0.1201

0.1005 0.0914 0.0858 0.0841 0.0886 0.0882 0.0975 0.0975

1.018 1.015 1.675 1.675 2.187 2.187 3.181 3.181

0.7006 0.6914 0.6858 0.6841 0.6887 0.6882 0.6975 0.6975

3.225 3.205 3.247 3.247 3.227 3.227 3.181 3.181

3.225 3.205 3.247 3.247 3.227 3.227 3.181 3.181

2.596 2.577 2.620 2.620 2.599 2.599 2.552 2.552

0.3342 0.3363 0.3268 0.3270 1.683 1.819

0.4793 0.4854 0.5369 0.5496 0.6169 0.6298

0.4341 0.4352 0.7259 0.7215 1.495 1.475

0.4033 0.4063 0.4154 0.4203 0.3716 0.3793

1.542 1.545 1.609 1.610 1.644 1.639

1.395 1.394 1.377 1.380 1.187 1.202

1.131 1.128 1.085 1.098 3.076 2.732

% 0.9717 0.8437

2

0.7918

3

~

+I

-0.9717 0.8437 0.7918

--

5

$ m z

r2 %

m

0.7027

a

0.6208

2

0.5627

5 0

TABLE

0.5

0.01

0

0

0.15 0.4 0.6 1 .o (a)

-0.2

0

0.15 0.4 0.6 1 .o (a)

-0.6

0

0.15 0.4 0.6

0.02

0

0

0.2

z

V (Continued)

N

0.4907 0.4893 0.6585 0.6583 0.7623 0.7622 0.9278 0.9278

0.3422 0.3466 0.4208 0.4207 0.4691 0.4691 0.4803 0.4803

0.3803 0.3791 0.4602 0.4601 0.5032 0.5032 0.5420 0.5420

0.3909 0.3907 0.6170 0.6167 0.7846 0.7844 1.147 1.147

0.2625 0.2603 0.2864 0.2861 0.2677 0.2676 0.3463 0.3463

1.337 1.332 1.340 1.339 1.203 1.202 1.437 1.437

1.111 1.104 1.118 1.117 1.044 1.044 1.147 1.147

0.8468 0.8389 0.8351 0.8342 Oi6484 0.6482 1.114 1.114

0.3647 0.3631 0 * 5333 0.5330 0.6367 0.6367 0.8126 0.8126

0.2392 0.2383 0.3093 0.3092 0.3420 0.3420 0.3815 0.3815

0.2554 0.2543 0.3321 0.3319 0.3688 0.3688 0.4156 0.4156

0.3593 0.3582 0.6430 0.6427 0.8506 0.8505

1.648 1.643 1.661 1.661 1.644 1.644 1.684 1.684

1.512 1.508 1.447 1.446 1.383 1.383 1.265 1.265

1.356 1.352 1.340 1.339 1.332

1.265 1.265

0.3822 0.3811 0.3963 0.3961 0.3935 0.3935 0.3793 0.3793

0.1898 0.1884 0.3400 0.3398 0.4407 0.4406 0.6138 0.6138

0.1104 0.1094 0.1553 0.1552 0.1854 0.1854 0.2280 0.2280

0.0850 0.0842 0.1383 0.1381 0.1697 0.1696 0.2147 0.2147

0.5001 0.4973 0.8302 0.8300 1.074 1.074 1.532 1.532

0.2865 0.2867 0.5243 0.5241 0.5031 0.5031 0.4475 0.4475

1.896 1.890 2.364 2.364 2.275 2.275 2.158 2.158

1.763 1.763 1.989 1.989 1.813 1.813 1.532 1.532

1.486 1.489 1.931 1.931 1.847 1.847 1 740 1 * 740

0.5292 0.5273

0.3517 0.3507

0.4055 0.4042

0.3575 0.3560

0.3505 0.3493

1.402 1.397

1.297 1 *292

1.175 1.170

aSolution independent of s.

0.5317 0.5174 0.5136

-_ 0.5317 0.5174

1.332

0.5136

1.378 1.378

--

-

0'

P

m

t;!

U N

5a

14

P

0.5317 0.5174 0.5136

-0.5515

0 P

Rrn

TABLE V (Continued)

0

s

0.5

0.3

f"

0

tw

01

0

0.4 0.6 1 .o (a)

-0.2

0

0.15 0.4 0.6 1 .,(a)

-0.6

0

0.15 0.4 0.6

1

0.6968 0.6935 0.7740 0.7731 0.9278 0.9278

0.4406 0.4391 0.4529 0.4525 0.4803 0.4803

0 -4964 0.4944 0.5108 0.5102 0.5420 0.5420

0.6075 0.6037 0.7875 0.7863 1.147 1.147

0.3695 0.3670 0.3615 0.3608 0.3463 0.3463

1.491 1.481 1.469 1.466 1.437 1.437

1.297 1.290 1.242 1.240 1.147 1.147

1.150 1.143 1.136 1.134 1.114 1.114

0.4973 0.4793 0.5732 0.5698 0.6537 0.6527 0.8126 0.8126

0.3495 0.3367 0.3424 0.3405 0.3543 0.3538 0.3815 0.3815

0.3760 0.3612 0.3691 0.3668 0.3836 0.3830 0.4156 0.4156

0.3989 0.3976 0.6606 0.6575 0 8653 0.8643 1.265 1.265

0.4394 0.4509 0.4252 0.4235 0.4141 0.4136 0.3793 0.3793

1.745 1.771 1.721 1.715 1.712 1.710 1.684 1.684

1.624 1.630 1.502 1.496 1.424 1.423 1.265 1.265

1.441 1.450 1.403 1.398 1.397 1.396 1.378 1.378

0.2762 0.2617 0.3682 0.3652 0.4530 0.4522 0.6138 0.6138

0.1779 0.1685 0.1813 0.1795 0.1971 0.1966 0.2274 0.2274

0.1587 0.1482 0.1644 0.1623 0.1815 0.1810 0.2146 0.2146

0.5437 0.5362 0.8431 0.8410 1.080 1.080 1.532 1.532

0.5998 0.5888 0.5466 0.5447 0.5128 0.5124 0.4477 0.4477

2.587 2.552 2.396 2.391 2.296 2.294 2.162 2.162

2.306 2.271 2.008 2.003 1.824 1.823 1.532 1.532

2.073 2.041 1.941 1.937 1.860 1.859 1.750 1.750

aSolution independent of s.

0.8437 0.7918

_0.9717 0.8437 0.7918

-0.9717 0.8437 0.7918

--

z

TABLE V (Continued)

1

0.01

0

0.9

0.6889 0.6884 0.9225 0.9232 1.080 1.077 1.216 1.216

0.3577 0.3577 0.3573 0.3578 0.2915 0.2906 0.0135 0.0121

0.5201 0.5199 0.5768 0.5779 0.6227 0.6196 0.6493 0.6505

0.2919 0.2913 0.5547 0.5551 0.7333 0.7319 0.9023 0.9026

0.2376 0.2342 0.3678 0.3683 0.3466 0.3455 0.3135 0.3136

0.4913 0.4804 0.567 1.569 1.598 1.593 1.597 1.600

0.9650 0.9574 1.179 1.180 1.092 1.090 0.9885 0.9885

0.7918 0.7837 1.041 1.041 0.8959 0.8952 0.4309 0.4292

0.4

1.825 1.975

0.8582 0.9369

1.720 1.878

0.1881 0.1740

0.1057 0.0978

0.4019 0.3703

0.3958 0.3655

0.3462 0.3192

0.4

0.8176 0.8162 0.9664 0.9660

0.4184 0.4179 0.4498 0.4496

0.4825 0.4817 0.5181 0.5179

0.4682 0.4675 0.6296 0.6294

0.3494 0.3488 0.3368 0.3366

1.424 1.422 1.434 1.433

1.180 1.179 1.106 1.106

1.187 1.185 1.192 1.192

0.6795 0.6715 0.8784 0.8780 1.021 1.021

0.4550 0.4555 0.5543 0.5580 0.6526 0.6553

0.4566 0.4521 0.5113 0.5119 0.5580 0.5589

0.3003 0.2981 0.5064 0.5064 0.6602 0.6601

-0.1013 -0.1471 -0.2382 -0.2592 -0.2510 -0.2595

0.3296 0.2247 -0.0073 -0.0498 -0.0737 -0.0905

0.0345 -0.0863 -0.8562 -0 9269 -1.835 -1.877

0.15 0.4 0.6 0.8

-0.2 0.05

0

1

0

0.6 0.1

0

0

0.15 0.4 0.6

0.2

P

1.247 1.183 0.9784 0.9441 0.5849 0 5598

-

-

0.5450 0.5249 0.5192 0.5158 0.5267 0.5811 0.5644 0.7384 0.6497 0.6210

0

0

0.5

1.686 1.686

0.4311 0.4312

0.6205 0.6207

0.5346 0.5347

0.0818 0.0816

1.309 1.310

0.4055 0.4051

0.9468 0.9469

0.7805

-0.5

0

0.05

0.4925 0.4496

0.2520 0.2315

0.2712 0.2458

0.2870 0.2749

0.5333 0.5099

2.307 2.216

1 * 879 1.791

1.845 1.764

1.0426

TABLE V (Continued)

1

0.3

0

0

0.4 0.6

0.9

0.15 0.6

2

0.01

0

0

0.6

0.8554 0.8512 0.9926 0.9915

0.4476 0.4458 0.4965 0.4964

0.5140 0.5117 0.5375 0.5372

0.4848 0.4824 0.6520 0.6511

0.3705 0.3687 0.2582 0.2566

1.489 1.482 1.370 1.362

1.226 1.220 0.9425 0.9388

1.235 1.229 0.7262 0.7192

0.7397 0.7351 1.031 1.035

0.3716 0.3695 0.2743 0.2750

0.5242 0.5215 0.5597 0.5682

0.3452 0.3430 0.7070 0.7088

0.3945 0.3920 0.3244 0.3262

1.583 1.573 1.486 1.501

1.339 1.330 1.061 1.063

1.169 1.161 0.8859 0.8842

0.8694

1.334 1.334

1.031 1.030

0.5433 0.5433

0.5452 0.5451

0.2251 0.2253

1.272 1.272

0.7623 0.7629

0.5427 0.5446

0.5136

0.8437 0.7918 1.080

LI, 0

2 2 0

5

$ w 0

2

tl

TABLE VI SOLUTION OF THE OUTER LIMIT EQUATIONS FOR

fi

+m

-*

~

0.5

0

0-1

0.3333

1

~~~

~~~

0.9

1

0.7

0.4 0.6

1.128 1.249

0.8388 0.9283

-2.487 -3.383

0.8330 0.8262

-3.856 -5.055

0.6067 0.5952

1

0

0.7

0.4 0.6

1.128 1.249

0.8388 0.9283

-2.487 -3.383

0 8330 0.8262

-3.856 -5.055

0.6067 0.5952

0.5

0.7

0.4 0.6

0.9489 1.050

0.8077 0.8939

-3 -008 -4.056

0.9803 0.9771

-4.641 -6.043

0.6971 0.6876

0.9

1

0.7

0.15 0.4 0.6

0.4837 0.6181 0.6840

0.4421 0.5650 0.6253

-0.0764 -0.5016 -0.8623

1.269 1.344 1.348

-0.1279 -0.9260 -1.529

0.9456 0.9786 0.9723

1

0

0.7

0.15 0.4 0.6

0.6535 0.8351 0.9242

0.5016 0.6410 0.7094

-0.0039 -0.3297 -0.6099

0.9708 1.011 1.007

0.0087 -0.6133 -1.085

0.7648 0.7824 0.7734

0.5

0.7

0.5495 0.7022 0.7772

0.4758 0.6080 0.6729

-0.0485 -0.4268 -0.7489

1.129 1.189 1.190

-0.0761 -0.7898 - 1.329

0.8555 0.8821 0.8750

0

--

0.15 0.4 0.6

0.7

0.15 0.4 0.6 1 .o

0.4154 0.5309 0.5875 0.6676

0.3019 0.2538 0.1795 0.0

1.082 1.140 1.143 2.128

0.7833 0.5792 0.3866 0.0

0.9215 0.9653 0.9666 1.954

0.5

--

0.4965 0.6345 0.7022 0.7979

0.15 0.4 0.6 1.0

0.4175 0.5336 0.5905 0.6709

0.3789 0.4842 0.5359 0.6089

0.3310 0.2783 0.1968 0.0

1.262 1.342 1.351 2.342

0.8588 0.6350 0.4239 0.0

1.010 1.058 1.060 2.046

0.7

-

m

n

U

H

W

n U

H

W N

n

U

H

W d

N U

h

-

aJ

W 0

n

0

-

W

)r

3

h

LL

U

M

-

I

a"b'

10

b

U

'us 3

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

0 0 0 H 0 0 0 d

In

0

r-

4 0 0 0 d 0 0 0

m

0

I-

0000

. . . . . . . . . . . . . . , . . . . . d. o. 0. 0.

0004

0

PI

d 0 0 0 0 0 0 0

0

r-

I I

0

ln

I

0

0:

I I

I

0

P

0

I I

0

tn

I

I

0004

?P???P?? ???? ?'P???P?? ?P??

ln

0 0 0 4 0 0 0 H

tn

0

f-

ln

r-

I I

In

I

0

m 0

rl

I

0

Y

4

0

ln

0

TABLE VI (Continued)

0.5

-0.4

0.7

0

1

0.9

--

0.7

0.15 0.4 0.6 1.0

0.2040 0.2804 0.3181 0.3714

0.2012 0.2760 0.3130 0.3652

0.1

0.9

1

0.7

0.6877 1.131 1.235

1

0

0.7

0.15 0.4 0.6 0.15 0.4 0.6

0.5

0.7

0.9

1

0.7

1

0

0.7

0.5

--

0.3333

1

0

0.5268 0.8412 0.9182

0.5451 0.4399 0.3066 0.0 -1.516 -2.449 -3.274

0.9729 0.8233 0.7998

-2.505 -3.780 -4.865

0.7355 0.6049 0.5833

0.9269 1.131 1.235

0.6899 0.8412 0.9182

-1.347 -2.449 -3.274

0.8417 0.8233 0.7998

-3.780 -4.865

-2.231

0.6287 0.6049 0.5833

0.15 0.4 0.6

0.8395 1.021 1.114

0.7152 0.8698 0.9486

-1.572 -2.771 -3.668

0.9169 0.9058 0.8835

-2.576 -4.256 -5.435

0.6679 0.6494 0.6290

0.15 0.4 0.6 0.15 0.4 0.6

0.5855 0.7091 0.7719

0.5355 0.6483 0.7025

-0.0812 -0.4520 -0.7596

1.209 .1.211 1.189

-0.1396 -0.8341 -1.345

0.8938 1.050 0 8608

0.6966 0.8468 0.9231

0.5357 0.6500 0.7081

-0.0101 -0.3338 -0.6048

-0.0036 -0.6207 -1.075

0.8116 0.7929 0.7707

0.7

0.15 0.4 0.6

0.6308 0.7649 0.8329

0.5467 0.6623 0.7209

- 0.4041

-0-6938

1.036 1.024 0.9998 1.129 1.126 1 104

-0.0920 -0.7475 -1.230

0.8497 0.8358 0.8147

0.7

0.15 0.4 0.6 1 .o

0.5450 0.6582 0.7153 0.7979

0.4598 0.5530 0.5999 0.6676

0.3313 0.2637 0.1831 0.0

1.180 1.185 1.168 2.128

0.8471 0.6000 0.3941 0.0

0.9966 1.0000 0.9853 1 954

-0.0559

2.202 2.255 2.246 3.207

1.431 1.008 0.6619 0.0

1.684 1.680 1.655 2.604

-

TABLE VI (Continued)

2

U

fw

0.7

0

-0.2

tS

1

1

a

Pr

tw

0.5

0.7

0.15 0.4 0.6 1 .o

0.4931 0.5943 0.6454 0.7191

0.4495 0 5406 0.5865 0.6526

0.3389 0.2698 0.1873 0.0

1.289 1.304 1.289 2.252

0.8666 0.6138 0.4032 0.0

1.019 0.8082 1.008 1.976

0.9

0.7

0.15 0.4 0.6 1 .o

0.3887 0.4677 0.5075 0.5649

0.3810 0.4582 0.4971 0.5531

0.3998 0.3183 0.2210 0.0

1.618 1.647 1.634 2.594

1.022 0.7242 0.4757 0.0

1.203 1.207 1.189 2.151

0

0.7

0.15 0.4 0.6 1 .o

0.4227 0.5355 0.5926 0.6751

0.3729 0.4663 0.5133 0.5811

0.3714 0.2898 0.1996 0.0

1.345 1.317 1.285 2.228

0.9575 0.6613 0.4304 0.0

1.126 1.102 1.076 2.030

0.5

0.7

0.15 0.4 0.6 1.0

0.3929 0.4941 0.5452 0.6190

0.3663 0.4576 0.5036 0.5699

0.3789 0.2958 0.2038 0.0

1.452

0.9768 0.6750 0.4394 0.0

1.149 1.125 1.098 2.052

0.15 0.4 0.6 1.0

0.3261 0.4053 0.4451 0.5026

0.3208 0.3983 0.4373 0.4934

0.4394 0.3441 0.2374 0.0

1.778 1.776 1.749 2.692

1.131 0.7847 0.5115 0.0

1.331 1.308 1.279 2.227

0.15 0.4 0.6

0.3149 0.4247 0.4807 0.5619

0.2941 0.3863 0.4328 0.5001

0.4182 0.3193 0.2182 0.0

1.540 1.467 1.416 2.338

1.089 0.7311 0.4710 0.0

1.281 1.219 1.177 2.114

0.9

-0.4

1

0

m

0.7

0.7

1 .o

-

1.435 1.406 2.351

TABLE VI (Continued) 2

t

fw

0.7

1

-0.4

0

0

c")

Pr

tw

gi(o)

ei(W

I2

I#)

11(2)

5(3)

m

0.5

--

0.7

0.15 0.4 0.6 1.0

0.3031 0.4025 0.4529 0.5258

0.2908 0.3810 0.4265 0.4923

0.4255 0.3252 0.2232 0.0

1.644 1.582 1.535 2.460

1.107 0.7445 0.4798 0.0

1.303 1.241 1.200 2.136

0.9

--

0.7

0.15 0.4 0.6 1 .o

0.2683 0.3469 0.3865 0.4437

0.2653 0.3421 0.3809 0.4368

0.4844 0.3727 0.2553 0.0

1.960 1.918 1.874 2.797

1.257 0.8523 0.5510 0.0

1.478 1.420 1.377 2.309

0.1

1

1

1

0 0.5

1 .o

0.9076 1.189 1.339

0.9076 1.189 1.339

-1.115 -2.871 -4.528

0.9629 0.7798 0.7030

-1.841 -4.278 -6.327

0.6826 0.5481 0.4929

0.1538

1

1

1

0 0.5 1 .o

0.8258 1.075 1.208

0.8258 0.075 1.208

-0.5459 -1.736 -2.885

1.051 0.8575 0.7753

-0.9319 -2.756 -4.265

0.7461 0.6034 0.5442

0.3333

1

1

1

0 0.5 1 .o

0.7063 0.9028 1.010

0.7063 0.9028 1.010

0.0364 -0.5265 -1.104

1.210 1.006 0.9161

0.1645 -0.9472 -1.832

0.8615 0.7098 0.6445

0.625

1

1

1

0.6326 0.7915 0.8793

0.6326 0.7915 0.8793

0.2647 -0.0202 -0.3317

1.331 1.130 1.037

0.7220 -9.0382 -0.6223

0.9504 0.7994 0.7317

1

1

--

0 0.5 1 .o

1

0 0.5 1.0

0.5898 0.7230 0.7979

0.5898 0.7230 0.7979

0.3567 0.1965 0.0

1.412 1.220 1.128

1.010 0.4327 0.0

1.010 0.8654 0.7979

1

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS TABLE VII SOLUTION OF THE INNERLIMITEQUATIONS FOR p +

~~~~

~

V

0.5

a (*I

0.2025 0.4050 0.4920 1.0000

0

1.1547 1.1672 1.1816 1.1885 1.2533

0.75

0 0.2025 0.4050 0.4920 1 .oooo

1.1547 1.1609 1.1679 1.1712 1.1991

1 .o

all

1.1547

*

441

C. FORBES DEWEY, JR.,

442

AND JOSEPH

F. GROSS

SYMBOLS A,, ..., Ah constants defined in Eqs. (106), (116), and (121) function defined in Eqs. (110), ( 154) function defined in Eq. (86)

Chapman-Rubesin constant limit of f ( a , t ) as t + m local skin friction coefficient ;

c,

=

2r/pm Urn2

local heat transfer coefficient ; qwlpm Um(Haw - Hw)

mass concentration of ith component specific heat of the ith component binary diffusion coefficient function as defined in Eq. (103) numerical difference function defined in Eq. (82) function as defined in Eq. (86) similarity function defined in Eq. (23) transformed fluid velocity as defined in Eq. (12) wall constant defined by Eq. (22) function defined in Eq. (83) dimensionless enthalpy function,

(HIH,)

transverse-velocity function, w / w , numerical difference function defined in Eq. (84) total enthalpy of mixture,

h + t(U2

+ d)

numerical difference function defined in Eq. (85) chemical enthalpy of ith component I,, Z2 integrals defined by Eqs. (75) and (76) 11(1),Zl(2), ZL(3) integrals defined by

(77)-(79)

,Jn(t)

j

k

function defined as the nth integral of the error function geometrical index in boundary layer equations ; also Reynolds analogy ratio 2C,,/C, thermal conductivity of the mixture

Lewis number pDI2cp/k Mach number m constant exponent defined in Eq. (34) P7 Prandtl number, c P p / k P fluid pressure 4 heat flow R gas constant r radial coordinate defined in Fig. 1 S Sutherland constant s c Schmidt number, p/pD12

Le

M

S

SIT0

transformed velocity function defined in Eq. (1 50) T temperature of the fluid TO free-stream stagnation temperature t transformed similarity variable defined in Eq. (150); transformedvariable defined by Eq. (109) taw dimensionless adiabatic wall temperature, t = t,, for #(Of = 0 Eckert reference temperature defined in Eq. (53c) dimensionless sweep parameter defined by Eq. (32), 1 ( Urn2/2H,)sin2 A t I" dimensionless wall enthalpy ratio defined by Eq. (31), T,,./To u r n free-stream velocity II flow velocity in the x direction flow velocity in the y direction V W constant defined in Eq. (141) W flow velocity in the z direction (transverse velocity) X coordinate in direction of flow Y coordinate normal to the surface Zi dimensionless concentration function defined in Eq. (15) z dimensionless function, (1 - 2,); also, coordinate transverse to the flow direction pressure gradient parameter defined in Eq. (34) Y adiabatic constant, c,/c,. 6* boundary layer displacement thickness defined by Eq. (80) SO

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS E

7

0 0 K

A h p

5

p u uI u2

asymptotic parameter defined in Eq. (98) similarity variable defined in Eq. (11) boundary-layer momentum thickness defined in Eq. (81) dimensionless enthalpy function, ( H - W / ( H e - Hw) trial function for solution of firstorder equation defined by Eq. (123) sweep angle dimensionless density-viscosity product, (l/C) ( p d p e iue) viscosity transformed x coordinate defined in Eq. (10) fluid density hypersonic parameter ( U , 2/2He) modified hypersonic parameter, (Um2/2He)(Uehrn)’ modified hypersonic parameter, (Um2/2He)[(ue/um)2COG A sin2 A]

+

4.43

total skin friction skin friction in spanwise direction skin friction in x direction T, CJ function defined in Eq. (106) x variable defined by Eq. (141) x(a) function defined in Eq. (112) xo constant defined in Eq. (141) Y ( x , y ) stream fun,ction defined by Eq. (9) ‘PI,. ..,Y, functionsdefined in Eqs. (1 18), (119), and (122) w exponent in the viscosity-temperature law, p TW w, constant defined in Eq. (53b) T

rZ

-

SUBSCRIPTS ( )e function at the edge of the boundary layer ( ) w function at the wall ( )rn function evaluated in the free stream

ACKNOWLEDGMENT T h e authors wish to express their gratitude to Susan Fredlund and Jeannine McGannLamar for their expert assistance in performing the numerical calculations of this chapter. The original numerical procedures were developed by Kathrine Purdom, presently with Lockheed Missiles and Space Division, Sunnyvale, California and James I. Carlstadt of The RAND Corporation. This research is supported and monitored by the Advanced Research Projects Agency under Contract No. SD-79. Any views or conclusions contained in this work should not be interpreted as representing the official opinion or policy of ARPA. One of us (C. F. Dewey, Jr.) was the recipient of a University of Colorado Faculty Fellowship during preparation of this manuscript. We have been privileged to receive a review of this manuscript by Nelson H. Kemp of the AVCO Everett Research Laboratory, Everett, Massachusetts.

REFERENCES 1 . G . Kuerti, T h e laminar boundary layer in compressible flow, Advan. Appl. Mech. 2 , 23-92 (1951). 2. W. D. Hayes and R. F. Probstein, “Hypersonic Flow Theory,” 1st ed., 292-312. Academic Press, New York, 1959. 3. I. E. Beckwith, Similar solutions for the compressible boundary layer on a yawed cylinder with transpiration cooling. NA SA Tech. Rept. TR-R-42(1959). 4. J. F. Gross and C. F. Dewey, Jr., Similar solutions of the laminar boundary-layer equations with variable fluid properties, in “Fluid Dynamic Transactions” (W. Fizdon, ed.) Vol. 2, pp. 529-548. Pergamon Press, New York, 1965.

444

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

5. N. H. Kemp, Approximate analytical solution of similarity boundary layer equations with variable fluid properties. Publi. No. 64-6. Muss. Inst. Technol. Fluid Mech. Lab., Cambridge, Massachusetts, September 1964. 6. T. Y. Li and H. T. Nagamatsu, Shock-wave effects on the laminar skin friction of an insulated flat plate at hypersonic speeds. J . Aerospace Sci. 20, 345-355 (1953). 7. C. B. Cohen and E. Reshotko, Similar solutions for the compressible laminar boundary layer with heat transfer and arbitrary pressure gradient. N A C A Rept. 1293 (1956). 8. E. Reshotko and I. E. Beckwith, Compressible laminar boundary layer over a yawed infinite cylinder with heat transfer and arbitrary Prandtl number. N A C A Rept. 1379 (1958). 9. I. E. Beckwith and N. B. Cohen, Application of similar solutions to calculation of laminar heat transfer on bodies with yaw and large pressure gradient in high-speed flow. N A S A Tech. Note TN-D-625 (1961). 10. C. F. Dewey, Jr., Use of local similarity concepts in hypersonic viscous interaction problems. A I A A J . 1, 20-33 (1963). 11. A. M. 0. Smith, Improved solutions of the Falkner and Skan boundary-layer equation. Inst. Aerospace. Sci. Sherman M. Fairchild Fund Paper FF-10, 1954. See also: A. M. 0. Smith and D. W. Clutter, Solution of the incompressibk laminar boundarylayer equations, A I A A J . 1, 2062-2071 (1963). 12. A. Busemann, in “Handbuch der Experimental Physik,” Vol. 4, Pt. 1, p. 336. Akad. Verlagsges., Leipzig, 1931. 13. Th. Von Karman and H. S. Tsien, Boundary layer in compressible fluids. .I. Aerospace Sci. 5 , 227 (1938). 14. L. Crocco, The laminar boundary layer in gases. Monogra5c Sci. Aeronaut. No. 3, December 1946 (Translated as North Am. Rept. APL/JHU CF-1038, 1948). 15. G. B. W. Young, and E. Janssen, The compressible boundary layer. J . Aerospace Sci. 19, 229 (1 952). 16. E. R. Van Driest, Investigation of the laminar boundary layer in compressible fluids using the Crocco method. N A C A Tech. Note TN 2597 (1952). 17. H. W. Emmons and D. Leigh, Tabulation of the Blasius function with blowing and suction. Rept. 15966. Fluid Motion Sub-Comm., Aeronaut Res. Council. 1953. 18. D. Meksyn, “New Methods in Laminar Boundary-Layer Theory.” Pergamon Press, Oxford, 1961. 19. L. Rosenhead (ed.), “Laminar Boundary Layers.” Oxford Univ. Press (Clarendon), London and New York, 1963. 20. K. Stewartson, “The Theory of Laminar Boundary Layers in Compressible Fluids”. Oxford Univ. Press (Clarendon), London and New York, 1964. 21. F. K. Moore (ed.), “Theory of Laminar Flows,” Vol. 4, “High-speed Aerodynamics and Jet Propulsion.” Princeton Univ Press, Princeton, New Jersey, 1964. 22. D. B. Spalding and H. L. Evans, Mass transfer through laminar boundary layers: 2, Auxiliary functions for the velocity boundary layer. Intern. ,I. Heat Mass Transjer 2, 199-221. 1961. 23. J. Pretsch, Die laminar Grenzschicht bei Starkem Absaugen und Ausblasen. Untersuch. Mitt. Deut. Luftfahrtforsch., No. 3091 (1944). 24. E. J. Watson, The asymptotic theory of boundary layer flow with suction. Brit. Aeronaut Rex. Council R and M No. 2619 (1952). 25. E. R. G. Eckert, P. L. Donoughe, and B. J. Moore, Velocity and friction characteristics of laminar viscous boundary layer and channel flow with ejection or suction N A C A Tech. Note TN 4102, 1957. 26. B. Thwaites, The development of the laminar boundary layer under conditions of

.

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

445

continuous suction: I, O n similar profiles Rept. 11,830. Brit. Aeronaut. Res. Council, 1948. 27. H. Schlichting and K. Bussman, Exakte Losungen fur die Laminare Grenzschicht rnit Absaugung und Ausblasen. Schriften Deut. Akad. Luftfahrtforsch. 76, No. 2 (1943). 28. W. Mangler, Laminare Grenzschicht mit Absaugen und Ausblasen. Untersucht. Mitt. Deut. Luftfahrtforsch. No. 3087 (1944). 29. H. Schaefer, Laminare Grenzschicht zur Potentialstromung U = u1 x“‘ mit Absaugung und Ausblasen. Deut. Luftfahrtforsch No. 2043 (1944). 30. J. C. Y. Koh and J. P. Hartnett, Skin friction and heat transfer for incompressible laminar flow over porous wedges. Intern. J . Heat Mass Transfer (Corrigendum) 5 , 593 (1962). 31. K. Stewartson, Further solutions of the Falkner-Skan equation. Proc. Cambridge Phil. SOC.50,454-465 (1954). 32. D. R. Hartree, On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer. Proc. Cambridge Phil. Sac. 33, 223-239 (1937). 33. J. H. Hufen and W. Wuest, Ahnliche Losungen bei kompressiblen Grenzschichten mit Warmeubergang und Absaugung oder Ausblasen. 2. Angew. Math. Phys. 17, 385-390 (1966). 34. I. Tani, On the approximate solution of the laminar boundary-layer equations. J . Aerospace Sci. 21 487-504 (1954). 35. F. B. Hanson and P. D. Richardson, Use of a transcendental approximation in laminar boundary layer analysis. J . Mech. Eng. Sci. 7, 131-137 (1965). 36. K.-T. Yang, An improved integral procedure for compressible laminar boundary-layer analysis. J . Appl. Mech. 28, 9-20 (1961). 37. A. Pallone, Nonsimilar solutions of the compressible laminar boundary-layer equations with application to the upstream-transpiration cooling problem. J . Aerospace Sci. 28, 449-456 (1961). 38. 0. M . Belotserkovski and P. I. Chuskin, T h e numerical solution of problems in gas dynamics. In “Basic Developments in Fluid Mechanics” (M. Holt, ed.), p. 1. Academic Press, New York, 1965. 39. A. M . 0. Smith and D. W. Clutter, Solution of the incompressible laminar boundarylayer equations. AZAA ./. 1, 2062-2071 (1963). 40. I. Flugge-Lotz and D. C. Baxter, Computation of the compressible laminar boundarylayer flow including displacement-thickness interaction using finite-difference methods. Tech. Rept. No. 131 (AFOSR2206). Stanford Univ., Stanford, California, January 1962. 41. D. C. Baxter and I. Flugge-Lotz,Z. Angew. Math. Phys. 9b,81 (1958). 42. L. Lees, Laminar heat transfer over blunt-nosed bodies at hypersonic flight speeds. ,let Propulsion, 26, 259-269 (1956). 43. A. M. 0. Smith, Rapid laminar boundary-layer calculations by piecewise application of similar solutions. J . Aerospace Sci. 23, 901-912 (1956). 44. N. H. Kemp, P. H. Rose, and R. W. Detra, Laminar heat transfer around blunt bodies in dissociated air. .I. Aerospace Sci. 26, 421-430 (1959). 45. B. Thwaites, Approximate calculation of the laminar boundary layer. Aeroit. Quart. 1, 245-280 (1949). 46. L. Lees and B. L. Reeves, Supersonic separated and reattaching laminar flows A I A A J . 2, 1907-1920 (1964). 47. B. L. Reeves and L. Lees, Theory of laminar near wake of blunt bodies in hypersonic HOW. AZAA J. 3, 2061-2074 (1965).

446

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

48. H. J. Merk, Rapid calculations for boundary-layer transfer using wedge solutions and asymptotic expansions. J . Fluid Mech. 5 , 460-480 (1959). 49. W. B. Bush, Local similarity expansions of the boundary-layer equations. AIAA J . 2, 1857-1858 (1964). 50. V. M . Falkner and S. W. Skan, Solutions of the boundary-layer equations. Phil. Mag. 12,865-896 1931. 51. W. B. Bush, A method of obtaining an approximate solution of the laminar boundarylayer equations. J . Aerospace Sci. 28, 350-351 (1961). 52. D. E. Coles, T h e laminar boundary layer near a sonic throat. Proc. Heat Trans. and Fluid Mech. Inst., pp. 119-137. Stanford Univ. Press, Stanford, California, 1957. 53. P. A. Lagerstrom, Laminar flow theory. In “High Speed Aerodynamics and Jet Propulsion,” Vol. 4 : “Theory of Laminar Flows” (F. K. Moore, ed.), pp. 125-129. Princeton Univ. Press, Princeton, New Jersey, 1964. 54. S. Kaplun and P. A. Lagerstrom, Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers; Low Reynolds number flow past a circular cylinder; Note on the preceding two papers. J . Math. Mech. 6 , 585-606 (1957). 55. P. A. Lagerstrom and J. D. Cole, Examples illustrating expansion procedures for the Navier-Stokes equations. J . Ratl. Mech. Anal. 4, 817-882 (1955). 56. M. Van Dyke, “Perturbation Methods in Fluid Mechanics.” Academic Press, New York, 1964. 57. T. Y. Li and J. F. Gross, Hypersonic strong viscous interaction on a flat plate with mass transfer. Heat Transfer and Fluid Mechanics Institute, 1961. Stanford Univ. Press, Stanford, California, 1961. 58. R. J. Whalen, Boundary-layer interaction of a yawed infinite wing in hypersonic flow. J . Aeron. Sci. 26, 839-851 (1959). 59. L. H. Back and A. B. Witte, Prediction of heat transfer from laminar boundary layers, with emphasis on large free-stream velocity gradients and highly cooled walls. /. Heat Transfer 88, 249-256 (1966). 60. R. Narasimha and S. S. Vasantha, Laminar boundary layer on a flat plate at high Prandtl number. Z . Angew. Math. Phys. 17, 585-592 (1966).

Author Index Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although - his name is not cited in the text. Numbers in italic show the page on which the complete reference is listed. Beckman, E. L., 90(27), 91,92,93,94(27),

A Abdalla, K. L., 228 Abramson, P., 213(98), 214(98), 227 Acrivos, A., 33,63 Adams, C. C., 98,239 Adams, D.E., 42,64 Adams, J. M., 206(85), 226 Adelberg, M., 157,158, 201,224,226 Advani, G . H., 90(28), 91(28), 92(28), 95

(28),109(28), 110(28), 238 T., 206(84), 226

95(33), 96,138

Beckmann, W., 5, 15,62 Beckwith, I. E., 318,325, 330,331, 334,

341 342,343,344,443,444 I

Belding, H. S., 93,238 Belotserkovski, 0. M., 334,445 Benedikt, E. T., 147(4), 222 Benzinger, T. H., 120(93), 240 Berenson, P. J., 98(41, 42), lOO(41, 43),

239,171,225

Alad’ev, I.

Albers,J.A.,210,211,212(96),213(96),227 Bevans, R. S., 108(53), 239 Blalock, A., 118(77), 240 Allingham, W.D., 206,227 Blankenship, V. D., 48,49(83), 64 Altman, P. L., 242

Altmos, D.A,, 228 Amenitskii, A. N., 206(84), 226 Anateg, R.,82(20), 238 Anderson, A. D., 35,63 Anderson, J. E., 269(32), 284(32),285,286,

287,288,289,324

Andracchio, C. R.,228 Arnett, C. D., 150,223 Arpaci, V. S., 49(85),64 Aydelott, J. C., 182(66), 183(66), 186(66),

195(66), 196(66), 225,227,228

B Bach, G. R., 241 (7),242(7),323 Back, L. H., 352,446 Bailey, R.V., 191 (69), 225 Baker, M.J., 50, 64 Barzelay, M.E., 238,323 Batchelor, G. K., 152,223 Bates, D.R.,251,252(15), 323 Baxter, D.C., 334,J45 Bayley, F. J., 56,64 Bazett, H.C . , 88(25), 91(25), 238

Blatt, T., 205(80), 206(80), 226 Blatz, W.J., 114(67), 240 Bobco, R.P., 25,63 Bond, G., 109(55), 110(55), 139 Bonilla, C. F., 166,224 Booda, L. L., 109(56,57),110(56), 239 Boinjakovid, F., 259(23), 260(24), 262(23),

263(23), 264(24), 313

Braginskii, S. I., 268,314 Brett, J. R., 95,138 Briggs, D.G . , 38,39,64 Brindley, J., 10,62 Brogan, T.R.,310(69),326 Bromley, L. A,, 173,174,225 Brown, D.D., 202(77), 203(77),226 Buhler, R.D., 283(48), 325 Bunche, C. M., 116(71), 1-10 Bundy, R.D., 206(82), 226 Burriss, W.L., 98,100,239 Burton, A. C., 88(25), 91(25), 238 Bush, W. B., 335,337,446 Buskirk, E. R.,120,I40 Bussman, A., 331,443 Bussman, K., 331,4J5

447

AUTHORINDEX

448 C

Cambel, A. B., 257(22), 258(22), 259(22), 260(22), 261 (22), 262, 265(22), 266 (22), 313 Campbell, J. B., 115(70), 140 Cann, G. L., 283(48), 315 Carlson, L. D., 92(30), 138 Carlson, W. O., 152, 223 Carne, J. B., 21, 63 Cess, R. D., 28, 63, 224, 228, 313 Chapman, S., 273(35), 314 Chen, F. F., 256(21), 313 Chianta, M. A., 129(110), 130(111), 135 (110), 141 Chiladakis, C., 170(53), 225 Chin, J. H., 153, 223 Ching, P. M., 35, 63, 230, 273,313 Choi, H. Y., 207(91), 227 Chu, N. C., 152, 223 Churchill, S. W., 36, 37, 63 Chuskin, P. I., 334, 445 Cichelli, M. T., 166, 224 Clark, J. A., 47,48,49(83, 85), 64,161,162, 169(39), 170(71), 171, 172(41), 173, 174, 206(88), 225, 226, 227 Clodfelter, R. G., 162, 163, 168, 191, 210, 224,227 Clutter, D. W., 331, 334, 444, 445 Cobine, J. D., 248(20), 255(20), 313 Cochran, T. H., 182, 183, 186, 195, 196, 225 Cohen, C. B., 330, 331, 444 Cohen, N. B., 279,314, 325, 330, 334, 341, 342, 343, 344,444 Colburn, A. P., 54, 64 Cole, J. D., 341, 446 Coles, D. E., 341, 342, 345, 446 Colombo, G. V., 228 Congelliere, J. T., 146, 222, 228 Corpas, E. L., 150(7), 222, 227, 228 Costello, C. P., 206(85), 226, 227 Cousteau, J. Y., 109(58), 139 Covino, B. G., 120(86), 140 Cowling, T. G., 273(35), 314 Crabs, C. C., 150(6), 211(96a), 222,227 Cramer, K. R., 97(38), 100(38), 139 Crandall, J. H., 53, 64 Cremers, C. J., 248(13), 303(65), 305(65), 313,315

Crocco, L., 331,444 Cummings, R. L., 210,214, 227

D D’Angelo, N., 251 (17), 313 Davis, A. H., 50, 54, 55, 59, 64 Day, R., 93(31), 94(31), 96(31), 138 Debolt, H., 283(49), 315 Delattre, M., 146, 222 Denington, R. J., 214, 227 Detra, R. W., 334(44), 445 Dewey, C. F., Jr., 318, 326(4), 330, 345, 354(4), 443, 444 Dillon, R. C., 228 Dittmer, D. S., 141 Donaldson, J. O., 153(24), 223 Donoughe, P. L., 331 (25), 444 Drake, R. M., Jr., 153,223, 270(33), 314 Drawin, H. W., 245, 251 (18), 313 Drufes, H., 284(52), 315 DuBois, E. F., 67(6, 8), 137 Dufton, A. F., 67, 137

E Ebaugh, F. G., Jr., 121(101), 140 Eberhart, R. C., 295, 315 Ebert, W. A , , 274(37), 314 Ecker, G., 295(62), 315 Eckert, E. R. G., 9, 16, 17, 31, 41, 42, 44, 51, 55, 59, 62, 63, 64, 152, 153, 157, 223, 224, 269(32), 270(33), 274, 284 (32), 285, 286, 287, 288, 289, 293(58), 294(58), 295, 296(64), 300(64), 301 (64), 303(64), 304(66), 315, 331, 444 Ede, A. J., 18, 33, 61, 63, 64 Edholm, 0. G., 94(34), 138 Ehlers, R. C., 206(86), 226 Eichhorn, R., 13, 15, 26, 27, 37, 62, 63 Eigenson, L. S., 21(31, 32), 59(32), 63 Eisenberg, M., 16(25), 63 Elenbaas, W., 17, 20, 63 Elsasser, W. M., 67, 138 Elsner, R. W., 92(30), 138 Elwert, G., 250, 259, 313 Emery, A., 152, 223 Emmons, H. W., 331, 444 Ende, W., 178,225 Enders, J. H., 150(6a), 222

AUTHORINDEX Erk, S., 152(14), 208(14), 223 Eshghy, S., 49, 64 Evans, D. G., 200, 201, 205, 226 Evans, H . L., 331,444

F Faber, 0. C., Jr., 164,165,224 Falkner, V. M., 336, 446 Fay, J. A., 271,278, 279, 281,314 Feider, M., 116(71), 140 Feldmanis, C. J., 205, 226 Felenbok, P., 245, 313 Feneberg, W., 268, 314 Feyerherm, A. M., 104(50), 106(50), 139 Finkelnburg, W., 232(3), 233(3), 234(3), 237(4), 240(4), 245, 248(3), 260, 293 (31, 313 Finston, M., 22, 59, 63, 64 Flage, R. A., 228 Flugge-Lotz, I., 334, 445 Foote, J. R., 23, 63 Forslund, R. P., 213(98), 214(98), 227 Forster, H. K., 178, 225 Forstrom, R. J., 274(37), 314 Fox, R. H., 94(34), 138 Frank-Kamenicky, D. A., 54, 64 Frea, W. J., 206, 227 Frederking, T. H. K., 173, 174, 225 Fritsch, C. A,, 30, 63 Fritz, W., 178, 181, 225 Frysinger, T. C., 182(66), 183(66), 186 (66), 195(66), 196(66), 225, 228 Fujii, T., 10, 16, 51, 57,61,62,64 Fullington, F., 92(30), 138

G Gagge, A. P., 67(9), 83, 88(25), 91 ( 2 9 , 104(47), 110, 111(63), 118(78), 138, 139,140 Gallagher, L. W., 153(24), 223 Gambill, W. R., 206, 226 Gartrell, H. E., 51 (92), 64 Gebhart, B., 32, 39, 40, 41, 42, 63, 64 Ginwala, K., 205, 206, 226 Goldman, K., 293(55), 315 Goldman, R. F., 93(33), 94(33), 95(33), 96(33), 138, 139 Goldstein, R. J., 16, 17, 38, 39, 41, 42, 63, 64, 157, 223

449

Goodell, H., 121(103), 141 Gorland, S. H., 211 (96a), 227 Graham, R. W., 201,206(86), 226 Greenberg, L., 67(5), 137 Greene, L. C., 119(81), 121(81), 123(81), 124(81), 125(81, 108), 127(81), 140, 141 Greene, N. D., 206, 226 Gregg, J. L., 12, 15, 17, 21, 23, 29, 42(18), 62,63 Grevstad, P. E., 210(93), 214(93), 227 Griem, H. R., 234(5), 242(5), 245, 313 Griffith, M. V., 135, 136,142 Griffiths, E., 50, 54, 55, 59, 64 Grigull, U., 152(14), 208(14), 223 Grober, H., 152,208,223 Grosh, R. J., 30, 63 Gross, J. F., 318, 326(4), 347, 354(4), 443, 446 Gruber, R. P., 227, 228 Gruszezynski, J. S., 280, 314 Guillemin, V., Jr., 75, 138 Guyton, Arthur C., 141

H Habip, L. M., 146, 148, 222 Haise, F. W., Jr., 150(6a), 222 Hale, F. J., 307, 309, 310, 316 Hall, A. L., 215, 218, 227 Hammel, R. L., 228 Hammond, W. H., 94(34), 138 Hanson, F. B., 334,445 Hara, T., 29, 63 Harder, R. L., 283(48), 315 Hardy, J. D., 67(6, 8), 69(12, 13), 70, 71 (13), 75(12, 17), 81(18), 82(19), 104 (47), 116(73), 117(73), 119(80, 83), 121(97, 99), 123(99), 125(108), 137, 138, 139, 140, 141 Harmathy, T. Z., 190, 225 Harper, E. Y., 153(24), 223 Hartnett, J. P., 331, 445 Hartree, D. R., 331,445 Haurwitz, B., 67, 138 Hausen, A. G., 33, 63 Hayes, W. D., 318,433 Hedgepeth, L. M., 164, 165, 224 Heilmann, R. H., 54, 64 Hellums. J. D., 36, 37, 63

AUTHORINDEX

450

Hendler, E., 121(97), 140 Hendricks, R. C., 206(86), 226 Henriques, F. C., Jr., 119(82), 125,140,141 Hermann, M., 283 (49), 315 Hermann, R., 51,64 Herrington, L. P., 67(9), 111(63), 118(78), 138, 139, 140 Hershey, F. B., 125(106), 141 Hervey, G. R., 94(34), 138 Hill, T., 67, 75, 137 Holm, F. W . , 90(28), 91(28), 92(28), 95 (28), 109(28), 110(28), 138 Holman, J. P., 51, 64 Hong, S. K.,90(26), 92(26,36), 120(86,87), 138, 140 Horton, G. K., 135, 136, 141 Hoshigaki, H., 278, 279,314 Hougen, 0. A , , 54, 64 Houghten, F. C., 104(46), 139 Howell, B. J., 120(86), 140 Howell, J. R., 169, 225 Hsieh, A. C . L., 92(30), 138 Hsu, Y. Y., 173, 201, 225,226 Hufen, J. H., 331, 445 Humphreys, C. M . , 104(49), 105(49), 139 Hunt, H., 90(27), 94(27), 138 Hurd, S. E., 153(24), 223

I Illingworth, C. R., 33, 63 Irvine, T. F., 97(38), 100(38), 139 Isoda, H., 215, 216, 217,227 Ivey, H. J., 206(83), 226

J Jackson, R. G., 228 Jackson, T . , 51, 55, 59, 64 Jakob, M., 54, 64, 155, 156, 223 Jansen, W., 205(80), 206(80), 226 Janssen, J. E., 113(64), 139, 331, 444 Jeffreys, H., 152, 223 Jennings, B. H., 104(49), 105(49), 139 Jerger, E. W., 44,46, 64 John, R. R., 283 (49), 315 Johnson, H. A., 43 (76), 64 Jones, M. C., 170(53), 225 June, R. R., 50, 64

K Kang, B. S., 90(26), 92(26), 95(26, 36), 120(86), 138, 140 Kang, D. H., 90(26), 92(26), 95(26, 361, 138 Kaplun, S., 341, 446 Kemp, N. H., 278, 279, 281,314, 318, 334, 444,445 Kenshalo, D. R., 121(102), 123(102), 141 Kerrebrock, J. L., 290, 305, 306, 307, 309, 310, 315, 316 Keshock, E. G., 145(5), 149(5), 162, 164, 175, 176, 179, 180, 181, 182, 183(60), 184, 185, 186,190, 199,222,225 Khabbag, G., 205(80), 206(80), 226 Kim, P. K., 90(26), 92(26), 95(26), 138 Kimzey, J. H., 217, 218, 227 Kincaide, W. C., 103, 139 King, C. D., 114(65), 139 Kingston, A. E., 251(14, 15, 16), 252(15), 313 Kinsey, J. L., 110(60), 139 Kirk, D. A., 151, 223 Kirkpatrick, M. E., 150, 223 Kitzinger, C., 120(93), 140 Knight, B. A , , 165, 166, 224 Knoche, K. F., 259(23), 262(23), 263(23), 313 Knoll, R. H., 228 Koch, W. B., 21, 63, 104(49), 105(49), 139 Koestel, A., 214(98a), 227 Koh, J. C . Y., 331,445 Kosky, P. G., 170(53), 225 Krantz, P., 111, 139 Kraus, W., 15, 63 Kreider, M. B., 123(104), 141 Krey, R. U . , 241 (7), 242(7), 313 Kudo, M., 26(44), 63 Kuerti, G., 318, 443 Kurnagai, S., 215, 216, 217, 227 Kurtz, E. F., 7(8), 53, 54(99), 62, 64 Kusko, A , , 283 (49), 315 Kutateladge, S . S., 167, 224

L Lagerstrorn, P. A , , 341, 446 Lancet, R. T., 213, 214, 227 Langmuir, I., 17, 14, 63

45 1

AUTHOR INDEX Lanphear, R., 109(55), 110(55), 139 Lee, D. Y., 120, 140 Lee, M. M., 119(85), 140 Lee, P. H., 120(87), 140 Lees, L., 334, 335, 347,445 Le Fevre, E. J., 7, 18, 56, 6.2, 63, 64 Lehninger, A. L., 141 Leigh, D., 331, 444 Lemlich, R., 26, 58,63 Lepper, R., 147(4), 222 Levy, S. E., 118(77), 140 Lewis, E. W., 170, 171, 172, 173, 174, 224 Li, T. Y., 330, 347,444,446 Licht, S., 141 Liebermann, R. W., 241(7), 242(7), 283 (49), 313,315 Lienhard, J. H., 167, 171, 199, 224, 225 Lin, S. H., 98(41,42), lOO(41, 43), 139 Linke, W., 54, 64 Lipkin, M., 119(80), 140 Lorenz, L., 5, 62 Low, A. R., 152,223 Lurie, H., 43(76), 64 Lyon, D. N., 170,225

Matsumoto, R., 26(44), 63 Meigs, P., 82(20), 138 Meksyn, D., 331, 335,444 Menold, E. R., 38, 63 Merk, H. J., 9, 18, 33, 62, 63, 335, 446 Merte, H., Jr., 161, 162, 169(39), 170(41), 171(41), 172(41), 173(41), 174(41), 206(88), 224, 226 Meryrnan, H. T., 123(105), 141 Metz, B., 120(92), 140 Michiyoski, I., 7, 26(44), 33, 62, 63 Mills, E. S., 228 Millsaps, K., 24, 63 Moncrief, J. A., 120(89), 140 Montagna, William, 141 Moore, B. J., 331 (25), 444 Moore, F. K., 331,334, 444 Moore, R. A,, 283 (48), 325 Morozkin, V. I., 206(84), 226 Morris, J. C., 241, 242(7), 313 Morrison, J. F., 120(90), 140 Moritz, A. R., 119(82), 125(82), 140, 141 Munro, A., 120(91, 92), 140 Munroe, L. R., 129(110), 135(110), 141 Murgatroyd, D., 121(97), 140 Murphy, D. W., 191(69), 225

M McAdams, W. H., 155, 156, 223 McArdle, J. G., 150(7), 222, 228 Macbeth, R. V., 204,226 McCook, R. D., 120(96), 140 McEntire, J. A., 206, 227 McGrew, J. L., 191, 225 Mackey, C. O., 67(4), 137 Macosko, R. P., 210, 21 1, 227 McPherson, R. K., 120(91), 140 McWhirter, R. W. P.,251(14, 15), 252(15), 313 Madden, A. J., 44, 61 Mader, P. P., 228 Maecker, H., 232(3), 233(3), 234(3), 243, 245, 248(3), 260, 262(25), 264, 293 (3, 56), 313, 311 Mahn, C., 265(10), 268(10), 313 Mahony, J. J., 156, 221 Malkus, W. V. R., 152, 223 Mangler, W., 331, 115 Mannes, R. L., 162, 164, 191, 194,221 Martin, J. H., 43(75), 61

N Na, T. Y., 33, 63 Nachtsheim, P. R., 53, 64 Nafe, J. P., 121(102), 123(102), 141 Nagamatsu, H. T., 330,444 Narnkoong, D., Jr., 210, 211, 212(96), 213 (96), 227 Nanda, R. J., 35, 63 Narasirnha, R., 355, 146 Nein, M. E., 150, 223 Nestor, 0. H., 295, 315 Neswald, R. G., 115(69), 110 Nevins, R. G., 90(28), 91(28), 92, 95, 104 ( S O ) , 106(50), 109(28), 110, 138, 139 Newburgh, L. H., I11 Ng, C. K., 119(85), 130 Niuman, F., 23, 63 Noyes, R. C., 167,221 Nunamaker, R. R., 150(7), 222, 228 Nusselt, W., 54, 61 Nussle, R. C., 146(2), 150(8), 222, 223

AUTHORINDEX

452 0

Oberbeck, A., 4, 62 Odum, T., 109(55), 110(55), 139 O’Neal, H. A., 109(55), 110(55), 139 Ostrach, S., 1, 6, 14, 15, 62, 64, 152, 223 Otto, E. W., 146(2), 150(8), 222, 223

P Pallone, A., 279, 314, 334, 445 Pannett, R. F., 114(67), 140 Papell, S. S., 164, 165, 202, 203, 204, 224, 226 Parczewski, K. I., 108(53), 139 Park, C., 280, 282,314 Patt, H. J., 284(52), 315 Pellew, A., 152, 223 Penski, K., 283,314 Peters, Th., 237(4), 240(4), 313 Petrash, D. A., 146(2), 150(8), 222, 223 Pfender, E., 248(13), 296(64), 300(64), 301 (64), 303 (64, 65), 304, 305 (65), 313,315 Piret, E. L., 44, 64 Plapp, J. E., 53, 64 Plesset, M. S., 178, 225 Pohlhausen, K., 23, 24, 63 Pomerantz, M. L., 206(87), 226 Pretsch, J., 331, 444 Prins, J. A., 9, 18, 62, 63 Probstein, R. F., 318, 443 Pugh, L. G. C. E., 94(34), 138 Pytte, A., 291, 315

R Rahn, H., 95(36), 138 Raithby, G., 304(66), 315 Randall, W. C., 118(96), 120(96), 140 Rapp, G. M., 104(47), 239 Reeves, B. L., 335, 445 Reeves, E., 90(27), 93(33), 94(27, 33), 95 (33), 96(33), 138 Rehm, T. R., 195, 196, 225 Reitz, J . G., 210(93), 214(93), 227 Rennie, D. W., 120, 140 Reshotko, E., 330, 331,444 Rex, J., 165, 166, 224 Richards, C. H., 69(12), 75(12, 17), 81 (18), 138

Richardson, D. L., 97, 98, 100, 130(112), 139,141 Richardson, P. D., 334,445 Riddell, F. R., 271, 314 Ringler, H., 265(10), 268(10), 313 Ritter, G. L., 170(53), 225 Robinson, J. M., 228 Rohles, F. H., 104(50), 106(50), 139 Rohsenow, W. M., 159, 189, 207(91), 224, 225 Romig, M. R., 28, 63, 230, 313 Rose, P. H., 281, 334(44), 314,445 Rosenhead, L., 331,444 Rosenthal, D., 296(63), 315

S Salgers, E. L., 114(67), 140 Satterlee, H. M., 153(24), 223 Saule, A. V., 214(98a), 227 Saunders, 0. A., 6, 15, 44, 59, 62, 63, 64, 152,223 Scala, S. M., 280, 314 Schaefer, H., 331,445 Scherberg, M. G., 46,47, 64 Schetz, J. A., 26, 27, 37, 63 Schlichting, H., 331, 445 Schmidt, E. H. W., 5, 15, 62, 155, 223 Schmidt, R. J., 152, 223 Schmitz, G., 284(52), 315 Schoeck, P. A., 293(57, 58, 59), 294(57, 58, 59), 295, 315 Schoenhals, R. J., 47, 64 Schuk, H., 6 , 62 Schwartz, S. H., 157, 158, 162, 164, 191, 194,224,228 Scott, C. J., 274(37), 314 Seckendorff, R., 118(75), I40 Senftleben, H., 156,223 Sharma, V. P., 35, 6.3 Shenkman, S., 205(80), 206(80), 226 Shepard, C. E., 283(46, 47), 314 Sherley, J. E., 161, 167, 224 Shure, R. I., 214(98a), 227 Siegel, R., 33, 42(601, 58, 63, 64, 147(5), 149(5), 157, 160(37), 162, 164, 167, 169, 175, 176, 179, 180., 181., 182 183 (60), 184, 185, 186, 190, 191, 209, 222, 224.225.227 .~ Simon, F. F., 202,226

AUTHOR INDEX Simon, H. A., 31, 63 Simoneau, R. J., 202(77), 203(77), 226 Singer, R. M., 228 Siple, P. A., 93(31), 94(31), 96(31), 138 Sissenwine, N., 82(20), 138 Skan, S. W., 336,446 Sliff, H. T . , 228 Smith, A . M. O., 331, 334,444,445 Smolak, G. R., 228 Soderstrom, G . F., 67(6), 116(73), 117(73), 137,140 Soehngen, E. E., 51, 64 Song, S. H., 90(26), 92(26), 95(26, 36), 120(86), 138, 140 Southwell, R. V., 152, 223 Spalding, D. B., 331, 444 Sparrow, E. M., 7,12, 15, 17,21,23, 25,29, 35,42(18), 44, 62, 63,64,209,227,274, 314 Spector, W. S., 118(76), 140 Spitzer, L., 264(26), 314 Springer, W., 104(50), 106(50), 139, 259 (23), 262(23), 263 (23), 313 Squire, H. B., 9, 62 Stambler, I., 114(66), 139 Stankevics, J. O., 281, 314 Steinle, H. F., 167, 225 Stenuit, R., 109, 139 Stevens, G . T . , 214(98a), 227 Stewartson, K., 331, 335, 444, 445 Stine, H. A., 283(45, 47), 314, 315 Stol1,A. M.,69, 71(13), 75(12, 17), 81(18), 82(19), 83, 117(72), 118(72), 119(81), 121(81, loo), 123(81, loo), 124(81), 125(81, 108), 127(81), 129(109, 110), 130(111), 135(110), 138, 140, 141 Stout, E., 51 (93), 64 Stuart, J. T . , 152,223 Sturas, J. I., 211,227 Sugawara, S., 7, 9, 24, 26, 33, 62, 63 Szewczyk, A. A., 16, 51, 52, 53, 63, 64

T Tani, I., 334, 445 Tanner, J. M., 94(34), 138 Taylor, R. B., 214(98a), 227 Thauer, R . , 121 (101), 140 Thompson, R. H., 120(88), 140 Thompson, W. D., 255(19), ,313

453

Thwaites, B., 331, 334, 444, 445 Tobias, C. W., 16(25), 63 Tribus, M., 25, 63 Tritton, D. J., 52, 64 Trusela, R. A., 210, 227 Tsien, H. S., 331, 444 Tsou, F. K., 7(8), 54(99), 62, 64

U Unsold, A., 245, 313 Unterberg, W., 146,222,228 Useller, J. W., 150(6a), 222 Usiskin, C . M., 160(37), 167, 169, 191, 224

v van Dilla, M., 93, 94, 96(31), 138 Van Driest, E. R., 331, 444 van Dyke, M., 341, 446 Van Tassell, W., 279, 314 Vardi, J., 26, 58, 63 Vasantha, S. S., 355,446 Veghte, J. H., 103, 139 Vendrik, A. J. H., 118(79), 140 Vernon, H. M., 67, 137 von Karman, T . , 8, 62, 331,444 Vos, J. J., 118(79), 140

W Wansbrough, R. W., 206(82), 226 Warren, W. R., 280(41), 314 Watanabe, K., 167, 171, 224 Watson, E. J., 331, 444 Watson, V. R., 283 (45, 46, 37), 284(51), 314,315 Webb, P., 116(74), 140 Weber, G . J., 114(67), 140 Weber, H. E., 283(50), 315 Westwater, J. W., 159, 173, 224, 225 Whalen, R. J., 349, 446 Whedon, G. D., 120(88), 140 Whitehouse, R. H., 94(34), 138 Wienecke, R . , 265(10), 267, 268(10, 31), 313,314 Wilke, C. R., 16, 63 Williams, A. R., 291, 315 Williams, C. G., 120(90), 140

454

AUTHOR INDEX

Wilson, J. S., 120(89), 140 Winslow, C.-E. A., 67(5, 9), 111(63), 118 (78), 137, 138, 139, 140 Witkowski, S., 265(10), 267(28), 268(10), 313,314 Witte, A. B., 352, 446 Wolff, H. G., 121 (103), 141 Wolff, H. S., 94(34), 138 Wong, P. T. Y., 199,225 Woodcock, A. H., 87, 138 Wright, L. T., 67(4), 137 Wuest, W., 331, 445 Wurster, R. D., 120(96), 140 Wyndham, C. H., 120(95), 140

Y Yaglou, C. P. J., 104(46, 48), 139 Yang, K. T., 24, 36, 38, 44, 46, 63, 64, 334, 445 Yos, J. M., 241(7), 242(7), 313 Young, G. B. W., 331,444

Z Zankl, G., 265(10), 268(10), 313 Zara, E. A., 164, 165, 224, 228 Zuber, N., 167, 178, 181,224, 225 Zwick, E. B., 228 Zwick, S . A , , 178, 225

Subject Index A

Acoustic wave effects, 47ff Airplane trajectory method of producing reduced gravity, 150 Arcs cathode axis parallel to a plane anode, 295 ff cylinder geometry with annular anode, 320ff free-hurning with plane anodes, 293 ff Asymptotic expansion of local similarity equation, 335ff

B

Bioclimate, 67 Bioclimatology, 67 Bioclimatological data, 84f Biotechnology, heat transfer in, 65ff artificial environments, 103ff indoors, 103 ff space vehicles, 111 ff underwater vessels, 109ff natural environments, 66ff extraterrestrial, 97 ff terraqueous, 90ff, 98f terrestrial, 66ff, 98f skin, its role in heat transfer, 115 ff characteristics, 115 ff functions, 115ff injury, 123ff protection, 128ff thermal sensation, 121 ff Blackbody radiation in a plasma, 238ff Blowing parameter, 322, 326, 330f Boiling in reduced gravity, see Reduced gravity Boltzmann distribution, 235, 245 Bound-bound radiation in a Plasma, 234f Bremsstrahlung in a plasma, 237ff Bubble growth, 174ff

C

Chapman-Ruhesin constant, 324 Clo, 91

Combustion in reduced gravity, see Reduced gravity Condensation in reduced gravity t see Reduced gravity Conservation of energy 4* 2853 308* 319 Conservation of mass equation, 3, 276, 285, 319 Conservation of momentum equations, 3f, 276f, 285, 319 “Critical thermal load”, 124 2779

D Debye length, 254 Dense plasmas, 242ff Displacement thickness, 329, 354 Diver’s wet suit, insulating properties of, 94 ff Drop tower method of producing reduced gravity, 146ff

E Eckert reference temperature, 326 Eckert-Schneider condition, 322 Effective temperature, 104ff Einstein relation, 239 Electrical conductivity, 231 Elenhaas-Heller differential equation, 263 Environmental temperature, 107 Equilibrium in a plasma, see Plasma

F

Fabrics, protective, 129ff Falkner-Skan parameter, 323 Free-hound radiation in a plasma, 235ff Free convection in reduced gravity, see Reduced gravity Free convection on vertical surfaces, 1 ff laminar flow, 4ff circular cylinder, 17 ff flat plate, 4ff differential equation method, 5ff experimental work, 11 ff integral equation method, 8ff 45 5

45 6

SUBJECT INDEX

Free convection on vertical surfaces-cont. laminar flow-cont. Grashof number, very low, 43 ff nonsteady conditions, 33ff experimental work, 41 ff negligible thermal capacity, 33 ff significant thermal capacity, 39ff physical properties effects, 28 ff large temperature differences, 29 f near-critical conditions, 30ff viscous dissipation, 32f radiative effects, 28 surface temperature effects, 22ff differential equation method, 22ff experimental work, 26 integral equation method, 24ff vibration, effects of, 47ff laminar flow instability, 51 ff experimental work, 51 f theoretical work, 52f turbulence, 5Off experimental work, 58 ff theoretical work, 54ff Free-free radiation in a plasma, 237f Fritz-Ende relation, 178f Froude number, 53

G Geometric scale factor, 171ff Grashof number, 3ff, 151

H Hall parameter, 308 Heat conductivity, 233

I Injection parameter, see Blowing parameter Inner limit similarity equations, 344f solutions, 441 Insulation. 87

J

Just noticeable difference (JND), 121

L Lewis number, 270f, 320 frozen, 275

Line radiation in a plasma, 234f Local similarity, 333 ff Local thermal equilibrium (LTE), 246ff

M Magnetic fields in plasmas, see Plasma Magnetic forces used for producing reduced gravity effects, 150f Mass-transfer parameter, see Blowing parameter Maxwell-Boltzmann distribution, 245 Melting, 33 Momentum thickness, 329

N Newton-Raphson scheme, 332 Non-Newtonian fluid, 33 Nusselt number, 3ff, 154ff, 270

0 Oberbeck equations, 5 Operative temperature, 83 Outer limit similarity equations, 341 ff solution, 435ff

P Panradiometer, 71 ff Peclet number, 296, 301 Planck function, 239, 245f Plasma, 229ff characteristic properties, 231 ff blackbody radiation, 238ff bremsstrahlung, 237f line radiation, 234f recombination radiation, 235 ff composition, 256ff heat transfer, 270ff in absence of an externally applied electric or magnetic field, 270ff basic transport equations, 272ff laminar boundary layer equations, 276ff results of reentry studies, 278ff in presence of a magnetic field, 304ff electrically insulating surface, 307ff electrode heat transfer, 305 ff

SUBJECT INDEX

' Reduced gravity-cont. Plasma-cont. --c condensation, forced flow, 209ff, 220f heat transfer-cont. flow behavior, 210f in presence of an electric current, 282ff noncondensable gas, 214 electrically conducting surface, 289ff pressure drop, 21 1ff experimental studies 292ff 'vapor-liquid interface, 213 f cathode axis parallel to a plane 2 condensation without forced flow, 206ff, anode, 295ff 220 cylinder geometry with annular ' laminar film condensation on a vertical anode, 302ff surface, 207f free-burning arcs with plane 4 laminar-to-turbulent transition, 208f anodes, 293ff '1, transient time to establish laminar electrically insulating surface, 282ff condensate film, 209 constricted arc, 283ff / experimental production of, 146 ff influence of on heat transfer, 231 ff 1 ) airplane trajectory, 150 plasma-wall boundaries, 252 ff ' drop tower, 146ff sheaths, 252ff : magnetic forces, 150f steady, dense plasmas, 242ff rockets, 150 thermodynamic equilibrium 244ff ' satellites, 150 local thermal equilibrium, 246ff .' forced convection boiling, 200 f excitation equilibrium, 249 .(,designs involving substitute body ionization equilibrium, 250ff forced, 205f kinetic equilibrium, 248f perfect thermodynamic equilibrium ' 5 two-phase heat transfer, 201 ff I free convection, elimination, 145 245 f .I free convection in, 151 ff, 218f temperature increase of plasma, 262ff fluid flow, 152ff reduction of heat conductivity by a magnetic field, 267ff I i boundary layer theory, 152 f reduction of wall heat fluxes by tran-'boundary layer transition, 153 spiration cooling, 269f Rayleigh numbers encountered in thermodynamic properties, 256ff low-gravity, 153f transport properties, 260ff u threshold of convective motion, 152 Pohlhausen transformation, 22 7 heat transfer, 154ff Pool boiling in reduced gravity, see Reduced :transient development times of boundgravity ary layers, 156ff gravity as an independent parameter, Prandtl number, 3ff, 1 5 1 , 271, 320 equilibrium, 275f 144f .> pool boiling, lSSff, 219f frozen, 275 f ibubble dynamics in saturated nucleate boiling, 174ff R diameter at departure, 181 f Radiant heat load, 83 experimental results, 181f Radiation effects, 67ff theoretical relations, 181 Rayleigh number, 3ff, 151 f forces acting during growth, 182ff modified, 151 ff growth rates, 177ff Recombination radiation in a plasma, 235ff 5 experimental results, 178ff Reduced gravity, 144ff , J theoretical relations, 178 i combustion, 215ff, 221 'i higher heat flux effects, 191ff . candle flame, 215 f nucleation cycle and coalescence, fuel droplets, 21 5 ff 175ff solid fuels, 217f " rise of detached bubbles, 190f

'

J

'

457

,

.

SUBJECT INDEX

45 8 1

Reduced gravity-cont. -2 pool boiling-coat. I bubble dynamics in subcooled boiling, ’ 194ff bubble growth, 195f L, forces acting on bubbles, 196ff .-: critical heat flux, 166ff experimental behavior, 167ff 14 theory, 167 ’; film, boiling, 173f L experimental results, 173 f :i theoretical relations, 173 minimum heat flux between transition boiling and film boiling, 171 f nucleate pool boiling, 158ff experimental results, 159 ff theory, 158f transition region for pool boiling, 171 7 space applications, 144 Reentry studies, 278ff Reynolds analogy, 346ff Reynolds number, 54ff, 208, 345 Rockets, their use in producing reduced gravity, 150 Runge-Kutta method, 331 f I

S Saha-Eggert equation, 238 Satellites, their use in producing reduced gravity, 150 Schmidt number, 16, 320 Self-emitted radiation, 234 Sheaths, plasma, 252ff Similar solutions, 356ff, 389ff, 401 ff Similar solutions for a power-law viscosity relation, 370ff Similar solution for a Sutherland viscositytemperature relation, 426ff Similarity for laminar boundary layers, 319ff

Similarity for laminar boundary layers -cont. conditions for, 322ff equations for solution, 324ff large values of the pressure-gradient parameter, 341 ff inner limit equations, 344f outer limit equations, 341 ff local similarity, 333ff asymptotic expansion of equations, 335ff determination of f, by successive approximations, 337ff numerical integration procedure, 33 1 ff numerical results, 330f range of solutions and parameters, 330 special classes of reduced equations, 327ff computation of boundary layer properties, 328 no mass transfer, 327f Prandtl number equal to unity, 328 pressure-gradient parameter equal to zero, 328 two-dimensional flow, 327 viscosity proportional to temperature, 328 Skin, properties of, 116ff Solar radiation measurements, 77ff Spitzer’s formula, 264 Stream function, 320 Sutherland constant, 326 Sutherland viscosity law, 326

T Thermal thickness, 329 Thermal variability factors (TVF), 107f Thermistor radiometer, 69 f Thermodynamic equilibrium in a plasma, see Plasma Transpiration cooling, 269 f