Advances in Heat Transfer, Volume 21

ADVANCES IN HEAT TRANSFER Volume 21

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Advances in

HEAT TRANSFER Edited by James €? Hartnett

Thomas E Irvine, Jr.

Energy Resources Center University of Illinois Chicago. Illinois

Department of Mechanical Engineering State University of New York at Stony Brook Stony Brook, New York

Associate Editor Young I. Cho Department of Mechanical Engineering Drexel University Philadelphia, Pennsylvania

Volume 21

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers San Diego New York Boston London

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PRINTED IN THE UNIIED STATES OF AMERICA 9 1 9 2 9 3 9 4

9 8 7 6 5 4 3 2 1

CONTENTS

Heat Transfer under Supercritical Pressures A . F. POLYAKOV

I. I1. I11. IV. V. VI .

Introduction . . . . . . . . . . . . . . . . . . . . . General Description of the Problem . . . . . . . . . . . Heat Transfer at Forced Flow in Round Pipes . . . . . . . Free Convection. . . . . . . . . . . . . . . . . . . . Special Problems . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . Nomenclature. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

1 2 5 37 44 48 49 50

Advances in Condensation Heat Transfer

ICHIRO TANASAWA I. I1. I11. IV. V. VI . VII .

Introductory Remarks . . . . . . . . . . . . . . . . . Fundamentals of Condensation . . . . . . . . . . . . . TransportProcess at vapor-LiquidInterfaces. . . . . . . Dropwise Condensation . . . . . . . . . . . . . . . . Film Condensation . . . . . . . . . . . . . . . . . . Enhancement ofCondensationHeat Transfer . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . Nomenclature. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

55 57 65

80

104 115 131 132 133

Hydrodynamicsand Heat/Mass Transfer near Rotating Surfaces

DAVID MOALEM MARON AND SHIMON COHEN I . Introduction . . . . . . . . . . . . . . . . . . . . . I1. State of the Art . . . . . . . . . . . . . . . . . . . . I11. Closure . . . . . . . . . . . . . . . . . . . . . . . V

141 143 176

vi

CONTENTS

Nomenclature. References . .

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

177 178

Modeling the Dynamics of lbrbulent 'hansport Processes

B . A . KOLOVANDIN I . Introduction . . . . . . . . . . . . . . . . . . . . . I1. Governing Equations . . . . . . . . . . . . . . . 111. Nearly Homogeneous Two-Point Correlations for Closely Spaced Points . . . . . . . . . . . . . . . . . . . . . IV. The Second-Order Model of Nearly Homogeneous Turbulence . . . . . . . . . . . . . . . . . . . . . . V. The Problems of the Evolution of Nearly Homogeneous Turbulence . . . . . . . . . . . . . . . . . . . . . . VI . Conclusion . . . . . . . . . . . . . . . . . . . . . . VII . Appendix . . . . . . . . . . . . . . . . . . . . . . Nomenclature. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

. .

185 188 191 193 202 231 232 234 236

Radiative Entropy Production-Heat Lost to Entropy

VEDATS. ARPACI I. I1. 111. IV. V. VI . VII . VIII. IX . X. XI . XI1.

Introduction . . . . . . . . . . . . . . . . . . . . . Thermodynamic Foundations . . . . . . . . . . . . . . Radiative Stress . . . . . . . . . . . . . . . . . . . . Local Entropy Production . . . . . . . . . . . . . . . Stagnant Gas . . . . . . . . . . . . . . . . . . . . . Qualitative Radiation . . . . . . . . . . . . . . . . . Heat Transfer . . . . . . . . . . . . . . . . . . . . . Flame Quenching . . . . . . . . . . . . . . . . . . . Microscales of Heat Transfer . . . . . . . . . . . . . . Radiation-Affected Turbulence . . . . . . . . . . . . . Lost Electromagnetic Work . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . Nomenclature. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

239 240 245 247 249 254 259 261 263 268 270 273 273 275

CONTENTS

vii

Heat, Mass,and Momentum "kansfer in a Multifluid Bubbling Pool

GEORGE ALANSON GREENE I. Introduction . . . . . . . . . . . . . . . . . . . . . 11. Enhancement to Heat Transfer between Stratified Immiscible Liquids by GasBubblingacross theInterface . . . . . . . 111. Onset of Entrainment between Immiscible Liquid Layers due to Rising Gas Bubbles . . . . . . . . . . . . . . . IV. Bubble-Induced Entrainment between Initially Stratified Liquid Layers. . . . . . . . . . . . . . . . . . . . . V. Heat Transfer between Stratified Liquids with Entrainment across the Interface . . . . . . . . . . . . . . . . . . VI. Heat Transfer from a Liquid Pool in Bubbly Flow to a Vertical Wall . . . . . . . . . . . . . . . . . . . . . VII. Heat Transfer from a Horizontal Bubbling Surface to an Overlying Pool . . . . . . . . . . . . . . . . . . . . VIII. Drag and Instability of Liquid Droplets Settling in a Continuous Fluid . . . . . . . . . . . . . . . . . . . IX. Concluding Remarks . . . . . . . . . . . . . . . . . Nomenclature. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

333 342 343 344

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

347

Index.. .

277 279 286 299 31 1 318 327

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ADVANCES IN HEAT TRANSFER, VOLUME 21

Heat Transfer under Supercritical Pressures A. F. POLYAKOV Institute of High Temperatures U.S.S.R. Academy of Sciences Moscow. U.S.S.R.

1. Introduction

The 1970 contribution of Petukhov [ 11 to Advances in Heat Transfer was devoted in part to heat transfer at supercritical pressures (SCP) of fluids. This problem is the most general and complicated case of heat transfer at varying physical properties of single-phase fluids. The term “under supercritical pressures” means the state of the medium with parameters near the thermodynamic critical point, but under pressures above the critical pressure, when the medium can be considered to be single phase. Since Petukhov’s paper [11 was published, investigations of heat transfer have advanced profoundly and versatilely, resulting in new data and generalizations of both theoretical and practical interest. The results obtained before 1978 are reflected in the review paper of Hall and Jackson [2], showing an important role of buoyancy forces for development of heat transfer peculiarities in forced turbulent pipe flow under SCP and giving empirical estimations for boundaries of the beginning of the buoyancy effect. As is noted by those authors, reliable data on hydraulic resistance, velocity, and temperature fields for SCP fluid were lacking, physical models were not especially elaborated, and the scope of numerical modeling results was insufficient at that time. This article presents an analysis both of the problems with the data in the mentioned publications and of the results of investigations mainly conducted during the past 10-12 years.

1 English translation copyright 01991 by Academic Press, Inc. All rights of reproduction in any form reserved.

A. F. POLYAKOV

2

11. General Description of the Problem

A. THERMOPHYSICAL PROPERTIES OF FLUID Specific features and peculiarities of heat transfer under SCP are determined by the character of the changes in the single-phase medium’s physical properties at subcritical state parameters. At present, many data on heat transfer to water, carbon dioxide, and helium are obtained; in addition, there are some results for nitrogen, hydrogen, freons, and hydrocarbonaceous combinations. Reliable data on thermophysical properties of fluid are needed to make analyses of and generalizations about experimental data, as well as to predict numerical solutions. However, such data are incomplete for a number of specific fluids, and work to verify data on thermophysical properties near the critical point continues for water, carbon dioxide, and helium. Naturally, the latter results will require further amendments of generalizations on heat transfer. By way of example, Fig. 1 presents some data on the change of physical properties versus temperature for water (P, = 22.12 MPa, T, = 647.3 K) and for carbon dioxide (P, = 7.38 MPa, T, = 304.2 K) near the pseudocritical temperature T,,, i.e., the temperature corresponding to the maximum value C, at the given pressure. Figure 1 (from Ref. [3], with some modifications) illustrates the complicated character of a medium’s physical property variations in the region under consideration, as well as their amendment. In com-

T (K)

FIG.1. Physical properties of (a) water (-,

c91; ---,csi).

T (K) [4,8]; ---, [S]) and (b) carbon dioxide (-,

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

3

paring later data on viscosity [4] with previous data [ S ] , it can be seen that the difference can reach 20% while the character of the variation (T) remains the same. As is indicated by Altunin [6] and Neumann and Hahne [7], the presence of a local maximum in the distribution k(T)for a single-phase region, near the critical point, is well established. Previously, such particularity of thermal conductivity dependence on temperature has not been indicated, as is clear from comparison of water data presented in Fig. l a [5,8]. Similar data are shown in Fig. l b for carbon dioxide. The distribution k ( T ) with local maxima have been calculated by Popov and Yankov [3] using curves k ( p ) at constant temperatures near the critical point [9]. The above-mentioned amendments of thermophysical properties refer to a narrow range of parameters near the critical point. They are not presented in all reference books on thermophysical properties used in analytical works or generalizations of data on heat exchange. The presentation of heat transfer data is not significantly changed by such amendments in the case of turbulent flow, whereas in the case of laminar flow analytical data on heat transfer, taking into account the different character of the distribution k(T),can vary substantially. This will be illustrated by examples in Sections III,E, IV,A, and IV,B.

B. GENERAL PREMISES AND APPROACHES TO PROBLEM SOLUTION Heat transfer in a single-phase flow near a wall, containing the very large changes in physical fluid properties versus temperature, is the subject of the following discussion. The principal focus is on medium motions that are stationary in the mean, i.e., quasistationary in turbulent fluctuation scales. For individual cases, experimental data on nonstationary and three-dimensional flows will be presented, whereas the basic material relates to momentum and heat transport, which is two-dimensional in the mean. The latter we shall interpret mathematically by means of the following system of equations written in the cylindrical coordinates as a boundary-layer approximation:

Equations (1)-(3) are written without taking into account physical property fluctuations, i.e., their variations are supposed to be in compliance with

A. F. POLYAKOV

4

changes of mean temperature (enthalpy), and their instantaneous variations caused by the fluctuating temperature are neglected. The principal difficulty in solving Eqs. (1) and (2) consists in a search for the most reliable approximations for correlations characterizing turbulent heat (n) and momentum (m) transport. In the study of heat transfer under SCP, the role of empirical data is decisive, and use of simpler and approved models as compared with developed models of high order for constant property fluid is justified in obtaining a numerical solution to this problem. The particular relations proposed for calculations in some cases of turbulent heat transfer under SCP, along with mathematical boundary conditions will be discussed later. One of the most important factors affecting the development of heat transfer particularities concerns the very large changes in the density (illustrated in Fig. lb). In the first place, an occurrence of regimes with a sharp local wall temperature maximum (“peak”) may be considered as such a particularity; these regimes were conventionally referred to by Petukhov [11 as “degraded heat transfer regimes,” contrary to “normal regimes” without the “peak” in the wall temperature distribution. Others [2,10] relate the local deterioration of turbulent heat transfer to a free convection effect, when wall temperature peaks are experimentally obtained in vertical, heated pipe upflows, but they are absent in downflows under the same conditions, as well as to thermally induced acceleration, which arises from the sharp decrease in fluid density along the pipe, when wall temperature peaks are discovered in experiments independently of the flow direction. However, the mechanism of buoyancy and acceleration effects, as well as quantitative correlations between the development of these effects, and heat transfer changes were not explained. Polyakov [l 13 proposed, apparently for the first time, to take into account the influence of buoyancy and acceleration effects for the analysis of heat transfer particularities under SCP, connecting them with density fluctuations by way of a turbulent energy balance equation in the following form: pu’d-au aY

+ p’u’

-(

+g

I

3+

+ u-

E

=0

(4)

I1

Two terms of Eq. (4) take into account the density fluctuations. Term I is connected with the acceleration due to gravity (the positive sign refers to the upward flow in heated pipes, the negative sign refers to the downward flow); I1 is connected with the individual particle acceleration in averaged motion and is written in supposition of the presence of mean fluctuating mass flux only along the pipe. The first and the last terms of Eq. (4) are turbulence production due to mean velocity gradients and to the dissipation of turbulence, respectively.

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

5

Such an attempt provided a clear and plausible explanation of the heat transfer peculiarities mentioned above and it was used as a basis for the further development of analyses, generalizations, and numerical modeling. Some more recent approaches have been made to use the “ K - E model” (the equations of balance for turbulence energy and for turbulence dissipation) for calculations of turbulent flow and heat transfer to SCP fluid. In particular, this model is used by Renz and Bellinghausen [121. 111. Heat Transfer at Forced Flow in Round Pipes A. LAMINARFLOW

A great importance is given to the study of heat transfer in turbulent flow, as it is necessary for practical purposes. However, for general analysis, it is interesting to consider the influence of variable physical properties, when only molecular momentum and heat transport determine heat transfer from the wall. In Shenoy et al. [131,the numerical solution of Eqs. (1)-(3) for laminar flow (&’ = tlV) = 0) was obtained without taking into account buoyancy forces with the following boundary conditions on velocity and temperature: u=v=O, T=Tw forr=r, u = const.,

u = 0,

for x = 0,

T=

T,, = const.

0 I r l ro

(5)

(aT/dr)= 0 for x > 0, I =0 The results of calculations for the hydrodynamic entry region, i.e., without a preliminary developed velocity profile, on water heating conditions (Tw2 Tpc> T,,) demonstrate a great increase in the heat transfer coefficient (Fig. 2a). Figure 2b shows dimensionless results of the prediction for carbon dioxide under other conditions, viz. for a thermal entry region (the parabolic velocity profile is set up at the start of heating) under boundary conditions of a second-order type (qw = const.). The other boundary conditions are in accord with those listed in Eq. (5). Data presented in Fig. 2b were obtained at the Institute for High Temperatures of the Academy of Sciences in Moscow by Medvetskaya and were gently proposed by her for this review paper. This numerical solution was carried out for upward flow in vertical, heated pipes with taking into account buoyancy forces, i.e., the term pg in Eq. (2) presented in the dimensionless form as (Ga/Re2), (p/pin)= gdp;Jm2. By neglecting buoyancy effects (Ga = 0), it was found that the heat transfer rate decreases with the growth of values qw and under ti, < tpc. The comparison of this result with data shown in Fig. 2a denotes a different character of SCP heat transfer change for different hyrodynamic and heat boundary conditions

(auldr) = 0,

u = 0,

6

A. F. POLYAKOV

FIG.2. Distribution of heat transfer along the pipe at Re,, properties.

=

lo3; (l), constant physical

and different temperature ranges, even in the simplest case of viscous flow. The case of mixed laminar convection is more complicate, with Archimedes force actions being interconnected with varying physical property actions. At low heat flux and under ti, .c tpc,these effects lead to the increase of heat transfer by 30-40% as compared with the case of constant physical properties. Whereas heat flux increases, buoyancy leads to the growth of heat transfer at constant properties (see, e.g., [14]); however, the effect of a significant decrease of fluid density and thermal conductivity in the wall region as compared with the flow core in this case dominates the first, manifesting itself in a reduction of the Nu number. As fluid heats at t, 2 tpc, medium physical properties vary with temperature similarly to gas properties. In this case the growth of thermal conductivity near the wall and the effect of Archimedes forces lead to increasing heat transport from the wall, the regions beyond the minimum value of Nu number on the curves Nu(X) corresponding to these phenomena. B. TURBULENT FLOWWITHOUT OF GRAVITY FIELD

SUBSTANTIAL INFLUENCE

As is clear from the previous discussion, even in laminar flow a great change of all physical properties at subcritical fluid parameters results in a specific character of heat transfer and in a realization of a large variety of cases as compared with heat transfer for constant property fluids. However, to solve this problem only mathematical obstacles are to be overcome, viz. it is necessary to find a solution of three-dimensional nonlinear equations with sharply varying coefficients in the complete formulation.

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

7

In the case of turbulent flow, major difficulties are connected with the determination of objective physical regularities of turbulent momentum and heat transport. The regimes with degraded local heat transfer cause a great anxiety in practice. As is indicated in Section I,B, at present it is known that in addition to physical property variability causing heat transfer decrease in some cases, buoyancy and thermal acceleration determine a high deterioration of heat transfer. All three effects are to be taken into account not only in Eqs. (1)-(3) for mean values, but also in the mathematical description of turbulent momentum and heat transport. The manifestation of Archimedes forces and thermal acceleration is coupled with a density change that intensifies more and more with the growth of the thermal load, and it is naturally accompanied by the increase of other physical property variabilities. While presenting the data, however, we shall try to separate some conditions that allow the consecutive consideration of all three cases noted previously. Let begin the analysis of theoretical and experimental data by means of presentation of results for a rather low heat load corresponding to the small difference of temperatures (t, - tb), when the effect of thermal acceleration and buoyancy forces can be neglected. Analysis of the development of the substantial influence of the gravity field is presented later in a separate section. The investigation of heat transfer in turbulent flow based on the system of Eqs. (1)-(3) demands the application of some dependences for correlations and The traditional approximations by the Boussinesq relation

m.

-

u / v I = -vT-

au

aY

and by a similar relation -

i’o’ = - vT Pr,-

ai aY

are widely used for prediction of SCP heat transfer. Petukhov et al. [IS] demonstrated that the use of relations for turbulent transport coefficients proposed for the forced flow with constant properties, without especially taking into account variable physical properties, Archimedes forces, and thermal acceleration, does not allow for a correct description of the behavior of heat transfer to a single-phase fluid with parameters near the critical point. Theoretical investigations are directed essentially to obtain some relations for turbulent transport coefficient fitted experimental data. The main volume of experimental material is formed from heat transfer data. Among the few experimental data on hydraulic resistance, velocity, and temperature fields, it is necessary to distinguish the complex results recently obtained for carbon

A. F. POLYAKOV

8

dioxide [16-181 as the most complete and reliable ones. Such experimental data provide many reasons to perfect the equivalent mathematical models. A computational model accounting for all the indicated effects was suggested by Petukhov and Medvetskaya [19,20]. This model makes use of the simplified equation of turbulent kinetic energy balance similar to Eq. (4) to find a turbulent momentum transport coefficient; to obtain the coefficient Pr, v,, the simplified enthalpy balance equation is applied:

(7)

The second term in the second equation shows mean heat transport of fluctuating motion by means of convection in the mean flow. By taking into account Eq. (6) and the following basic approximations borrowed from the general theory of turbulence (see, e.g., [Zl]),

the relations given by Eq. (7) were transformed in the expressions for coefficients of momentum and heat turbulent transport as follows:

5 V =

{p); (--)

c2lZ

-

a,-C p, Pr, 1 ar ai[ +g

;;

+ (u- + u-

;)]}1/2

(9)

In Eqs. (9) and (10) the terms (vT/v)*and Pr,, are determined taking into account a variability of physical properties, but in the absence of density fluctuations ( p = 0) and therefore the influence of buoyancy forces and acceleration on turbulent transport. Here we shall not dwell in detail upon expressions for additional functions and constant values presented by Petukhov and Medvetskaya [20]. It is evident that the mathematical model under consideration possesses a lot of gross approximations. In particular, every approximation written in Eq. (8) may be criticized, and under the conditions of Archimedes force influence, the coefficient a, in the last approximation is dependent on the value

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

9

of the buoyancy parameter, as has been shown in experiments using air [141. Moreover, an objection of principle can be brought up relating to the possibility of using the gradient presentation of turbulent transport [Eq. (6)] under certain conditions; however, this concerns all the theoretical investigations on turbulent SCP heat transfer. Together with the above-stated discussion, it is necessary to note that the described model has a series of specific and advantageous features as compared with the others: by introducing into the equation of turbulence kinetic energy balance the terms associated with gravity acceleration and with mean individual particle acceleration, the influence of density fluctuations on turbulent transfer is taken into account; by use of the equation for temperature fluctuation balance the influence of the same effects on turbulent heat transport (turbulent Prandtl number) is also taken into account. Adopted approximations and constant values were verified to be acceptable by correspondence of preliminary calculations of hydraulic resistance and local heat transfer to theexperimental data obtained for turbulent flow of water and air in pipes under significant influence of a gravity field. Some results of calculations according to a described mathematical model [20] for the cases without buoyancy effects will be presented in the following discussion. For such classification it is necessary to know the limits of realization of the considered regimes. Polyakov and Petukhov [11,14], on the basis of Eq. (4), obtained the following estimations of the boundaries, below which it is possible to neglect heat transfer change in vertical pipes by means of variations of turbulent momentum transport induced by effects of buoyancy and thermal acceleration:

IfGr,

k J ( c 4 x 10-4Re2.8E= Bth

where the acceleration parameter is

md

Re = fib

+

where 5 is evaluated at (Tw &)/2.

(11)

A. F. POLYAKOV

10

The positive sign in front of the Grashof number Gr, in Eq. (11) is for upward (downward)flow in heated (cooled)vertical pipes, and the negative sign is for the opposite case; the positive sign in front of the parameter J is for the case of fluid heating (T, > Tb), and the negative sign is for the cooling (Tw < Tb). The analysis [22] of experimental data on heat transfer to water and carbon dioxide in a heated pipe upflow resulted in the following estimations of boundaries of the absence of Archimedes forces effects: (GrJRe’) < 0.01

(14a)

or

c1 - ( p ~ / p b ) l ( ~ ~ p
(14b)

where Grp

= g(pb - pw)d3/pbvt

Transforming Eq. (11) to a form similar to Eq. (14) at J = 0, we can obtain (Grp/Re2)< 4 x 10-4(Reo.*/Nu)(c/pb)h (15)

0.01[1 + (pw/pb)] Thus Eq. (15), corresponding to Eq. (1l), is similar to the Eq. (14) at different degrees of approximation. Protopopov [22] does not prefer Eqs. (14) over one another, and for his determination he uses both the data for vertical pipes and for horizontal ones. Jackson and Hall [23], on the basis of a series of gross physical suppositions, in particular, relationships between the thickness of the buoyancy, thermal, and hydrodynamic layers and the constancy and magnitude of the dimensionless thickness of specific hydrodynamic layers, suggested the following relation: x 0.02(p”lpb)

%

(GrJRe’.’) <

(16)

where Grp

=

c1 - (p/pb)lgd3/vz

Thus, comparing Eqs. (1 l), (14), and (16), we shall use Eq. (11) for an estimation of the boundaries of the gravity field influence on heat transfer in vertical pipes. From Eq. (11) it is clear that the combined action of buoyancy and thermal acceleration at upward flow in vertical, heated pipes is the greatest (the signs in front of Gr, and J are the same) and leads to a decrease

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

11

of turbulent transport [see Eq. (9)], and thus to a decrease of heat transfer. Under high heat loads, when the temperature over the pipe arises considerably and ti, < t,, < tout, the influence of these effects can manifest itself at most in some logitudinal region of the pipe, resulting in local heat transfer deterioration. When the condition Gr, + J < &, is satisfied, the degraded local heat transfer is not observed, and such regimes are “normal regimes” according to conventional classification [ 11. In Fig. 3 the results of calculations [20] are compared with experimental data [161 on the friction factor ( f ) and the complete hydraulic resistance factor (fd) as well as for distributions of wall temperature and bulk fluid temperature over the heated pipe (4, = const.) for upward carbon dioxide flow. In this regime t , < t,, < t,; however, the heads of temperature (to,, - ti,) and (t, - t b ) are relatively low, and despite the fact that (JIB,,) = 10-12 at (Grq/B,,) = 0.9-0.65, the local maximum t , is absent, and thus the local heat transfer decrease is also absent. The last is confirmed by the fact that in practice the value (J/&,)i, = 10 and (J/&)out = 12 is constant. Its magnitude reveals a possible noticeable (and uniform over the pipe) decrease of turbulent transport, friction resistance, and heat transfer due to thermal acceleration, which is confirmed by the data on f/foand the data on velocity distribution distortion shown in Fig. 3b. The absence of an Archimedes force effect [Grq/Blb)< 1) is confirmed by the comparison of experimental data on wall temperature values at upward and downward flows. According to classification [l], this regime is also referred to as a “normal regime” by the form of the t,(x) distribution. It is to be noted that in the considered case the Reynolds number changed over the range (3.4-5.6) x lo5. With the increasing heat load, the character of heat transfer distribution changes over the pipe, as is illustrated in Fig. 3c. The relative parameter of thermal acceleration acquires the values (J/&)in = 280 and (J/Blh)oul= 78, which are considerably greater, changing largely along the pipe. The substantial decrease (JIB,,) toward the outlet leads to the growth of friction factor f,to the fall in t,, and thus to the increase of heat transfer. Thus the distribution t,(x) with a local maximum is formed. Such regime in Ref. [l] is classified as a “degraded heat transfer regime,” i.e., a regime with a local “peak” of wall temperature. Data on velocity profile change presented in Fig. 3d confirm an important role of thermal acceleration in that regime. In this experiment, Reynolds numbers were in a range of (0.41-1.36) x lo6 and the relative parameter of buoyancy had the values (Grq/Bth)in = 2.1 and = 0.3, thereby confirming the impossibility of appreciable in(Grq/Blh)oul fluence of the gravity field on heat transfer. Cornelissen and Hoogendoorn [25] present a specific development in numerical investigations of turbulent flow and heat transfer for SCP fluid. Their work carried out a comparative calculation of heat transfer to helium

A. F. POLYAKOV

12

u I;40

0.5

0

1

p -7.7 M Pa d-0.8cm

____ t m

tb

tin= i9.yc

0

V I

I

50

100

I

l

a

1.4 -

I

-

I

50

100

150

05

i

P = 27MPa d=O.Bcm

200

\

t,n'190C

I

50

0 I..

I00 I

I

C

150

I 0

2.0

Y

m =4034ug/m2

I

- 1.01 I

0

I

I

100

50

I

150

x/d

FIG.3. Comparison of predicted results(lines)[20] with experimental data ( 0 ,upward flow; 0 , downward flow) [16]. (c) For experimental downward flow rn = 4100 kg/m2 sec, and qw = 1150 kW/m2. (After Ref. [24].)

according to three models of turbulence: the model of the mixing length

( I model), the model using the equation for turbulent energy (K model), and the K--Emodel, The K--Emodel was found to give the worst results. The results of calculations according to two other models fit experimental data only for low heat flux. Known empirical relations for constant physical proper-

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

13

ties with consideration of local values of physical properties are used in the I model. Approximationsfor eddy viscosity and turbulence dissipation similar to those written for Eq. (8) supplemented by functions of damping near the wall are used in the K model. Gravity field influence is not considered. The results of the work allow the following conclusions: (1) The simple models of turbulence of the I-type model satisfactorily describe heat transfer to different SCP fluids under low heat loads, as was demonstrated by Petukhov and Hall and Jackson [1,2]. (2) The use of complex models, such as the K--E model, needs a special perfection of the approximations and constants for a prediction of the flow of media with very large changes in physical properties. Special attention is merited by the complex theoretical and numerical investigations being developed by V. N. Popov in terms of a hypothesis of mixing length dependenceon ow,p, p, y, and 6 for variable properties, identical to constant properties, and of preservation of the form of this dependence: 1 = F(ow,P,

~9

Y, 61,

1

3 = Fi(y+, Y)

(17)

where y+ = ( c ~ , p ) ' / ~ y /Yp ,= y/6, and physical properties correspond to local values. As result, from Eq. (6) and the Prandtl formula vT = l 2 au/dy, taking into account Eq. (17) in Ref. [26], the formula for eddy viscosity at variable properties was obtained:

~ determined by the relations for constant where 1o/0,1~ and ( V ~ / V ) are properties, but using their local values. The term with (a/aw)is interpreted as taking into account flow acceleration. Equation (18), in conjunction with Eqs. (1)-(3), has been widely used for analyses of heat transfer and hydraulic resistance over a wide range of defining parameters and boundary conditions at turbulent motion of uniform media, with a different dependence of thermophysical properties on temperature (liquid and gas at pressures below critical; water, carbon dioxide, helium, and nitrogen at supercritical state parameters). Satisfactory results have been obtained in many cases. A series of these results will be described in the following sections. The calculations using Eqs. (1)-(3) make it possible to follow local heat transfer development immediately from the start of heating. At the same time, in practice, the empirical formulas for heat transfer in the pipes do not take into account the dependence on x. Also, two questions occur: (1) Under which conditions is such an approach acceptable?(2) What is the degree of the effect of the flow's history on the heat transfer development along the pipe? In Popov et al. [27], the effect of the flow's history was investigated using Eqs. (1)-(3) and (18). The graph calculated for single-phase helium near its

A. F. POLYAKOV

14

'0"

, roo

50

t5o v'd

FIG.4. Influence of upstream flow history on the heat transfer of helium at P = 0.29 MPa, d = 0.109 cm, rn = 178 kg/m2 sec, and qw = 8.75 kW/m2.

critical point (P, = 2.3 MPa, T, = 5.2 K) in Fig. 4 [27] shows the maximum possible difference between heat transfer for different fluid enthalpies at the start of heating. As can be seen from the figure, the greatest difference between the heat transfer coefficients h at small x/d was 50%. When x / d > 50, the value of h varies by _+ IS%, this variation decreasing with increasing x/d. Thus the heat transfer in a turbulent flow can be considered fully developed for x / d > 50, i.e., it does not depend on the thermal conditions at the start of heating or on the history of the flow. Together with the development of theoretical and numerical investigations, the improvement of empirical formulas on heat transfer continues. The 18 empirical formulas presented in Ref. [23] are variations of power functions N~

~ ~ 0f 0..1 8 ~ ~ 0f .0 .6 2 ~

(19)

n

or, in alternate form, NU

-

Nuo&

where F, and F,,, are different combinations of simplexes of physical properties. This work [28] suggests another approach to the construction of an interpolation formula directed to describing SCP heat transfer far from the start of heating (x/d > 50). This formula is based on the relation for constant physical properties [29], which may be presented as follows: St = ( f / 8 ) / [ k ,

I

+ k2(f/8)'/2(PrZi3- l)] I1

(20)

111

The denominator of this expression, known as the Reynolds analogy factor, may be interpreted as a sum of thermal resistances of the near-wall layer (term

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

15

11) and turbulent core (terms I and 111). The influence of variable physical properties, as well as the buoyancy effect and the thermal acceleration effect, is taken into consideration by way of corrections to corresponding terms of the Reynolds analogy factor. In order to predict SCP heat transfer, the Stanton number is determined by a difference of enthalpies, i.e., St = q,/rn(i, - ib), and the Prandtl number by some averaged effective value. Equation (20) does not predict heat transfer in the entry region and may be used at ( x / d ) > 50. Equation (20) has the friction factor, which is to be determined taking into account the variability of fluid physical properties. Experimental data [171 on the friction factor for carbon dioxide were obtained and analyzed together with different analytical dependences. This analysis has shown that in the region of states near the region of the pseudophase transition under the conditions of an absence of buoyancy effect, the experimental data were described in the best manner by the following formula:

f/fO

= (PW/P~)’‘~

(21a)

Equation (21a) was established [30] as a result of generalization of calculated data. It is evident that the variability of molecular viscosity, taken into account in a number of empirical relations for fluids, together with density variations, affects the friction factor. As a result of a generalization of the great volume of calculated data for different media in a wide range of defining parameters, the authors of Ref. [27] have recently obtained an interpolation equation, which is written here with their amiable consent. The analysis [17] of experimental data of a number of authors on heat transfer to water, carbon dioxide, and helium for “normal regimes” resulted in the suggestion of a relationship similar to Eq. (20): St = (f/8)/[1

+ 900/Re + 12.7(f/8)’/2(E2/3- l)]

(22)

where Pr=-- iw - ib

pb

t w - tb kb

and f is calculated from Eq. (21). All indicated data, as well as the relations obtained on the basis of such data, strictly speaking, are related to constant (or weakly varying) wall heat flux under the conditions of heat supply to fluid. Most of the experimental data concerns heat transfer to water and carbon dioxide. Recent interest in the improvement of new engineering techniques has encouraged acquisition and development of additional data on heat transfer to other fluids.

A. F. POLYAKOV

16

0

40

80

120

160

(80

yd FIG.5. Relative change in the heat transfer coefficient along a pipe for helium: P

qn= 4.25 K, and G = 0.25 kg/sec.

= 0.25 MPa,

In Bogachev et al. [31], special attention is given to the conditions of heat transfer growth during a turbulent motion of helium, when a free convection effect can be neglected. The experiment was made with a vertical pipe of diameter 0.18 cm at qw = const. Figure 5 illustrates typical curves of relative heat transfer variation over the pipe. With the initial growth of q,, the ratio Nu/Nu, increases over the entire length of the pipe. The number Nu, was calculated using Eq. (22) for constant physical properties, the values of Re = Pr being evaluated at the bulk temperature of the fluid in a given and cross-section of the pipe. Beginning from some heat load in the distribution Nu/Nu,, a maximum appears at the coordinate after the pipe section where T, = Tpc(coordinates corresponding to this equality are illustrated by arrows directed downward). With the increase of q,, the maximum is displaced upstream in the flow; however, its value remains practically constant. In a regime with the highest heat load, a minimum of Nu/Nuo is observed for the coordinate corresponding to the condition Tb x Tpc.The decrease of heat transfer, especially at the values Nu/Nuo 1, seems to be connected with the above-mentioned effects. In the region Nu/Nu, > 1, the relative heat transfer distribution approximately follows the course of Change c p / c p b [ c p = (i, - ib)/(Tw- T b)]. Arrows upward directed indicate the coordinates of maximum values of the simplex c p / c p b for some regimes. The values Nu/Nu, > 1 are described with an accuracy of about &20%by following the interpolation equation:

-=

(23) Nu/Nu0 = ( c p / c p b ) 0 ' 3 5 Equation (23) is analogous to well-known power expressions for the description of heat transfer to water and carbon dioxide [23]. The character of the heat transfer change shown in Fig. 5 in terms of quality is similar to the behavior of heat transfer to water and carbon dioxide.

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

17

Additional problems of SCP heat transfer may occur during the use of hydrocarbonaceous fluids, their specific feature being very strong viscosity dependence on temperature, changing by two to three orders of magnitude over the temperature range of interest. However, such investigations are only at the initial stage of development [32].

C. TURBULENT MIXEDCONVECTION In this section we shall turn to a consideration of the features of heat transfer under the essential influence of a gravity field. Under intensive thermal conditions at nearcritical parameters of a single-phase medium, the buoyancy effect manifests itself in one way or another for the majority of parameter combinations typically realized in practice. Naturally, in this case the character of the flow and heat transfer will depend on the orientation of flow direction with respect to the direction of gravity acceleration. At first we shall analyze the results of recent investigations for vertical pipes, then for horizontal pipes. The second case is much more difficult for mathematical modeling and generalization and is much less studied. To some extent it may be connected also with the fact that constructions with vertical channels are used more often, generally. All presented data are related to the case of fluid heating at qw = const. In accordance with the fact that buoyancy effects (Archimedes forces) occurring with the differences of densities in nonisothermal fluids are considered in this discussion, we shall occasionally use the expressions “thermogravitational forces” and “thermogravity.” 1. Vertical Pipes Analysis of experimental data [33] showed that at Gr,/Re2 > 0.6

(24)

heat transfer is higher than that at constant physical properties, or (Nu/Nuo) > 1, and with the growth of heat load, or, more exactly, the value of the parameter Gr,/Re2, heat transfer increases at x/d > 50. As is illustrated in Fig. 6, generalized results [141 for conventional fluids (air, water) at low nonisothermality obtained on the basis of experimental and theoretical research show the pattern of buoyancy influence on heat transfer. The condition given by Eq. (24) relates to rising branches of curves after the values St = St, in Fig. 6 for the case of heated fluid upflow. The Stanton number St, corresponds to the limit Gr + 0 (horizontal dash-dot lines). As is seen from Eqs. (11), (14), and (15), the parameter Gr,/ReZ determines a boundary of the beginning of thermogravity influence on heat transfer in

A. F. POLYAKOV

18

2 1 o-2

6 St 4 2 t 0-3

6

(Grq/Pr Re4)"2 FIG.6. Stanton number versus buoyancy parameter for upward (solid line) or downward (hatched line) flow in heated vertical pipes at x/d > 50.

forced turbulent flow. However, there is no reason to believe all processes of mixed forced and free convection are determined by the only combination of Gr, Re, and Pr. Namely, data in Fig. 6 allow to such judgement. Petukhov and Polyakov [141 propose interpolational equations describing heat transfer over every range from the beginning of buoyancy influences on forced turbulent flow to their dominant manifestation in the limiting case for both upward and downward fluid motion. Here we shall confine ourselves to writing a relation only for the case of upward flow for heating and qw = const., -(f/8)

St

- (1 + 0.83e2)(1 + 0.042e2[E'/410g(Re/8)]-'}-'

+ 12.7(f/8)1/2{Pr2/3[1+ 0.72e3(1 + 0.28e'/2)/(1 + 0.43e4)] - (1 + 0.58e2)/(1 +0.83e2)} (25) where

e = 103Gr,/Pr Re2.75 E = Gr,/Pr Re4 Gr, = gpqwd4/v2 f = ((1

+ 0.83eZ)/[1.8210g(Re/8) + 0.076e2E'/4]}2

For the case of the absence of a buoyancy effect, i.e., at Gr + 0 (E = 0, e = 0), Eqs. (25) and (26) reduce to equations for forced flow at constant physical properties Sto, fo. At the initial stage of manifestation of a buoyancy effect,

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

19

the heat transfer equation may be presented in the following relative form (see Ref. [14]): St/St,

= Nu/Nu, =

1 T 112 Gr,/Re2." Pr1.lS

(27)

where the upper sign belongs to upward flow and the lower one, to downward flow. In the limiting case Gr + co,i.e., for a free convection regime, Eq. (25) takes the form: Stfree= (Gr,/Pr Re4)'/4[1.33 log(Re/8) + 3.3(Pr2I3- 0.7)]-'

(28)

For downward flow and Gr + co,i.e., for a regime of thermal generation of turbulence [14], the following formula is suggested: Stgen= (Gr,/Pr Re4)1'40.5[Pr''4(1

+

PI-'/^)]-'

(29)

As is seen from Eqs. (25)-(29), the thermogravity effect over the whole range from Gr + 0 to Gr + 00 cannot be taken into account by only one of the combinations, Gr/Re"Pr", with constant exponents n and m. It is more justifiable to present the appreciation of the considerable influence of both variable physical properties and buoyancy forces on heat transfer in the form of relations such as Eq. (25), rather than in the form of an amendment to the ratio of the dimensionless heat transfer coefficients Nu/Nu, (or St/St,). As follows from Eqs. (28) and (29),in the limiting case of a dominant effect of gravity field for both upflow and downflow, the relationship between Nusselt number and the Grashof number corresponds to free turbulent convection Nu Gr:I4; however, dependences on Pr are different. Also, in the case of upward flow in Eq. (28) for a free convection regime, a weak dependence on Re (geometrical size) remains, which is connected with free convection development in a constricted volume of pipe, where boundary layers developed over the perimeter interact in the central region of flow. Equations (25)-(29) were obtained for weakly varying physical properties of moving fluids. Let us first consider interpolation correlations for a calculation of SCP heat transfer. Then we shall analyze additional data on the physics of the process-experimental results on velocity and temperature fields, mathematical models, and their efficiency. As follows from Eq. (1 l), the thermogravitational effect is substantial at

-

I( k Gr,

+ J)/Re2.' fil > 4 x

Gr, 2 J

(30)

or, without taking J into consideration, (Gr,/Re2.'

E)> 4 x

lop4

(304

and there are regimes with degraded heat transfer in heated upflow, whereas there are no such regimes in downflow.

A. F. POLYAKOV

20

In the experiments carried out using pipes of relatively large diameter [33,34], the peaks of wall temperature, conditioned by thermogravity at upward flow were found to be placed near the start of heating, i.e., at x/d 50, when the condition given by Eq. (24) is satisfied. For local heat transfer calculations [14] taking into account thermogravity and thermal acceleration for the regimes (including regimes with degraded local heat transfer) and corresponding to the conditions given by Eq. (30) and

-=

(Gr,/ReZ) < 0.6

(31)

at x/d > 50, the following formula is suggested: St = q,/m(i,

- ib)

=b(l 8 +ReE +12.7($)1’2{[Prf(1

+i)]”’-O.1E2

- l})-’

(32)

where f, = 0.184/Ret2, Re, = md/p,, Pr = 2 Pr, Pr,/(Pr, + Pr2), and Re = md/pb, p,, pi,8, and Pr, are evaluated at the temperature T, (enthalpy i,), which can be considered as the first approximation to the wall temperature. qw/m; Pr, and pz are The value i , is calculated from i, = ib + 43.5 evaluated at the ttmperature T, = + (il - ib)&/kb. The parameter k is

-k

= lo3(J2/Pr:

Re4)’I3 + B(Gr,/Ei Re4)’I3

where

+

Gr, = 984 d4 kbVb

-

Pr, =

iZ ~

TZ

- ib - Tb

pb

kb

The coefficient B = 800 is for an upward flow in a heated pipe and B = - 100 is for a downward flow; /4p2/pb

at Re I2.5 x lo5

As distinct from all other formulas for calculation of SCP heat transfer, Eq. (32) does not need to be solved iteratively, i.e., at predetermined q,, none of the parameters of Eq. (32) requires a knowledge of T,, thereby complicating its evaluation but significantly simplifying calculations.

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

i b (K V

21

9 )

FIG.7. Comparison of wall temperature distributions calculated using Eq. (32) (lines) with experimental data (points) [35].

Equation (32) is obtained on the basis of a generalization of experimental data on water and carbon dioxide. It has as the limit the relation for St,, whereas in the absence of a buoyancy effect (Gr = 0) but at variable physical properties, i.e., for “normal regimes,” calculations using Eq. (32) may bring great deviations of the value Nu as compared with experimental data, especially at low differences (T, - Tb), as was shown by Petukhov et al. [17]. For these cases the use of Eq. (22) is recommended. An impressive example of the use of Eq. (32) is shown in Fig. 7. In order to generalize experimental data on heat transfer to helium upflow in a vertical, heated pipe with a diameter of 0.18 cm, the above-mentioned relations along with other relations for x/d > 30 and for the relatively small change in physical properties (0.7 I p,/pb < 1; 0.7 I Cp/Cpb I 1.8; 0.7 I k,/kb I 1.6; 0.9 I pw/pb s 1.2) are analyzed [36]. To give a representation of the heat transfer coefficient in the relative form Nu/Nu,, Bogachev et al. [36] have selected the complex Gr,/Pr Re3 as a buoyancy parameter, this selection being considered as giving the most successful generalization of experimental data and as being close to the parameter well-founded physically and included in Eqs. (25)-(32). The same parameter is used in Ref. [37] for a generalization of experimental data for helium downflow obtained with the same setup. The character of heat transfer change established [36,37] both for upward and downward flows corresponds to the behavior of curves shown in Fig. 6. The expression Gr,/PrRe3 = 4 x corresponding in

22

A. F. POLYAKOV

practice to Eq. (30) for values of the Reynolds numbers Re = lo4 that are specific to the work of Bogachev et al. [36], is recommended as the boundary of the beginning of the buoyancy influence on heat transfer. As is indicated in Ref. [36], Eq. (32) can be used to describe heat transfer in mixed convection of SCP helium in the region (Gr,/Pr Re3) c 0.25, which exceeds even the limit given by the Eq. (31). The data examined above are obtained for pipes of relatively small diameter. Naturally, when the pipe diameter increases, the free convection influence is augmented more and more. As was noted previously, in the work of Petukhov et al. and Watts and Chou [33,34] (carbon dioxide, d = 2.9 cm; water, d = 2.5 and 3.22 cm, respectively),the local heat transfer deterioration was found to be disposed near the start of the heating, whereas intensive heat transfer, with the value for upward flow being close to that for downward flow, i.e., corresponding to regimes of free convection and thermal turbulence under the condition given by Eq. (24),was observed far from the start of the heating at x/d > 50. Different empirical equations generalizing their results are suggested in both papers. Nevertheless, it is notable that for very large Gr, the dependence of Nu on Gr, being similar to that for turbulent free convection, is always observed. In particular, it is necessary to distinguish the results of generalization for the minimum Nusselt number (Numi,),i.e., for the value of the wall temperature peak as determined by Watts and Chou [34]. The shape of the curves of dependence on the buoyancy parameter for upflow is quite similar to that shown in Fig. 6. A generalization for the averaged heat transfer downstream beyond the peak T, does not reveal any appreciable “hole,” i.e., values Nu/Nu, < 1. It is evident that empirical relationships from Watts and Chou [34] require a special verification to extend them over other liquids and ranges of parameters. In particular, the relation for Numi, in the entry region of the pipe does not include the influence of x/d and upstream flow history. In the numerical analysis of mixed convection development in pipes [12,20,38-401, the use of different mathematical models of turbulence allows us to find the correlation between heat transfer, resistance, velocity, and temperature fields; to try to construct the most general method of calculations; and to give a more profound analysis of physics of the process. With the purpose of explaining the experimentally observed velocity profile deformation, which results in the M-shaped distribution at upward fluid motion, Hauptmann and Malhotra [38] used the K--E model for closing Eqs. (1)-(3). In the equations for turbulent characteristics, the buoyancy effect was not taken into consideration. As a result, M-shaped velocity profiles are obtained, but data on heat transfer are qualitatively contrary to experimental data, i.e., heat transfer was higher in upflow as compared with that in downflow. Such a pattern of velocity and heat transfer changes is typical for laminar mixed convection (see, e.g., Ref. [141).

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

23

In Renz and Bellinghausen [12], the model used previously [38] was supplemented by some terms defining the buoyancy effect in the equations for turbulent kinetic energy and dissipation. A result was obtained that is qualitatively correct: the distribution of wall temperature along the pipe in downflow is monotonic, and the values of temperature are lower as compared with upward fluid motion; in the second case, in the distribution T,, zones with a sharp increase of T, were obtained, these zones displacing toward the inlet of the pipe when the heat load grew. At the same time there was no correspondence between analytical and experimental data, especially in the region of local heat transfer deterioration. The work of Bellmore and Reid [39] merits great attention, wherein, in order to solve the problem of SCP heat transfer, density fluctuations are taken into account in the mean energy, momentum, and continuity equations. Density fluctuations are supposed to be induced only by fluctuations of enthalpy, whereas enthalpy and velocity component fluctuations are also supposed to be determined by gradients of mean values and corresponding scales, viz.: p' x (ap/di)i' = -p/?*i'

i'

=

-li(ai/dy)

u' = l,(du/dy)

u'

=

(33)

-z,(du/ay)

The expressions for turbulent shear stress and for turbulent heat transport, accounting for density fluctuations, have been written as

Using Eqs. (33) and (34) and the approximation of Boussinesq [Eq. (6)], a relation for the coefficient of turbulent viscosity, allowing for density fluctuations, has been obtained:

By means of Eq. (35), Bellmore and Reid [39] carried out calculations of heat transfer at an upward flow of hydrogen (P, = 1.29 MPa, T, = 33.0 K ) in pipes with diameters from 0.56 to 1.29 cm. Heat transfer deterioration is connected with the generation of M-shaped profiles of velocity. Calculations presented for four regimes satisfactorily describe the experimentally obtained distribution of local heat transfer along the pipe.

24

A. F. POLYAKOV

The suggested method is not universal, evidently, and its improvement requires a wider verification for different conditions; however, the idea to present momentum and heat transport in the form of Eq. (34) and to allow for density fluctuations in the mean via Eqs. (1)-(3) appears fruitful and merits further development. The most complete numerical modeling of flow and heat exchange in mixed convection in vertical pipes has been made by Popov and Valuyeva [40-431. Methods of prediction and results of calculations for water [40] and carbon dioxide [41] have been presented. Equations (1)-(3) are solved under boundary conditions [Eq. (5)] with two differences: x = 0 and uin = uo(r,Re) for a developed turbulent velocity profile; r = ro and (k/c,)(ai/i%) = qw for the boundary condition of a second-order type. On the basis of a simplified balance equation of turbulent stresses [taking into account the buoyancy effect, a number of approximations, and suppositions similar to those used to write Eqs. (17) and (18)], there have been found some relations for calculations of turbulent stress and turbulent viscosity:

vT., vT, = [2.24(/?gl4/C,~r)lai/ar(] 1/2

is the turbulent viscosity in the absence of and in the dominant effect of the gravity field. When the thermogravity effect is so low as to be neglected, Eq. (37) can be reduced to Eq. (18)written for vT = vT*. Equation (6) is used for turbulent heat transport, and the value of the turbulent Prandtl number is assumed to be one. Before considering the results of mixed turbulent convection numerical modeling [40-431, for a more complete analysis of the heat transfer character in vertical pipes it seemsjustified to present Fig. 8 [14] and experimental data on the behavior of local heat transfer in air flowing upward through a heated pipe at practically constant physical properties. As can be seen from Fig. 8, the thermogravity effect in the thermal entry region (x/d < 50) is weaker than it is far from the start of heating. Data from Fig. 8 for x/d > 50 have been used to plot generalized graphs in Fig. 6. At the initial stage of the buoyancy effect, a sharp decrease in heat transfer takes place essentially over the whole length of the pipe, and is caused by a partial relaminarization of the entire flow. Some minimum level of heat transfer for the given values of Re and Pr is established over the whole length. When the heat load continues to augment, heat transfer

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES c4

P, .r = 0 7

I -\\

25

R e = 5100

20

Nu

to

FIG.8. Experimental data for air at various values of buoyancy parameters Gr,/B,, = a and (Gr,/Pr Re4)’” x lo4 = b.

far from the start of heating begins to increase intensively, due to the beginning of the development of free turbulent convection (in Fig. 6 , the branches of the curves going upward beyond the minimum values). Near the start of heating (x/d < 30), heat transfer varies weakly even in the case when at x/d > 50 the values of Nu exceed Nu,. In Fig. 9 results of calculations [40] are presented in comparison with experimental data on the distribution t, for water flow in pipes of relatively large diameters. A remarkable feature of Fig. 9a and b is the fact that wall temperature variations for single-phase flow have the same character at 350 300

2 50

-

200.z 1.8 c.’ 3 0

Z

1.0

50 x/ d

0

50 x/ d

0

30

xld

60

30

60

0.6

L 2

X/d

FIG.9. Comparison of calculated results (solid lines) [40] with experimental data for upward water flow (points, dashed lines). Data for a and b from Ref. [44],d = 2 cm: (a) Re = (1.2-4.7) x lo4, Gr,/Re2 = 0.10-0.25; (b) Re = (3.5-6.5) x lo4, Gr,/Re2 = 0.13-0.30. Data for c-e from Ref. [34], d = 3.22 cm,P = 25 MPa: (c) Re = (0.84-1.07) x lo5, Gr,/Re2 = 0.05 + 0.07; (d) Re = (5.2-7) x lo4, Gr,/ReZ = 0.20-0.38; (e) Re = (2.2-3.9) x lo4, Gr,/Re2 = 1.5-2.1.

26

A. F. POLYAKOV

subcritical and supercritical pressures. The equality of values of approximate buoyancy parameters Gr,/Re2 for given regimes can be used as a confirmation of the preceding statement. It follows from Eqs. (14) and (15) that the value Gr,/Re2 for these regimes is 5-10 times larger than the threshold value, i.e., the boundary of the beginning of the buoyancy influence on heat transfer. Data shown in Fig. 8 reveal the fact, that in this case, heat transfer in the entry thermal region is substantially lower than without the influence of body forces, and at x/d > 50 the values Nu failed to achieve the value Nu,. Such qualitative analysis is confirmed by distributions Nu/Nu, calculated by Popor and Valuyeva [40] shown in Fig. 9a and b. Data in Fig. 9c-e illustrate heat transfer distributions along the pipe under different degrees of buoyancy force influences. In Fig. 9, the first case (c) is related to the initial stage of thermogravity (see the line for which a = 1 in Fig. 8). The increase of the buoyancy parameter by four to five times leads to a result that has been already shown in Fig. 7. Here it is necessary to emphasize once more that the process at the initial stage is developing with an extremely large intensity-with the buoyancy parameter changing only by two-three times, heat transfer assumes a minimum value along the whole length of the pipe (see the line for which a = 2.4 in Fig. 8). In this connection there is evidence that it is unjustified to consider that heat transfer deterioration due to an Archimedes forces effect is localized only in the thermal entry region. Such conclusions are based on data obtained for the flow of SCP fluid in pipes of relatively large diameters, when, in practice, this narrow region of parameters is missed. The results shown in Fig. 7 are in favor of this opinion. With a further decrease of flow rate (in experiments by Watts and Chou [34], the change of degree of thermogravity influence is realized by the change of in under q , = const.), a substantial increase of buoyancy parameter results in heat transfer along the whole length of the pipe and surpasses the initial level, i.e., Nu/Nu, > 1. The last depends on the development of intensive free turbulent convection over the entire surface of heat transfer. The results of calculations for helium [43] are compared in Fig. 10 with experimental data [45] obtained for a pipe having a 1.8-cm diameter. The character of the distribution, T, and Nu, is similar to that shown in Fig. 9b and d for water. Kurganov et al. [18] obtained reliable experimental data on velocity and temperature (enthalpy) fields, on the local hydraulic resistance of the motion of carbon dioxide in a vertical pipe with a 2.27-cm diameter. Parts of these data shown in Fig. 11 are compared with the results of calculations [41,42]. The results presented in Fig. 11 have been obtained for very high values of Re w lo6. However, an appreciable influence of thermogravity is observed even for such great values of Re, its contribution being evaluated by comparison of t, calculations for both upward and downward flows. A local

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

0

0

20 4 0 X/d

20 40 Xjd

27

0

FIG. 10. Comparison of calculated results (lines) [43] with experimental data (points) [45]. (a) Re = (8.7-9.1) x lo4, Gr,/Re2 = 0.27-0.64; (b) Re = (9.1-9.7) x lo4, Gr,/Re2 = 0.30-0.95.

300

-

6.0

200

W 0

5.0

100

* * *

0 A

y" 400

A x 300

n W

co,

*

0, v

4.0 0

0

/-.

4

c 3

3.0

.-.c_ 200 I

100

2.0

0 1.0

>< 0.6 0.2

FIG. 11. Comparison of predicted results (lines) [41,42] with experimental data for upward flow(points)[18]. Re = (8.7-18) x 105,Gr,/Re2 = 0.1-0.2(solidlines,upwardflow;dashedline, downward flow; hatched line, one-dimensional approach).

A. F. POLYAKOV

28

deterioration of heat transfer with wall temperature “peak” is realized near the region with t b x t,, in upflow, but is lacking in downflow. As can be seen, in this case the buoyancy effect is manifested in a local manner at tb x tpc, and, moreover, far from the start of heating. If we reanalyze Fig. 6, we shall state that the conditions of the downward branch of the beginning of heat transfer deterioration correspond to this regime, and after the region t b x t,, has been passed, we shall return to conditions without buoyancy influence. This case is close to results shown in Fig. 7 and differs from data presented in Figs. 9 and 10, illustrating the increasing development of the thermal convection effect. The development of velocity profiles along the pipe takes place in such a manner that the profiles with the most strongly manifested M-shaped form are related to cross-sections near the highest buoyancy effect, degraded local heat transfer. Such a deformation of velocity profile is typical for mixed convection in upward flows of heated fluids in vertical pipes and under weakly varying physical properties (see, e.g., Ref. [14]). The M-shaped character of the velocity distribution in the pipe cross-section becomes more distinct under a strong influence of free convection. Distributions of enthalpy along the pipe for different radius coordinates are of special interest, demonstrating their difference from the distribution ib(X). The regime presented in Fig. 1 1 is characterized by the great acceleration of flow, determined by large values of a local inertia factor:

f, = 4r0-Pb

-[

2j01puz(r/ro)d(r/ro)]

m dx

In addition, in this regime there are low values of the friction factor

f

=

- 8awpb/m2

and low values of the heat transfer coefficient along the whole pipe length. The hydraulic drag is equal to the sum of the inertia factor and the friction factor For the regime under consideration fd

2 fo x 0.316/Re1l4 x 0.184/Re0.’

(40)

In a one-dimensional approximation, the inertia factor can be presented as

The last equality has been obtained.for the condition qw = const.

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

29

The results of calculations of the inertia factor according to the onedimensional model presented in Fig. 1 1 show that for the regimes with degraded heat transfer, one-dimensional approximations can lead to essential errors in calculations of hydraulic resistance. Such conclusions have been made as a result of both experimental investigations [17,18] and numerical predictions [20,27]. For regimes for which the influence of thermogravity and thermal acceleration is not important, the one-dimensional approximation is correct. In fact, Eq. ( 1 1) finds its confirmation in the estimations of boundaries of the buoyancy effect and boundaries of local heat transfer deterioration due to thermal acceleration, as suggested by Petukhov et al. [17] on the basis of analysis of experimental data on components of hydraulic resistance. In particular, it is indicated that for values of mass velocities of carbon dioxide (m 2 1000 kg/m2),the regimes of “normal heat transfer” satisfy the condition of weak influence of buoyancy forces Gr,/B,, < 1. O n a basis of analysis of experimental data [17,47] and data on heat transfer to gases, Petukhov et al. [171 have suggested the following condition of the existence of “normal heat transfer” under heating of SCP fluids and gases: (fu/fd)

5

-

(42)

1.3

Using Eq. (21) as recommended [17] for the value fd, Eq. (41) for f, = f u l , and Eq. (40) for fo, we can write Eq. (42) in the following form: (f;/fd)

= (44pqwd/CppRe0’8)(Pb/Pw)

-

1.3

(43)

On the other hand, if the threshold acceleration parameter is written according to Eqs. ( 1 1)-( 13), we obtain J/Bth

= (lOOpqwd/qpb

Reo’g)(Pb/p)<

(44)

Comparison of Eqs. (43) and (44) shows not only their qualitative correspondence, but also their quantitative proximity at the values of Re z lo5, typical for experimental data. After the generalization of the considered data has been made, we can make the following presentation of the development of mixed turbulent convection. In the initial stage of gravity field influence on upward turbulent fluid motion in vertical, heated pipes, an intensive decrease of heat transfer takes place due to the suppression of turbulence. Such a heat transfer deterioration under SCP can have greater local manifestations connected with the very large change in the density near the pseudocritical temperature. As the thermogravity effect increases far from the start of heating, turbulent free convection begins to develop, thereby leading to the increase of heat transfer. Incidentally degraded heat transfer envelops some area of the thermal entry region, the

30

A. F. POLYAKOV

dimension of which becomes more and more shortened with the intensification of the buoyancy effect, approaching the start of heating. At very strong influences of free convection, “the entrance wall temperature peak” actually disappears, and in these regimes Nu/Nu, 2 1. In addition to the considered picture of development of mixed turbulent convection, thermal acceleration (e.g., the parameter J) leading to the suppression of turbulence under heating, and hence to a decrease of heat transfer (see Fig. 3), as well as to a large change in density, heat capacity, viscosity, and other physical properties, affects the character of heat transfer in media under SCP. Various combinations of the indicated factors depending on the parameters of the regime can lead to the specific alternation of zones with “deteriorated” and “improved” heat transfer along the pipe. In particular, a realization of such a distribution of t, with two local maxima seems to be real in a long pipe: in the thermal entry region, at tb c t,, and t , It,,, the distribution similar to that shown in Fig. 9b and d is set; at further advancement along the pipe, when tb x t,, is attained, a development of a second local deterioration of heat transfer is possible, and, consequently, a second wall temperature peak can appear. There are no peaks caused by a buoyancy effect in downflow. It is necessary to note that the considered cases of relaminarization caused by suppression of turbulence by thermogravity and thermal acceleration refer to the conditions of nonperturbed fluid motion, i.e., experimental data have been obtained in the presence of hydrodynamic entry isothermal regions before the start of heating, and theoretical results have been obtained under the boundary condition for x = 0, given as the fully developed isothermal turbulent velocity profile. The distribution of local heat transfer appears to have qualitatively distinguishing features without the hydrodynamic entry isothermal region, viz. a local deterioration of heat transfer in the entry region at mixed convection is not realized.

2. Horizontal Pipes The development of turbulent mixed convection in horizontal pipes has a three-dimensional pattern, thereby complicating theoretical investigations as well as generalizations of experimental data. Local heat transfer changes along the length, as well as around the perimeter, of the pipe. Hall and Jackson [2] presented experimental data showing the possibility for the wall temperature maximum to arise at coordinate x,when tb !z t,,, near the upper generatrix. In addition to that, the difference between the wall temperatures at the top and bottom generatrices can reach 200-250°C when water flows in pipes with diameters of 3.0 cm [46]. Even in pipes of small diameters the nonuniformity of temperature around the perimeter can have a substantial value, and has

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

31

attained 80°C in experiments [47] with water in a steel pipe with a diameter of 0.444 cm. Petukhov and Polyakov [14] give a relation describing the boundaries of the beginning of the buoyancy effect on local heat transfer near the top (bottom) generatrix, far from the start of heating ( x / d > 50),

Gr,

= Grth=

3x

(

Re2.” 1 + 2.4

Pr2I3 - 1 Re’/8

where Gr,, E,and Re are defined by Eq. (13), in which i, and t , can be calculated from Eq. (22) without taking into account the influence of free convection. When there is the condition J 5 Gr,, the experimental data on the difference between the temperatures at the top and bottom generatrices under 4, = const. and x/d > 50 can be generalized in the coordinates

where

%

cth3 =

and

pr2/3- 1 Re2.” 1 + 2.4 Re’’’

(

x

B = 281B2/(Pl +

rv

Pr

P2)

= 2Pr1Pr2/(Prl

+ Pr,)

FIG. 12. Generalization of experimental data on temperature differences between top and bottom generatrices of horizontal steel pipes at qu = const.

32

A. F. POLYAKOV

F'

I

0

On

0

'

72.2 2.6

FIG. 13. Comparison of calculated results using Eq. (32) at Gr -t 0 with experimental data on averages around the perimeter heat transfer coefficient for water flow [46].

Here 8, Pr,, b2,and Pr, are evaluated at the temperatures T, and T, defined in Eq. (32). In Fig. 12 the experimental data [46,48,49] are plotted using the coordinates given by Eq. (46).There is a good deal of scatter. However, they suggest that there is a distinct relationship between the maximum wall temperature difference around the pipe circumference and the magnitude of the body forces. This relation could be used to obtain, without iteration, the estimations of this difference and the limits at which gravity begins to have an effect on local heat transfer. Even though the difference between the wall temperatures at the top and bottom generatrices may become quite large, the average heat transfer around the perimeter remains practically unchanged. Calculations of St, using Eq. (32), neglecting body-force effects (Gr = 0), are compared in Fig. 13 with experimental values of 5 [48]. But the ratio of % to the values of the Stanton number 5, at constant physical properties gives deviations from experimental data of substantially greater values.

D. TURBULENT HEATTRANSFER AT HEATFLUX VARYING LENGTHWISE Local heat transfer at upward flow of carbon dioxide (PIP,z 1.3) has been investigated [SO] experimentally under the conditions of linear growth or decrease of wall heat flux qw along a pipe with a diameter of 0.546 cm and with was from 6.6 to 9.4 length 37d. Heat flux change over the length qvX/q:'" at the values qy over the range 0.25-1.75 MW/m2. The Reynolds number and the temperature have been changed in experiments over the following ranges: Re = (0.57-3.7) x lo5, TWIT,,,= 0.94-2.7, and T,/T,, = 0.92-1.03.

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

33

It has been shown [ S O ] to be possible to use the Duhamel integral (superposition principle) for calculations of heat transfer in single-phase subcritical regions in the form

where Nu,, is the relation for heat transfer in SCP fluid at qw = const., qw(0)is heat flux at the start of heating, and el is a correction for heat transfer increase in the thermal entry region for qw = const. and constant physical properties. The following equation for E' has been suggested [SO]: =

1

+ 2.35

Pr- 0 . 4 ( ~ / d ) - 0 .exp( 6 - 0.39 Re-'.' x / d )

(48)

Thus the superposition principle can be used as an empirical procedure to allow for the variability of qw, its influence on heat exchange being assumed to be the same at constant physical properties of the fluid. The influence of variability of physical properties and body and inertia forces is allowed for by the scale number Nu,, . In Fig. 14 the results of calculations of wall temperature using Eqs. (47),(48), and (32) for Nu,, are presented. The combinations of experimental parameters satisfy the boundaries of validity of Eq. (32). The presented data

X/d

x/d

FIG. 14. Wall temperature distribution and values of normalized Nusselt number at linear (a) growth of qw or (b) decrease of qw along the pipe on the basis of experimental data [SO].

34

A. F. POLYAKOV

correspond to the regimes with the greatest manifestation both of thermal acceleration effect and of thermogravity effect. Values of relative parameters JIB,,,, Grq/Bthare indicated for the coordinate x/d = 32 near experimental points for the Nusselt number. The curves [(Nu/Nu,), (xld)] for the growth and decrease of qw are calculated from Eq. (47) using Eq. (32) for Nu,, in which the parameters J and Gr are ignored. The experimental values Nu/Nu, for the conditions satisfying Eq. (1 1) are disposed near these predicted curves. Such points are not shown in Fig. 14. Heat flux variations over the length realized in experiments lead to some increase of heat transfer with the growth of 4;, on the contrary, with the fall of qw it has a more appreciable manifestation by diminishing heat transfer significantly. In the both cases the influence of acceleration and buoyancy substantially decreases the values of Nu/Nu,, as well as under boundary conditions qw = const. considered above. As is seen from Fig. 14, the suggested approximative method correctly describes the distribution t, and consequently the local heat transfer at q,, which changes along the length of the pipe.

E. TURBULENT HEATTRANSFER UNDER COOLING It is much more difficult to obtain experimental data on local heat transfer under cooling conditions as compared with the case of heating. Experimental data have been published on local heat transfer in pipes under conditions involving cooling of water [Sl], carbon dioxide [52,53], and helium [54]. Systematic numerical investigations have been carried out [55,56] for carbon dioxide and water under cooling conditions, wherein the system of Eqs. (1)-(3), (6), and (18) is used with the boundary conditions of Eq. (5), in which the following changes have been inserted: at x = 0 a developed velocity profile has been given for constant physical properties uin = u&); on the inside surface of the pipe the first-order boundary condition T,(x) or the secondorder boundary condition qw(x) has been given. The range of calculation parameters for carbon dioxide corresponds to experimental regimes [52,53]. To obtain q,, the distributions T,(x)have been predetermined from experimental data, and to obtain T, the experimental distribution of heat flux along the channel has been specified. Physical properties were taken from data [6] allowing for the peak of thermal conductivity on the basis of other data [3]. Calculations were carried out without taking into account the influence of free convection, which is considered unimportant according to the obtained estimations. The comparison of predicted data [values of qw(x) were taken from the experiment] [55] with experimental data on heat transfer [53] for the cases T, < Tpc-= Tbis presented in Fig. 15. Calculated data on the friction factor

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES 6 4

35

Y

4

50 3

0-

e A

I-

0.6

40

30

0

3

1.0 R

0 R

1.0

3z 2

0 1.0 x/d

05

0

R

FIG.15. Predicted results [55] and experimental data [53] for P rn = 2400 kg/m2 sec, and Re = (4.8-4.5) x 10’.

=

8.0 MPa, d

=

0.412 cm,

1; the inertia resistance coefficient ju, the total drag coefficient fd, and the velocity and temperature distributions are also shown in this figure. While under heating conditions, the M-shaped velocity profiles are connected with the sharp decrease of shear stress and with the passage of its values through zero into the negative region at the coordinate of maximum velocity, according to Eq. (6); on the contrary, under cooling conditions, such peculiarities are not observed. The flow decelerates, velocity profiles are monotonic, and the distribution of shear stress is close to that at constant physical properties of the fluid. Physical property variations under cooling conditions significantly increase friction resistance. Local inertia resistance is commensurable with local friction resistance, but they have different signs. At some combinations of parameters, this can lead to a sharp decrease of total hydraulic drag, which attains negative values, thereby resulting in the appearance of zones with pressure increasing along the pipe. The results of numerical modeling demonstrated that with cooling, for all the investigated ranges of parameters, the inertia resistance coefficient can be calculated from the one-dimensional Eq. (41) with an error less or equal to f 10%. The results of calculations for pressure that is closer to the critical value than was found in Baskov et al. [53] are compared in Fig. 16 with experimental data [52] for the heat transfer coefficient. For this case, predicted data, taking into account the heat conductivity “peak,” are compared with data that do not consider the peak (see Fig. 1b). It can be seen that the results of calculations fit the experiment, and the consideration of the thermal conductivity peak

36

A. F. POLYAKOV

I

32

36

34 tb

(“4

38

FIG. 16. Comparison of predicted results (lines) [ 5 5 ] with experimental data (points) [52]: (1) qw = 36 kW/m2, m = 990.5 kg/m2 sec; (2) qw = 14.4, m = 495.2 (solid lines, with peak of

thermal conductivity;dashed lines, without peak of thermal conductivity).

leads to a change in the heat transfer coefficient by approximately 10%. Calculations for P = 8.0 MPa (with the thermal conductivity peak) plotted in the same graph illustrate a significant influence of pressure on heat transfer at the values close to P,. While moving from P, with the increase in pressure, the local maximum in the distribution k( T) loses itself rather quickly, as follows from Fig. lb. Calculations of turbulent heat transfer [ S S ] with different data on thermal conductivity resulted in concluding that taking into account the thermal conductivity peak leads to some variations of the Nusselt number under pressures close to the critical one, but at P/P, > 1.1, the difference in the data is not routinely observed. Special multiparametric calculations for different boundary conditions (qw = const. and T, = const.) showed that, in fact, the type of boundary conditions has no influence on turbulent heat transfer and resistance. The generalization [56] of calculated data on the friction factor for carbon dioxide, water, and helium suggests an interpolation equation,

where f U l is calculated from Eq. (41). The approach [28] based on a two-layer model has been used to suggest a heat transfer formula. Analysis of the entire volume of calculated data has

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

37

convinced Petrov and Popov [56] that an equation such as Eq. (20) describes data more successfully than does an equation such as Eq. (19). From generalizations [56] of calculated data on coefficients of heat transfer for water, helium, and carbon dioxide, the following formula has been obtained: Nu = (f/8)Re Pr/( 1.07 + 12.7({)1’2{A“3(2)1’2 x

[

1-A,

(‘7’)”’] [ -

-

1 - A,(!$$-)”’]})

(50)

where for water A, = 1.1, A, = 1.0; for helium A, = 0.8, A, = 0.5; and for carbon dioxide A , = 0.9, A, = 1.0. The value f is calculated from Eq. (49).

IV. Free Convection A. VERTICAL SURFACES

Processes of heat transfer and motion in a free-convective boundary layer on the vertical plate are described by the system of Eqs. (1)-(3), which are written not for cylindrical coordinates (x, r), but for Cartesian coordinates (x, y); thus, naturally, the term in the momentum equation taking into account the change of static pressure at forced flow is eliminated. Then the problem is specified by the following boundary conditions: when y = 0 u = 0,

when y

u =0,

.

I

.

= t,

or

(aildy), = -qwcpw/kw (51a)

+ 00

u=o,

.

t=t,

.

1. Laminar Flow

Laminar heat transfer in SCP fluid, moving along a vertical surface, was studied experimentally and theoretically. Experimental investigations have been carried out for water [57], carbon dioxide [58-601, and helium [61] with vertical plates [57-59,611 and cylinders [60]. Some theoretical investigaations have been undertaken for these fluids. Among these studies, the work of Popov and Yankov [3] can be distinguished; in this work, experimental and predicted data from other studies were analyzed and generalized along with a great volume of Popov and Yankov’s calculations for water, carbon

A. F. POLYAKOV

38

(T,-T-

) (K)

(T,-L)(k)

FIG.17. Calculated results [3] for laminar free convection using physical property data [5,6] and comparison with results of calculations [62,63] and experiments [57]. [3]' shows calculated results [3] using values K from Refs. 4,8, and 9 (see Fig. 1).

dioxide, helium, and nitrogen. Results of these calculations [3] are shown in Fig. 17. The relations q,x114 = Fl(T,,AT);

hx'14

=

F,(T,,AT)

(52)

following from self-similarity of the solution under the boundary condition Tw= const., are justified by numerical calculations. In conjunction with this, data in Fig. 17 are plotted in the coordinates given by Eq. (52). Calculations [3] carried out using different data on physical properties confirm that a difference in physical properties for laminar flows can lead to differences in heat transfer coefficients reaching 30-40%. Consequently, the coincidence of prediction and experiment concerning laminar free convection is determined not only by experimental precision, but also is greatly dependent on the accuracy of physical property data (primarily on transport properties). In the relation h(AT),a maximum appears at the values of T, close to Tpc. The value of this maximum appreciably increases if calculations are made taking into account the thermal conductivity peak. The influence of boundary conditions (T, = const. and qw = const.) on heat transfer is illustrated in Fig. 17. When qw = const. and T, and AT = T, - T, have the same values that they have at T, = const., heat transfer is greater by approximately 15%. The calculations have not revealed any objective regularities for the ratio Nu,/Nu,, so, according to Popov and Yankov [3], this ratio is assumed to be constant and equal to some mean value: Nu,/Nu, = 1.15

(53)

An interpolational equation for heat transfer coefficient at T, = const. has been proposed [3]:

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES hx NU = - = N U ~ ( C , / C , , ) ~ ~ ~ ~ ( ~ ~ / ~ , ) " k,

Nu, = 0.75[2E/(5 n={

+ 10Pr'" + 10fi;)]1/4Ra'/4

0.32, 0.1,

39

(54)

-

Cp/CPwI1 C,/C,, > 1

-

Ra = ( g p A T x 3 / v i )Pr, -

Pr

= pwq/kw

= (i,

-

i,)/(T, - T,)

Equation (54)describes the entire set of experimental and predicted data on heat transfer to water, carbon dioxide, nitrogen, and helium better than other known empirical and interpolational equations. Reliable data on thermal conductivity close to Tpcis lacking, thus the generalization has been made using calculated data obtained both with and without taking into account the thermal conductivity peak.

2. Turbulent Flow

A number of experimental [60,61,64-661 and theoretical [67-691 investigations of heat transfer at turbulent free convection near vertical plates [60,6 I ,64,67-691 and cylinders [65,66] under heating [60,61,64,65,67-701 or cooling [66] conditions have been published recently. In order to estimate boundaries of transition from a laminar to turbulent boundary layer, the following approximate equation has been proposed [70] : Ra,

=

5 x 10'o(~/Cp,)-'.66

(55)

Numerical modeling of turbulent heat transfer at free convection of carbon dioxide near a heated plate has been made [67,68] for regime parameters corresponding to experimental data [60,64]. Popov and Yankov [68] have used Eqs. (36) and (37),which were proposed by Popov and Valuyeva [40,41] for turbulent momentum transport. The influence of the thermal conductivity peak on the calculated heat transfer coefficient and a comparison with data [60,64] in the form h = h(AT) are depicted in Fig. 18. The lines show the results of calculations for the crosssection x = 9 cm corresponding to experimental points [64]. Taking account of peak "k" gives a noticeable increase (up to 30% at P = 7.5 MPa and up to 20% at P = 8.0 MPa) in the heat transfer coefficient in the region of its maximum values and is more in agreement with experimental data. Results of

40

A. F. POLYAKOV A

3 < 2,

1.5

m 1

0

1.0

JZ

0.5

4

6

8

10

12

14

16

FIG. 18. Comparison of calculated results (-, with peak of thermal conductivity; -.-, without peak of thermal conductivity [68]; [67]) with experimental data (A, x [64]; 0,. [60]).

--.

calculations for P = 7.85 MPa and AT> 6 K appear to be essentially lower (by 30-50%) than experimental data [64]. It is important to pay attention to a certain disparity among experimental data. The points [64] for P = 7.85 MPa are close to data [60] for P = 7.5 MPa and greatly exceed the values h for P = 8.0 MPa. Probably, they should in fact be closer to the latter values. In work aimed at finding turbulent viscosity (Seetharam and Sharma [67]), which is determined by Eqs. (6) and (8), Eq. (4) has been used, supplemented with convective and diffusive terms. Results of predictions are compared with experimental data and the empirical relation [60] for carbon dioxide: NU = 0.135 Ra'/3(Cp/Cpm)0.75(pw/pm)0~4 (56) It is shown that heat transfer at turbulent free convection is not dependent on the type of boundary condition (T, = const. or qw = const.) in the limits of validity of the empirical Eq. (56). In Fig. 18, the maximum value of the heat transfer coefficient can be revealed at T, % Tpc.It is shown distinctly in Fig. 19 by an example of the results of numerical modeling of turbulent free-convectional heat transfer to helium [69], which has been investigated experimentally [61]. Calculations [69] have been made using the same mathematical model applied in earlier work [68]. As can be seen from Fig. 19, the maximum h becomes less pronounced with an increase of PIP,. Popov and Yankov [69] showed that Eq. (56) gives a bad description of experimental and calculated data for helium. They have suggested the following equation to describe heat transfer to the turbulent motion of helium along the vertical plate: Nu = 0.12 Ra1~3(~/Cpm)n(pw/pm)o~15 n = 1.0 T, > Tpc n = 0.5,

T, ITpc

(57)

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

41

I

0.8

0.9

1.0

1.1

T . / Tpc

FIG.19. Heat transfer coefficient at turbulent free convection of helium: experiment (points) [61]; theory (lines) [69].

Temperature distributions in the boundary layer (Fig. 20) have some particularities distinguishing them from monotonic distributions at constant physical properties. The shape of the curves of dimensionless temperature (T - T,)/(T, - T , ) varies substantially depending on the coordinate when turning from one regime to another. For regimes with T, > Tpc> T, there are some sections along the plate with negative values of the second-order derivative in cross-sections of the boundary layer, when local temperature is close to the pseudocritical value. Such particularity has been denoted in the case of laminar free convection of SCP helium [3]. As experimental data on carbon dioxide [66] showed, the character of the dependence h = h(T,) for turbulent free convection under cooling conditions is typical for heat transfer under SCP: near the pseudocritical temperature a heat transfer coefficient attains the maximum value, which increases when the pressure approaches the critical value. Maximum values of h versus T, correspond to the temperatures T,, which are slightly higher than Tpc.At the

I

lo-'

4

10"

4

10'

0

Nu b / X )

FIG.20. Temperature distribution in turbulent free convection boundary layer of helium [69].

A. F. POLYAKOV

42

same time, experimental data on cooling are not described by the relation for the heat transfer coefficient at free convection in carbon dioxide under heating conditions [Eq. (56)]. The following empirical equation for the coefficient of local heat transfer to carbon dioxide under cooling conditions has been suggested [66]: hx Nu, = - = N~ow(c,/c,w)"(P~/Pw)

(58)

k W

where NuOw= 0.135Ra:l3, -

( s x 3 / v 3Prw,C, = (i,

Raw = [(p,

- i W ) /(~ T,) ,

q/Cpw< 1,

n = 0.4

C,/Cpw> 1,

n = 0.75

-

- p,)/pw]

B. HORIZONTAL WIRES In this section, in accordance with the results of the work of Neumann and Hahne [7], we shall discuss heat transfer from a horizontal heated thin wire under free convection in single-phase carbon dioxide with nearcritical parameters. Typical results for the heat transfer coefficient from thin platinum wire are illustrated in Fig. 21. When the temperature of the medium far from the wire, t, ,is appreciably lower than the pseudocritical value, the heat transfer coefficient sharply increases, when the wire temperature t , approaches t,, . At

i0 20 30 40 5060 70 tw

R a:

@)

FIG.21. Heat transfer from horizontal platinum wires [7].

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

43

t , > t,, > t,, the coefficient h decreases, adopting always the same values, which are practically constant. When t, is close to the pseudocritical temperature, changes of h have a sharp character, and maximum values of the heat transfer coefficient become rather large at t , x tpc. For t , > tpc, heat transfer has a relatively low level and varies slightly. Thus at t , < tpc, the coefficient h is dependent both on t , and t,, while at t , > tPc, it remains dependent only on the temperature t , . Special attention has been given [7] to the analysis of the contribution of the thermal conductivity anomaly (see Fig. 1) in heat transfer at free convection from horizontal wires. To distinguish a sharp increase in thermal conductivity near the pseudocritical temperature, the latter is presented in the form

k = ko + Ak

(59)

where Ak is an anomalous increment of thermal conductivity and ko is an interpolation “without the peak.” When the conditions t, < t , < (t,, - 2K) or (t,, + 2K) < t , < t , are satisfied and the variation of the medium physical properties is relatively small, experimental data are described with an accuracy of f 10% by the known relation of Van der Hegge Zijnen [71], Nu

= 0.35

+ 0.25 Ra,‘/* + 0.45 Ra!I4

(60)

which is valid for heat transfer from horizontal cylinders and wires over the < Rai < lo9. Here range

(

Rai = Gri Pri = 2, g d 3 P,; ~

“i

~ i ) ( z ~ iii - ia)

ki

t, - t,

(all the other values with subscript i are determined similarly). Equation (60) describes with the same accuracy, f lo%, the experimental data [7] over the indicated range of temperatures, with even a simpler averaging of physical properties, when they are determined not as the integral mean values, but over the arithmetic mean temperatures ( t , + t,)/2. The variability of physical properties has its most important manifestation over the range of temperatures (t,, - 2K) < t , < t, < (t,, + 2K). To reveal the degree of influence of the thermal conductivity peak, the experimental data on heat transfer are presented [7] in modernized coordinates Nu:(Ral), which are calculated by use of “k” from Eq. (59), where Ak is ignored, i.e., the values k, are used in Nu: and R a l . The curves in Fig. 21 b are plotted over the data relating to this range of temperatures. The deviations from Eq. (60)using

44

A. F. POLYAKOV

the dimensionless numbers Nu? and R a l , which have been observed, are found to begin when the wire temperature approaches the pseudocritical value (compare with Fig. 21a), and to attain the maximum value when the thermal layer around the wire is at a pseudocritical state. At the same time, when the diameter increases, the deviation begins at greater values of R a l and has a lower value. With a growth of the intensity of free convection, the contribution of the thermal conductivity layer near the wall in heat transfer decreases. The dependence of the studied deviation on ( t , - tpc),(tpc- tm),P i s established in experiments, and in view of the generalizations of data, Neumann and Hahne [7] suggested the following modification of Eq. (60): Nui = (0.35 + 0.25 Ra,"8

+ 0.45 RafI4)(l + a,a,a,)-'

(61)

where a, = zexp(-z),

z = 4.5[(Tw- Tpc)/TJ''2

a2 = tanh(30 TpcT, - Tm

)

( 'iCPC)

a, = 1 - 0.3 tanh 1 5*

The empirical Eq. (61) describes all the experimental data [7] with an accuracy of f 10%. The comparison of Eq. (61) with experimental data is shown in Fig. 21a.

V. Special Problems A. SOMEDATAON NONSTATIONARY HEATTRANSFER

In this review we have considered the data generally related to stationary or quasistationary (i.e., stationary in the mean) conditions. The latter determine both developed turbulent flow, when the averaging interval corresponds to the scale of turbulence, and flows with fluctuations of hydrodynamic and thermal values, which are developed, for example, in some regimes of mixed convection, but when the conditions of quasistationary approximation for mean values are satisfied. Some of the work devoted to studying nonstationary processes in SCP fluid is examined in the following discussions. 1. Nonstationarity Determined b y External Conditions

Smirnov and Krasnov [72] presented results of experimental investigations on nonstationary heat transfer in upflowing water in vertical, electri-

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

45

800 2.

'z

600 400 200

0

a

10

b

200

0

20

30

FIG.22. Temperature of outside surface of the pipe and normalized coefficient of nonstationary heat transfer under (a) decreased or (b) increased heat load.

city heated 109-cm-long pipes with internal and external diameters of 0.408 and 0.815 cm, respectively. Nonstationary conditions were created by step changes of heat power in the wall by means of variation of the electrical load. The experiments have been carried out over the ranges P = 25-30 MPa, m = (0.4-1.2) x lo3 kg/m2 sec at heat flux on the inside surface of channel reaching 1 MW/m2. At mass velocities (m = 400-500 kg/m2 sec), regimes with the local degraded heat transfer have been observed. In all the experiments the fluid temperature remained lower than the pseudocritical one. Figure 22 illustrates the character of changes of wall temperature and the normalized Nusselt number Nu/Nu, for three cross-sections of a pipe for the regime with locally degraded heat transfer under stationary conditions, which is determined by the value Nu,. In the initial stage of the experiment (z = 0), some value qws= const. was given and the stationary distribution t,, was set. In Fig. 22a, three values t,, for the coordinates x/d = 16.5, 215, and 240 are shown. The coordinate x/d = 215 corresponds to a local maximum of wall temperature. In the moment t = 0, heat power in the wall of the pipe drops stepwise to some constant level of heat load. At this level of load a new distribution t, corresponding to Nu/Nu, = 1 establishes during 20 sec. This distribution t, is monotonic. Then the heat load increases stepwise to its initial value qws,and during 30 sec the transition process develops, as is shown in Fig. 22b. After 30 sec, at the end of second transition process, the initial distribution t,, with a local maximum, again establishes. When the heat load decreases, the zone with local degraded heat transfer moves to the outlet; this is reflected in the sharp fall of value Nu/Nu, at the coordinate x/d = 240. At that

46

A. F. POLYAKOV

time, the rate of change of wall temperature sharply decelerates. Such behavior of the relative coefficient of heat transfer is also observed under an increasing heat load. As is shown by the example in Fig. 22, variable physical properties of SCP media lead to substantially greater peculiarities of heat transfer under nonstationary conditions as compared to constant physical properties. Under the real conditions, the development of transition processes in fluid is connected with changing temperature fields in the wall and with the necessity to solve the conjugated problem. Even under stationary conditions, the influence of the wall on heat transfer to SCP fluids can have a significant value in the realization of wall temperature distributions with local maxima. Watson [73] has showed that the interaction of wall thermal conductivity along the flow and local heat transfer dependence on wall temperature can lead to a disturbance of the stationary state and to a realization of distributions t , with several peaks. 2. Thermoacoustic Perturbations As the experimental investigation of Daney et al. [74] showed, under thermal conditions (ti, .c t,, < t,,,) in flows of SCP fluid (helium) in heated channels, when a transition from pseudoliquid state to pseudogas one takes place, the development of pulsation regimes and the propagation of density waves along the pipe are possible. The amplitude of thermal perturbation moving from inlet to outlet of the heated channel was established to increase. Labountsov and Mirzoyan [75] have theoretically investigated the problem of the thermal stability of helium flow in heated channels of small diameter (i.e., without manifestation of buoyancy forces). The analysis carried out on a basis of the basic theory of hydrodynamic stability resulted in the determination of a diagram of flow regimes. Experimental data [74] are plotted in Fig. 23 over the coordinates APi,,/APou,- u,,,/uin (APi, and AP,,, are the pressure losses at the inlet and outlet; uin and u,,,are specific volumes of fluid in input and output cross-sections). In general, theoretical and experimental data are in good agreement qualitatively and quantitatively. Thermoacoustic oscillations generated in flows of fluid under nearcritical parameters and in the presence of heat transfer are of special scientific and practical interest. In turn, thermoacoustic oscillations have an important influence on the character of heat transfer. As a result of theoretical investigations [76-791, it was established that under the temperature condition t , > t,, > t b , the anomalous improvement of heat transfer, which occurs at a certain combination of parameters and is accompanied with high-frequency fluctuations of pressure, has an acoustic character. This has been determined by the presence of the wall layer of fluids

HEATTRANSFER UNDER SUPERCRITICAL PRESSURES

47

‘JoutlVi n

FIG.23. Diagram of stability of pipe flow of SCP helium on the basis of experimental data (points)[74] and theoretical results (line) [75].

with a low sound velocity and distortion of the acoustic wave profile in the region of high gradients of sound velocity, thus resulting in the sharp growth of velocity in standing sound waves and the intensification of heat transfer in turbulent flows. Such regimes are realized, provided that some conditions are satisfied, in particular, certain acoustic properties of the channel. Some results on amplitude-frequency features of oscillations have been obtained. Estimations of the limits of the existence of regimes with acoustic oscillations and of the degree of heat transfer increases have been made. Theoretical data have agreed with experimental data. When conducting the analysis and generalization of data for regimes with acoustic oscillations, it is necessary to allow for the anomalous minimum value of sound velocity, as was discovered by Erohin an Kalyanov [80], under the temperature that slightly exceeds the pseudocritical temperature. Experimental data on the development of thermoacoustic oscillations in turbulent flow of SCP water in a heated pipe (d = 0.44cm) and in an annular channel ( d , : d , = 1:0.6 cm) have been presented by Maevskiy et al. [Sl]. In this work the lower boundaries of existence of the thermoacoustic oscillations have been stated; the oscillations had frequencies of 100 and 200 Hz and amplitudes up to 0.6 MPa in the annular channel. B. HEATTRANSFER AUGMENTATION UNDER CONDITIONS CORRESPONDING TO DEGRADED HEATTRANSFER

The complex investigation of heat transfer and hydraulic resistance is an advantage of the work [82] in comparison with other works on heat transfer intensification for SCP fluid.

48

A. F. POLYAKOV

In previous studies, investigations have focused on the influence on heat transfer of swirled flow using helix ribbons and internal helix finning of the pipe. The result of applications of such techniques was the elimination of wall temperature peaks at the same qw and rn, as for a smooth pipe. The possibility of a twofold increase of admissible heat flux (under the same maximum value of wall temperature) is indicated. The installation of a ribbon in horizontal pipes makes it possible to equalize heat transfer over the circumference of the pipe. However, note that when heat transfer degrades due to thermal acceleration, in swirled flow the trend to heat transfer deterioration is conserved under sufficiently high heat loads. A method for heat transfer augmentation based on turbulization of the wall region of flow has been studied experimentally [82]. Augmentation has been accomplished using spiral wire inserts; the wires had diameters from 0.02 to 0.06 cm and were inserted on the inner surface of a heated pipe with a diameter of 0.8 cm. Experiments have been carried out with carbon dioxide under a pressure of 7.7 MPa, Rein= (2.3-3.4) x lo5, and heat flux on the wall up to 1.5 MW/m2. It was stated that all the above-considered positive results of augmentation can be achieved by using spiral inserts; under conditions corresponding to degraded local heat transfer in smooth pipes, spiral inserts with a relatively big step(s/d = 3-5) are the most effective ones. The limitation of data obtained in this way did not allow suggesting general recommendations on the optimization of augmentation devices. Another type of augmentation device has been subjected to a test [83]: transverse annular bosses were directed into the pipe. Such devices have been widely tested for gases and liquids with weakly varying properties and have demonstrated their efficiency. Experiments [83] have been carried out with hydrocarbonaceous fluids under SCP in heated pipes with a diameter of 0.4 cm. Local heat transfer and hydraulic resistance were measured. It was found that at low values [Re = (1.1-5.2) x lo3], the increase of the normalized heat transfer coefficient Nu/Nu,, attains large values (> 5), and the effiexceeds ciency of augmentation determined by the ratio (Nu/Nu,,)/( jd/fd,,,) one, attaining the value 1.5. For ordinary fluids under the same conditions, the are the values of maximum value of efficiency was 1.01. Here Nu,, and id,,, the Nusselt number and of drag coefficient for smooth pipes. VI. Concluding Remarks

Based on the material discussed in this review, some conclusions and proposals about directions of further investigations can be made. During the past 10-12 years, the studies of heat transfer grew beyond the fully empirical state. Physically based analyses of the nature of the develop-

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49

ment of some peculiarities of turbulent flow and heat transfer (in particular, for regimes with degraded local heat transfer and for regimes with pulsating flow and with thermoacoustic perturbations) have been carried out. Methods of mathematical modeling of turbulent heat transfer in vertical pipes have been developed, and numerical calculations for water, carbon dioxide, and helium have been carried out under different boundary conditions over a wide range of change of the definitive parameters. Systematic experimental investigations of hydraulic resistance, velocity, and temperature fields have been undertaken. The importance of a number of new problems has been demonstrated, and some preliminary data have been obtained on the temporal and spatial nonuniformity of boundary conditions, and on problem solution in conjugated formulation of “wall fluid.” A number of other scientific and practical problems are being solved: refinement of fluid thermophysical properties at supercritical state parameters, specifically, on thermal conductivity, and their application for refinement of analysis and generalization of data on heat transfer; search for optimum methods to augment heat transfer, viz. in order to eliminate regimes with local wall temperature peaks; accumulation of experimental data on heat transfer to helium under different conditions to satisfy the needs of cryogenic engineering; and obtaining more perfect computational dependences. The achievements indicated above do not completely resolve the enumerated problems, nor do they encompass the diversity of problems of heat exchange and hydrodynamics under SCP. Current investigations may develop in the following directions: experimental study of turbulent flow structure and heat transport; development of methods of three-dimensional mathematical modeling of heat transport, including a conjugate formulation of problem; experimental and theoretical computational investigations of nonstationary heat transfer, including generation of internal (in particular) thermoacoustic oscillations; accounting for the refined peculiarities of the thermophysical properties of media and their contamination with solute components; development of generalized methods of calculations based on a combined analysis of hydrodynamic and thermal features of flow; and development of effective procedures of heat transfer augmentation and a search for its optimization, with elimination of regimes with unfavorable thermomechanical actions on the surface. NOMENCLATURE EL,, C,

d

parameter of threshold value, Eq. (11) specific heat at constant pressure diameter of the tube ( = Zr,) or diameter of the wire

f f* f. Y

Ga

friction factor [u,/(m2/8p)] drag coefficient, Eq. (39) inertial resistance factor, Eq. (38) acceleration due to gravity Gallileo number (gd3/v2)

A. F. POLYAKOV Gr Gr, h i J

k

K m

Nu,% P Pe Pr

Grashof number (gpATd3/v2) Pr, modified Grashof number q r (eP4wd4/kv2) heat transfer coefficient (qw/Tw- Tb) r0 R specific enthalpy dimensionless parameter of accelerRa ation defined in Eq. (12) Re thermal conductivity St turbulence energy t mass velocity averaged through T tube cross-section U CfS:(Pu)R dR1 V local and average Nusselt number V pressure X Peclet number (= Pr Re) Prandtl number (pC,/k) Y

turbulent Prandtl number heat flux radial coordinate tube radius dimensionless radial coordinate W O )

Rayleigh number (= Pr Gr) Reynolds number (md/p) Stanton number (= Nu/Re Pr) temperature ("C) absolute temperature (K) axial velocity radial velocity specific volume ( = l/p) axial coordinate radial distance (= ro - r)

Greek Symbols

P E

0

volume expansion coefficient C-(1/P)(aP/at),I dissipation of turbulent energy dimensionless temperature defined in Eq. (46)

P V

P u T

viscosity kinematic viscosity (p/p) density shear stress time

Subscripts

b

W

bulk critical value inlet or outlet (section), respectively pseudocritical temperature value for turbulent flow threshold value value at the wall

+

dimensionless value

C

in, out PC

T th

0

* a,

scale, reference, characteristic, initial, or axial value variable physical properties value taken far from the surface in the flow (outside the boundary layer) or far from the start of heating when in a pipe

Superscripts

Abbreviation SCP

supercritical pressure

REFERENCES 1. B. S . Petukhov, Ado. Hear Transjer 6, 503 (1970). 2. W. B.Hall and J. D. Jackson, Heat Transjer 1978: Proc. Int. Heat Tran.$er Conf., bth, Toronto 6,377. Hemisphere, New York, 1978. 3. V. N. Popov and G. G. Yankov, Teplofz. Vys. Temp. 20,1110 (1982). 4. A. A. Aleksandrov, A. I. Ivanov, and A. B. Matveev, Teploenergetika No. 4,59 (1975).

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5. S. L. Rivkin, “Thermophysical Properties of Water in Critical Region. Handbook,” p. 635. Izdatelstvo Standartov, Moscow, 1970. 6. V. V. Altunin, “Thermophysical Properties of Carbon Dioxide,” p. 551. lzdatelstvo Standartov, Moscow, 1975. 7. R. J. Neumann and E. W. P. Hahne, Int. J. Heat Mass Transfer 23, 1643 (1980). 8. A. A. Aleksandrov, Teploenergetika No. 4.70 (1980). 9. Le Neindre, R. Tufeu, P. Bury, and J. V. Sengers, Ber. Bunsenges. Phys. Chem. 77,262 (1963). 10. H. Tanaka, A. Tsuge, M. Hirata, and N. Nishiwaki, lnt. J. Heat Mass Transfer 16,1267 (1973). 1 1 . A. F. Polyakov, Teplofz. Vys. Temp. 13,1210(1975). 12. U. Renz and R. Bellinghausen, Heat Transfer 1986: Proc. Int. Conf.,8th. San Francisco, Calif. 3,957. Hemisphere, New York, 1986. 13. S. U. Shenoy, B. S. Jagadish, and G. K. Sharma, Proc. Natl. Heat Mass Transfer Conf., Bombay Pap. No. HMT-C19-85 (1975). 14. B. S. Petukhov and A. F. Polyakov, “Heat Transfer in Turbulent Mixed Convection,” p. 216. Hemisphere, New York, 1988. 15. B. S. Petukhov, V. D. Vilensky, and N. V. Medvetskaya, Teplofz. Vys. Temp. 15,554 (1977). 16. B. S. Petukhov, V. A. Kurganov, V. B. Ancudinov, and V. S. Grigoriev, Teplofz. Vys. Temp. 18, 100 ( 1 980). 17. 8. S. Petukhov, V. A. Kurganov, and V. B. Ancudinov, Teplofiz. Vys. Temp. 21, 92 (1983). 18. V. A. Kurganov, V. B. Ancudinov, and A. G. Kaptilniy, Teplofz. Vys. Temp. 24, 1104 (1986). 19. B. S. Petukhov and N. V. Medvetskaya, Teplofz. Vys. Temp. 16,781 (1978). 20. B. S. Petukhov and N. V. Medvetskaya, Teplofiz. Vys. Temp. 17,343 (1979). 21. P. Bradshaw, ed., Top. Appl. Phys. 12, 350(1978). 22. V. S. Protopopov, Teplofiz. Vys. Temp. 15,815 (1977). 23. J. D. Jackson and W. B. Hall, in “Turbulent Forced Convection in Channels and Bundles”(S. Kakas and D. B. Spalding, eds.), Vol. 2, p. 563. Hemisphere, New York, 1979. 24. V. A. Kurganov and N. V. Medvetskaya, Int. Semin. Near-Wall Turbul. Dubrounik, Yugosl. Sess. 8 ( 1988). 25. M. C. M. Cornelissen and C. J. Hoogendoorn, Numer. Methods Laminar Turbul. N o w , lnt. Conf., 3rd. Seattle. Wash.p. 832 (1983). 26. V. N. Popov, Teplofz. Vys. Temp. 15, 795 (1977). 27. V. N. Popov, V. M. Belyaev, and E. P. Valuyeva, Teplofz. Vys. Temp. 16, 1018 (1978). 28. B. S. Petukhov, A. F. Polyakov, and S. V. Rosnovskiy, Teplofz. Vys. Temp. 14, 1326 (1976). 29. B. S. Petukhov and V. V. Kirillov, Teploeneryetika No. 4 (1958). 30. B. S. Petukhov and V. V. Popov, in “Teplo-Massoperenos,” Vol. 1, p. 50, ITMO, Minsk, 1965. 31. V. A. Bogachev, V. M. Eroschenko, and L. A. Yaskin, Ingenerno-Fiz. J. 44,544 (1983). 32. L. S. Yanovskiy, E. V. Kuznetsov, A. S. Myakichev, and A. I. Tichonov, in “Sovremenniye problemi Hydrodin. i teploob. v elementah energetich. ustanovok i kriogen. techn.,” p. 129. Vses. Zaochn. Mashinostroit. Inst., Moscow, 1987. 33. B. S. Petukhov, N. P. Ikryannikov, and V. S. Protopopov, Teplofiz. Vys. Temp. 11,1068 (1973). 34. M. J. Watts and C. T. Chou, in “Heat Transfer 1982,” Vol. 3, p. 495. Hemisphere, New York, 1982. 35. Y. D. Barulin, Y. V. Vihrev, and B. V. Dyadyakin, Ingenerno-Fiz. J. 20,929 (1971). 36. V. A. Bogachev, V. M. Eroschenko, and L. A. Yaskin, Teplofz. Vys. Temp. 21, 611 (1983). 37. V. A. Bogachev, V. M. Eroschenko, 0.F. Snittina, and L. A. Yaskin, in ‘~TeplomassoobmenVII,” Vol. 1, part I, p. 19. ITMO, Minsk, 1984. 38. E. G. Hauptmann and A. Malhotra, J. Heat Transfer 102, No. 1 (1980). 39. C. P. Bellmore and R. L. Reid, J. Heat Transfer 105,536 (1983). 40. V. N. Popov and E. P. Valuyeva, Teploenergetika No. 4 , 2 2 (1986).

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41. V. N. Popov and E. P. Valuyeva, in “Hydrodinamika i teploobmen v odno- i dvuhfaznih potokah,” Proc. Moscow Power Inst., No. 113, p. 35. MEI, Moscow, 1986. 42. V. N. Popov and E. P. Valuyeva, in “Heat/Mass Transfer-MIF,” Sect. 1, Part 2, p. 101. ITMO, Minsk, 1988. 43. V. N. Popov and E. P. Valuyeva, Teploenergetika No. 7,54 (1988). 44. N. S. Alferov, R. A. Ribbin, and B. F. Balunov, Teploenergetika No. 12,66 (1969). 45. D. J. Brassington and D. N. H. Cairns, lnt. J. Heat Mass Transfer 20,207 (1977). 46. I. I. Belyakov, L. Y. Krasyakova, A. V. Giukhovskiy et al., Teploenergetika No. 11,39 (1971). 47. S. Ischigai, 1. Kagii, and M. Nakamotto, in “Teplomassoobmen-V,” Vol. 1. ITMO, Minsk, 1976. 48. Y. V. Vichrev, A. S. Konkov, and I. T. Sinnitsin, Elektr. St. No. 7,35 (1970). 49. G. A. Adeboyi and W. B. Hall, Int. J. Heat Mass Transfir 19,715 (1976). 50. V. S. Grigoriev, A. F. Polyakov, and S . V. Rosnovskiy, Teplofz. Vys. Temp. 15, 1241 (1977). 51. M. E. Schitsman, Teploenergetika No. 1,83 (1962). 52. H. Tanaka, N. Nishiwaki, and M. Hirata, Semi-lnt. Symp. ISME, Tokyo 2,77 (1967). 53. V. L. Baskov, I. V. Kuraeva, and V. S. Protopopov, Teplofz. Vys. Temp. 15,96 (1977). 54. V. M. Eroschenko, E. V. Kuznetsov, and S . V. Harchenkov, in “Heat/Mass Transfer-MIF,” Sect. I , Part 2, p. 46. ITMO, Minsk, 1988. 55. V. N. Popov and N. E. Petrov, Teplofz. Vys. Temp. 23,309 (1985). 56. N. E. Petrov and V. N. Popov, Teploenergetika No. 10,45 (1988). 57. C. A. Fritsch and R. J. Grosh, J. Heat Transfer 85, 1 (1963). 58. K. Nishikawa, T. Ito, M. Kinoshita et al., Tech. Rep., Kyushu Uniu. 40,986 (1967). 59. H. Kato, N. Nishiwaki, and M. Hirata, Bull. JSME 11,654 (1968). 60. V. S. Protopopov and G. K. Scharma, Teplofz. Vys. Temp. 14,781 (1976). 61. V. I. Deev, A. K. Kondratenko, V. I. Petrovichev et al., Int. Heat Transfer Conf., 6th, Toronto NC-4 (1978). 62. S. Hasegawa and K. Yoshioka, Mem. Fac. Eng.. Kyushu Univ. 26, 1 (1966). 63. K. Nishikawa and T. Ito, lnt. J . Heat Mass Transfer 12, 1449 (1969). 64. S. P. Bestchastnov, P. L. Kirillov, and A. M. Saykin, Teplofiz. Vys. Temp. 11,346 (1973). 65. I. V. Kurayeva, N. N. Klimov, V. S. Protopopov, and N. E. Ferubko, in “TeplomassoobmenMI,” Vol. 1, Part 1, p. 116. ITMO, Minsk, 1984. 66. N. N. Klimov, I. V. Kurayeva, and V. S . Protopopov, Vestsi Akad. Navuk BSSR, Ser. FizEnerg. Nauuk No. I, 106 (1986). 67. T. R. Seetharam and G. K. Sharma, Heat Transjer 1982: Proc. Int. Heat Transfer Conf., 7th, Munich, 2, NC 36,329. Hemisphere, New York, 1982. 68. V. N. Popov and G. G. Yankov, in “Teplomassoobmen-VII,“ Vol. 1, Part 2, p. 143. ITMO, Minsk, 1984. 69. V. N. Popov and G. G. Yankov, Teploenergetika No. 3,30 (1985). 70. G. K. Sharma and V. S . Protopopov, Proc. Natl. Heat Mass Transfer Conf., 3rd, Bombay Pap. NO. HMT-75 (1975). 71. B. G. Van der Hegge Zijnen, Appl. Sci. Res., Sect. A 6, 129 (1956). 72. 0.K. Smirnov and S. N. Krasnov, Teploenergetika No. 4,86 (1978). 73. A. Watson, Int. J. Heat Mass Transfer 20,65 (1977). 74. D. E. Daney, P. R. Ludtke, and M. C. Jones, J . Heat Transfer 101,9 (1979). 75. D. A. Labuntsov and P. A. Mirzoyan, Teploenergetika No. 3,2 (1983). 76. A. T. Sinitcin, Izu. Sib. Otd. Akud. Nauk SSSR 2(8), 81 (1979). 77. V. V. Sevastyanov, A. T. Sinitcin, and F. L. Yakaytis, Teplofz. Vys. Temp. 18, 546 (1980). 78. A. T. Sinitcin, Teplofz. Vys. Temp. 18, 121 1 (1980). 79. V. V. Sevastyanov, A. T. Sinitcin, and F. L. Yakaytis, in “Teploobmen, Temperatur. Regim i Hydrodin. pri Generatcii Para,” p. 48. Nauka, Leningrad, 1981.

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80. N. F. Erohin and B. 1. Kalyanov, Teploenergetika No. 11, 50 (1980). 81. E. M. Maevskiy, V. 1. Vetrov, and V. G. Razumovskiy, in “Teploprovodnost i Convect. Teploobrnen,”p. 63. Nauk. Durnka, Kiev, 1980. 82. V. B. Ancudinov and V. A. Kurganov, Teplojiz. Vys. Temp. 19, 1208 (1981). 83. E. P. Fedorov, L. S. Yanovskiy et al., in “Sovrernen. Probl. Hydrodin. i Teploobmena v Element. Energet. Ustan. i Cryogen. Techn.,” p. 10. Vses. Zaochn. Mashinostroit. Inst., Moscow, 1986.

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ADVANCES IN HEAT TRANSFER, VOLUME 21

Advances in Condensation Heat Transfer ICHIRO TANASAWA Institute of Industrial Science Unioersity of Tokyo Roppongi, Minato-ku. Tokyo 106, Japan

I. Introductory Remarks Condensation is still one of the most important heat transfer processes in many energy conversion systems, such as electric power generation plants. As is well known, the industrial revolution in the eighteenth century owed much to improvements of steam engines by James Watt and other engineers; the greatest achievement that James Watt performed was that he greatly improved the efficiency of the steam condenser, which was a direct-contact type. Although the surface condenser, for which James Watt had taken a patent in 1769 [l], was not in practical use during Watt’s lifetime, because of immature manufacturing technology, it has subsequently played a most significant role as a heat-removing device in power generation, refrigeration, or air conditioning systems. The theory of heat transfer by condensation appeared much later than the development of condensers. Nusselt’s pioneering paper [2] on film condensation of steam was published in 1916, a century and a half after Watt’s work. A study on dropwise condensation by Schmidt et al. [3] was reported in 1930. Like most of the research in heat transfer, theories followed industrial developments. However, the situation seems to be changing. Although steam and other vapors of pure substances have been employed as working fluids of the energy conversion cycles, use of multicomponent media is developing, especially in high-performance heat pump systems. Because condensation of multicomponent vapors is much more complex in nature, fundamental studies are indispensable before designing an industrial system, both from thermodynamical and from heat transfer points of view. Cooperation between fundamental research and design engineering becomes more important. 55 Copyright 01991 by Academic Press, Inc. All rights or reproduction in any lorn reserved.

56

ICHIROTANASAWA

In this review an attempt has been made to put more emphasis on the areas of condensation heat transfer that have made progress in the past 15 years. Some of the fundamentals of condensation, such as nucleation and growth of a liquid droplet in bulk or surface condensation, are omitted. Basic mechanisms of dropwise condensation and direct contact condensation are not mentioned in detail. These are described in detail in other volumes of Advances in Heat Transfer. For example, homogeneous nucleation has been described by Springer [4], some of the mechanisms of dropwise condensation have been described by Merte [S], and direct contact condensation has been extensively discussed by Sideman and Moalem Maron [ 6 ] . In the present work, the various types of condensation are introduced in Section 11. Then the wettability of the surface, which determines the mode of surface condensation, film or dropwise, is discussed. In Section 111 the transport process at the vapor-liquid interface and the arguments on whether the condensation coefficient takes the value of unity are discussed. In Section IV dropwise condensation is dealt with, focusing attention on the results of recent investigations. Also described are how to promote dropwise condensation and how to maintain it in condensers for industrial use. Film condensation is discussed in Section V. However, the classical theory of film condensation is skipped, and only the results of recent achievements are introduced. Special emphasis is put on condensation of multicomponent vapors. Use of multicomponent working fluids has become more important to increase efficiencies of thermodynamic cycles for advanced heat pump systems. However, there remains much to be done to predict and enhance heat transfer performance of condensers in which multicomponent vapors condense. Section VI is assigned to the techniques of enhancement of condensation heat transfer. In principle, there are not so many choices for the augmentation of condensation. Use of surface tension force is one of the most sophisticated ways for it, because it does not require extra energy. However, it has been confirmed lately that an electric field is very efficient for this purpose, although the method requires the consumption of some electric power. Examples of both of the techniques are discussed in some detail. Concluding remarks include the author’s personal view on the future trends in research on condensation heat transfer. More than 100 publications are provided as references. However, this is not a complete list of publications on condensation heat transfer. On the contrary, a considerable number of excellent papers are not cited. Works done by our predecessors and colleagues are selectively quoted only whenever it proved necessary. When similar papers had been published on a certain subject, sometimes only one of them was quoted, although an attempt was made to

CONDENSATION HEATTRANSFER

57

choose the paper published earlier. The author would hope that this writing will make some contribution to the further development of condensation heat transfer.

11. Fundamentals of Condensation

A. TYPESOF CONDENSATION

In this work the word “condensation” represents the change of phase from the vapor state to the liquid state due to cooling. When the vapor is cooled strongly enough, the liquid phase changes further to the solid phase. Or, in some cases, the vapor, when cooled, turns directly into a solid. This last process is the reverse process of sublimation. Condensation that proceeds to solidification is out of the scope of this text. From the microscopic point of view, condensation takes place when the molecules in the vapor phase impinge on the liquid surface, which is in contact with the vapor, and remain there in the liquid. However, there are always some molecules escaping from the liquid surface into the vapor, freed from the intermolecular force field acting in the liquid phase. This is microscopic description of the evaporation process. Net condensation occurs when the number of molecules crossing a unit area of the vapor-liquid interface per unit time into the liquid phase is larger than the number crossing toward the vapor phase. When the number of impinging moleculesexceeds the number of escaping molecules, net evaporation is said to occur. Usually the adjective “net” is omitted and net condensation and net evaporation are simply called condensation and evaporation. The microscopic process of condensation will be discussed in more detail in another section of this article. As mentioned, condensation is the reverse process of evaporation. However, condensation is often discussed in contrast to boiling. It is not always exact because boiling is merely a type of evaporation in which the liquid vaporizes into the vapor phase formed in the bulk of the liquid. Condensation is classified into homogeneous condensation and heterogeneous condensation. The former occurs in a space where no foreign substance exists. It takes place stochasticallyas the result of fluctuation of motion of the vapor molecules. Such a process rarely plays an important role in actual heattransferring devices. On the other hand, the latter occurs with the aid of foreign materials, liquid or solid, forming the liquid phase on its surface or by using it as a nucleus. Heterogeneous condensation is divided further into condensation in space (volume or bulk condensation) and condensation on a surface (surface

58

ICHIROTANASAWA

condensation). Volume condensation takes place by making use of the small particles of liquid or solid, which are floating in space, as the nuclei. An example is the formation of clouds, mist, or fog. Apart from its phenomenological interest, volume condensation was rarely considered to be important so far as industrial applications were concerned. However, deposition of sodium vapor on a solid wall occurring in the liquid-metal-cooled fast-breeder nuclear reactor, which is now being developed, offers a problem of practical importance. The process is as follows. Liquid sodium, used as the primary coolant in the fast-breeder reactor (FBR) core, evaporates into argon, which is used as the cover gas. Then it is carried upward to a rotating plug or to the other solid walls by diffusion and/or convection and is deposited there. If the deposited sodium is solidified, it causes an operational problem. Recent investigation of this phenomenon has revealed that the rate of evaporation and, consequently, the rate of deposition of sodium increase to a great extent by formation of sodium mist in the bulk of argon gas. The higher the rate of volume condensation, the larger the probability of occurrence of trouble. Although similar phenomena are observed in devices such as refrigerators operated in low temperatures, fewer studies have been made on volume condensation than on surface condensation. Surface condensation is subdivided into condensation onto liquid surfaces and condensation onto solid surfaces. The former includes direct contact condensation on drops, jets, or films of liquid and the process of collapse of the vapor bubble during subcooled boiling. The number of practical applications is considerable. Solid surface condensation is literally condensation on a solid wall kept at low temperature. However, if we observe this process more closely, the vapor phase is not always in touch directly with the solid wall, because a layer of liquid is usually formed on it. Nevertheless, this type of surface condensation is different from vapor-liquid direct-contact condensation in the sense that the solid wall plays an important role in transfer of heat. When we notice the form of the liquid phase on the solid surface, we can distinguish dropwise condensation from film condensation, These two typical types of condensation will be discussed in the following section. The classification of condensation as mentioned above is illustrated in Fig. 1. Speaking very roughly, most of the research in heat transfer by condensation up to the present has been performed with regard to heterogeneous, surface,solid surface,and film condensation (categories B, b, p, and ii in Fig. 1). However, the recent tendency seems to indicate an interest in homogeneous, liquid surface, and dropwise condensation (a, a, and i) as the new areas of research. A more detailed discussion is not provided here, because it is not the objective of this article.

CONDENSATION HEATTRANSFER

A. Homogeneous Condensation

59

B. Heterogeneous Condensation

I a. Volume Condensation

b. Surface Condensation

A p.

a. Condensation on Liquid Surface

Condensation on Solid Surface

i. Dropwise Condensation

I

ii. Film Condensation

FIG. 1. Classificationof condensation phenomena.

B. FILMCONDENSATION AND DROPWISE CONDENSATION When condensation takes place on a solid surface, it occurs in one of two ways: dropwise or film condensation. Which of these two processes occurs is determined by the wettability of the solid surface. If the liquid formed by condensation does wet the surface, film condensation is observed. In film condensation, the liquid condensate forms a continuous film over the surface. This film flows down the surface under the actions of gravity, shear force due to vapor flow, or other forces. The latent heat released at the vapor-liquid interface is transferred through the condensate film and then through the solid wall, and is finally removed by a coolant, which in many cases is flowing on the other side of the condensing surface. A steady state is established when the rate of condensation is balanced with the rate of flow of the condensate. As is obvious from the above description, the rate of heat transfer by film condensation is determined by the thickness of the condensate film, which forms, in most cases, the greatest portion of the thermal resistance. The situation is different in the case of dropwise condensation. Shown in Fig. 2 are sketches of film and dropwise condensation processes and a picture of dropwise condensation. Dropwise condensation takes place when the liquid condensate does not wet the solid surface. The condensate does not spread, but forms separate drops. The process of dropwise condensation consists of a combination of several random processes. It is more easily understood if we view it as a cycle composed of four elemental subprocesses as shown in Fig. 3. After the vapor impinges on a surface cooled to a temperature below the saturation temperature, numerous minute droplets (initial droplets) are

ICHIROTANASA WA

60

FIG.2. Film and dropwise condensation.

formed, releasing the latent heat of condensation. These droplets grow very rapidly due to the continuing direct condensation of vapor onto them. Because the distances between neighboring droplets are very small, some of the droplets touch each other and coalesce to form larger drops by the action of surface tension force. At each coalescence, the drops shift their positions a little, leaving open areas on the surface where initial droplets are generated again. These droplets also grow by direct condensation and coalescence, but

Departure from surface

Growth due to direct condensation

CONDENSATION HEATTRANSFER

61

most of them vanish when they are absorbed by larger neighboring drops. Because the process of drop growth is the repetition of two steps, frequent condensation and coalescence as described above, it is almost impossible and does not make sense to attempt to treat these two steps separately. There have been a number of studies in which such a mistake has been made in discussing the process of drop growth. At any rate, when the drops grow to a certain limiting size, which we call “departure size,” they are swept off the surface by the actions of gravity, vapor shear force, and other external forces. As the drops depart the surface, they take in other droplets within their path, sweeping clean a portion of the surface, where again droplets generate anew. Thus, one cycle of dropwise condensation is completed. To fully understand these four fundamental subprocesses, which comprise the entire cycle of dropwise condensation, and to know the interrelation between these subprocesses as well, are the essential tasks in investigating the mechanism of dropwise condensation. As will be discussed in Section IV, such tasks await further elucidation.

C. WETTABILITY When vapor condenses on a cooled surface, the type of condensation, film or dropwise, that does occur is dependent upon whether the liquid condensate wets the surface. In the present case, the word “wet” means a phenomenon illustrated in Fig. 4.When we place a drop of liquid on a solid surface, either the drop spreads over the surface to form a thin film or it does not spread, but rather forms a cap-shaped droplet. In the former case we say the liquid wets the surface, and in the latter case we say the liquid does not wet the surface. As everyone may understand at once, this phenomenon is related to the surface tension of the liquid. However, things are not so simple. If, then, we try to put a small amount of liquid into a capillary tube, as illustrated in Fig. 5, the liquid either proceeds into the tube or it will not go

FIG.4. Wettability of a solid surface.

c---

FIG.5. Wettability in a capillary tube.

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ICHIROTANASAWA

S

FIG.6. Balance between interfacial forces.

into the tube. We may say that the liquid wet the tube in the former case, but that it does not wet the tube in the latter case. The criterion of wettability in this situation is different from that of the case shown in Fig. 4. Let us now discuss the situation drawn in Fig. 4. When a pure vapor condenses on a solid surface, three phases appear: solid, liquid, and vapor. Three kinds of interfaces exist between these three phases: liquid-solid (Is), solid-vapor (sv), and liquid-vapor (lv) interfaces. If we denote the interfacial energies attributed to the three interfaces as cis, csvand oIv,respectively, wettability is determined from the interrelation between these three interfacial energies. In the first place, if the liquid does not wet the surface in the sense demonstrated in Fig. 4, the balance of three interfacial forces (which are identical to the corresponding interfacial energies) leads to the following relation (see Fig. 6): OS"- Ols = 0,"cos 8

(1)

where 8 is called the contact angle, which is the angle made by the liquid-solid interface and liquid-vapor interface at the point where the three interfaces meet. It is important that the angle should be measured inside (not outside) the liquid phase. Equation (1) is usually called Young's equation. Although Eq. (1) seems quite reasonable, it is not so simple. In the first place, it seems that the contact angle is determined uniquely if three substances (solid, liquid and vapor) and their thermodynamical conditions (temperature, pressure, etc.) are specified. This is not true. The contact angle takes a seemingly arbitrary value between two extremes called the receding contact angle and the advancing contact angle, respectively (Fig. 7). The change of the contact angle between these two values is called hysteresis, the cause of which is not fully understood. Second, from a microscopic point of view, the three phases do not form a point (or line) of contact as shown in Fig. 6. A balance of forces considered within an area a few molecular thicknesses from the geometrical point of

CONDENSATION HEATTRANSFER

63

A

FIG.7. Hysteresis of contact angle.

contact may be different from that given by Eq. (1). It should be understood that Eq. (1) holds for a finite virtual volume taken around the point of contact. Let us proceed with our discussion, suspending these issues for the present. When a cap-shaped drop resides on the surface as shown in Fig. 6, the adhesive force Fa between the liquid and solid is derived from a thought experiment as illustrated in Fig. 8:

(2) Fa = osv + ~ l v 01s where the first two terms on the right-hand side represent the interfacial energies corresponding to newly formed interfaces and the third term is for the interfacial energy corresponding to the vanished interface. If we substitute Eq. (1) into Eq. ( 2 )we obtain Fa = clv(1+ C O S ~ )

FIG.8. Adhesive force.

(3)

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ICHIROTANASAWA

FIG.9. Cohesive force.

This equation shows that Fa = 0 when 0 = 180", which corresponds to the situation that the drop has no affinity at all with the solid surface. A drop of mercury on a glass plate shows a characteristic just like this. On the other hand, the cohesive force F, due to the intermolecular attraction is derived by considering a situation as shown in Fig. 9. If the adhesive force holding the liquid against the solid surface is too strong to release the drop from the surface, the drop may be broken inside, as illustrated. If this happens, two liquid-vapor interfaces are formed anew. Thus, the cohesive force is F, = 261,

(4)

The criterion for the spreading of the liquid drop depends on which one of the two forces, Fa or F,, is the larger. From Eqs. (2) and (4), the spreading force F, is defined as follows: F, = Fa - F, = oSv- 01"

- 01,

(5)

The spreading force also represents the energy needed to reduce the unit area of the solid-liquid interface. Substitution of Eq. (1) (Young's equation) into Eq. ( 5 ) gives F, = O,,(cose - 1)

(6) When the drop does not spread over the surface, F, < 0, because lcos 01 < 1 and 0 < 8 I 180". If 8 = 0, then cos0 = 1 and F, = 0, representing the situation that the liquid spreads over the surface without having a finite contact angle. Although any contact angle 8fails to yield F, > 0, it is possible if the solid-vapor interfacial energy oSvis sufficiently large in Eq. (5). This situation also permits the liquid to spread over the surface. In the ideal situation, the liquid spreads until it forms a monomolecular layer. As stated above, whether the liquid wets the surface or not is dependent upon the size of the three interfacial energies. However, there are a couple of issues to be noted. First, two out of the three interfacial energies, Q,, and olS,are difficult to measure. Both of these are energies on the solid surface. But the characteristicsof the solid surface change often, depending on how the surface

CONDENSATION HEATTRANSFER

65

is processed (machined, ground, rolled, heat treated, etc.) and on adsorption of foreign substances. To specify the proper ch.aracteristicsof the surface is very difficult. Second, in connection with the above, the characteristics of the interface are determined by the state of the molecules within the thickness of a monolayer or, at the maximum, a few molecular layers. This fact is important with regard to the method of promotion of dropwise condensation, which will be discussed later. To sum up, the interrelation between the interfacial energies, which makes the liquid wet the surface or not, is theoretically clear, but in reality it is not. In many cases, it is difficult to know in advance, when the vapor of a certain substance condenses on the surface of some material, which one of the two types of condensation occurs. To speak very roughly, vapor of pure water condenses filmwise on clean metal surfaces. To cause dropwise condensation of steam on the metal surface, we have to cover the surface with a thin layer of a foreign substance that has a low interfacial energy. To obtain dropwise condensation of an organic vapor, such as freon, is much more difficult because of the very low surface tension of the liquid phase. A substance that causes dropwise condensation is called a promoter. Substances that are effective as promoters will be mentioned in Section IV.

111. Transport Process at Vapor-Liquid Interfaces A. INTRODUCTION

When film condensation occurs on a solid surface, the most dominant resistance to heat transfer in most cases is the thermal resistance of the condensate film. The thermal resistance of the condensate film was first analyzed by Nusselt [2] more than 70 years ago, and his work still retains its usefulness. On carrying out the analysis, Nusselt made, both explicitly and implicitly, a lot of assumptions for simplification.His assumptions are that the liquid condensate flows smoothly and steadily due to gravity in laminar flow; there are no noncondensable gases in the vapor phase; no vapor shear force acts on the liquid-vapor interface; momentum terms are negligible; the temperature distribution in the condensate film is linear; transfer of the sensible heat in the liquid film is also negligible, with only the heat transported across the liquid film being the latent heat of condensation released at the liquid-vapor interface; and the temperature of the liquid-vapor interface is equal to the saturation temperature of the vapor. The last assumption is to be discussed later.

66

ICHIROTANASAWA

Under those assumptions, Nusselt derived formulas for condensation on a vertical flat plate and a horizontal circular tube. The result for the former is

where h is the local heat transfer coefficient at a distance z from the top of the plate (see nomenclature at the end of this article for the rest of the symbols). Forty years after Nusselt's work, Rohsenow et al. [7,8] modified the analysis, including the effect of the buoyancy force acting on the liquid film, transfer of the sensible heat in the condensate flow, and nonlinear distribution of the temperature in the condensate layer, with the following result:

where AT = T, - T,. Comparison of Eqs. (7) and (8) indicates that the effect of the buoyancy force is a change from p1 to p1 - p v , which is significant only near the critical pressure. The effects of the transport of sensible heat and the nonlinear temperature distribution cause a change of h,, to h,,[l + 0.68(c1ATlh,,)]. This also is a small effect for most applications, because the additional term in the blanket, 0.68(c,AT/hf,),remains small as long as the Prandtl number of the liquid is not too small. For low-Prandtl-number liquids, to which liquid metals correspond, both the effects of the momentum change in the liquid film and the nonlinear temperature distribution become important, and the deviation from the modified Nusselt equation becomes larger, as shown in Fig. 10 [9,10]. However, in most applications the magnitude of (cIAT/hfg)is in the lower range and the simple Nusselt formula is usable. Other factors that change the heat transfer coefficient are the vapor shear force and the noncondensable gases. The former tends to increase and the latter to decrease the heat transfer coefficient (for discussion, see Section V). Generally speaking, the simple Nusselt analysis agrees rather well with experimental data obtained for film condensation of nonmetal vapors, if proper account for various effects is taken whenever necessary. However, the situation is quite different insofar as condensation of a metal vapor is concerned. This is related to the assumption in defining the heat transfer coefficient that the temperature at the liquid-vapor interface equals the saturation temperature of the vapor. This assumption, of course, is false, and leads to a big error when condensation of a metal vapor at low pressures is to be dealt with. The assumption that the temperature at the liquid-vapor interface exactly equals the saturation temperature of the vapor seems doubtful if we imagine a

CONDENSATION HEATTRANSFER

67

1.4 -

-

0.4

I

0.002

l

I

l

0.01 0.02

l

l

0.05 0.1 0.2

,

0.5

I

1.0 2.0

Ce A T /hf, FIG.10. Effect of the Prandtl number on the heat transfer coefficient of film condensation. (-), Stagnant vapor, vertical plate. From Chen 101; (--), z = 0, vertical plate and horizontal tube. From Sparrow and Gregg [9].

situation wherein a vapor and a liquid layer, both of which are at the saturation temperature, coexist. In this case, both of the phases are in thermal equilibrium and no heat should be transferred between the two. If we observe microscopically, there might be transport of molecules crossing the liquidvapor interface in both directions, as discussed in Section 11. However, this does not cause any net condensation or evaporation, and the state of thermal equilibrium continues. The situation is similar when the condensate surface with the saturation temperature faces the saturated vapor. No net condensation must take place under such a condition. For net condensation to occur there should be a finite amount of difference between the saturation temperature of the vapor, T,, and the temperature of the liquid-vapor interface, T . As will be made clear, the difference between T, and is not so large for nonmetal vapors except at very low pressures, but for metal vapors the difference can be large. This is schematically depicted as in Fig. 11. If T, differs from appreciably, the condensation heat transfer coefficient based on ( T , - Tw),as in Eq. (7) or (8), will not predict a true value.

B. THEORYOF INTERPHASE MASSTRANSFER According to the kinetic theory of gases, the statistical behavior of a gas at a certain temperature is described by an appropriate velocity distribution function. If the gas is in a uniform steady state, so that its macroscopic properties do not vary with time or position, then the function known as

ICHIROTANASAWA

68

T

FIG. 11. Temperature distribution in vapor and liquid film in condensation of metal vapor.

the Maxwell velocity distribution is derived. If we denote the three velocity components of the gas molecule in Cartesian space by U, V , and W, the Maxwell velocity distribution function is expressed as

where R is the gas constant, and u, V, and W are the components of the mean linear velocity due to global mass motion of the total system. Schrage [111 used this Maxwellian distribution kinetic theory to analyze the mass transfer process at the liquid-vapor interface. As will be mentioned later, Schrage’s analysis is rather easy to understand, but is based on a few assumptions for simplification. Some modifications have been made by other researchers. An article by Rohsenow [l2] gives a compact summary of those works. If the velocity of vapor molecules obeys the Maxwell distribution, the rate of net condensation m per unit area is expressed by the following relation:

where r = i +

m

pv dmm

Here, the first term on the right-hand side of Eq. (10)(without acT)represents the mass flux of molecules from the vapor phase that impinge on the surface.

CONDENSATION HEATTRANSFER

69

The coefficient o,, the condensation coefficient, represents the fraction of molecules captured by the liquid. The term r corresponds to the fact that the vapor as a whole progresses toward the surface as long as net condensation takes place. This “progress” velocity should be superimposed on the Maxwell velocity distribution. In Eqs. (10) and (1l), P, is the vapor pressure at the liquid surface and is assumed to be the saturation pressure corresponding to T, and to be identical with the pressure of the system. The second term of Eq. (10) (without oe) represents the emission from the liquid surface at a uniform temperature T . Pi here is the saturation pressure corresponding to T . The coefficient oe,the evaporation coefficient, represents the fraction of molecules that actually leave the surface. It is assumed that the emission of molecules from the liquid surface is not affected by the higher pressure in the vapor when net condensation is occurring. Substitution of Eq. (11) into Eq. (10) gives

If we assume that the condensation coefficient is equal to the evaporation coefficient and is simply denoted o,Eq. (12) becomes

It should be noted here that there is no theory that has proved or disproved that the assumption that 0, = oe is correct whenever net condensation or evaporation takes place. Of course, the assumption must be valid at an equilibrium state when T, = T , because no mass transport should occur. However, let us suspend our doubt for the present. Equation (13) can be put into the following form if we assume (T, - q)/ T, << 1, which is valid except when the vapor temperature is very low:

. 20 m=-

(14)

When the two terms in the parentheses on the right-hand side of Eq. (14) are compared, the first term, (P, - Pi)/P,, is usually much larger than the second Even for pressure as low as 0.1 kPa for both metal vapors term, (T, - T)/(2T,). and for water vapor, the second term is around 3-4% of the second. At higher pressures this relative magnitude is much smaller. Thus, if we neglect the second term, Eq. (14) is written as

.

m=-

20 2-0

P, - Pi

JZ

70

ICHIRO TANASAWA

Further, from the Clausius-Clapeyron relation

If this approximation is employed, Eq. (15) becomes

Here, let us convert the rate of condensation m into the heat flux using the relation

4 = hfgrit Then Eq. (17) becomes

Or we can derive from Eq. (19) an apparent heat transfer coefficient at the liquid-vapor interface, hi:

We should note that this interfacial heat transfer coefficient is defined for the temperature difference (T, - T). It means that the total resistance to condensation heat transfer can be written as 1

1 +hf hi

-=-

h

1

where (l/h,) is the thermal resistance due to the liquid film and h, is expressed by Eq. (7) or (8). The interfacial heat transfer coefficient, or the temperature “jump” at the liquid-vapor interface (T - T )can be obtained from Eq. (20),if we assume the value of 0. Shown in Table I are the interfacial heat transfer coefficients hi and the vapor-to-interface temperature differences, calculated for three vapors at different pressures under the condition of a constant vapor-to-wall temperature difference, i.e., T, - T, = 10 K. The condensation coefficient is assumed to be unity. It is known that the interfacial heat transfer coefficient of water vapor at atmospheric pressure is extremely high, being almost three orders of magnitude higher than the heat transfer coefficient of film condensation. This is the cause of the high heat transfer rate of dropwise condensation. The

TABLE I

INTERFACIAL HEATTRANSFERC O ~ C I E N T AND S TEMPERATURE DROPSFOR VARIOUSVAPORS" Substance Property Molecular mass M (kg/kmol) Gas constant R (J/kg K) Vapor pressure Pv (atm) Vapor temperature T, (K) Vapor density P" (kg/m3) Interfacial heat transfer coefficient hi (MW/m2K) Interfacial temperature drop T, - T(K) Temperature drop in condensate - T,(K) (I

Water

Sodium

18.052

22.99

461.5 1.o

0.1

373.15

319.24

0.598

Potassium 39.098

361.7 0.01

1.o

0.1

280.32

1154.6

949.1

212.65

0.0 1 806.7

1.o 1029.7

0.01 697.4

0.0690

0.00783

0.270

0.0315

0.00361

0.504

0.00699

2.57

0.383

5.74

0.771

0.117

3.71

0.100

0.007

0.037

0.18

0.33

2.3

7.3

0.30

6.3

9.99

9.96

9.82

9.67

7.7

2.7

9.70

3.7

15.7

Calculations are carried out under a vapor-to-surface temperature difference T, - T, = 10 K. Thermal properties are from Refs. [128-1321.

72

ICHIROTANASAWA

interfacial temperature drops for water vapor are relatively small, whereas those for metal vapors are much larger, especially when the pressure is low.

c. MODIFICATION OF SCHRAGE’S THEORY There have been a number of attempts to improve Schrage’s result. Some of these are referred to in the following discussion. Huang [13] has considered that molecules impinging on the surface at low grazing angles are not captured. His analysis has led to the following result:

If Eq. (22) is compared with Eq. (14), the only difference is that 2042 - cr) in Eq. (14) is replaced by n/2 in Eq. (22).This means that Eq. (22)is identical with Eq. (14) only if cr = 2n/(4 + n) = 0.880, which is equivalent to say that the upper limit of cr in Schrage’s equation should be 0.88. However, Huang’s assumption that molecules at a small grazing angle are not captured has not been verified either theoretically or experimentally. Springer and Patton [14] presented an analysis using the double-flow distribution function and the moment method and obtained the following result : 1

P”

/ crt r Ai Pl ( l 1+ AT 4/15Kn 2-ac. 1 + 2 + 8/15Kn 20,

m=m ‘L

where Kn is the Knudsen number, which is the ratio of the mean free path of the molecules to some characteristic length. Because all available data for the condensation coefficient have been taken at pressures where the Knudsen number is almost zero, Eq. (23) is identical with Schrage’sequation in most of the cases. Labuntzov [151 reported an analysis also using the double-flow distribution function. His result, which is valid for near-equilibrium condensation, can be written as m=

2a P” - Pi 2 - 0.798a JZIIRT,

The term involving the coefficient cr is a little different from that in Eq. (15). If the cr coefficients in Eqs. (14) and (23) are distinguished by subscripts S and L,

CONDENSATION HEATTRANSFER

73

respectively, and equating the both terms, we get as = 0 . 7 9 8 ~ ~

(25)

This means that Schrage’s coefficient a, is about 80% of Labuntzov’s a,. Labuntzov and Kryukov [I61 have revised the analysis, taking into consideration a unique character of the nonequilibrium state of vapor near the liquid-vapor interface, where the Knudsen number is finite. They used a model of the interface region in which the liquid surface and the uniform gas are separated by a “Knudsen layer” and a “gas-dynamic flow region.” They treated the former region by solving approximately the Boltzmann equation and the latter region on the basis of the continuum mechanics, and derived a formula that described the process of intensive condensation as follows:

In deriving this formula, the condensation coefficient is taken as unity. A big difference between this equation and the equations mentioned before is that the term multiplied by (P, - P i ) / J m is not constant in Eq. (26) whereas it is constant in the other equations, such as Eqs. (15) and (24).It seems that, at present, Eq. (26) is the most reliable equation for describing mass transfer at the liquid-vapor interface, but only if the condensation coefficient is actually very close to unity.*

D. EXPERIMENTAL DETERMINATION OF CONDENSATION

COEFFICIENT

Both the condensation and the evaporation coefficients have been measured in a variety of experiments. Table I1 summarizes the previous experimental values for the condensation coefficient of water vapor. Very low (0.006)to very high (1.0)values have been reported.’ Shown in Fig. 12 [20] is the dependence of the condensation coefficient of metal vapors on the system pressure. Although the degree of scatter of the data is considerable, it looks as though there is a tendency for the condensation coefficients to be close to unity when the pressure is low, and to decrease as the pressure increases. Wilcox and Rohsenow [20] considered that

* The condensation mass flux for the arbitrary value of the condensation coefficient is given in Labuntzov [lS], but is not shown here. * Most of Table I1 is from Mills and Seban [17]. The only exceptions are the results by Narusawa and Springer [lS] and Hatamiya and Tanaka [19].

ICHIROTANASAWA

74

TABLE I1 EXPERIMENTAL VALUES FOR CONDENSATION/EVAWRATION COEFFICIENT OF WATER Temperature Investigators‘ AltY ( 1 )

18

Alty and Nicoll(2)

18

AltY (3)

u

(“C)

-

60

0.006

60

0.01

-8-

+4

-

Nature of the experiment 0.016

0.02

0.04

15

0.036

100

0.02

Hammecke and Kappler (6)

20

0.045

Hammecke and Kappler (7)

?

0.100

Alty and Mackay ( 4 )

Pruger (5)

Delany et al. (8) Narusawa and Springer [18]

0 19

Hickman (9) Navabian and Bromley (10) Jamieson ( 11 )

-

43

0.0415

27

0.038

0

10 0

Berman (12) Hatamiya and Tanaka [19]

-

-

-

0.0265 0.2

0.42

50

0.35 5 u < 1.0

70

0.35

Near 1.0

?

7

-

27

0.2 I u 5 0 . 6

evaporation from a suspended drop Evaporation from a suspended drop Evaporation from a suspended drop Evaporation from a suspended drop Evaporation from a horizontal surface Evaporation from a horizontal surface Evaporation from a horizontal surface Evaporation from a horizontal surface Evaporation from a horizontal surface Evaporation from a tensimeter jet Film condensation on a fluted tube Condensation on a tensimeter jet Film condensation on a horizontal cylinder Dropwise condensation on a flat plate

References provided here are not given in the general reference list: ( 1 ) Proc. R. SOC.London, Ser. A 131, 554(1931);(2)Can.J. Res.4,547(1931);(3)Philos.Mag. 15,82(1933);(4)Proc.R.Soc. London,Ser.A 149,104 (1935);(5) Z. Phys. 115, 202 (1940);(6) Z. Geophys. 19, 181 (1953);(7) Verk. Minist. N. Rhein-Wesu.3, 125 (1955); (8)Chem. Eng. Sci. 19, 105 (1964);(9) Ind. Eng. Chem. 46,1442 (1954);(10) Chem. Eng. Sci. 18,651 (1963); (11) Symp. Thermophys. Prop., 3rd, Purdue Unio.. Lafayette, Indiana p. 230 (1965); (12) Khim. Mashinostr. No. 36,66 (1961).

the apparent decrease of the condensation coefficient with the increase of pressure was due to a lack of precision of measurement. In determining the condensation coefficient 0, the method most commonly employed is to measure the heat flux and the surface temperature at the condensing surface, then to derive the temperature of the liquid-vapor interface from Eq. (8) (by replacing T, by TJ, and finally, using Eq. (19) or another

75

CONDENSATION HEATTRANSFER 20 1.0 c

.-5 V

i!

C

.-+

B %

0.1

V

0.01

0 I0 1

I

0.00 1

I

I

I

0.01

0.1

1.0

Vapor pressure, P, [ a t m l

FIG.12. Condensation coefficient data.

appropriate equation, to calculate 6.The major cause of error is in the determination of T,, - T . At a given heat flux q, the temperature difference T, decreases rapidly to a very small value as pressure increases. As illustrated in Table I, the vapor-to-interface difference T,, - T is of the order of, or less than, 1 K when the pressure exceeds 0.1 atm. Because some inaccuracies in the determination of heat flux and temperatures T, and 7.; are inevitable, it may be very difficult to obtain T,, - with high precision at higher pressures. Wilcox and Rohsenow [20] discussed the accuracy of measurement in determining the condensation coefficient of metal vapor. In most experiments the heat fluxes and the surface temperatures are obtained by measuring the temperature distribution in the heat transfer block and extrapolating the readings of several positions to the surface. They have postulated a Gaussian distribution in error that is related to the thermocouple hole size at each location. They have presented a relation for the standard deviation in the determination of the value of the surface temperature T,. This standard deviation is dependent on the hole size, the number and spacing of the holes, and the thermal conductivity of the block. The standard deviation is reduced by using a highly conductive solid as the block material, a smaller diameter, and a greater number of thermocouple holes. Shown in Table I11 is an example of the standard deviation in the determination of T, for three different block materials [12]. The standard

ICHIROTANASAWA

76

TABLE 111 STANDARD DEVIATIONS IN THE DETERMINATION OF WALLTEMPERATURE' Material of the condensing block

Number of holes

Diameter of holes (mm)

Standard deviation

Copper Nickel Stainless steel

6 3 3

1.2 1.6 1.6

0.13 1.6

a

("C)

14.4

q = 150 kW/mz.

deviation (T'. - Tw)is larger, if the number of thermocouple holes is reduced and the block material has poorer thermal conductivity. Comparison of Tables I and I11 suggests that we could hardly trust the values of condensation coefficients obtained at pressures higher than, say, 0.3 atm. Wilcox and Rohsenow have concluded that, taking into account the limitation of precision of measurement, the magnitude of the condensation coefficient is quite high-very close to unity. The present author considers that the manner in which Wilcox and Rohsenow reached their conclusion was intuitive rather than logical, though the conclusion might still be somewhat valid. However, very recently stronger experimental evidence has been presented by Ishiguro and Sugiyama [2 11, who, as a result of a very elaborate experimental study, have determined that the value of the condensation coefficient of potassium vapor is substantially unity, in the sense that Labuntzov and Kryukov [16] have defined it. Figure 13 shows their experimental apparatus. They took great care to avoid inaccuracies due to (1) errors caused by measurement of the heat flux and the surface temperature of the condensing surface, (2) errors in measuring the vapor pressure, (3) the effect of noncondensable gases, and (4) inconsistencies between the analytical model and the actual process. Shown in Fig. 14 is the result of the measurement. The variable 5 on the ordinate is defined as follows:

Comparison of Eq. (27) with Eq. (15) gives 1

20-

CONDENSATION HEATTRANSFER

77

to second

to

I

Thermocouple

1

we'1s

/condenser

Test condenser

second condenser

i Boiler

I

to distillation vessel

FIG.13. Experimental apparatus by Ishiguro and Sugiyama [21].

<

In this case takes a constant value. On the other hand, if comparing with Eq. (26),it follows that

where the condensation coefficient is assumed to be unity.

ICHIROTANASAWA

78 I

1

I

I

I

I

I

0.6 -

I

I

Tv= 633 K - 0.8 11.1

Y."

E

1.2

1.2

1.c

1.1

E

0.f

1.0

0.9 0.f

I

0.10 I

0.05

I 5

&(kg/m2s) (-),

FIG. 14. Dependence of ( [defined by Eq. (27)] on condensation mass flux. (0), Present result; Labontsov and Kryukov (u = 1); ( O ) ,Subbotin et al.

CONDENSATION HEATTRANSFER

79

The experimental results shown in Fig. 14 indicate that they all agree fairly well with the theory by Labuntzov and Kryukov. It means that the variable 5 is dependent on the condensation mass flux m, while Schrage’s theory yields a constant 5 as expressed by Eq. (28).If a state very near to equilibrium, that is to say, ti^ E 0, is considered, the Labuntzov-Kryukov theory gives a value g = 0.67, whereas 5 = 0.8 by Schrage’s theory, assuming 0, = 1. This means that the rate of condensation derived by Schrage’s equation is 1.19 times larger than that by the Labuntzov-Kryukov equation. The cause of this difference lies in the fact that the collision between the vapor molecules and the molecules emitted from the liquid phase is taken into consideration in the Labuntzov- Kryukov analysis, whereas such an interference is not considered by Schrage. Note in Fig. 14 that the variable 5 increases with ri2. This can be explained by considering the interference between the oncoming molecules and the departing molecules as described above. As the temperature of the vapor phase increases, the average momentum possessed by the emitted molecules becomes relatively smaller than the average momentum of the condensing vapor molecules, which prevent the molecules in the liquid layer from leaving. On the other hand, the effect of such an interaction becomes smaller as the temperatures of the phases converge. Similar conclusions were also reported by Necmi and Rose [22] and Niknejad and Rose [23] as the result of experiments on the condensation of mercury vapor. However, their results differed a little from the analysis by Labuntzov and Kryukov. It seems that most of the experimental results published recently indicate that the condensation coefficient is close to unity in the case of condensation of a metal vapor, though there is a little difference among the formulas describing the process of interfacial mass transfer. No studies have reported that the condensation or evaporation coefficient of water vapor is unity. Even very recent experiments have reported the value to be significantly smaller than 1.0. As shown in Table 11, Narusawa and Springer [18] reported a value of 0.2 as the evaporation coefficient. Hatamiya and Tanaka [I91 have estimated that the value of the condensation coefficient is in the range between 0.2 and 0.6, the most probable value being 0.4, from the result of an experiment on the dropwise condensation of low-pressure steam. That the condensation coefficient of water vapor is significantly smaller than unity is supported by some and doubted by others. Fujikawa and Maerefat [24] have discussed this problem on the basis of the transition-state theory of statistical mechanics. They insist that the condensation coefficient of a polyatomic fluid should be significantly smaller than 1.0 because the rotational motion of molecules in the liquid state is constrained. On the other hand, Stylianou and Rose [25] and Rose [26] suggest, on the basis of work done by Le Fevre [27], that, if polyatomicity of

80

ICHIROTANASAWA

the water molecule is taken into account, the data on the condensation coefficient should be amended to be closer to unity. For example, the values estimated by Hatamiya and Tanaka, 0.2-0.6, should be shifted to 0.38-0.93. The present author is not capable of judging which one of these theories is most reasonable. It seems that a much deeper understanding about molecular kinetic processes at the liquid-vapor interface is needed to reach a conclusion. IV. Dropwise Condensation A. MICROSCOPIC MECHANISMS

As described in Section 11, dropwise condensation is composed of several processes that are combined randomly both in space and in time. Figure 3, which shows what the author calls the “life cycle” of dropwise condensation, helps to understand the basic nature of the process. To understand each individual process of the life cycle of dropwise condensation and to know about the interrelationship between them are the most essential aspects of the study of the mechanism of dropwise condensation. It is likely that elaborate work done within the past 25 years has thrown sufficient light on those processes. But on closer examination of each of those studies, it can be concluded that there still remain surprisingly many unanswered questions. Table IV provides a summary of the nature of the problems that remain unresolved. Detailed descriptions of each of the problems will not be given here because of the space limitation. Only a few brief comments will be made below. Interested readers are invited to refer to the author’s monographs published elsewhere [28].

1. Initial Droplet Formation

Two opposite hypotheses have been proposed as to the mechanism of initial droplet formation: one is the film fracture hypothesis, which was first proposed by Jakob [29] and later supported by Welch and Westwater [30], and the other is the nucleation theory. But since Umur and Griffith [31] have insisted that the continuous growth of film is impossible on a nonwetting surface, and McCormick and Westwater [32] have made supporting microscopic observations, the nucleation hypothesis has been supported by most people. The present author makes no objection to it so long as the nucleation occurs on a completely dried surface. However, in the actual process of dropwisecondensation, it might be possible that a very thin film of condensate is left behind in the wake of a departing drop, and from there the initial drops are generated in a somewhat different way. But it does not matter to the macroscopic process of heat transfer whether or not the drops originate by nucleation.

CONDENSATION HEATTRANSFER

81

TABLE IV THINGS

NOT KNOWNABOUTTHE MECHANISM OF DROPWISE CONDENSATION

Classification of problems Item

Subitem

Supplementary explanation

Initial droplet formation

Mode of formation

Film fracture or nucleation? (Nucleation hypothesis is supported by the most of the researchers, but decisive proof has not yet been given) What parts of the surface can be the nucleation sites? Pits, grooves, extrudes, or defects of promoter layer? (Proof by observation is yet lacking) What is the magnitude? What is the relationship with the surface condition? What is the relationship with the surface subcooling? (Possible relationship with the heat transfer coefficient) Is the critical radius given by the Kelvin-Helmholtz equation correct? (This equation holds only for the thermodymic equilibrium process, whereas the drop growth process is nonequilibrium. Actual observation is necessary) How many drops are formed per unit area and per unit time? (Relationship with the nucleation site density and the surface subcooling) Theoretical analysis of the drop growth process due to condensation is necessary. (Heterogeneity of the surface temperature and its fluctuation should be taken into consideration) Can it be confirmed by the actual observation? How much is the drop growth augmented by coalescence? Dynamics of coalescence should be made clear (rate of coalescence, behavior of drop at coalescence) Surface temperature fluctuation due to coalescence should be measured. What is the cause of the hysteresis of contact angle? Can it be estimated theoretically? Can the departure size be decided theoretically? Departure size should be measured under various conditions What i s the relationship between the static and the dynamic departure size? Behavior of the departing drop, i.e., deformation, sliding velocity, and sweeping process, should be measured and analyzed

Nucleation site

Population density of nucleation site

Critical size for drop growth

Rate of drop formation Growth of drop

Growth by condensation

Growth by coalescence

Departure of drop

Mechanism of drop departure

Dynamics of drop departure

82

ICHIROTANASAWA

2. Drop Growth As already mentioned, it is fruitless to make an attempt to analyze the drop growth process due to condensation exclusive of coalescence. What remains for the future is to measure the rate of drop growth, simultaneously with the local surface temperature fluctuation.

3. Drop Departure To derive theoretically the departure size seems difficult for the time being, because of the yet unsolved phenomenon of “contact angle hysteresis,” as shown in Fig. 7. At present, we have no definite theory that can give an explanation for this. More extensive measurement is necessary, and the effects of external forces, surface conditions, and heat flux should be made clear. Once the relationship between the departure size and the external conditions is established, the heat transfer coefficient becomes predictable, as will be discussed later.

B. MEASUREMENT OF HEATTRANSFER COEFFICIENTS Experimental studies on the heat transfer by dropwise condensation have been carried out mostly under very limited conditions. It is only recently that the data on the heat transfer coefficient of dropwise condensation of steam at atmospheric pressure have been put in order. Those data for other vapors and measurements under different thermal conditions are still scarce. Therefore, the description that follows is mostly concentrated on dropwise condensation of steam at near-atmospheric pressure. 1. Results of Measurements on Steam at Near-Atmospheric Pressure

Since the first observation was made by Schmidt et al. [3], the heat transfer coefficient of dropwise condensation of steam near 1 atm has been measured by many other researchers. The results have been summarized by Tanner et al. [33], Graham [34], and Tanasawa [28], and show a considerably wide scatter. However, because the experimental techniques have shown great progress, the reproducibility and reliability of the experimental data have been established to some extent. We owe much to the work by Le Fevre and Rose [35,36] and Citakoglu and Rose [37]. At present, we can say that the heat transfer coefficient of dropwise condensation of steam at 1 atm, under normal gravitational acceleration and on a vertical copper surface, is about 250 i50 kW/m2 K, provided that there is no effect of noncondensable gases, the height of the condensing surface is less than about 50 cm,the heat flux is of the order of 0.1-1 MW/m2, and the steam velocity is less than 10 m/sec.

CONDENSATION HEATTRANSFER

83

The variation of the magnitude of the heat transfer coefficient mentioned above is mostly due to the effects of steam velocity and surface characteristics. As several authors have pointed out, the reasons that most of the past measurements reported much lower heat transfer coefficients may be (1) inaccuracy in the measurement of surface temperature and heat flux and (2) carelessness regarding the buildup of noncondensable gases on the condensing surface.

2. Measurements on Low-Pressure Steam Fewer results are available on the heat transfer coefficient of dropwise condensation of steam under pressures lower than 1 atm. Figure 15 shows the data reported by five different groups of researchers [34,38-411. There is a little difference among groups, but every result shows a tendency of decreasing heat transfer coefficients with decreasing pressures. 3. Effect of Drop Size

The heat transfer coefficient of dropwise condensation is dependent upon the distribution of drop sizes on the condensing surface. And what largely

2"

I

100

I

,

I

I

5

10'

5

1

1

1

1

1

102

Steam pressure. [ kPa 1

FIG.15. Dropwise condensation of low-pressure steam. ( O ) ,Tanner et al. 1381; (A),Graham [34]; (a), Wilmshurst and Rose, [39]; ( A ), Tsurta and Tanaka [40]; (0), Hatamiya and Tanaka ~411.

ICHIRO TANASAWA

84

determines the drop size distribution is the maximum drop diameter or the departing drop diameter, together with the fraction of the area covered by sliding drops. Thus, the author would like to emphasize that the relationship between the drop size and the heat transfer rate should be studied more intensively. Some of the factors that have an influence upon the heat transfer coefficient of dropwise condensation are closely related to the maximum drop diameter. They are, for example, the vapor velocity, the physical and chemical characteristics of the surface (such as roughness and wettability), and the inclination and the height of the surface. A number of reports have been published in which the effects of vapor velocity or the height or the inclination of the condensing surface are measured. Drop size measurements were not carried out for the most part, however. Tanasawa et al. [42] measured the dependence of the heat transfer coefficient on the departing drop diameter, using the gravitational, centrifugal, and steam shear forces to change the departing drop diameter. The result is shown in Fig. 16. It has been found that, no matter what kind of force may be used to prompt drop departure, the heat transfer coefficient is proportional to the departing drop diameter to the power of about -0.3. It is very interesting to note that Tanaka [43,44] and Rose [45] have theoretically derived quite similar conclusions.

?. Y B \ 3

zc

0.5

-

0.4 -

0.3 -

o 0.2

fi

2

X

gold plated silicone resin gold plate oleic acid

0.2

0.5

0

7 0

Y

vapor shear vapor shear centrifugal force inclined surface

0.1 0.1

0

1.0

2.0

Departing drop diameter, D Im 1

FIG.16. Dependence of heat transfer coefficient of dropwise condensation on departingdrop diameter.

CONDENSATION HEATTRANSFER

85

4. Condensation Curves In pool boiling, it is well known that the mode of heat transfer shifts from convection to nucleate boiling, then to transition boiling, and finally to film boiling, with the increase in the degree of surface superheat. The heat flux also varies with the superheat. The curve that represents the change of heat flux with the superheat is called the boiling curve and was first obtained by Nukiyama [46]. In dropwise condensation, a similar curve, which may be called a “condensation curve,” must be obtained. In the case of the condensation curve, however, it was very difficult to realize a large surface subcooling. The first measurement of condensation curves on steam at atmospheric pressure was carried out by Takeyama and Shimizu [47]. However, there was some doubt about the accuracy of their measurement. Tanasawa and Utaka [48] made measurements with much greater accuracy using a condensing surface with a special configuration of a concave sphere (Fig. 17). The results are shown in Fig. 18, with the result obtained by Takeyama and Shimizu included as a thin, solid line. It is seen that the shapes of the condensation curves are similar to those of the boiling curves, whereas the maximum heat fluxes are higher, exceeding 10 MW/m2. Dropwise condensation is maintained from a lower degree of surface subcooling up to the maximum heat flux. At a much larger surface subcooling, the condensate begins to accumulate on part of the condensing surface and a state of pseudofilm condensation is observed. When the surface temperature is lowered below O”C,a layer of ice is formed on the surface and film condensation occurs on the layer (on-ice condensation). One of the important conclusions derived from this experiment is that dropwise condensation continues up to a considerably larger heat flux. This

Condensing surface ro = 4.52mm Copper block (Purrty>999%)

Constantan wires 0 08 4 Positions of junctions r I = 5 08mm r 2 = 7 11 mm r 3 = 1005mm r ( = 14 l o r n per wire Cooling surface r5=60.0mm

FIG. 17. Heat transfer block with a concave condensing surface.

86

ICHIROTANASAWA

Degree of surface subcooling I K 1

FIG.18. Condensation curves.

suggests that the heat transfer performance on lower tubes among a vertical tube array may not be deteriorated much by impingement of liquid dripping from the upper tubes. In other words, the condensate inundation would not cause much of a problem in the case of dropwise condensation. This superposition has been verified by Tanasawa and Saito [49]. 5. Efleect of Material Thermal Properties

The surface temperature of the condensingsurface was once assumed tacitly (or carelessly) to be uniform and constant when defining the heat transfer coefficient by dropwise condensation. However, because there are drops of various different sizes on the surface, and because those drops are moving randomly due to coalescence and departure, the temperature and the heat flux on the condensingsurface must be nonuniform and incessantly fluctuating. As a matter of fact, such a temperature fluctuation has been observed by a number of investigators. Fluctuation and nonuniformity of the surface temperature are inherent to the finite thermal conductivity of the surface material. Its possible effect on the heat transfer coefficient was first predicted theoretically by Mikii: [SO].

CONDENSATION HEATTRANSFER

87

The thermal resistance (denoted R ) between the vapor and the condensing surface must be nonuniform over the surface because drops of different sizes are distributed. Owing to that, the distribution of the heat flux on the surface must also be nonuniform. If the thermal conductivity of the surface material is finite, the local surface temperature T, depends on the local heat flux. If the average heat flux over the entire surface is denoted as 4, and the corresponding surface temperature is denoted T,, then T, takes on a much larger value than T, where R is smaller. On the other hand, because the mean heat flux is an average of the local heat flux, it follows that

For an ideal condensing surface having infinitely high thermal conductivity, T, = T,, and then

4 = (T, - TYRO where 1 RO

-=

(31)

a jA

1 R

--dA

For an actual surface material with finite thermal conductivity, T,varies on the surface. If we define an average surface temperature T,, as

then Eq. (30)can be written as

In the experiment, we usually measure the average heat flux 4 and the surface temperature T,, and derive the heat transfer coefficient h from the following relation: = 4/(T -

T,)

(35)

By rewriting this equation, we get

This equation indicates that T, is not equal to T,, on a surface having a finite thermal conductivity, and that there exists another thermal resistance R, on top of the resistance R o . This thermal resistance R , is the constriction resis-

ICHIRO TANASAWA

88

tance, i.e., R , = (K* - T,)/4 (37) According to the MikiC theory, the heat transfer coefficient by dropwise condensation must be lower on a condensing surface made of a poor conductive material. That is to say, the heat transfer coefficient on a stainlesssteel surface must be lower than that on a copper surface. Several groups of researchers have carried out measurements to find out the effect of surface thermal conductivity on the heat transfer coefficient by dropwise condensation. Some of those results are plotted in Fig. 19. The figures shows two opposite tendencies. The results obtained by Aksan and Rose [Sl] (line A) make us consider that the heat transfer coefficient is substantially independent of the surface thermal properties. The results of experiments done afterward by Rose [52] and Stylianou and Rose [53] (in which the overall heat transfer rates are compared) have shown similar tendencies. On the contrary, Tanner et al. [54], Wilkins and Bromley [55], and Hannemann and MikiC [56] (linesB, C,and D) insist that the heat transfer coefficients on the poorly conductive materials are lower. Regarding such a discrepancy among the results, discussion continued for nearly 20 years. However, recent papers by Tsuruta and Tanaka [57] and Tsura and Togashi [58] seem to settle the controversy. In the first place,

h

I

\

B E

v

4

100

300

200

Thermal conductivity

(W/m

400

K)1

FIG.19. Dependence of heat transfer coefficient of dropwise condensation on the thermal Askan and Rose [Sl]; (A),Hannemann and MikiC [56]. conductivity of surface materials.(0), (0)Tanner et a/. [54]; ( A ) , Wilkins and Bromley [SS].

CONDENSATION HEATTRANSFER a

\

Conductor path width= 10 if m

p

0.5mm

89

,

Detail of sensing element

b Leading

Conductive

, s

-.

~ 0 7~0 m N ~

Quartz glass

FIG.20. Thin-film surface thermometer. (a) Pattern of aluminum resistor. (b) Make-up of thinfilm thermometer.

Tsuruta and Tanaka [57] derived theoretically a differential formulation, which describes the constriction resistance as the function of the surface thermal conductivity, the maximum and minimum drop radii, and the interfacial mass transfer resistance. Following this, Tsuruta and Togashi [SS] carried out an experiment to verify the theory. They used thin-film temperature sensors (Fig. 20) made of aluminum to measure the surface temperature directly and accurately. They measured the heat transfer coefficient of dropwise condensation on the surfaces of quartz glass (thermal conductivity A, = 1.3 W/m K), stainless steel (A, = 16.0 W/m K), and carbon steel (2, = 51.6 W/m K), under different reduced pressures of 1-10 kPa. Shown in Fig. 21 are their experimental results, together with the theory by Tsuruta and Tanaka. Results of measurement on a copper surface by Hatamiya and Tanaka [41] are also plotted. Agreement between experiment and theory is fairly good. It shows that the heat transfer coefficient by dropwise condensation of steam on the quartz glass plate is about one-tenth that on the copper surface, when compared at a pressure of 0.1 atm.

90

;

0

ICHIRO TANASAWA

0

5

3b.51 '

I

I ' 1

I

I

t

1

1

1

,

I

I

I

t

,

10

1

1

1

1

100

Steam pressure Ps, (kPal FIG.21. Heat transfer coefficient of dropwise condensation of steam on four different surface materials. Experimental data: (0),Quartz glass; (A), stainless steel; ( O ) , carbon steel; (A), copper. From Hatamiya and Tanaka [41].

In addition to theoretical considerations and experiments with homogeneous surfaces (which are assumed to be sufficiently thick), Tsuruta [59] has coped with a case in which a layer of poorly conductive material covers the surface of another material. Such a case is important when considering the use of coating by a hydrophobic material, e.g., poly(tetrafluoroethy1ene)(PTFE) as the promoter for dropwise condensation. One of the results is shown in Fig. 22. As is expected, the thicker the PTFE coating, the lower the heat transfer coefficient. 6. Heat Transfer Coeficients Obtained under Other Conditions

a. Measurements on High-pressure Steam. Heat transfer coefficients of steam at pressures higher than 1 atm have beem reported by several investigators. However, we will not introduce them here because the reliability of those results are doubtful.

CONDENSATION HEATTRANSFER

91

Steam pressure P,, IkPal

FIG.22. Heat transfer coefficient of dropwise condensation on surfaces covered with poly(tetrafluoroethy1ene)film. (-), PTFE surface, rmax= 1.5 rnm;(...), Nusselt theory, AT = 2 K.

b. Dropwise Condensation at Small Vapor-to-Surface Temperature Differences. In Section IV,B,4, the change of the heat flux along with the increase in the surface subcooling AT, i.e., condensation curve, is discussed. O n the contrary, the phenomenon that might occur when ATis reduced is also of interest. At smaller AT, the rate of drop nucleatjon may be reduced because the critical radius of drop formation must be larger. Because measurements under smaller ATaccompany technical difficulties, most of the results [33,34,60-621 reported up to now involve, more or less, some inaccuracy. More efforts will be needed in the future. c. Dropwise Condensation of Organic Vapors. When considering applications to condensers to be used in chemical plants (e.g., distillation plants) and energy conversion devices (e.g., heat pumps and refrigerators), dropwise

92

ICHIROTANASAWA

condensation of organic vapors should be investigated extensively. However, this has not been the case, mainly because a method of surface treatment to maintain dropwise condensation has not yet been established. Up to now, experiments have been performed with vapors of ethylene glycol (ethanediol), propylene glycol, aniline, nitrobenzene, and methanol, for example. Dropwise condensation has not been observed for vapors of freons and their substitutes. d. Dropwise Condensation of Metal Vapors. Reports have been published on dropwise condensation of mercury vapor. Among them, papers by Rose and colleagues [22,23,63,73,74], Kollera and Grigull [64,65], and Ivanovskii et al. [66] have yielded apparently reliable results. No reports have been published on dropwise condensation of other metal vapors, such as sodium, NaK, and potassium, though the liquid phases of these substances have rather large surface tensions. It should be noted that, as discussed in Section 111, the interfacial resistance to mass transfer at the liquid-vapor interface plays an important role in the condensation of metal vapors. The total resistance to condensation should be calculated using an equation similar to Eq. (21). C. HEATTRANSFER THEORY

The theory of heat transfer by film condensation was established by Nusselt [2] more than 75 years ago, but no complete theory has yet been proposed for dropwise condensation. This is due to the very complicated nature of the dropwise condensation phenomenon. In addition to the randomness of the local and instantaneous distribution of drop size, the locations of the drops change very frequently due to coalescence and drop detachment, resulting in fluctuation and nonuniformity of the surface temperature. Therefore, how to handle such random processes becomes the most important problem to be solved. Of the theories of dropwise condensation proposed up to the present, only two seem to be near completion: one proposed by Le Fevre and Rose [67] and Rose [45,68,88] and one proposed by Tanaka [43,44,69-711. 1. Le Fevre-Rose Theory

This theory [67], first proposed in 1966 and revised afterward, attempts to derive the heat transfer rate of dropwise condensation of a pure vapor from the combination of the following two properties: (1) the heat transfer resistance of a single drop of given size, and (2) the average heat transfer coefficient on the surface, talking into consideration the drop size distribution.

CONDENSATION HEATTRANSFER

93

Nonuniformity and fluctuation of the surface temperature are not considered in the analysis. On deriving the heat transfer resistance of the single drop, the following three items are accounted for: (1) the influence of surface curvature on the phase equilibrium temperature, (2) the mass transfer resistance at the liquidvapor interphase, and (3) the heat conduction resistance through the drop. (The thickness of promoter layers was involved in the original paper [67], but later its effect was neglected because it was considered small.) The relation between the heat flux (iB, which goes from the base of a hemispherical drop into the condensing surface, and the surface subcooling AT is expressed as

where T is the saturation temperature of the vapor, r is the drop radius, o is the surface tension, u, and u, are the specific volumes of the condensate and the vapor, respectively, h,, is the latent heat of condensation, k, is the thermal conductivity of the condensate, and K is the ratio of specific heats. The first term in the right-hand side of Eq. (38) represents the reduction of the vaporto-surface temperature difference due to condensation on the convex liquid surface. The first term in the parentheses represents the conduction resistance of the drop with radius r, where K , is a constant relating the temperature drop in the drop AT, to (iB and is defined as GB =

kl(ATd/Klr)

(39)

This equation means that the drop with radius r is replaced by a liquid layer with uniform thickness K,r. Le Fevre and Rose assumed K , = 2/3 because a circular cylinder having an equal volume with a hemisphere is (2/3)r. The second term in the parentheses comes from the interphase mass transfer resistance. Here, Le Fevre and Rose use an expression based on a result by Schrage [l 13 (which is introduced in Section 111) for a monoatomic gas, and the condensation coefficient is taken as unity. The presence of a ratio ( K + I)/(K - 1) is due to a correction for polyatomicity. This correction has a relatively small effect for steam but is important for complex molecules when K is closer to unity. Another constant K , appearing here is the ratio of the base area of a drop to its curved surface area, i.e., for hemispherical drops K , = 1/2. Thus, the right-hand side of Eq. (38) represents a series connection of temperature drops due to three different causes. To obtain the average heat transfer coefficient over a whole condensing surface, the distribution of drop size is to be considered as the next step. If the fraction of surface area covered

94

ICHIROTANASAWA

by drops with radii between r and R,,, (the maximum drop radius) is denoted f, its functional form is assumed by Le Fevre and Rose as

f = 1 - (r/Rmax)"

(40)

Equation (40) involves an unknown exponent, n, and has the properties that no area is covered by drops greater than the largest, and that if the distribution extended to drops having zero radius, then the whole surface would be covered. The surface heat transfer coefficient is then given by

because the fraction of surface area covered by drops having radii between r and r + dr is (-df/dr)dr. If Eqs. (39) and (40) are substituted into Eq. (41), we get

For the lower limit of integration, Rmin,the radius of the smallest viable drop was taken, i.e.,

On the other hand, the maximum drop radius, R,,,, was obtained from the dimensional analysis as follows:

In most of the cases py is much smaller than pI and can be omitted. Equation (44) involves another constant K 3 , and Le Fevre and Rose assumed K , w 0.4

for dropwise condensation of steam on vertical plates. Substituting the values of 0, p I , g, and K , (=0.4) into Eq. (44) we get RmaX= 1.0 mm. To calculate the integral in Eq. (42), the value of the index n must be determined. Le Fevre and Rose evaluated Eq. (42) for various values of n and found that excellent agreement was obtained, as may be seen from Fig. 23, between experiments and the theory if n was assigned a value of 1/3. Afterward, the theory was compared favorably with data for low-pressure steam [39,53], using the constants given above. Also, the theory was compared with data for ethylene glycol at various pressures [72] and with data for mercury

61

I

I

I

I

I

I

I

+

A

P

A

/+

I

1

I

0.3

0.4

0.5

Heat flux

4

I 0.6

+ a

I

I

I

0.7

0.8

0.9

[MW'dl

FIG.23. Comparison of Le Fevre-Rose theory with experimental resuits. (-), Le Fevre and Rose [67]; (a), Le Fevre and Rose, [35,36]; (0), Tanner et al. [33]; (+). Citakoglu and Rose [37]; (x), Graham [34]; (A),Wimshurst and Rose [39].

I

1.0

ICHIROTANASAWA

96

on stainless steel [73,74]. However, in those cases the constants used had to be somewhat different from those employed for steam at 1 atm in order to get good agreement between the theory and the experiment. It is interesting to note that the value of the index, n = 1/3, leads to a relation showing that the heat transfer coefficient of dropwise condensation is approximately proportional to R;fL3. This relation is very close to the experimental result obtained by Tanasawa et a/. [42] (Fig. 16). Another interesting thing is the drop size distribution. If the number of drops having a radius between r and r + dr is denoted N ( r )dr, it is related to f defined by Eq. (40) as follows: f =

lrRmax Ar2 N ( r )dr

(45)

or

where A represents the ratio of the base area of the drop to the radius (of curvature) squared, and when the drop is a perfect hemisphere, then A = 71. If the value of n = 1/3, which is adopted by Le Fevre and Rose, is used, Eq. (46) suggests that the drop size distribution function is proportional to r to the power -8/3. On the other hand, the distribution of the drop radius has been measured by several researchers (e.g., Graham [34] and Tanasawa and Ochiai [75]). According to Tanasawa and Ochiai, the function N ( r ) is expressed as follows: N ( r ) = 0.159r-2.54,

0.005 I r < 0.25

(474

N ( r ) = 0.0316r-3.87,

0.25 5 r < 0.96

(47b)

where r is expressed in millimeters. It should be noted that the exponent to r, - 2.54,in Eq. (47a)is very close to - 8/3. Compared in Fig. 24 are the drop size distributions obtained by experiments [34,75], by numerical simulations [76] (which are not mentioned here), and by theory [67,77]. All of these agree quite well. In addition, a similar result has been obtained by Tanaka [69-711, which is to be introduced later. This might suggest that the distribution of the drop size is determined almost uniquely, irrespective of the randomness of the process of drop formation or coalescence. 2. Tanaka’s Theory

A theory proposed by Tanaka [43,44,69-711 in 1973 and 1974 is the most elaborate and sophisticated in coping with the mechanism of dropwise

CONDENSATION HEATTRANSFER

10-2

10-1

97

1

Drop radius r hd

FIG. 24. Drop size distribution.(-), Rose and Glicksman [77]; (----), Le Fevre and Rose [67]; (---),Glicksman and Hunt [76].(Figures represent sitedensities per cm2.)(0,O),Graham [34]; ( 0 ,+,x), Tanasawa and Ochiai [75].

condensation. The theory considers the transient change of local drop size distribution, taking into account the processes of growth and coalescence of drops. In the following discussion, only an outline of Tanaka’s theory is introduced. For more detail the original papers should be referred to. Let us denote the number of drops per unit area having radii between rand r dr as N(r, t)dr. Here, t is the time elapsed after the area of the condensing surface of interest was swept off by a departing drop. One of the important features of Tanaka’s theory is that it focuses attention on the transient growth

+

ICHIRO TANASAWA

98

and coalescence process of drop behavior. The drops formed on the bare surface grow by direct condensation (see Fig. 3) with a rate of i,(r), but, because coalescence between drops takes place at the same time, the apparent rate of growth is, as an average, ia(r,t). In such a situation, the relation below is derived for drops with radius r by considering the balance between growth and loss (by coalescence and sweep).

1-

rE(r,t ) = 2

SrRmax

n p 2 N ( p , t )d p

In these equations, the product N(r, t) ia(r,t ) represents the “flux” of the number of drops per unit area and unit time that grow exceeding the radius r. Then the first term in the right-hand side of Eq. (48) signifies the difference between the “afflux” from drops smaller than r and the “efflux” into drops larger than r. The second term in the right-hand side of Eq. (48) represents the number of drops lost by coalescence, and subtraction of this number from the first term is equilibrated with the left-hand side, which is the rate of increase of the number of drops. The third term is for correction of the reduction of the area to be considered due to the departure of drops. Further, the function Y(r, t ; N) is necessary to take into account the requirement that every drop does not overlap each other. The variable rE defined by Eq. (50) is the “equivalent diameter for clearance’’and is the area that is not covered by drops divided by the sum of the peripheral length of the drops. It is a measure of how closely packed the drops are. The equation as follows is derived for the volume increase of a drop with radius r when the drop grows by coalescence:

r

2 -np32nr[ia(r)+ i , ( p ) ] Y ( p ; N ) N ( p ) d p= 2nr2(ia- i,) SRmim 3

(51)

where it is assumed that the drops are hemispheric. The left-hand side of Eq. (51) gives the volume increase due to coalescence and the right-hand side is

CONDENSATION HEATTRANSFER

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the difference between the volume increase by apparent growth and the increase by the growth due to direct condensation. If i,(r) is known and the initial drop size distribution N ( r ,0) is given, and, further, if one more restrictive condition is settled, then the set of Eqs. (48) and (51) can be solved to yield N ( r , t ) and ia(r,t).Here, Tanaka adopted the rate of drop growth obtained by Fatika and Katz [78] as for i,, and assumed as a restrictive condition that the equivalent diameter for clearance is equal to the average minimum distance between nucleation sites, D, i.e., rE(Rmint

t)= D

(52)

and solved Eqs. (48) and (51) numerically. Figure 25 shows an example of the result, where T, = 100°C, AT = 1"C,R,,, = 1.0 mm, Rmin= 0.006 mm, and

I

Drop radius, r I m m )

FIG.25. Timewise change of drop size distribution.

100

ICHIRO TANASAWA

D = 0.018 mm. Shown in the figure is a change of the drop size distribution from t = 0 to c = 40 sec. A main feature of this result is the fact that the drop size distribution at every instant has a small “hill” near the right end of the curve, and the hill moves down to the right as the time proceeds. The shape of the curve becomes similar after t = 0.3 sec, and simply moves along the straight line with a gradient of - 3. On the other hand, the drop size distribution to the left of the hill is approximated by a straight line, of which the gradient is - 2.68, if the fraction of area covered by all the drops is assumed unity. This value of -2.68 almost coincides with that in Eq. (47a) and also with one obtained by Le Fevre and Rose, i.e., - 813. It should be added here that the distribution function, N(r), which appeared earlier, is the average of N ( r , t ) with respect to time and area. Furthermore, Tanaka derived the heat transfer coefficient using ia(r,t) and N ( r , t) obtained from Eqs. (48) and (51), and derived the following relation: hR,,,/k, = 5.3(R,,,/0)0.7

(53)

where k, is the thermal conductivity of the condensate. Equation (53) shows that the heat transfer coefficient is proportional to R,,, to the power -0.3, which agrees well with the experimental result by Tanasawa et al. [42] and with the theory by Rose [45], both of which are introduced in Section IV,B,3. There are a few doubtful points in Tanaka’s theory. For instance, the theory employs as the rate of drop growth the result by Fatika and Katz [78], which is now proved to be incorrect. At the same time, the fluctuation of local surface temperature is not considered, and the way of using the distance between nucleation sites, D, is questionable. In spite of all of these discrepancies, this theory seems to describe the process of dropwise condensation most accurately. D. METHODS OF MAINTAINING DROPWISE CONDENSATION FOR A LONGPERIOD OF TIME Although there remain quite a few yet-unsolved problems on dropwise condensation, there is little doubt that the heat transfer coefficient of dropwise condensation is remarkably high. If the presupposition of film condensation, which the design procedures of most of condensers are usually based on, could be changed to that of dropwise condensation, a substantial economy of materials, dimensions, and, possibly, costs might be achieved. Of course, it is solely the vapor-side heat transfer that is improved by dropwise condensation, because the overall heat transfer is dependent also upon the thermal resistance of the solid wall and the coolant-side heat transfer. And still, a rough estimation reveals that if dropwise condensation is available on the vapor side, the overall heat transfer coefficient is doubled and hence the heat transfer

CONDENSATION HEATTRANSFER

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surface area is reduced to half, in the case of water vapor. This will result in a considerable reduction of the cost of a condenser. An objection that might be raised here most commonly may be that the vapor-side thermal resistance can be reduced without employing dropwise condensation. As a matter of fact, and as will be dicussed later, the so-called high-performance condensing surfaces of a variety of types have been developed in recent years. Among them are microstructured surfaces that make use of the effect of the capillary force on top of the effect of extended area. These high-performance condensing surfaces have indeed considerably small vapor-side heat transfer resistance at their optimum operating condition. However, such an excellent performance is usually limited to a relatively narrow range of operating conditions. Especially, at higher condensing rate, cavities on the microstructured surface are flooded with condensate and the capillary force becomes ineffective. In addition, fouling by various causes may easily deteriorate the high performance. On the other hand, in the case of dropwise condensation, a very high heat transfer rate is available even at a very high heat flux. In addition to this, there is a possibility that the low-energy surface for dropwise condensation can be manufactured less expensively, if an excellent technique for surface treatment is developed in the future. In reality, however, to find out techniques most effective in promoting dropwise condensation for a sufficiently long period of time is the most important and serious problem to be solved before the practical application of dropwise condensation ceases to be a mere dream. The methods (both on a laboratory scale and an industrial scale) employed up to present to maintain dropwise condensation are classified into the following five categories: 1. Application of an appropriate nonwetting agent, i.e., organic promoter, to the condenser surface before operation. 2. Injection of the nonwetting agent intermittently (or continuously) into the vapor. 3. Coating the condensing surface with a thin layer of an inorganic compound such as metal sulfide. 4. Plating the condensing surface with a thin layer of a noble metal such as gold. 5. Coating the condensing surface with a thin layer of an organic polymer such as poly(tetrafluoroethy1ene) (PTFE). The first method has been used most commonly to obtain dropwise condensation in laboratory apparatus. A solution of organic promoter may be wiped, brushed, or painted on the surface, or it can be sprayed directly on the surface. Sometimes the surface is immersed into a solution of promoter. In every case the promoter chemisorbs on the substrate in question. It seems

102

ICHIROTANASAWA

there is little evidence indicating that one method is more effective than another. The adsorbed layer thus obtained is proved to be nonwetting as long as it is thicker than a few molecular layers. Hence, the amount of promoter needed to produce dropwise condensation is very minute. What matters is how long such a surface can sustain dropwise condensation. Tests have been carried out for a number of such substances. The most systematic study among them has been done by Blackman et al. [79]. They discussed in detail the necessary characteristics that the promoters should possess and synthesized 37 compounds that would serve the purpose. Condensation tests were performed for all these compounds and it was proved that a compound called glycerol tri-[11-ethoxy(thiocarbonyl)thiundecanoate] exhibited the longest lifetime; perfect dropwise condensation was observed for 3530 hr (about 21 weeks). This is a very encouraging report and we are inclined to believe that when an adequate promoter is chosen to be applied to a carefully cleaned surface, dropwise condensation lasts for a sufficiently long period. But this may happen only in the most ideal case in which every condition is kept satisfactory. The same may not be expected in industrial systems, for degradation of promoter due to impurities deposited out of the vapor and stripping off by some mechanical causes are very likely. The second method, namely, periodic (or continuous) injection of promoter into the vapor, could be a useful technique, because the stripped-off promoter layer will be replenished at once. Blackman et al. [79] have reported that the minimum amount of promoter needed for injection method is about 0.020.1% of the vapor, though it depends on the substrate and the promoter. This amount will not matter economically, if some fraction of it is recovered afterward. Problems with the injection method are possible contamination and corrosion of the surface. However, once these problems are solved, this method seems more promising than the first. Erb and Thelen [SO] have published the result of their systematic tests on the third method. They have paid attention to the fact that sulfides of copper and silver exhibit extremely low solubility in water. The hydrophobic nature of these mineral sulfides has also been known. Tests were done for a number of sulfided metal surfaces. Among them, a sample of sulfided silver on mild steel showed excellent dropwise condensation after more than 10,000 hr (about 60 weeks) of exposure to continuous condensation. Although this result seems very promising, there are a couple of things that are not clear. In the first place, the reproducibility of their test results seems not so good. Considerably different lifetimes have been observed for the same surfaces with similar treatments. Second, the authors did not give a strong recommendation to the use of sulfided surfaces for dropwise condensation. Instead, they recommended the use of noble metal-coated surfaces as described below. It is not obvious if this is owing to the expenses involved in sulfiding the metal surfaces or owing to other reasons.

CONDENSATION HEATTRANSFER

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In the condensation study mentioned above, Erb and Thelen [80] found that the unsulfided silver surface had even better dropwise condensation than the sulfided silver sample. By comparison of the position in the periodic table (Cu- Ag- Au), they predicted that gold also would produce dropwise condensation. That some noble metal-plated surfaces, especially gold-plated surfaces, produce dropwise condensation had been known from experience by quite a few researchers. Experiments have been carried out using gold-plated condenser surfaces without any organic promoter. Extensive condensation studies were made by Erb and Thelen [80] on silver, gold, rhodium, palladium, chromium, and platinum surfaces. Three of these surfaces (gold, palladium, and rhodium) exhibited good dropwise condensation for more than 12,500 hr (1.43 years). It was found that, of the three, gold was the most reliable because it was not adversely affected by a period of shutdown in air as the other two sometimes were for a period of time. A question has been raised, however, whether the gold surface possesses genuine nonwettability. Wilkins et al. [81] have reported that the steam condenses as a film on a very carefully cleaned gold surface. They have concluded that, in Erb and Thelen’s experiment, some organic compounds adsorbed onto the gold surface must have promoted dropwise condensation. Against this, Erb [82] has insisted that a thin oxide layer was formed on the gold surface in the case of Wilkins and others’ experiment and it rendered the surface wettable. A definite conclusion to this debate has not yet been reached. However, judging from results of recent works (e.g., Refs. [83,84]), it seems that uncontaminated surfaces of noble metals always result in filmwise condensation. However, electroplated gold (and silver, too) surfaces are powerful adsorbers of trace organic materials from the surroundings and the absorbed organics cause dropwise condensation. But considering practical use, it does not matter whether some impurities are adsorbed on the surface. Of greatest significance is whether the surface can really produce long-term dropwise condensation. In this respect, the work by Erb and Thelen should be appreciated. Another important thing to be considered is the economical feasibility. Erb and Thelen have compared by a simple analysis the surface coating cost with the overall condensation cost savings due to gold plating for the case of a distillation plant for saline water conversion. They have reached the conclusion that the gold-plating approach to dropwise condensation is economically feasible,provided that the coating cost is reduced to some extent. As a countermeasure they have proposed what they call low-cost multilayer systems. Nevertheless, it seems still too early to have an optimistic outlook before technical problems are thoroughly solved. Seemingly, many people consider that the fifth method, i.e., polymer coating, is more promising. Since World War I1 a great number of highly

104

ICHIROTANASAWA

polymerized compounds having various physical and chemical properties have been manufactured. Some of them have very low surface energies and cannot be wetted by liquids such as water. The most typical of them is poly(tetrafluoroethy1ene) (PTFE, or Teflon). Attempts have been made to produce long-term dropwise condensation using PTFE-coated surfaces. A couple of difficulties have to be overcome before the polymer film approach to the industrial use of dropwise condensation attains success. The first is to form a film with good adhesion to the substrate, few voids, and sufficiently high mechanical strength. The second is to make the film thin enough not to excessively retard heat transfer with its low thermal conductivity. Because the thermal conductivity of a polymer such as PTFE is as low as 0.25 W/m K, the allowable maximum thickness of the film may be the order of 1 pm, taking into account both the conduction and the constriction resistances (see Section IV,B,5). These two difficulties are related each other, and a solution to the one (for example, thinning the film) results in enhancing the other (reducing the mechanical strength). However, several new processes of forming very thin polymer coatings on the surface have been developed recently: Two examples of those are the glow-discharge method and the electrophoresis method, in both of which polymers are formed by polymerization directly on the substrate from monomers. At any rate, the use of polymer coatings to promote dropwise condensation seems to have the advantage of lower cost when compared with the noble metal-plating method. A little more than 10 years ago, the present author considered that the most promising of the five methods described above was the use of polymer coating, the next being the injection method. The author’s opinion has changed a little. Now the author believes that the best way is to use a metal coating, not necessarily of noble metals, but of metals such as chromium. Although there is no firm theoretical basis that can explain why some metal coatings promote dropwise condensation for a considerably long period, no one can deny the fact that dropwise condensation can be maintained long enough on such surfaces. In the not so distant future a theoretical explanation will catch up with the fact.

V. Film Condensation A. INTRODUCTION

As stated in the beginning of Section 111, the theory of heat transfer by film condensation on plane and horizontal cylindrical surfaces was established first by Nusselt [2] in 1916, more than 70 years ago. The theory is useful even today. In the world of engineering, which is continually developing, it is very seldom

CONDENSATION HEATTRANSFER

105

that a theory proposed more than 70 years ago is still effective. The Nusselt theory is one of such a rare example. Its long-standing effectivenessmeans that the theory grasps the very essential nature of film condensation. However, the Nusselt theory was founded on many idealizations in the modeling of the actual phenomena. Whenever the assumptions for idealization fail, modification to the theory is more or less necessary. Some of such modifications are mentioned in Section 111. The other improvements are only briefly reviewed in the following discussion, focusing attention mainly on laminar film condensation outside a horizontal tube. Turbulent film, or wavy condensation, and condensation on a tube bundle are not discussed.

B. FILMCONDENSATION OF A SINGLECOMPONENT VAPOR Among the assumptions on which the Nusselt theory is based, the one that may have a considerable effect in industrial condensers is that the vapor shear force acting on the condensate surface is neglected. Considering the vapor velocity in most condensers, the effect of the vapor shear force on condensate film thickness should be examined. Fundamentals of film condensation, including this problem, are described in textbooks (e.g., Rohsenow and Choi [85]). Reviews by Fujii and Uehara [86] and Rose [87,88] are also very instructive. Different from the single-phase flow, the vapor flow with condensation around a horizontal circular tube is influenced both by the flow of the condensed liquid and by the suction of vapor at the vapor-liquid interface. Fuji and Honda [89] have evaluated the shear force at the interface using an approximate solution for the boundary layer, examined theoretically the effects of the vapor velocity and the thermal boundary conditions upon the heat transfer rate, and compared the result with experiment (Fujii et al. [90]). They found that the experimental result agreed fairly well with the theory, assuming uniform heat flux on the wall, but that the azimuthal distribution of the wall temperature differed considerably at the point of separation of vapor flow and near the location where the flows of condensate merged. They estimated that it was because they neglected the effect of heat conduction in the wall. The correlation for the heat transfer coefficient that the authors have proposed for laminar film condensation of steam is as follows: Nu,/,/&

= 0.96(PrL/FrH,)0.2

for 0.03 5 Pr,/Fr H,

-= 600

(54)

Nu, = 0.69(Ga Pr,/H,)0.25 for 600 < Pr,’Fr H, (55) where Nu, is the average Nusselt number ( = h,do/AL), Pr, is the Prandtl number of the liquid, Re, is the two-phase Reynolds number (= U,&/V,), Fr

ICHIRO TANASAWA

106

is the Froude number (= U,2/gd0), H, is the average phase change number ( = CpLAT/L), Ga is the Galileo number ( =gdo3/vz), CpLis the specific heat of the liquid, do is the outer diameter of the circular tube, g is the acceleration of gravity, L is the latent heat of condensation, U, is the approaching velocity of steam, h, is the average heat transfer coefficient, vL is the kinematic viscosity of the liquid, 1, is the thermal conductivity of the liquid, and ATis the area-averaged subcooling of the condensing surface. As is well known, the Nusselt theory for natural convection film condensation on the horizontal tube with uniform wall temperature gives, in nondimensional form, Nu, = 0.728(Ga Pr,/H,)0.25

(56)

The only difference between Eq. (55), which is for a low steam velocity region, and Eq. (56) is in the value of the constant on the right-hand side. And the difference is only about 5%. After this, Honda and Fujii [91] have carried out an analysis taking into consideration the heat conduction through the tube wall. They have confirmed that the heat transfer by condensation is affected not only by the condition of steam but also by the size and the material of the tube and by the condition of cooling inside the tube. However, the effects of size, material, and cooling condition may be negligible in the case of condensation of steam, because the steam-side coefficient of heat transfer is usually much higher than the heat transfer coefficient of the coolant side. On the other hand, in the case of condensation of refrigerants like freons, the vapor-side heat transfer coefficient is rather lower than that of the coolant side. It may bring about a different situation. Fujii et al. [92] have applied the result of their preceding analysis (Honda and Fuji [91]) to condensation of freon vapors and compared the result with the result of an experiment using R113 vapor. They have found that the heat transfer coefficient by film condensation of freon vapor approaches the solution for the uniform wall temperature, while that of steam shows the characteristic close to the uniform wall heat flux solution. The formula for the average Nusselt number that the authors have proposed is as follows: Nu,/&

= 0.73(Pr,/FrH,)0.25-

x [1

+ 2.6(1 + Pr,/RH,)0~3(Pr,/FrH,)-0~45]0.3 for 0.03 I PrJFrH,

(57)

(=,/a),

where R is the so-called pp ratio i.e., the square root of the ratio of the product of the density and the viscosity between the liquid and the vapor. The other nomenclature is the same as those in Eqs. (54) and (55).

CONDENSATION HEATTRANSFER

107

The right-hand side of Eq. ( 5 7 ) approaches that of the Nusselt equation [Eq. (56)] as the variable Pr,/(FrH,) increases. According to the authors, Eq. (57) gives values a little higher than predicted by the experimental results for R22 by Gogonin and Dorokhov [93], but gives a little lower values than those of their own experiment for R113 vapor. However, agreement in general seems fairly good. OF A MULTICOMPONENT VAPOR C. FILMCONDENSATION

Until recently, not much work has been accomplished on the film condensation of a multicomponent vapor consisting of more than three components. Most of the work has coped with the condensation of a twocomponent vapor. Condensation of a two-component vapor is classified into two categories. One is the case wherein one of the two components is a noncondensable gas, and the other is the case wherein both components condense. However, such a classification is relative to the degree of cooling, because any vapor can be condensed at a sufficiently low temperature. Condensation of a two-component vapor, one component of which is a noncondensable gas, which is in many instances air, has been studied for the purpose of predicting deterioration of the heat transfer performance of condensers used in power plants. Because such condensers are operated at a pressure lower than 1 atm, air from outside is likely to leak inside the devices. However, recently the methodology of theoretical analysis was established and accurate and reliable results of numerical computation are available. Figure 26 illustrates how a noncondensable gas acts to reduce the heat transfer rate of film condensation. In the case of condensation of a pure saturated vapor, the vapor pressure, P, (which is identical with the saturation pressure, P,), is equal to the total pressure of the gas phase, P. The temperature of the vapor is the saturation temperature, T,, corresponding to the vapor pressure. Vapor-to-liquid phase change occurs at this temperature, provided that the interfacial mass transfer resistance discussed in Section 111 is negligible. Let us assume, then, some amount of noncondensable gas is contained in the vapor. The total pressure of the gas phase is a sum of the partial pressure of vapor, P,, and the partial pressure of a noncondensable gas, P,, i.e., P = P, + P , . When the vapor containing a noncondensable gas touches a cold surface, only the vapor condenses and the noncondensable gas remains there. When a steady state is reached, distributions of partial pressures of both the vapor and the noncondensable gas are established as shown in Fig. 26 by dashed lines. The partial pressure of the noncondensable gas is highest at the liquid-vapor interface and decreases toward the bulk of vapor-gas mixture. On the contrary, the partial pressure of the vapor is lowest at the liquid surface. The vapor condenses at the saturation temperature corresponding to

ICHIROTANASAWA

108

,

I

I

FIG.26. Effect of noncondensable gas on condensation heat transfer.

the partial vapor pressure at the surface. This interface temperature, denoted IT.; in Fig. 26, is more or less lower than T,. The driving force for heat transfer across the condensate film decreases from T, - T, to - T,. It results in a reduction of the condensing rate. The important thing here is the fact that the partial pressure of the noncondensable gas at the liquid-vapor interface (not the partial pressure in the bulk) decides the interface temperature IT.;. Even if the concentration of the noncondensable gas in the bulk vapor is very small, the gas accumulates on the condensate surface when the rate of condensation is high. In order to know the value of Pgat the liquid-vapor interface, the mass transport process in the binary mixture must be analyzed. Because such an analysis is rather complicated, only a brief and simple description is given here. Usually the heat transfer coefficient of condensation, h, is defined using the difference between the saturation temperature of vapor T, and the surface temperature of solid T,. If we use the temperature of the bulk vapor-gas

CONDENSATION HEATTRANSFER

109

mixture, T,, in place of T,, to define the heat transfer coefficient, we can write the following relations, which are similar to Eq. (21):

l / h = l/h,

+ l/hf

h, = 4/(T,

-

T)

hf = 4/(T - T w )

(58)

(59) (60)

Equation ( 5 8 ) indicates that the overall resistance to heat transfer is a series connection of the two resistances: the one in the concentration boundary layer and the other in the liquid film. In the case of pure vapor, the former equals zero because T, = T . If we denote the heat transfer coefficient of condensation of pure vapor by h,, the ratio h/h, represents the degree of degradation of condensation heat transfer due to noncondensable gas. To obtain this ratio, the mass transfer resistance in the concentration boundary layer, within which the partial pressure of vapor decreases, and the interfacial temperature, q ,must be known. A very complicated task is needed for doing this, because transports of momentum, heat, and mass are coupled together. A number of papers have been published regarding this problem. For example, Minkowycz and Sparrow [94] employed a local similarity solution to obtain effects of the variation of physical properties, the condensation coefficient, vapor superheat, thermal diffusion, and diffusion thermos. Rose [95] derived an approximate solution using the Karman-Pohlhausen method. Shown in Fig. 27 is a comparison of the result obtained by Minkowycz and Sparrow [94] with the results of measurement by Slegers and Seban [96] for laminar film condensation of a steam-air mixture on a vertical flat plate. Taken on the horizontal axis is the mass concentration of noncondensable gas in the bulk mixture. Although many of the experimental data are a little lower in value than the analytical results, agreement in general is not so bad. It should be noted that the heat transfer coefficient is reduced to 20-35% of that of the condensation of pure vapor, even when the bulk concentration of air is 1%. Mori and Hijikata [97] considered that formation of mist in the concentration boundary layer would take place as the first step, and analyzed the transport process in the boundary layer, which contained minute droplets, using the Karman- Pohlhausen method. Their result has revealed that those preceding solutions that did not take into consideration the formation of mist give reasonable results if the bulk concentration of air is not large. Recently, the use of a large-scale computer has enabled us to obtain a more rigorous solution to this problem (e.g., Fuji and Kato [98] and Hijikata et al. [99]). Although details are not mentioned here, it should be noted that formulas for condensation of a binary vapor, recommended by F u j i and Kato [98] (which will be shown later), is applicable also to vapor-air mixtures.

ICHIROTANASAWA

110 1.0 I

"."\

TCU=

Too =

-64.6"C -.-

0

47.8"C 285°C

0 A

65.6"C 461°C 26.7"C

<

0

c

--.- htl

'0

0.002

0.004

0.006

0.008

C

wg-

FIG.27. Comparison of Minkowycz-Sparrow theory with experimental results. From Minkowycz and Sparrow [94]; Slegers and Seban [96].

Practical research has been performed on film condensation of a multicomponent vapor, which contains more than two condensable gases, in connection with the distillation process in chemical engineering industries. A more recent trend is that the two-component fluid has been used as the working substance in heat pump or refrigeration systems, thus stimulating research and development of the phase-change heat transfer of such a fluid. That is to say, when a pure substance is used as a working fluid for a heat pump or refrigerator, evaporation and condensation take place almost at constant temperatures, making the temperature differences at the inlets and outlets of the heat exchangers different (see Fig. 28). Because a minimum temperature difference (the so-called pinch-point temperature difference) of a few degrees is required for heat exchange, most of the heat exchange process is to be performed at larger temperature differences. The larger the temperature difference for heat transfer, the larger the irreversibility of the process, thus resulting in larger loss of the available energy. Such a loss can be reduced by the use of a nonazeotropic multicomponent substance. When a mixture of a two-component vapor is condensed, the mixture is called azeotropic if the condensate has exactly the same molar concentration ratio as the vapor phase. If a liquid of such an azeotropic mixture is evaporated, we can get a vapor phase that has exactly the same molar concentration as the liquid. A solution of 96.5% ethanol with water is

CONDENSATION HEATTRANSFER

111

I

T

Condensation

Entropy

FIG.28. Condensation and evaporation in thermodynamic cycle.

an example of an azeotropic mixture. A mixture that is not azeotropic is called nonazeotropic mixture. Condensation of a vapor of an azeotropic mixture can be treated as condensation of a pure vapor. Discussion of this is not within the scope of this section. Figure 29 is a typical phase diagram of a nonazeotropic binary fluid composed of species A and B. The temperature is taken on the vertical axis and the concentration of component B is taken on the horizontal axis. Two curves are drawn in the figure. The upper curve is called the vapor-phase line and the lower one is called the liquid-phase line. These lines take different shapes depending on combinations of components and pressure. Temperatures q,A and q,Bare the saturation temperatures of pure components A and B, respectively. When a binary mixture, of which the concentration of the B component is cp, at temperature To is cooled down to TI, condensation initiates. However, the concentration of the condensed phase is different from

ICHIROTANASAWA

112 T

I I I

I

I

I

I

‘S,

B

I

I C -

Concentration of B

Component

FIG.29. Phase diagram of a binary mixture.

that of the vapor. The liquid phase formed at this stage corresponds to the point Q, of which the concentration of B component is cQ, where cQ cp, meaning that the liquid formed is richer in the A component. When a finite amount of vapor is condensed at a constant pressure, the concentration of the A component tends to decrease by condensation. Then, the vapor-to-liquid phase change occurs from the point P along the vapor-phase line down to the point B. The condensate changes its concentration along the line Q-P’-B, if cooling continues. This means that the condensation temperature of the binary mixture changes from TI to T, to G,Balong with the change of the concentration ratio of both components during the progress of condensation. Such a phenomenon can be utilized to reduce the loss of available energy in a thermodynamic cycle by decreasing the average temperature difference for

-=

CONDENSATION HEATTRANSFER

113

heat exchange (Fig. 28). If an appropriate binary substance is chosen, and if the phase-change temperature is varied continuously during evaporation or condensation so as to keep constant the difference between the temperature of the heat source (or sink) and the evaporation or condensation temperature, the irreversibility caused by heat transfer can be reduced. In Japan, a project named “Super Heat Pump and Energy Accumulation Systems,” promoted by the Ministry of International Trades and Industries, started in 1984. In this project, success or failure of the development of a highperformance heat pump, aiming at a coefficient of performance as high as eight, is largely dependent upon the effective use of multicomponent working fluids. A problem pertinent to the use of two-component fluids is in the heat transfer performance. The two-component media to be chosen for practical application are nonazeotropic mixtures, in most cases from a standpoint of improvement of the efficiency of the thermodynamic cycle, as stated previously. When a vapor of such a mixture is condensed, one of the components that has a lower dew point behaves like a noncondensable gas, and thus possibly brings about deterioration of heat transfer performance, as in the case of condensation of steam-air mixtures. Therefore, a couple of important issues concerning the research objectives of the condensation of a two-component vapor are raised: (1) prediction of characteristics of condensation heat transfer and (2) a technique for avoiding deterioration of (and, if possible, enhancing) condensation heat transfer. As to the first item, an analytical procedure similar to the one employed for a vapor-noncondensable gas mixture can be used. However, obtaining a proper solution is more difficult, because the phase equilibria as shown in Fig. 29 should be taken into consideration for each combination of components. Fujii and Kato [98] have proposed formulas to calculate heat transfer and condensation rates for the case of condensation of a binary vapor on a vertical flat plate. Flows of both vapor and condensate are assumed to be laminar and the liquid phases of both components are considered to be miscible.

1. Forced Convection Condensation

If the total pressure P of the vapor mixture, the bulk mass concentration W,, of component 1, and the wall temperature T, are given, the interface temperature T , the mass concentration Wli of component 1 at the interface, the mass concentration W,, of component 1 in the liquid phase, and the nondimensionalized rates of condensation of both components &fl and &f2, are derived from Eqs. (61)-(63) using two equations expressing the

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vapor-phase and the liquid-phase lines in the phase diagram. Then, the average heat flux q and the average mass flux of condensation m are obtained from Eqs. (64) and (65).

m = 2(hlI

+ h2,)-Rej'(2 PI 1

(65)

where pL and A, are the viscosity and the thermal conductivity of the liquid mixture, respectively, Ah is the latent heat of condensation, R is the pp ratio that appeared before, Scis the Schmidt number, 1 is the length of the plate, Re,, is the two-phase Reynolds number (= Um,/vL),and h is the dimensionless mass flux ( = ml/pL(&). 2. Natural Convection Condensation

CONDENSATION HEATTRANSFER

C(Sc) = -

115

sc 2.4

+ 4.9& + 5Sc

where M is the molecular mass, Ga is the Galileo number, and M is another dimensionless mass flux ( = ml/pLGa:I4). Modification of the above equations for application to a horizontal tube has been given by Goto and Fuji [IOO]. Also, the above equations are applicable to the case when one of the two components is a noncondensable gas. In that case, we must put &,(or &,(or Ml) = 0 and W,, = 0. As mentioned, the heat transfer coefficient of condensation of a binary (or multicomponent) vapor is lower than that of pure vapor, because of the accumulation of the more volatile component on the liquid-vapor interface. The effect is more prominent in the case of natural convection condensation than in the case of forced convection condensation. Therefore, to raise the vapor velocity is an effective means to avoid deterioration of the condensation heat transfer of a binary vapor.

VI. Enhancement of Condensation Heat Transfer A. INTRODUCTION

In principle there are only a couple of ways of enhancing condensation heat transfer, one is to increase the area of the condensing surface, and the other is to reduce the thermal resistance due to the condensate film formed on the surface. The former is realized by the use of extended surfaces, such as finned or fluted surfaces, and has been employed for a long time. However, because the heat transfer coefficent of condensation is in itself rather high, the use of high fins is not so efficient when the fin efficiency is considered. Low fins are used in most of the cases. As for reducing thermal resistance, some effort is required to reduce the thickness of the condensate film by various means, as will be shown in this section. In practice, a combination of these two methods, i.e., the use of flow fins and techniques of reducing the film thickness, is employed simultaneously to develop high-performance condenser surfaces.

B. ENHANCEMENT BY SURFACE TENSION FORCE 1. Basic Principle Surface tension is rather a small force compared with other forces affecting heat transfer processes. It has not been positively taken into consideration

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when condensation heat transfer is discussed. In reality, however, surface tension force is one of the most important items to be considered when enhancement of condensation heat transfer is at issue. Utilization of surface tension force is the most sophisticated way to enhance condensation, because no extra energy is needed to reduce the thickness of the condensate film. Use of dropwise condensation, if possible, might be the most desirable among the surface tension-aided means of enhancement of condensation. If the condensing surface is not wetted by the condensate, the condensed liquid gathers together to form separate drops, leaving bare parts of the solid surface where the thickness of the liquid film is zero, and an extremely high heattransfer rate is available, as discussed in Section 111. To our great regret, however, dropwise condensation has never been applied to industrial condensers. The reason is, as stated previously, that maintaining dropwise condensation for a long period of time has not yet been successful.It seems that to manufacture a permanent hydrophobic surface for the dropwise condensation of steam is not entirely impossible. However, to promote dropwise condensation of refrigerants, such as freons, seems difficult. In the following discussions, film condensation is considered as the mode of condensation occurring on the condensing surface. Probably Gregorig [loll was the first person to propose that surface tension could be effective in enhancing heat transfer by condensation. Gregorig’s work was on film condensation on a vertical fluted tube, of which the cross-section is illustrated in Fig. 30. When a vapor condenses on such a surface, condensate forms a thin film over the profile of the solid wall. As is well known, when a liquid surface has a curvature, there arises a pressure difference between the fluids at both sides of the vapor-liquid interface. This pressure difference AF is described by the Laplace equation below: AF = o(l/R,

+ l/RJ

FIG.30. Vertical, fluted surface studied by Gregorig.

(72)

CONDENSATION HEATTRANSFER

117

where 0 is the surface tension of the liquid and R , and R , are the principal radii of curvature at the point. A positive AP in Eq. (72)signifies that the pressure in the liquid is higher than the pressure outside. In the case of Fig. 30, the pressure in the liquid film at the tip of the fin is higher than the ambient vapor pressure, because AP in Eq. (72)is positive since R , takes a finite positive value and R , is infinity. On the other hand, the pressure in the liquid film at the bottom of the groove between the fins is lower than the ambient pressure, because R , is negative there. Therefore, the liquid on the fin tends to move from the tip to the groove because of the pressure gradient along the film. As a result, the thickness of the condensate film at the fin tip is reduced and the liquid acccumulates in the bottom of the groove. The flow rate of the condensate toward the bottom depends upon the pressure difference and the distance between the tip and the bottom of the fin. Gregorig described a shape of the fin for a convex profile, which provides a constant film thickness over the arc. The required shape of the vapor-liquid interface is 1/R = 0.75n/L[1 - (x/L)']

(73)

where R is the radius of curvature of the interface, x is the distance along the interface measured from the tip, and L is the arc length. Eq. (73)shows that R is smallest at x = 0 and grows to infinity at x = L . It should be noted that, although the flow of condensate considered here is restricted to the one occurring on the horizontal plane, the gravitational force drains the condensate vertically downward in the actual situation. The ratio of gravity to surface tension force is represented by a nondimensional parameter called the Bond number. If R is taken as the characteristic length, the Bond number is Bo = RZ(Pl- P")g/a

-

-

(74)

If the physical properties of water are assumed, Bo when R = 0.1 mm, and Bo 0.1 when R = 1 mm. This means that the flow of condensate at the tip of a fin is governed mainly by surface tension force, as long as the radius of curvature there is less than 1 mm. 2. Optimum Fin Geometry for Vertical Condenser Tube More elaborate work on the enhanced condensation on fins was carried out by Hirasawa et al. [102-1051. They solved the equations of momentum and energy for the condensate film both at the fin tip, with a small radius of curvature, and in the groove, for fins with three different geometries, as shown in Fig. 31. Using the results they obtained, they determined the optimum fin

118

Triangular fin

Wavy fin

Flat-bottomed groove

FIG.31. Geometries of fins.

profile and the optimum interval of the drainage disk fitted to a vertical condenser tube as shown in Fig. 32. The role of the drainage disk here is to remove the condensate, which fills up the grooves and reduces the effective area for condensation. Shown in Fig. 33 is one of their results. The heat transfer coefficient averaged over the interval of the drainage disk is plotted against the fin pitch for three different geometries. It is assumed that R113 is condensed under a vapor-to-surface temperature difference of 10 K. Height of the fin is 0.61 mm for all the geometries. It is seen that the maximum value of the heat transfer coefficientexists for every geometry at a fin pitch of 0.5-0.6 mm. At a lower fin pitch, the grooves are filled with condensate, while at a larger fin pitch, the relative area of the bottom of groove, which is ineffective for condensation, becomes larger than the area of the fin tip. Such a tendency is more remarkable for the fin with a flat bottom. The maximum heat transfer coefficient is more than 10 times that of a smooth tube calculated by the Nusselt theory. It is estimated that the factor of about 2.5 out of 10 is owing to the increase of surface area and the rest is due to the contribution of the surface tension and drainage disk. A variety of high-performance condenser tubes, which have minute fins with sharp tips, have been developed and used in refrigerators and airconditioning devices. The mechanism of condensation enhancement in those tubes is basically the same as discussed here, though alignments, geometries, and dimensions of fins are somewhat different among different manufacturers.

FIG.32. High-performance longitudinal fin tube with drainage disks. Flat-bottomed groove

( r o= 10-5m)

Nusselt solution for smooth tube

I

I

I

I

0.5

1.0

1. 5

Fin spacing p ( m m l I,

FIG. 33. Experimental results using a longitudinal fin tube. R-113, T, - T, = 10 K, = .61 mm, h = 50 mm.

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3. Horizontal Low-Fin Tube Use of horizontal tubes instead of vertical tubes seems much more frequent in condensation of vapor in industrial devices. Also, in this case, low-fin tubes (integral-fin tubes) are employed to enhance condensation. As mentioned, extension of surface area by fins and reduction of the thickness of condensate film by surface tension are simultaneous goals. However, these two effects sometimes are not compatible. That is to say, if one reduces too much the pitch between fins to increase the surface area, the condensate is flooded in the grooves, resulting in reduction of the effective surface area. It is known that the tendency for the condensate to collect in the interfin space depends on the surface tension a and the density p I . The larger the ratio alpI,the more likely the condensate accumulates. The degree of condensation heat transfer enhancement compared with the smooth surface decreases for substances having larger alp, values. If we measure the position up to which the space between the fins is flooded with the condensate by the azimuthal angle from the bottom centerline of the tube, and call it the retention angle (denoted a),this 0 must be determined from the surface tension and the density of condensate and the dimensions and geometries of the tube and fins. Several authors, e.g., Honda and colleagues [106,107] and Ruby and Webb [lOS], derived, under some assumptions, an expression for 0.For example, in the case of rectangular shaped fins, their expression yields

where D is the tube diameter and p is the spacing between the fins. It is seen, by examining Eq. (75), that the values of alp, and the fin spacing p are very important values for flooding. Equation (75) is plotted in Fig. 34 for a tube with a diameter D of. 14.7 mm and for three values of alp,. Clearly, a highsurface-tension liquid such as water floods very well a finned tube at relatively small fin spacings, whereas for a low-surface-tension liquid such as freon, the fin spacing can be decreased considerably before complete flooding occurs. Masuda and Rose [1091 measured the dependency of the enhancement ratio on fin spacing and condensate flooding. Shown in Fig. 35 is their result. The enhancement ratio on the ordinate is the one taken under the condition that the vapor-to-wall temperature difference AT is the same. Thirteen integral-fin tubes with a root diameter of 12.7 mm were used. Fins were rectangular shaped, having the same fin thickness (0.5 mm) and height (1.6 mm). Tests were carried out at near-atmospheric pressure and downward vapor flow with velocities between 0.5 and 1.1 m/sec. The fin spacing at which the maximum enhancement ratio was obtained differs depending on fluid. It is 1.5 mm for steam, 1.0 mm for ethylene glycol, and 0.5 mm for R113. The

CONDENSATION HEATTRANSFER

121

7

180 160 -

,-.

140 44 (ethylene glycol

M

0)

'CI Y

e

120 -

al

M ?J

100-

C .-

E

3

2

8060 40 -

"

210

4b0

'

6hO

8bO

1,dOO

l,iOO

1,400

F i n s per meter

FIG.34. Dependence of retention angle on fin spacing.

values of the maximum enhancement ratio decrease in the order of R113, ethylene glycol, and steam. The percentage numbers for each data point correspond to the percent flooding calculated using Eq. (75). As expected, the maximum enhancement ratio increases and the optimum fin spacing decreases with the decrease of u/p, value. A rather complicated mathematical treatment would be required to make a theoretical prediction of the performance of a horizontal integral-fin tube. Equations of flow and heat transfer should be solved simultaneously under complexed geometrical, thermal, and fluid dynamical conditions. However, a prediction method proposed by Honda and Nozu [llO] and Honda et al.

ICHIROTANASAWA

122

a 36 %

+

6-

3-

+20%

ethylene glycol

100%

steam

1

2

3

4

F i n spacing [ m m l FIG.35. Dependence of enhancement on fin spacing and condensate flooding.

[ t 111 gives a little less complicated procedure for obtaining the performance of a low-fin tube. They distinguished a finned tube by two regions: one is the region where the interfin space is not flooded with the condensate and the other is the region flooded with the condensate [Fig. 361. The two regions can be divided by the retention angle mentioned before. Then, they made models simulating the flow of condensate in the interfin spaces and they calculated the heat transfer rates. Comparison of their results of calculation with experimental data (e.g., Masuda and Rose [1091)showed fairly good agreement, and the effect of the fin spacing could be predicted correctly for fluids having different alp, values. Honda et al. [ 1061 and Honda and Nozu [1071 have proposed a technique for enhancing condensation of vapors, the liquid phases of which have large values of alp,. Their idea was to attach a porous drainage plate to the bottom of the horizontal finned tube, reducing the condensate retention between the fins and increasing the effective area for condensation, thus enhancing the condensation heat transfer by the actions of surface tension and gravitational forces.

CONDENSATION

HEATTRANSFER

123

FIG.36. Physical model of condensation on a low integral-fin tube.

In the experiment the authors prepared finned tubes of 19 mm outer diameter and four different fin profiles. Their heights were 0.92- 1.46 mm and the pitches were 0.50-0.98 mm. The drainage plates were 1.9 mm thick and 14-19 mm wide and were made of porous nickel of 0.8 mm pore diameter and 490 kg/m3 bulk density. One of those plates was inserted in a 2-mm-wide and 1- to 1.5-mm-deep groove machined along the bottom of the finned tube. Figure 37 illustrates how effective such a drainage plate is. The fluids used are (a) methanol and (b) the refrigerant R113. For both fluids, aspects of condensation on finned tubes without porous drainage plates are shown. The degree of surface subcooling is kept constant at about 8 K. Horizontal arrows on the left-hand sides of the pictures indicate the positions at which the thickness of the condensate film changes greatly and the curvature of the film turns from convex to concave. It means that the grooves between the fins beneath this position are flooded with condensate. The pictures suggest that this position moves toward the bottom of the tube by attaching the drainage plate. Comparison between methanol and R113 reveals that flooding occurs at a higher position for methanol than for R113, and the effect of the drainage plate is also larger for methanol, which has a greater o/p, value. These results clearly indicate the effectiveness of drainage plates. The issues to be discussed before they are used in industrial condensers may be the cost of manufacturing. 4. Other Methods of Enhancement Although details will not be mentioned, there are some other methods of enhancement of condensation heat transfer utilizing surface tension force.

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FIG.37. Effect of porous drainage plates for enhancement of condensation on a low-fin tube. (A), Methanol; (left),without porous drainage plate; (right), with 14 mm high drainage plate.

(B), R113; (left), without porous drainage plate; (right), with 15 m m high drainage plate. [Courtesy of Prof. Hiroshi Honda, Kyushu University.]

One of them is to wind a wire around the tube. This method seems to have been proposed first by Thomas [112] in 1967. The wires, with a circular or rectangular cross-section, when wrapped around the tube, attract and retain the condensate, thus reducing the thickness of the liquid film on the surface in between them. There must be an optimum value of the pitch of winding, because too large or too small a pitch would reduce the effect. Fujii et al. [1131 carried out an experiment and analysis for the case of natural convection condensation and found that the maximum rate of enhancement was attained when the ratio of the wire diameter to the pitch was about 1:2. The enhancement ratio obtained was about three when compared with condensation

CONDENSATION HEATTRANSFER

125

on the bare tube, though the value was dependent upon the liquid and the diameter ratio. As another effort to reduce the average condensate thickness, use of non wettable strips has been proposed. For example, Glicksman et al. El141 conducted measurements on condensation of steam on a single horizontal tube on which various strips of PTFE were placed. According to their result, enhancements as large as 100%were observed at low heat fluxes when the strip was placed longitudinally along the bottom of the tube. Kumagai and colleagues [115,116] manufactured special condensing surfaces on which hydrophobic and hydrophilic parts exist alternately. They measured the heat transfer characteristics on these surfaces, where dropwise and film condensation occurred simultaneously. As a matter of course, the local heat transfer coefficient is higher on the parts where dropwise condensation occurs and lower where film condensation takes place. However, they found that, under a certain condition, the heat transfer coefficient of the film condensation area was increased, due possibly to a disturbance caused by drops absorbed into the condensate film from the dropwise condensation area. These authors have suggested that this mechanism could be employed for high-performance condenser surfaces. There still remain a variety of proposed techniques for enhancement of condensation heat transfer. Also, there are a number of review papers regarding this subject. Interested readers are advised to refer to some of them [88,117-1201.

c. ENHANCEMENT BY ELECTRIC FIELD 1 . Basic Principle According to Bergles [117], the techniques of heat transfer enhancement in general can be classified into three categories: (1) passive technique, (2) active technique, and (3) combined technique. The passive technique needs no extra energy to enhance heat transfer. Use of extended surfaces is one of the typical examples of this technique. The active technique requires an extra power supply for heat transfer enhancement. As proposed, though not always successfully, by some people, use of a wiper or scraper to remove the condensate film is an example of the active technique. Also, utilization of an electric field, to be described in the following discussion, is another example. The combined technique is, literally, the combination of the two methods. If a finned surface is used under an electric field, both the passive and and active techniques are employed simultaneously. Generally speaking, as Bergles has insisted, the active technique is less preferable in many cases, because the consumption of extra power for heat

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transfer enhancementis likely to mar the gain. However, in case the amount of power consumed is much less than the advantage expected by the active enhancement, it is worthwhile to consider. Enhancement of condensation by an electric field is very effective when the condensed liquid has a low electrical conductivity. When a fluid with dielectric constant E is placed under an electric field E (which is a vector quantity), the body force f acting on the fluid is expressed as below [121]:

1 f = p,E - -E’VE 2

+

The first term on the right-hand side of this equation represents the Coulomb force acting on the free electric charge p e ,the second term is the force due to the spatial variation of dielectric constant E, and the third term is generated from the spatial change of the electric field intensity. In most liquids, the sum of the second and the third terms, representingthe force acting on the electric charges by polarization, is important because pe is almost negligible. The basic principle of the heat transfer enhancement by electric field is to utilize the body force mentioned above to modify the flow and the temperature distribution in the vicinity of the heat transfer surface. How to use the electric field may change depending upon the mode of heat transfer, fluid, geometry, purpose of heat transfer enhancement, and other conditions. Theoretical treatment of this techique is rather complex, because the interaction between electrical, thermal, and flow fields must be considered. Sometimes this technique is called electrohydrodynamic(EHD) enhancement. A detailed description of the EHD technique applied to the other modes of heat transfer is given by Jones [122]. If we focus attention on the EHD enhancement of condensation heat transfer, the applied electric field can be effective in two ways. One is that the electric field induces convective motion in the layer of condensate liquid, and the other is that the vapor-liquid interface is rendered unstable by the electric force. However, the former does not seem to be very effective. Perhaps it is because of the difficulty in inducing strong convection, sufficient to increase appreciably the heat transfer rate, in so thin a liquid film. The latter principle seems more promising. 2. Enhancement of Condensation by EHD Instability of Vapor- Liquid Interfaces

When an external electric field is applied to a body of fluid, the distribution of electric charges is altered. The transient process for redistribution of electric charges is described by an equation as below, which is derived from the

CONDENSATION HEATTRANSFER

127

Maxwell equations for electromagnetic fields: DpelDt

+ (ae/E)Pe

(77)

=0

where oe is the electrical conductivity of the fluid. Here, both a, and E are assumed to be constant. Equation (77) tells us that the electric charge density p e decreases with time in proportion to exp[ -(a,/~)t]. That is to say, the characteristic time of the electric charge distribution due to a change of the electric field distribution is (78)

z = &/be

This time constant z is called the relaxation time and is a measure of the electric field tendency to be steady. Approximate values of z for liquids are as follows: water, 1 psec; ethanol, 10 psec; R113, 10 sec. When an electric field is imposed on a liquid for periods much longer than the relaxation times, the electric charges in the liquid are distributed so that the electric field inside may become substantially zero except at the liquid surface. In case one of the electrodes is placed in the liquid and the other is in the gaseous phase, as shown in Fig. 38, the electric potential in the liquid is everywhere homogeneous, being nearly equal to the potential on the electrode immersed in the liquid. If an electric potential is applied between these two electrodes, a stress called Maxwell stress acts on the gas-liquid interface. The Maxwell stress in the case of Fig. 38 is normal to the interface and is equal to (1/2)&,E2,where E , is the dielectric constant of the gas phase. Let us consider here the stability of the gas-liquid interface. The forces acting on the interface are gravitational, capillary, and electrical forces, if the liquid is quiescent. The first two forces tend to stabilize, and the last one to

-

-

N

Electrode /// / / // / / / / / / / / /// //// / / / /

'

Maxwell stress E

1/2

liquid

/ / / / / / I / // // / / / / / / / / / , / /

////

Electrode

FIG.38. Stability of a liquid surface under an electric field.

ICHIROTANASAWA

128

destabilize, the interface. If, by some disturbance, the liquid surface deforms as shown in Fig. 38, the Maxwell stress at the peak of the surface increases, because the distance from the upper electrode is reduced. At the same time, the gravitational potential and the capillary force on the curved interface act to pull down the surface. The interface becomes unstable when the attractive force by the upper electrode exceeds the pull-down forces. The criterion for onset of instability can be obtained by solving a modified Rayleigh-Taylor instability problem, which includes the contribution of electric field. The equation to be solved to derive the instability criterion is as below. Detailed explanation is omitted here but is given elsewhere (e.g., see Ref. 1231).

c

+

c m z - 8.85 x 10-'2~,(E/6)2mcoth(m6) ( p , - p,)q = 0,

m = 2 ~ / 1 (79)

where E, is the dielectric constant of the vapor phase and 6 is the distance between the liquid surface and the electrode above the liquid. The wavenumberm obtained from Eq. (79)represents the unstable wavenumber at which the liquid-vapor interface becomes unstable. It should be noted that in Eq. (79)an assumption is made that the gravitational force is in the direction of vertically downward, whereas the liquid surface faces horizontally upward. Equation (79) has two asymptotic solutions for two extremes, m --t co and m 0. For m O(2 --t 00)

-

-

and for m --t co (A

-

0)

A=

2x0 8.85 x 10-2~,(E/S)z

The value of E expressed by Eq. (79) represents the electric field potential above which the liquid surface with inifinite expanse becomes unstable. The surface with a finite expanse needs a stronger electric field to be destabilized. Solutions for other values of m, between 0 and infinity, are obtained numerically. Figure 39 shows the dependence of the unstable wavelength A on the applied electric voltage. In general, the unstable wavelength tends to decrease with the voltage. It means that the liquid surface becomes unstable more easily under a stronger electric field. However, if we observe Fig. 39 more closely, we find that there appear minimum values of electric voltage where the distance 6 is large (in Fig. 39, 6 > 3 mm). These minimum values make determination of the unstable wavelengths a little complex. For example, if the dimension of the liquid surface is 20 mm, the intersections of the horizontal line at 2 x lo-' m (drawn in Fig. 39) with the curves obtained by calculation

CONDENSATION HEATTRANSFER

129

Voltage E I V I

FIG.39. Dependence of unstable wavelength on an applied electric field. Distance 6: A, 0.5 mm;

B, 1 mm;C, 2 mm; D, 3 mm; E, 5 mm;F, 7 m m ;G , 10 m m ;H, 16 mm.

give the critical electric voltage over which the liquid surface becomes unstable, as long as the curves have no minimum values of E. Then, the critical values of E are above 500 V for 6 = 0.5 mm, 1040 V for 6 = 1 mm, and 2800 V for 6 = 2 mm. However, for 6 = 3 mm, the critical value of E is not 6500 V (corresponding to the intersection) but 5500 V corresponding to the minimum value of E . At any rate, the liquid surface can be unstable under an electric field with sufficient intensity. When the electric field intensity exceeds a certain critical value, a disturbance wave appears on the surface and it grows if the electric field is intensified. The principle described here is applicable to enhancing heat transfer by film condensation, unless the condensate liquid has a high electrical conductivity. Choi [I241 attempted to enhance film condensation by destabilizing the condensate film with a strong electric field. He observed that waves were generated by application of an electric field and enhancement of condensation heat transfer as high as three times the normal value was obtained. Yabe et al. [125] confirmed that the liquid film of condensate was pulled toward an electrode when a nonuniform electric field was applied between the cylindrical surface of a vertical condenser tube and the electrode facing it because of an instability of the liquid layer. They used this phenomenon to

ICHIROTANASAWA

130

Condensate (R 1 1 3 )

L L

W C a,

P

0

0.4 Ocm3/S

0.30 cm3Is

57 % -

43% -

FIG.40. Enhancement of condensation heat transfer by an electric field.

remove a portion of the condensate from the surface. Shown in Fig. 40 is the concept of such an idea. In this case wire electrodes are placed at a distance 1.8 mm from the condenser tube. An electrical potential of 8.3 kV was applied between the tube and the wire. As expected, the condensate was drawn toward the wire electrodes and was removed through a conduit placed behind the electrodes. This reduced the thickness of the condensate film, resulting in enhancement of condensation heat transfer. If electrodes and conduits for drainage were placed at a certain interval along the condenser surface, a considerable enhancement could be achieved. What makes this method more efficient is that the electrical power consumed with this process can be very small (less than 1% of the amount of heat transmitted), because in the case of liquids with low electrical conductivity, such as freons, the current induced remains very small even if a high voltage is applied. On the basis of their fundamental study, Taketani et al. [1261developed this technique further for industrial applications with the use of thin, helical electrodes (Fig. 41). Furthermore, Yabe et al. [127] attempted to reduce still more the thickness of the condensate film on the areas of the condensing surface that did not stand opposite the helical electrodes. The means that the authors employed was a planer electrode, with which many protuberances of the condensate were formed on the liquid film when an appropriate electric field was imposed. The authors called this phenomenon pseudodropwise condensation, because the appearance resembled dropwise condensation. In the case of condensation of freon R114 on a vertical tube, an enhancement ratio of about 6 has been observed with the simultaneous use of the helical electrodes for pulling out the condensate and the planer electrodes for generating pseudodropwise condensation.

CONDENSATION HEATTRANSFER

131

I

I ItFIG.41. Helical electrodes. (a), Helical wire with lattice electrode; (b), helical wire with plate electrode.

It should be noted that the results mentioned above were obtained under a nonuniform electric field. However, the present author believes that the electric field does not need to be nonuniform. As shown in Fig. 39, a uniform electric field with same intensity can have almost the same effect. VII. Concluding Remarks In closing this review the author would like to mention briefly some personal comments on the future trends in the progress of the condensation research. The order does not coincide with the order of sections. 1. Film condensation of single-component vapors: Numerical computation will be performed extensively assuming complicated boundary and thermofluid-dynamic conditions. More accurate and detailed results will be published on a variety of combinations of fluids and geometries. However, these will not contribute much to deepen our knowledge and understanding of the essential characteristics of the phenomenon. 2. Film condensation of multicomponent vapors: As stated previously, the need for research on film condensation of multicomponent vapors will increase in connection with the technological development of thermal energy conversion devices such as heat pumps. Numerical computation will be an efficient tool for investigation, though importance of experimental work will not decrease. Analytical approaches will be rather limited in this area. Obtaining experimental formulas, which are valid for a wide range of parameters,

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will not be easy either, because there are too many variables to be taken into account. One of the important issues of this field is to obtain by some means the reliable and accurate values of relevant thermophysical properties of multicomponment fluids. From a practical point of view, to develop a technique for preventing deterioration and, if possible, enhancing condensation will be most urgently in demand. 3. Enhancement of condensation heat transfer: This will remain a subject that will interest many researchers. However, no great progress will be expected as long as the so-called passive techniques are concerned, because almost every mechanism of enhancement that we can imagine has already been developed. Thus, what seems to be left for us will be to pursue active (or combined) techniques such as the use of electric fields. 4. Dropwise condensation: As mentioned previously, the most important objective of dropwise condensation research is to manufacture surfaces on which dropwise condensation can be maintained for a sufficiently long period of time, even under industrial conditions. The other issues concerning heat transfer mechanisms are of minor significance,because a great deal of work on these issues has been performed during the past 25 years. 5 . Direct contact condensation: This is an area not mentioned in this review. An insufficient number of studies has been carried out in this field. There remain a variety of interesting phenomena to study, such as are observed in direct contact condensation of immiscible components. However, the importance of any such investigation is largely dependent upon its practical application. 6. Microscopic aspects of condensation research: Understanding the condensation phenomenon from a microscopic view point will become more important in the future. The author’s personal opinion is that one of the possible ways that condensation research can expand its territory is to pay attention to phenomena of molecular scale. By broadening our field of vision, we shall be able to get a deeper insight into the conventional process of condensation. NOMENCLATURE A Bo C C

D E

F Fr

f

area, area ratio Bond number specific heat molar concentration, specific heat drop diameter, tube diameter, distance between nucleation sites electric voltage, electric field intensity force Froud number distribution function

Galileo number gravitational acceleration phase change number heat transfer coefficient, enthalpy latent heat of condensation Knudsen number liquid thermal conductivity length, latent heat of condensation characteristic length molecular mass

CONDENSATION HEATTRANSFER M M

dimensionless mass flux dimensionless mass flux m wavenumber mass flux m N ( r ) drop size distribution function Nu Nusselt number n number density, index P pressure Prandtl number Pr fin spacing P heat flux 4 heat flux through the base of a drop 4B gas constant, thermal resistance, R pp ratio, radius of curvature

Re Rmax Rmin

r sc T t

U V V

W X

z

133

Reynolds number maximum drop radius minimum drop radius radius Schmidt number temperature time velocity velocity specific volume velocity, mass concentration coordinate coordinate

Greek Symbols

r s E

8 K

I P V

r

variable defined by Eq. (1 1) distance dielectric constant contact angle specific heat ratio unstable wavelength viscosity kinematic viscosity

P P. U

oc 7

Q,

variable defined by Eq. (27) density electric charge density surface tension, condensation coefficient electrical conductivity relaxation time retention angle

Subscripts

a C

e f g I

L

adhesive condensation, constriction, cohesive evaporation, electrical liquid film gas interface liquid

1, I m S V W

m

liquid, characteristic length mean saturated, solid, surface vapor wall bulk, infinity

REFERENCES 1. R. Donaldson, ed., “Bicentenary of the James Watt Patent.” Univ. of Glasgow, Glasgow, 1969. 2. W. Nusselt, Die Oberllachenkondensation des Wasserdampfes. Z. VDI 60,541,569 (1916). 3. E. Schmidt, W. Schurig, and W. Sellschop, Versuche iiber die Kondensation von Wasserdampf in Film- und Tropfenform. Tech. Mech. Thermodyn. I, 53 (1930). 4. G. S. Springer, Homogeneous nucleation. Adu. Heat Transfer 14,281 (1978). 5. H. Merte, Jr., Condensation heat transfer. Adv. Heat Transfer 9, 181 (1973). 6. S. Sideman and D. Moalem Maron, Direct contact condensation. Adu. Heat Transfer 15, 228 (1982). 7. W. M. Rohsenow, Heat transfer and temperature distribution in laminar film condensation. Trans. ASME 78, 1645 (1956).

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8. W. M. Rohsenow, J. H. Webber, and A. T. Ling, Effect of velocity on laminar and turbulentfilm condensation. Trans. ASME 78,1637 (1956). 9. E. M. Sparrow and J. L. Gregg, A boundary layer treatment of laminar-film condensation. J. Heat Transfer 81, 13 (1959). 10. M. M. Chen, An analytical study of laminar film condensation: Part 1-Flat plate. J. Heat Pansfer 83,48 (1961). 11. R. W. Schrage, “A Theoretical Study of Interface Mass Transfer.” Columbia Univ. Press, New York, 1953. 12. W. M. Rohsenow, Film condensation of liquid metals. Prog. Heat Mass Transfer 7, 469 (1973). 13. Y .S. Huang, Heat transfer by condensation of low pressure metal vapors. Ph.D. Thesis, Case Western Reserve Univ., Cleveland, Ohio, 1971. 14. G. S. Springer and A. J. Patton, A kinetic theory description of liquid-vapor phase change. lnt. Symp. Rarefied Gas Dyn., 6th p. 1497 (1969). 15. D. A. Labuntzov, An analysis of evaporation and condensation process. High Temp. (Engl. Transl.) 5, 579 (1967). 16. D. A. Labuntzov and A. P. Kryukov, Analysis of intensive evaporation and condensation. Int. J . Heat Mass Transfer 22,989 (1979). 17. A. T. Mills and R. A. Seban, The condensation coefficient of water. Int. J. Heat Mass Transfer 10,1815 (1967). 18. U. Narusawa and G. S. Springer, Condensation coefficient of water. AEC Rep., USAEC COO-2032-6 (1972). 19. S. Hatamiya and H. Tanaka, A study on the mechanism of dropwise condensation (2nd

20. 21. 22. 23. 24. 25. 26. 27. 28.

29. 30.

31.

Report: Condensation coefficient of water at low pressures) (in Japanese). Trans. JSME. Ser. B 52,2214 (1986). S. J. Wilcox and W. M. Rohsenow, Film condensation of potassium using copper condensing block for precise wall temperature measurement. J. Heat Transfer 92,359 (1970). R. lshiguro and K. Sugiyama, An experimental study on potassium condensation (in Japanese). Trans. JSME, Ser. B 54,967 (1988). S. Necmi and J. W. Rose, Heat-transfer measurements during dropwise condensation of mercury. Int. J. Heat Mass Transfer 20,877 (1977). J. Niknejad and J. W. Rose, Interface matter transfer: an experimental study of condensation of mercury. Proc. R. SOC.London, Ser. A 378,305 (1981). S. Fujikawa and M. Maerefat, A study of the molecular mechanism of vapor condensation (in Japanese). Trans. JSME, Ser. B 56, 1376 (1990). S. A. Stylianou and J. W. Rose, Dropwise condensation of ethanediol. lnt. J. Phys.-Chem. Hydrodyn. 3, 199 (1982). J. W. Rose, On interphase matter transfer, the condensation coefficient and dropwise condensation. Proc. R. SOC. London, Ser. A 411,305 (1987). E. J. Le Fevre, Personal communication to J. W. Rose (1966); see also Appendix 6 in J. W. Rose, Ph.D. Thesis, University of London, London, 1964. 1. Tanasawa, Dropwise condensation-The way to practical applications. Proc. lnt. Heat Transfer Conf., 6th 6, 393 (1978); see also Dropwise condensation-Progress toward practical applications. In “Heat Transfer Science and Technology” (B.X.Wan& ed.), p. 43. Hemisphere, New York, 1987. M.Jakob, Heat transfer in evaporation and condensation-11. Mech. Eng. 58,729 (1936). J. F. Welch and J. W. Westwater, Microscopic study of dropwise condensation. lnt. Deu. Heat Transfer, Proc. 1961-1962 Int. Heat Transfer Conf. p. 302 (1963). A. Umur and P. Griffith, Mechanism of dropwise condensation. J. Heat Transfer 87, 275 (1965).

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32. J. L. McCormick and J. W. Westwater, Nucleation sites for dropwise condensation. Chem. Eng. Sci. 20, 1021 (1965). 33. D. W. Tanner, C. J. Potter, D. Pope, and D. West, Heat transfer in dropwise condensationPart I. The effects of heat flux, steam velocity and non-condensable gas concentration. Int. J . Heat Mass Transfer 8,419 (1965). 34. C. Graham, The limiting heat transfer mechanism of dropwise condensation. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, 1969. 35. E. J. Le Fevre and J. W.Rose, Heat-transfer measurement during dropwise condensation of steam. Int. J. Heat Mass Transfer 7,272 (1964). 36. E. J. Le Fevre and J. W. Rose, An experimental study of heat transfer by dropwise condensation. Int. J. Heat Mass Transfer 8, 11 17 (1965). 37. E. Citakoglu and J. W. Rose, Dropwise condensation-Some factors influencing the validity of heat-transfer measurements. Int. J. Heat Mass Transfer 11,523 (1968). 38. D. W. Tanner, D. Pope, C. J. Potter, and D. West, Heat transfer in dropwise condensation at low steam pressures in the absence of non-condensable gas. Int. J . Heat Mass Transfer 11, 181 (1968). 39. R. Wilmshurst and J. W. Rose, Dropwise condensation-Further heat-transfer measurements. Proc. Int. Heat Transfer Conf., 4th 6, Cs. 1.4 (1970).

40. T. Tsuruta and H. Tanaka, Microscopic study of dropwise condensation (in Japanese). Trans. JSME. Ser. E 49, 1828 (1983). 41. S. Hatamiya and Tanaka, A study on the mechanism of dropwise condensation (1st Report: Measurement of heat-transfer coefficient of steam at low pressures) (in Japanese). Trans. J S M E , Ser. B 52, 1828 (1986). 42. I. Tanasawa, J. Ochiai, Y. Utaka, and S. Enya, Experimental study on dropwise condensation (Effect of departing drop size on heat-transfer coefficients) (in Japanese). Trans. JSME 42,2846 (1976). 43. H. Tanaka, A theoretical study on dropwise condensation. J. Heat Transfer 97, 72 (1975). 44. H. Tanaka, Measurement of drop-size distribution during transient dropwise condensation. J . Heat Transfer 97,341 (1975). 45. J. W. Rose, Further aspects of dropwise condensation theory. Int. J. Heat Mass Transfer 19, 1363 (1976). 46. S. Nukiyama, The maximum and minimum value of the heat Q transmitted from metal to boiling water under atmospheric pressure (in Japanese). J . JSME 37, 367 (1934); Engl. transl., Int. J. Heat Mass Transfer 9, 1419 (1966). 47. T. Takeyama and S . Shimizu, On the transition of dropwise-film condensation. Proc. Int. Heat Transfer Conf.,5th 3,274 (1974). 48. 1. Tanasawa and Y.Utaka, Measurement of condensation curves for dropwise condensation of steam at atmospheric pressure. J . Heat Transfer 105,633 (1983). 49. I. Tanasawa and M. Saito, Film and dropwise condensation of steam on a vertical bank of horizontal circular tubes. Proc. ASME-JSME Therm. Eng. J . Conf.,2nd, Honolulu 5, 143 ( 1987). 50. B. B. Mikic, On mechanism of dropwise condensation. Int. J . Heat Mass Transfer 12, 1311 (1969). 51. S. N. Aksan and J. W. Rose, Dropwisecondensation-Theeffect of thermal properties of the condenser material. Int. J . Heat Mass Transfer 16,461 (1973).

52. J. W. Rose, Effect of condenser tubematerial on heat transferduringdropwise condensation of steam. Int. J . Heat Mass Transfer 21,835 (1978). 53. S. A. Stylianou and J. W. Rose, Dropwise condehsation on surfaces having different thermal conductivities. J. Heat Transfer 102,477 (1980). 54. D. W. Tanner, C. J. Potter, D. Pope, and D. West, Heat transfer in dropwise condensationPart 11, Surface chemistry. Int. J . Heat Mass Transfer 8,427 (1965).

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55. D. G. Wilkins and L. A. Bromley, Dropwise condensation phenomena. AlChE J. 19, 839 (1973). 56. R. J. Hannemann and B. B. MikiC, An experimental investigation into the effect of surface thermal conductivity on the rate of heat transfer in dropwise condensation. lnt. J. Heat Mass Transfer 19, 1309 (1976). 57.' T. Tsuruta and H.Tanaka, A theoretical study on constriction resistance in dropwise condensation (in Japanese). Trans. JSME, Ser. B 54,281 1 (1988). 58. T. Tsuruta and S. Togashi, A study on the constriction resistance in dropwise condensation (in Japanese). Trans. JSME, Ser. B 55,2852 (1989). 59. T. Tsuruta, A study on the constriction resistance in dropwise condensation (in Japanese). Ph.D. Thesis, University of Tokyo, Tokyo, 1989. 60. 1. Tanasawa and Y. Shibata, Dropwise condensation at low heat flux and small surface subcooling. Condens.Heat Transfer. Nat. Heat Trander Conf., 18th. San Diego, 79 (1979). 61. K. Kaino, M. Izumi, S. Shimizu, and T. Takeyama, Heat transfer near the initiation of dropwise condensation (in Japanese). Trans. JSME, Ser. B 48, 1339 (1982). 62. I. Tanasawa and S. Nagata, Dropwise condensation heat transfer of steam under small surface subcooling. Proc. lnt. Heat Transfer Conf., 8th 4, 1665 (1986). 63. J. W. Rose, Dropwise condensation of mercury. lnt. J . Heat Mass Transfer 15,1431 (1972). 64. M. Kollera and U. Grigull, Uber das Abspringen von Tropfen bei der Kondensation von Quecksilver. Wiirrne-Stofiiebertrag. 2,3 1 (1969). 65. M. Kollera and U. Grigull, Untersuchung der Kondensation von Quecksilverdampf. Wiirme-Stofliiebertrag.4,244 (1971). 66. M. N. Ivanovskii, V. I. Subbotin, and Y. V. Milovanov, Heat transfer with dropwise condensation of mercury vapor. Teploenergetika 14,81(1967). 67. E. J. Le Fevreand J. W. Rose, A theory of heat-transfer by dropwise condensation. Proc. Int. Heat Transfer Conf., 3rd 2,362 (1966). 68. J. W. Rose, On the mechanism of dropwise condensation. Int. J . Heat Mass Transfer 10, 755 (1967). 69. H. Tanaka, A theoretical study on dropwise condensation (in Japanese). Trans. JSME 39, 3099 (1973). 70. H. Tanaka, A theoretical study on dropwise condensation (2nd Report: A theory that excludes overlap between drops) (in Japanese). Trans. JSME 40,2283 (1974). 71. H. Tanaka, A theoretical study on dropwise condensation. J . Heat Transfer 97,72 (1975). 72. S. A. Stylianou and J. W. Rose, Dropwise condensation of ethanediol. Int. J . PhpChern. Hydrodyn. 3, 199 (1982). 73. J. Niknijad and J. W. Rose, Dropwise condensation of mercury. Proc. Int. Heat Transfer Conf., 6th 2,483 (1978). 74. J. Niknijad and J. W. Rose, Comparison between experiment and theory for dropwise condensation of mercury. Int. J . Heat Mass Transfer 27,2253 (1984). 75. I. Tanasawa and J. Ochiai, Experimental study on dropwise condensation (in Japanese). Trans. JSME 38,3193 (1972); Engl. trans]., Bull. JSME 16, 1184 (1973). 76. L. R. Glicksman and A. W. Hunt, Jr., Numerical simulation of dropwise condensation. Int. J . Heat Mass Transfer 15,2251 (1972). 77. J. W. Rose and L. R. Glicksman, of drop sizes. lnt. J. Heat Mass Transfer 16,411 (1973). 78. N. Fatika and D. L. Katz, Dropwise condensation. Chem. Eng. Prog. 45,661 (1949). 79. L. C. F. Blackman, M. J. S. Dewar, and H. Hampson, An investigation of compounds promoting the dropwise condensation of steam. J. Appl. Chem. 7,160 (1957).

80. R. A. Erb and E. Thelen, Dropwise condensation characteristics of permanent hydrophobic systems. U.S.Of, Saline Water Res. Den Rep. No. 184 (1966).

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81. D. G. Wilkins, L. A. Bromley, and S. M. Read, Dropwise and filmwise condensation of water vapor on gold. AIChE J. 19, 119 (1973). 82. R. A. Erb, Dropwise condensation on gold. Gold Bull. 6,2 (1973). 83. D. W. Woodruff and J. W. Westwater, Steam condensation on electroplated gold: Effect of plating thickness. Int. J. Heat Mass Transfer 22,629 (1979). 84. C. A. Nash and J. W. Westwater, A study of novel surfaces for dropwise condensation. Proc. ASME-JSME Therm. Eng. J . Conf., 2nd 2,485 (1987). 85. W. M. Rohsenow and H. Y. Choi, “Heat, Mass and Momentum Transfer.” Prentice-Hall, Englewood Cliffs, New Jersey, 1961. 86. T. Fujii and H. Uehara, Film condensation heat transfer (in Japanese). In “Progress in Heat Transfer,” Vol. 1, p. 1. Yokendo Book Co., Tokyo, 1973. 87. J. W. Rose, Fundamentals of condensation heat transfer: Laminar film condensation. JSME lnt. J., Ser. 2 31,357 (1988). 88. J. W. Rose, Condensation heat transfer theory. lnt. Commun. Heat Mass Transfer 15, 449 (1988). 89. T. Fujii and H. Honda, Forced convection condensation on a horizontal tube (1st Report: Theoretical treatment) (in Japanese). Trans. J S M E , Ser. B 46,95 (1980). 90. T. Fujii, H. Honda, and K. Oda, Forced convection condensation on a horizontal tube (2nd Report: Experiments for horizontal flow of low-pressure steam) (in Japanese). Trans. JSME, Ser. B 46, 103 (1980). 91. H. Honda and T. Fujii, Effect of wall conduction upon forced convection condensation on a horizontal tube (in Japanese). Trans. JSME, Ser. B 46,2420 (1980). 92. T. Fujii, H. Honda, K. Oda, Y. Kato, and S. Kawano, Film condensation of fluoro-carbon refrigerant vapors flowing across a horizontal tube (in Japanese). Trans. JSME, Ser. B 47, 197 (1981). 93. I. I. Gogonin and A. R. Dorokhov, Experimental investigation of heat transfer with condensation of moving vapor of R-21 on horizontal cylinders. J. Appl Mech. Tech. Phys. 17, 252 (1976). 94. W. J. Minkowycz and E. M. Sparrow, Condensation heat transfer in the presence of noncondensables, interfacial resistance, superheating, variable properties and diffusion. lnt. J. Heat Mass Transfer 9, 1125 (1966). 95. J. W. Rose, Condensation of a vapour in the presence of a non-condensing gas. Int. J . Heat Mass Transfer 12,233 (1969). 96. L. Slegers and R. A. Seban, Laminar film condensation of steam containing small concentrations of air. Int. J. Heat Mass Transfer 13, 1941 (1970). 97. Y. Mori and K. Hijikata, Free convective condensation heat transfer with noncondensable gas on a vertical surface. Int. J. Heat Mass Transfer 16,2229 (1973); also Trans. JSME, Ser. B 38,418 (1972). 98. T. Fuji and Y. Kato, Laminar film condensation of a binary miscible vapour on a flat plate (in Japanese). Trans. JSME, Ser. B 46,306 (1980). 99. K. Hijikata, Y. Mori, and K. Utsunomiya, Nonsimilar solution of condensation heat transfer containing noncondensable gas (in Japanese). Trans. JSME, Ser. B 46, 1514 (1980). 100. M. Goto and T. Fujii, Film condensation of binary refrigerant vapours on a horizontal tube. Proc. lnt. Heat Transfer Conf., 7th 5,71 (1982). 101. R. Gregorig, Film condensation on finely rippled surfaces with consideration of surface tension. Z. Angew. Math. Phys. 5,36 (1954). 102. S. Hirasawa, K. Hijikata, Y. Mori, and W. Nakayama, Effect of surface tension on film condensation (A study on the liquid film in small grooves) (in Japanese). Trans. J S M E , Ser. B 44,2041 (1978).

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103. S. Hirasawa, K. Hijikata, Y. Mori, and W. Nakayama, Effect of surface tension on laminar

104.

105. 106.

107.

108. 109.

110.

111.

film condensation along a vertical plate with a small leading radius. Proc. Int. Heat Transfer Conf., 6th 2,413 (1978). S. Hirasawa, Y. Mori, K. Huikata, and W. Nakayama, Optimum surface geometry of vertical condenser tubes (Effect of thermal conduction in fins) (in Japanese). Trans. JSME. Ser. B 48, 527 (1982). Y. Mori, K. Hijikata, S. Hirasawa, and W. Nakayama, Optimized performance of condensers with outside condensing surface. J . Heat Transfer 103.96 (1981). H. Honda, S. Nozu, and K. Mitsumori, Augmentation of condensation on horizontal finned tubes by attaching a porous drainage plate (in Japanese). Trans. JSME. Ser. B 49, 1937 (1983); Engl. transl., Proc. ASME-JSME Therm. Eng. J . Conf., l s t , Honolulu 3,289 (1983). H. Honda and S. Nozu, Augmentation of condensation on horizontal finned tubes by attaching a drainage strip(Effect of material and height of the drainage strip). Trans. J S M E , Ser. B 51,3191 (1985). T. M. Rudy and R. L. Webb, An analytical model to predict condensate retention on horizontal integral-fin tubes. J. Heat Transfer 107,361 (1981). H. Masuda and J. W. Rose, An experimental study on condensation of refrigerant 113 on low integral-fin tubes. In “Heat Transfer Science and Technology”(B. X.Wang, ed.), p. 480. Hemisphere, New York, 1987. H. Honda and S. Nozu, A prediction method for heat transfer during film condensation on horizontal low integral-fin tubes. Trans. J S M E , Ser. B 51, 572 (1985); Engl. transl., J. Heat Transfer 109,218 (1987). H. Honda, S. Nozu, and B. Uchima, A generalized prediction method for heat transfer during film condensation on horizontal low integral-fin tubes (in Japanese). Trans. JSME, Ser. B 53,

1329 (1987). 112. D. G. Thomas, Enhancement of film condensation rates on vertical tubes by vertical wires. Ind. Eng. Chem., Fundam. 6,97 (1967). 113. T. Fujii, W. C. Wang, S. Koyama, and Y. Shimizu, Heat transfer enhancement for gravity controlled condensation on a horizontal tube by a coiled wire (in Japanese). Trans. JSME, Ser. B 51,2436 (1985). 114. L. R. Glicksman, B. B. Mikit, andD. F. Snow, Augmentationoffilmcondensationon the outside of horizontal tubes. AIChE J. 19,636 (1973). 115. S. Kumagai, A. Yamauchi, H. Fukushima, and T. Takeyama, Condensation heat transfer on various dropwise-filmwise coexisting surfaces. Proc. ASME-JSME Therm. Eng. J . Conf., 2nd, Honolulu 4,409 (1987). 116. S. Kumagai, S. Tanaka, H. Katsuda, and R. Shimada, On the effect of co-existence of dropwise condensation to filmwise condensation heat transfer. Proc. Int. Symp. Condensers Condens.. 2nd. Bath, Engl. p. 483 (1990). 117. A. E. Bergles, Enhancement of heat transfer. Proc. Int. Heat Transfer Conf., 6th 6,89 (1978). 118. R. L. Webb, Special surface geometries for heat transfer augmentation. In “Developments in Heat Exchanger Technology- 1” (D. Chisholm, ed.), p. 179. Appl. Sci., London, 1980. 119. R. L. Webb, Enhancement of film condensation. Int. Commun. Heat Mass Transfer 15,475 (1988). 120. P. J. Marto, Recent progress in enhancing film condensation heat transfer on horizontal tubes. Proc. Int. Heat Transfer Conf., 8th 1, 161 (1986). 121. J. A. Stratton, “Electromagnetic Theory.” McGraw-Hill, New York, 1941. 122. T. B. Jones, Electrohydrodynamically enhanced heat transfer in liquids-A review. Ado. Heat Transfer 14, 107 (1978). 123. K. Takano, I. Tanasawa, and S. Nishio, Enhancement of evaporation of a droplet from a hot surface by electric field. Int. Heat Transfer Conf., 9th, Jerusalem, 4, (1990). 124. H. Y. Choi, Electrohydrodynamic condensation heat transfer. J. Heat Tranger 90,98 (1968).

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125. A. Yabe, K. Kikuchi, T. Taketani, T. Yamanishi, Y. Mori, and K. Hijikata, Augmentation of condensation heat transfer by applying non-uniform electric field (in Japanese). Trans. JSME, Ser. B 48,2271 (1982). 126. T. Taketani, A. Yabe, K. Kikuchi, Y. Mori, and K. Hijikata, Augmentation of condensation

127.

128. 129. 130. 131. 132.

heat transfer by applying non-uniform electric field (2nd Report: Augmentation of condensation outside a vertical tube by use of helical wire electrodes) (in Japanese). Trans. JSME, Ser. B 52,2202 (1986). A. Yabe, T. Taketani, K. Kikuchi, Y. Mori, and H. Maki, Augmentation of condensation heat transfer by applying electro-hydro-dynamical pseudo-dropwise condensation. Proc. Int. Heat Transfer Conf., 8th 6,2957 (1986). 0. J. Foust, ed., “Sodium-NaK Engineering Handbook,” Vol. 1. Gordon & Breach, New York, 1972. “JSME Data Book: Heat Transfer,” 3rd Ed. Jpn. SOC.Mech. Eng., Tokyo, 1975. “JSME Data Book: Heat Transfer,” 4th Ed. Jpn SOC.Mech. Eng., Tokyo, 1986. “1980 SI JSME Steam Tables.” Jpn. SOC.Mech. Eng., Tokyo, 1981. “Mechanical Engineering Handbook. A6: Thermal Engineering,” New Ed. Jpn. SOC.Mech. Eng., Tokyo, 1985.

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ADVANCES IN HEAT TRANSFER, VOLUME 21

Hydrodynamics and Heat/Mass Transfer near Rotating Surfaces

DAVID MOALEM MARON AND SHIMON COHEN Department of Fluid Mechanics and Heat Transfer School of Engineering University of Tel Aviv Ramat Aviv, Tel Aviv 69978, Israel

I. Introduction Induced rotational flow, imparted on axial flow through annular passages, has many uses in engineering. The resulting helical flow is particularly applied in rotating electrical machines, swirl nozzles, rotating disks, and other chemical and mechanical mixing equipment. As such, rotational flow, with and without axial flow, has been subjected to many experimental and analytical studies. These studies established the flow modes, stability, and heat/mass transfer characteristics associated with the rotation of either of the bounding surfaces of the annular space. The reported results indicate a significant enhancement of heat and mass fluxes from rotating surfaces. This important characteristic is a valuable factor in circumventing some of the crucial difficulties in chemical reactor design. A number of industrial chemical reactions are referred to as low-yield processes, due to their thermally sensitive products at the required high temperature levels in the reactor chamber. One can look on pyrolysis oxidation reactions as representative examples. In general, thermal instability problems are greatly associated with exothermic reactions, wherein the thermal control is difficult. Consquently, the reaction conversion and yield decrease with temperature. Moreover, the thermal instability as well as the possible appearance of multiplicity of steady states may cause unwanted by-products to appear in catalytic reactions. For instance, temperature nonuniformities and residence time distributions are believed to be possible causes of molecular weight dispersion in chain-type polymerization; diffusion effects in both 141 Copyright 01991 by Academic Press, Inc. All rights of reproduction in any form reserved.

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the bulk fluid and the catalyst are known to have a profound influence on the overall reaction rates in enzyme-catalyzed reactions. Prior methods of improving the thermal stability, which in turn affects the conversion of such processes, allow for short residence time and include, among others, reaction in the vapor phase, high velocities, and high recycle ratios. However, as the chemical species flow rapidly over the reaction surface, diffusional rate limitations are encountered. These are particularly significant in fast heterogeneous reactions, in which the limiting step is the rate at which energy and mass are transport to or from the catalytic surface. A pertinent problem is that of determining the true reaction rate in kinetic studies. Due to the coupling between kinetics and the transport rates, only apparent kinetic rates are obtained. Hence, interpretation of experimental data leads to falsification of rate constants. Unlike the situation in other model geometries, such as flat plates or porous pellets, the boundary layers around rotating surfaces are approximately of uniform thickness. Owing to the well-defined flow in the vicinity of the resulting equiaccessible surfaces, diffusional aspects of heat and mass transfer are greatly simplified, particularly regarding chemical kinetics and thermal stability of catalytic surfaces. Practically, in utilizing the transport characteristics of induced rotational flow, a better control of concentration profiles and residence time distributions of diffusion-controlled reactions may be achieved, compared to those obtained in stationary systems. To the authors’ knowledge, none of the available investigations on the transport phenomena around rotating surfaces relates to a direct application in chemical reactors. An extensive study undertaken by the authors recently represents a preliminary attempt to develop a new type of reactor based on a rotating reaction surface, whereon the main reaction takes place. To explore the associated operational characteristics, the work incorporated an experimental phase followed by a theoretical analysis (Cohen, 1984; Cohen and Moalem Maron, 1984; Yacoub and Moalem Maron, 1983). It was shown that the enhanced heat and mass transfer at the reactor surface, associated with the vortex and turbulent flow regimes, significantly improved the reactor performance, conversion, and yield. However, it was suggested that in some cases centrifugal effects (high rotations) on the various chemical species can be used to further improve the reactor performance (Moalem Maron, 1983; Yacoub and Moalem Maron, 1983). In what follows, a survey is presented, which introduces the fundamental work carried out in the experimental and theoretical research of the induced rotational flow between coaxial cylinders. Included in this review is the state of knowledge pertaining to the associated heat and mass transport.

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TRANSFER NEARROTATING SURFACES

143

11. State of the Art

Convective transport phenomena in rotating systems are intimately related to the flow characteristics. This feature, together with the three-dimensionality of the boundary layer flow, coupled with centrifugal and Coriolis forces, demonstrates a very complicated system. Extensive research concerning flow and heat transfer in rotating systems is available. Dorfman (1963, 1972) and later Kreith (1968) prepared comprehensive reviews of the state of knowledge of flow and heat transfer near rotating surfaces. These include axially symmetric bodies of revolution such as disks and cones as well as cylinders and spheres. Mass transfer studies, together with catalytic kinetics research in electrochemical systems such as rotating rods and rotating disks, have been summarized by Riddiford ( 1 966) and Gabe ( 1 974). The following survey is intended to introduce the fundamental work concerning the nature of induced rotating flow, and the associated heat and mass transfer between coaxial cyclinders, primarily when the inner cylinder is rotating and the.outer is at rest. Other works by Cole (1969) relating to eccentric rotating systems are only briefly discussed. Prevailing flow regimes and transition and stability aspects are treated in detail in Section II,A, whereas reference to heat transfer is made in Section II,B. Section II,C deals with mass transfer phenomena with and without reactions. A. STABILITY, TRANSITION, AND FLOWREGIMES 1. General

Because of its symmetry and the high accuracy at which the rotational Reynolds number (based on the inner cylinder rotation rate) can be controlled, the induced flow between the concentric rotating cylinders has been the subject of extensive experimental and theoretical studies. Unlike the direct and catastrophic turbulence appearance, when the motion of the outer cylinder predominates, the flow, when the inner cylinder is rotating and the outer one is at rest, is known to exhibit a sequence of instabilities in the transition from laminar to fully developed turbulence. This kind of instability was already indicated in the early experimental studies by Mallock (1888, 1896) and Couette (1890). About 30 years later, in a classic study of hydrodynamic stability, Taylor (1923) observed and calculated the critical angular velocity at which the laminar flow (known now as Couette flow) becomes unstable. Since then, extensive work based on theoretical and experimental studies has been done in determining the evolution of infinitesimal disturbances and the

144

DAVIDMOALEM MARONAND SHIMON COHEN

transition flows as the speed of the inner cylinder is increasing while the outer one is at rest. A stability margin has been defined later for isothermal as well as for nonisothermal induced rotational flows with and without axial flow, driven through the annular duct. Theoretical analysis almost always assumes the cylinders to be of infinite length. In an attempt to fulfill this assumption, most experiments have been carried out in long cylinders. In what follows, theoretical and experimental results for the infinite cylinder are first considered. Then, the effect of finite cylinder length on the transition and the uniqueness of the flow is referred to.

2. Isothermal Flow Experimental evidence supported by theoretical considerations indicates various possible flow modes existing in the annular passage of coaxial cylinders. For zero axial flow in a rotating inner cylinder system, these are characterized as (a) Couette (laminar) flow, (b) Taylor vortex (periodic)flow, (c) wavy vortex (doubly periodic) flow, (d) turbulent vortex flow, and (e) turbulent (aperiodic)flow. Transition criteria between the flow modes were determined experimentally, including hot wire anemometry, laser-Doppler measurements, and various power spectra techniques. Experimental observations, based on flow visualization and high-speed photography, were the most prominent techniques in determining the flow pattern in the first and second transitions. Also, torque measurements and heat and mass transfer studies confirm qualitatively the flow transitions occurring when the rotation speed of the inner cylinder is increased from a stationary state while the outer is at rest. a. Couette Flow. Before considering in detail the experimental and theoretical findings of various flow transitions, for sake of clarity, the basic laminar flow will be introduced. Let I , 8, z be the cylindrical coordinates of the flow geometry shown in Fig. 1. The radii and angular velocities of the inner and outer cylinders are respectively denoted by ri ,r,, and mi, w,. The exact globally stable unique solution of the Navier-Stokes equation for steady, viscous, and incompressible axisymmetric flow (V, = V, = 0) is given in Schlichting’s (1960) monograph as

V, = Ar

+ B/r

where & denotes the circumferential velocity and A and B are as follows:

(1)

HEATIMASS

TRANSFER NEARROTATING SURFACES

145

FIG.1, Schematic presentation of circular Couette flow.

The laminar velocity profile in Eq. (1) is a linear combination of a rigid body rotation (Ar) and circulation of a potential vortex ( B / r ) .The relative proportion of each is determined by the angular velocities assigned to the two cylindrical surfaces. The corresponding tangential shear stress, zrO,is rrO = -2p(B/r2)

(3)

146

DAVIDMOALEMMARONAND SHIMON COHEN

where p stands for the dynamic viscosity. As is evident from Eq. (3), constant torque (rr0r2)is exerted on flow surfaces in the annular gap. The basic flow described by Eqs. (1) and (2) constitutes the early study made by Mallock (1888, 1896) and later on by Couette (1890), to test the Newtonian stress approximation in the Navier-Stokes equations. Equations (1) and (3) characterize a flow system in which a dynamic equilibrium exists between the centrifugal forces and the radial pressure gradient. Where it becomes impossible for the radial pressure gradient and viscous forces to dampen and restore changes in centrifugal forces caused by small disturbances in the flow, the unstable motion results in a secondary flow. Based on this physical argument, a simple criterion for inertial instability was worked out by Rayleigh (1916). According to this criterion, an inviscid circulatory flow is stable whenever the square of the circulation is increasing outward. Mathematically, it is formulated as (d/ar)(r2w)> O

(4)

Applying Eq. (4)to the solution in Eq. (1) requires A > 0, and according to Eq. (2) this is equivalent to wi > 0

and

wo/wi > (ri/ro)2

(for stability)

(5 1

FIG.2. Secondary flow: counter-rotating toroidal vortices between circular cylinders (Taylor, 19231.

HEAT/MASS TRANSFER NEARROTATING SURFACES

147

In other words, an inviscid flow is unstable whenever the mean velocity distribution is such that the product V,r decreases outward with conservation of angular momentum about the axis of rotation. b. Taylor Vortex Flow. Early experiments revealed that the elementary laminar flow was always replaced at a sufficiently large speed of rotation by some complicated eddying flow. Exceeding a critical angular velocity leads to a new steady axisymmetric secondary flow in the form of regularly spaced vortices in the axial direction, as shown in Fig. 2. Pictures of a Taylor vortex flow and other transitions in a fluid contained between concentric cylinders, wherein only the inner cylinder is rotating, are shown in Fig. 3.

FIG.3. Photographs of the flow between concentric cylinders, with the inner cylinder rotating = 1 (Taylor vortex flow); (b) Re/Re,, = 10.4 (wavy vortex flow);

(4 = 0.87, ji = 0). (a) Re/Re,,

(c) Re/Re,, = 12.3 (first appearance of randomness in wavy vortex flow); (d) Re/Re,, = 23.5 (disappearance of azimuthal waves, and the flow is turbulent with axial periodicity [a, b, and c, from Coles (1965);d, from Fenstermacher et a/.(1979)l.

DAVID MOALEM MARONAND SHIMON COHEN

148

Taylor (1923) formulated the stability problem, predicted by assuming an axisymmetric infinitesimal disturbance. He also solved the dynamic condition under which instability occurs in a viscous flow. Whereas Rayleigh’s analysis [Eq. (5)] shows, for the case when the outer cylinder is at rest, that the azimuthal flow of an inviscid fluid is unstable at infinitesimally small rotations of the inner cylinder, Taylor’s linear stability analysis of the viscous fluid shows that viscosity delays the onset of the secondary flow to finite rotations. Assuming that the annular spacing d is small compared to the mean cylinder radius (“small gap approximation”), Taylor’s solution for the critical angular velocity is

where S = 0.0571[-

1-p

+ 0.652( 1 -

$1 -!)I-’

+ 0 . 0 0 0 5 16 [ 3,i+ i 0.652( 1

For the case where the outer cylinder is stationary, reduce to the form

(7)

p = 0, Eqs. (6) and (7)

S = 0.0571(1 - 0.652d/ri) + 0.00056(1 - 0.652d/ri)-l

For the case of a narrow gap, ?+ (d/ri -t j2 = 0, Eqs. (8) and (9) reduce to the form

Where (Ta& defined as

(9) 0) and stationary outer surface,

(Tam),,= 1697

(10)

S = 0.0577

(11)

is the critical Taylor number based on the mean radius,

The critical angular velocity is for this case

HEAT/MASS

TRANSFER NEARROTATING SURFACES

149

For the case where the outer cylinder is stationary, ,ii = 0, and the gap is finite, one obtains 4 r = (%),2Fg (14) in which ( w ~is the ~ ) critical ~ velocity given in Eq. (13). For ji = 0 and d/r, -+ 0, Fg is a geometrical factor defined as

Fg = (n4/1697)(1 - d/2r,)-2S-’

(15)

An alternate form for S is given by 1 -dirm d/2r,

)+

0.00056( 1

- 0.652 1 - d/2r,

Theoretical predictions, compared to experimental results for marginal stability in “small gap approximations,” as well as Rayleigh’s inviscid instability criterion, are presented in Fig. 4, from which the flow is stable when ji > Taylor’s instability criterion is in excellent agreement with measurements. This can be inferred very clearly from the pictures of Taylor vortices obtained by Schulz-Grunow (1963) and other experiments undertaken by Donnelly (1958),Donnelly and Fultz (1960),Donnelly and Simon (1960),and Donnelly and Schwarz (1965). Apart from the critical angular velocity, Taylor also Ifor the instability in the azimuthal flow, which for calculated the wavelength , a narrow gap was found to be approximately 2d. The wide gap case was subsequently considered by Davey (1962), Walowit et al. 1964), and Roberts (1965).

c’.

-----

EXPERIMENTAL VISCOUS STABILITY BOUNDARY (TAYLOR, 1923) THEORETICAL INVISCID STABILITY BOUNDARY (RAY LEIGH, 1916)

E 32000 -

--___I

- 4000

I--------_-

- 2000

0

2000

4000

WJU

FIG.4. Stability limit in flow between two rotating cylinders (Taylor, 1923).

150

DAVIDMOALEM MARONAND SHIMON COHEN

Besides the above-mentioned expressions for the instability criterion, other mathematical studies were also undertaken. These were intended to determine the stability margin for various ji and 4. Some comprehensive solutions of the stability problem and experimental studies are summarized in Tables I and 11, where a = 2 x d / l is the dimensionless wavenumber of the critical disturbance and Tacr. The critical Taylor number for the appearance of vortices is (see TABLE I THEORETICAL STABILITY STUDIES IN COUETTE FLOW Reference Walowit er al. (1964) Roberts (1965)

Sparrow et al. (1964) Walowit et al. (1964) Roberts (1965) Walowit et al. (1964) Roberts (1965) Walowit et al. (1964) Roberts (1965) Sparrow et al. (1964) Walowit et al. (1964) Kirchgassner (1961) Roberts (1965) Walowit et al. (1964) Roberts (1965) Sparrow ef al. (1964) Walowit et al. (1964) Kirchgassner (1961) Chandrasekhar and Elbert (1962) Walowit et al. (1964) Roberts (1965) Sparrow et al., (1964) Walowit et al. (1964) Roberts (1965) Sparrow et al. (1964) Roberts (1965) Walowit et al. (1964) Sparrow et al. (1964) Sparrow et al. (1964) Walowit et al. (1964)

-

5

p range

a

1 0.975 0.9625 0.95 0.95 0.95 0.925 0.9 0.9 0.875 0.85 0.80 0.75 0.75 0.70 0.667 0.65 0.60 0.50 0.50 0.50 0.50 0.50 0.40 0.36 0.35 0.30 0.28 0.25 0.25 0.20 0.20 0.15 0.10 0.10

0 0 0 - 3.0-0.85 -0.25-0.9025 0 0 - 0.25-0.8 I 0 0 -0.25-0.64 0 -2.0-0.53 -0.50-0.49 -0.40-0.44 0 -0.25-0.36 0 - 1.5-0.235 -0.5-0.25 -0.50-0.25 -0.40-0.25 -0.25-0.16 0 -0.85-0.116 -0.125-0.09 0 0-0.06

3.12 3.13 3.13 3.13 3.14 3.12 3.13 3.13 3.13 3.13 3.13 3.13 3.14 3.14 3. I4

0 -0.30-0.04

-0.15-0.02 1 0-0.01 -

-

3.14 3.15 3.16 3.16 3.16

Tar!z 41.18 41.79 42.09 42.44 42.44 42.45 43.13 43.87 43.88 44.66 45.50 47.37 49.52 49.53 52.04 -

55.01 58.56 68.19 68.19 68.18

-

-

3.17 3.19 3.21 3.20 3.22 3.24 3.33 3.25 3.23 3.31 3.36 3.30

-

83.64 92.12 95.38 I 18.89 120.43 136.40 122.91 176.26 176.33 250.10 423.48 422.79

HEAT/MASSTRANSFER NEARROTATING SURFACES

151

TABLE 11

EXPERIMENTAL STABILITY

STUDIES IN COUETTE

-

Reference Taylor (1923) Coles (1965) Nissan et a!. (1963) Donnelly and Fultz (1960) Nissan et al. (1963) Taylor (1923) Nissan et al. (1963)

Donnelly and Schwarz (1965) Lewis (1927)

FLOW

( 2 0.5

p range

0.942 0.880 0.873 0.854 0.848 0.763 0.743 0.698 0.584 0.9625.0.95 0.925,0.90 0.875,0.85,0.50

-2.89-0.864 -3.25-0.765 - 3.0-0.76 -0.41-0.41 -0.68- 0 - 0,705-0.498 -2.31-0.552 -0.975-0.435 -0.555 - 0.294 - 1.5-0.25 0 0

Fig. 3) based on the inner radius: Ta = w:rid3/v2 = Re2(d/ri)

(17)

with Re denoting the rotational Reynolds number, defined as Re = wirid/v

(18)

As indicated in the tables, the information relating to the instability criterion, predicted theoretically and experimentally in the above-mentioned studies, is restricted to a relatively narrow range of the parameters, ji and f (strictly dealing with the “small gap” stability problem). Sparrow et al. (1964) were concerned with the general case of the finite gap or arbitrary radius ratios, f, ranging from 0.95 to 0.1 for both conditions, where ji 2 0 or ji 5 0. The results for marginal instability are summarized in Fig. 5. The curves are seen to rise monotonically with increasing absolute values of (oo/wi) 0, with a marked spreading of the curves as a function of radius ratio. On the other hand, for (wo/wi) > 0, the curves for the various radius ratios are dense and approach infinity asymptotically as (wo/oi)(ro/ri)2approaches unity. For ji = 0.5, good agreement is obtained with the values of Chandrasekhar (1958) as well as with those of Chandrasekhar and Elbert (1962), in which f = 0.5 in the range -0.5 5 ji < f’. Some other expressions for the critical Taylor number, derived theoretically for the small as well as for the finite gap, were obtained in Walowit’s study (1966).An excellent agreement is obtained with the theoretical results of Chandrasekhar and Elbert (1962) and Kirchgassner (1961) as well as the experimental results of Taylor (1935), Lewis (1927), Donnelly and Caldwell (1962), and Donnelly (1958). For 4 = 0.8, Taylor’s

-=

DAVIDMOALEM MARONAND SHIMON COHEN

6 (+) C

INVISCID FLOW RAYLEIOH PARAMETER,

FIG. 5. Dimensionlesscritical Taylor number for various radius ratios, f, and rotational speed ratios, ji(Sparrow et a!., 1964).

correction formula for the finite gap begins to deviate rapidly upward from Walowit’s results.

c. Wavy Vortex Flow. A more extended experimental investigation undertaken by Coles (1965) has shown that many regular flow configurations can exist after the first transition has been observed. Taylor vortices, periodic in the axial direction, are only the first mode that exists when ji c 0. A pattern of traveling waves, periodic in the azimuthal direction, appears just above the j r s t critical Taylor number, which denotes the departure from the basic laminar Couette flow. The transition is not unique and reveals strong hysteresis. For a system geometry of 4 = 0.874 and height to gap ratio of 27.9, as many as 20 or 15 different states (each state being defined by the Taylor cells and the number of waves axially spaced) have been observed at a given speed. A specified state depends upon the manner in which the final speed was reached. As the speed of the inner cylinder was increased beyond the onset of

HEAT/MASS TRANSFER NEAR ROTATING SURFACES

153

wavy flow, the angular wave velocity decreased from about 0.50, at the first appearance of the wavy vortices to an approximately constant value of 0.340, at Ta/Ta,, 22. Flow visualization studies of the onset of wavy vortex flow have been reported by Schwarz et al. (1964), Snyder (1969, 1970c), and Coles (1965). A linear stability analysis analogous to that carried out by Taylor is not possible for the instability of the Taylor vortex flow. Davey et al. (1968) have resorted to a nonlinear analysis of the growth of Taylor vortices and examined their stability against small perturbations, periodic in both axial and azimuthal directions. In this analysis, which is strictly limited to angular velocities not greatly different from the critical one, the predicted torque is in close agreement with experimental measurements taken by Donnelly and Simon (1960). More complete calculations were carried out by Eagles (1974) to predict the torque necessary to maintain wavy vortex flow for different numbers of azimuthal waves. The theoretical results for an azimuthal wavenumber of four agree well with torque measurements of Donnelly (1958) and Donnelly and Simon (1960). The second critical Taylor number in which the axisymmetric Taylor vortex flow becomes unstable was found to depend strongly on the radius ratio, 4. Snyder and Lambert (1966) and Snyder (1962) show that the ratio of angular velocity at the second transition to that at the first is about 1.05-1.10 for f = 0.95 and is 10 for f = 0.5.

-

d. Turbulent Flow. Light-scattering measurements and spectral analysis of the time-dependentlocal radial velocity, as carried out by Gollub and Swinney (1975) and Fenstermacher et al. (1979), reveal three distinct transitions to turbulence for the flow in rotating cylinders. Each transition adds a new frequency to the velocity spectrum, and at a higher sharply defined Reynolds number (rotational), all discrete spectral peaks suddenly disappear. Hence, chaotic flow is preceded by two periodic flow modes. First, at the onset of wavy vortex flow the velocity spectra contains a single fundamental frequency that corresponds to traveling azimuthal waves. As the Reynolds number is increased, a second fundamental frequency appears at Re/Re,, = 19.3, which is identified as a modulation of the waves, and multiple values have been observed. The frequencies ratio increases continuously and monotonically with increasing the Reynolds number. Hence, the two frequencies appear to be incommensurate (not harmonics). This regime with the two modes exhibits a quasiperiodic behavior. In addition to the sharp spectral components, a broad band that characterizes weakly turbulent flow appears at the flow spectrum at Re/Re,, = 12. The flow spectrum has no spectral peaks at Re/Re,, = 21.9, in which the first fundamental frequency disappears, leaving a spectrum with only the broad component with a frequency of 0.44. The two fundamental

154

DAVID MOALEMMARONAND SHIMON COHEN

frequencies in the quasiperiodic regime are assigned the average values of 1.3 and 0.9, respectively. This sequence of time-dependent phenomena is in agreement with recent experiments by Walden and Donnelly (1979) using entirely different experimental methods. Theoretical simulations by Yahata (1978a,b)and Alziary de Roquefort and Grillavd (1978) reproduce well aspects of the experimental findings. The relevant theoretical studies were recently summarized by Fasel and Booz (1984). The major experimental studies of the transition to turbulence in the flow between concentric cylinders with the inner cylinder rotating and the outer one at rest are given in Table 111(Di-Prima and Swinney, 1981). The steady turbulent organized structures observed at Ta/Ta,, >> 400 were experimentally investigated by Smith and Townsend (1982) and Townsend (1984).The work reported aimed to trace the significant changes in the largescale motion with increase in rotation of inner cylinder (4 = 152.4/229), L/d = 180/(229- 152), beyond the critical Taylor number. The measurements, taken in presence of a slow axial flow (V,/V,< 0.01) to convect the large-scale motion, intended to map the velocity pattern of the toroidal eddies and the associated distribution of small-scale turbulent intensity at low and large Taylor numbers (or, interchangeably, Reynolds numbers). The considerable changes observed, for Reynolds numbers over the range from 300 to 1000 times the critical value, are attributed to curvature effects on the motion. However, at sufficiently high Taylor numbers, a large part of the wall layers is unaffected by the curvature. A logarithmic distribution of mean velocity does exist in vicinity of the walls, similar to that observed in unidirectional turbulent flow. Smith and Townsend (1982) suggest the Monin-Obukov similarity to estimate the characteristic length and the mean velocity profile, which is actually logarithmic by analogy to surface layer similarity of horizontally stratified fluids. Claussen (1984) identified the counterpart terms on which the mean temperature and mean velocity depend, respectively, in stratified shear flow and Couette flow. The Taylor-Couette flow in the annular regime was found (Barcilon et al., 1979; Barcilon and Bridley, 1984)to be characterized by the coexistence of steady coherent motion on two widely separated scales. These are gap width and boundary layer thickness, where each are identified as the consequence of a centrifugal instability and are described as Taylor vortices and Gortler vortices, respectively. The mutual interdependence and the features of the flow structures, based on widely differing scales, were readily simulated by a model proposed by Barcilon and Bridley (1984). As is indicated by Fenstermacher et al. (1979), the experimental observations of only two flow modes, with distinct frequencies that precede turbulent flow, disagree with the early Landau (1944; Landau and Lifshitz, 1959) conjectures, that transition to turbulence would occur as an infinite sequence of instabilities, each adding a new frequency to the motion.

TABLE I11 EXPERIMENTS ON TRANSITION TO TURBULENCE BETWEEN Reference ~~

Technique ~~

Schultz-Grunow and Hein (1956) Coles (1965) Gollub and Swinney (1975); Fenstermacher et a/. (1979) Walden and Donnelly (1979) Bouabdallah and Cognet (1980) Koschmieder (1979) Barcilon et al. (1979)

a

~~

~

CONCENTRIC CYLINDERS

(4

-

718, p = 0)"

rJr0

L/d

Tal,Z (LID --t x)

Re*/Re,,

Re**/Re,,

Re***/Re,,

0.84 0.874 0.889 0.877

50 27.9 30 20

105.1 117.7 125.1 119.1

200 15 25 45

10 -11

18:38

12

23 21.9

0.875 0.909

18-80

40

118.2 137.8

67 70

12

21:25 19.5

0.896

123 65

129.1 137.1

62 320

-I1 --

~~~

Flow visualization Flow visualization Hot wire anemometry Laser-doppler spectra; flow visualization Ion current spectra Electrochemical current spectra; flow visualization Flow visualization Hot film spectra; flow visualization

0.908

-

Re*/Re,,, maximum ratio studied; Re**/Re,,, for appearance of noise; Re***/Re,,, for disappearance of azimuthal waves.

10 5.5

26 21

156

DAVIDMOALEMMARONAND SHIMONCOHEN 2.51

A =0.0478-0.1190 'i

"f4d'

Tacr=,Y

1.00

rm

I

50 LENGTH/RATIO, L/d

I

I00

FIG. 6. Critical speeds for onset of Taylor and wavy vortices (Cole, 1976). Tae,, denotes the appearance of the Taylor vortex flow.

e. Finite Annulus Length Effect. As the inner cylinder rotation is increased, the circulatory motion that exists in the boundary layers at the ends of the annulus develops into a cellular structure, until a complete array of Taylor vortices is established. Critical speeds were determined by torque measurements for the radius ratio of 0.894-0.954 by Cole (1976).It has been found that the first transition to vortex flow is insensitive to the annulus length. However, as depicted in Fig. 6, the wavy critical speed is substantially affected by annulus length, rising appreciably as the length to gap width ratio is reduced below 40. Similar results were found for all radius ratios examined. Other aspects of the various steady states of Taylor vortex flow, where end effects are significant, were summarized by Benjamin (1981). The reported experimental observations (Benjamin and Mullin, 1982; Mullin, 1982; Lorenzen et al., 1983) and theoretical simulations (Aliziary de Roquefort and Grillaud, 1978) are principally focused on the evolution of the primary flow as the dynamical parameters: the rotational Reynolds number and the aspect ratio ( = L / d ) are gradually changed. It was found that mutation of the primary flow involves various distinct steady flows that are extremely sensitive to length effects for moderate values of the dynamical parameters. Anomalous modes in Taylor flow, which exist only for sufficiently high Reynolds numbers, were numerically calculated and compared with experimental observations by Cliffe and Mullin (1985). f. Axial Flow Effect. Hot wire anemometry and flow visualization employed by Kaye and Elgar (1958) imply the existence of four possible flow modes, when induced rotational flow is superimposed on an axial flow in a coaxial cylinder system. These are characterized as (a) laminar flow, (b) laminar flow with Taylor vortices, (c) turbulent flow, and (d) turbulent flow with Taylor

HEAT/MASSTRANSFER NEARROTATING SURFACES

157

I

''K

TURBULENT FLOW

0 c,

3 z

I

/ L A M I N A R FLOW\ PLUS VORTICES

01 / SPEED OF ROTATION OF INNER CYLINDER,aj(

FIG.7. Schematic representation of different flow regimes in an annulus with axial flow and the inner cylinder rotating (Kaye and Elgar, 1958).

vortices. These four flow regimes are related to the axial flow through the annular gap and the speed of rotation of the inner cylinder as shown schematically in Fig. 7. As shown by Chadrasekhar (1960a), the Rayleigh inviscid criterion for rotational instability remains valid in the presence of axial flow and fully developed tangential velocity. To account for viscosity effects, Chandrasekhar (1960b) and Di-Prima (1960) employed linear stability theory to predict the onset of instability in a narrow gap (4- 1). Considering the tangential velocity to be fully developed and assuming uniform axial velocity, the onset of instability is found to increase monotonically with the axial velocity as TaX(Re,) = Ta,:(Re, = 0) + 26.5 Re:;

Re,

+0

(19)

where the Taylor number is defined as Ta* = o z r z d 3 / v 2 r , (20) and Re, is the axial Reynolds number based on the mean axial velocity V, Re,

(21) Equation (18) agrees to within 5% of the experimental results obtained by Snyder (1962)for 5 < Re, < 40. Above Re, > 40 the discrepancy increases to 15%. Below Re, < 5 a separate calculation using the actual axial profile, initially made by Chardrasekhar, shows that the Taylor number initially increases as Re:. Snyder's experiments are in good agreement with this result =

V,d/v

158

DAVIDMOALEM MARONAND SHIMON COHEN

up to Re, 5 2. Also, the relative velocity with which the Taylor vortices move axially, as well as the size and form of the stable disturbance, were found to be consistent with Chandrasekhar’s theoretical prediction, for the range 0 Re, < 20. Above Re, > 20, the axisymmetric disturbance (usually assumed in theoretical predictions) was observed not to be the stable form of the secondary flow for Re, > 20. Actually, experimental results in this range differ considerably from the values deduced by theoretical predictions. For 0 < Re, < 20, the relative drift velocity V,/V, and the wavenumber were found to be independent of Taylor number. The same parabolic dependence of the critical Taylor number on the axial flow rate was observed by Schwarz et al. (1964), as depicted in Fig. 8, for a narrow gap system and 0 < Re, 25. For Re, 2 30, the cell pattern becomes spiral rather than toroidal when the Taylor number exceeds the critical value. Other experimental studies, concerning the instability of the developing tangential across developed axial flow at the annulus entrance, were undertaken by Astill (1964), Martin and Hasoon (1976), and Gravas and Martin (1978). As shown by Astill et al. (1968) and Martin and Payne (1972), theoretical prediction of the stability criterion for the onset of vortices in tangential developing flow accord well with measurements. Two length effects were observed and predicted. These relate to the distance to the point where vortices occurred, and the distance required for

-=

-=

0

5 10 15 2 0 25 AXIAL REYNOLDS NUMBER, Rez

FIG.8. Dependence of the critical Taylor number on axial Reynolds number, Re, (Schwartz et al., 1964).

HEAT/MASSTRANSFER NEARROTATING SURFACES

159

vortices to develop. The point in a developing flow where vortices originate is wiriSo So

1’2

Y(;I)

100 < Re, < 850 1.38 x lo4 < Ta* < 3.12 x lo5

’24;

(22)

where So is the displacement thickness of the tangential boundary layer. By combining Eq. (18) with a prediction of So over the ranges 0.5 5 ri/ro I0.98, 0.01 II* I0.15, to within *20% of all computed values, transition occurs if w2r2d3 1150 2 - 7 ; v2r,

Z

I * = - d Re,

where I* is a dimensionless length parameter and z an axial coordinate. g. Eccentricity Efect. Torque measurements and flow visualization techniques made by Cole (1976) show progressive rise of critical speeds for the onset of Taylor vortices as well as the wavy vortex flow as a function of the eccentricity ratio, E (displacement of cylinder centers divided by gap width). As depicted in Fig. 9, the first critical speed at an eccentricity ratio of 0.25 is 10%

N

- 60-

a W

m

40 0

I

0.1

I

I

0.2 0.3 ECCENTRICITY RATIO, c/d

I

0.4

FIG.9. Variation of the dimensionless critical Taylor number with the eccentricity ratio (Cole, 1976). Tacr, denotes the appearance of Taylor vortex flow; Tacrl denotes the appearance of azimuthal waves.

160

DAVIDMOALEM MARONAND SHIMON COHEN

higher than the concentric value. As with length effects(see Section II,A,2,e), reducing the length to gap ratio has an appreciable effect on the critical speed for the onset of the wavy vortex flow. No effecthas been noticed on the onset of the Taylor vortex flow. 3. Nonisothermal Flow Stability in a rotating annular flow is affected by density variation. Radial temperature gradient across the annular gap gives rise to radial convection through the interaction of the density gradient with centrifugal force ( p V ; / r ) , and free convection due to the interaction of the density gradient with gravity (pgz). Also, the temperature dependence of viscosity is of inherent importance due to Taylor number dependence with kinematic viscosity. Solutions of this problem with the use of the small gap approximation were obtained by Becker and Kaye (1962b). The finite gap problem was also treated by Walowit et al. (1964).The theoretical analysis accounts only for the radial density variations but neglects the temperature dependence of viscosity and thermal conductivity. Effects of free convection, which may change the nature of the basic flow pattern, are overlooked as well. In general, results in these studies indicate that negative temperature radial gradients tend to stabilize, whereas positive ones tend to destabilize the flow. Accordingly, a larger value of critical Taylor number is required than is required for isothermal flow when the rotating inner cylinder is hotter than the outer one. Inclusion of both the effects of radial convection and variable viscosity is found in a study by Walowit (1966). Computations are restricted to a small gap and free convection because gravity is not considered. The effect on stability due to the interactions of the density gradient with centrifugal accelerationwas found to be extremely small. Unlike the effect of radial convection, which depends on the direction of heat transfer, the effect of viscosity variation has no directional dependence and particular effects will depend on the overall shape of the viscosity profile of the fluid under consideration. In any case, the critical Taylor number was always found to be lower than the isothermal Taylor number at the mean temperature. However, instability is more highly influenced by the lower viscosities and is enhanced if the Prandtl number is high. These theoretical results were also predicted and reproduced by Li (1977, 1978) for the variable density case, the variable viscosity case, and the combined effects of variable density and viscosity. It was clearly shown that variable viscosity plays almost an exclusive role in the effect of temperature gradient on marginal stability. The effect of gravity on the onset of Taylor vortices still remains to be analyzed. Small temperature differences, small spacing between cylinders, and high kinematic viscosity, assumed in the above studies, are expected to satisfy Kraussold’s (1964) criterion. It was experimentally found that convective

HEAT/MASSTRANSFER NEARROTATINGSURFACES

161

effects may be neglected for small values of Rayleigh number (less than lo4), defined as Ra

=

Prp(T, - T ) d 3 g:

P r = - CPP

VL

K

Cpdenotes the specificheat capacity, IC is the thermal conductivity, pis the bulk coefficient of thermal expansion, and g is the acceleration due to gravity. The radial temperature difference across the annular gap is T, - T . A different critical dimensionless group over which natural convection dominates the motion was established by the numerical study by Leonardi et al. (1982). Convection with differentiallyheated cylindrical walls and adiabatic end walls in a finite annulus was theoretically investigated. Based on the computation of the heat transfer rates, it was found that at a critical value of Ra/Pr Re2 equal to 14, natural convection becomes important and heat transfer rates are independent of rotation. The correlated heat transfer rates were found in agreement with those established by Thomas and de Vahl Davis (1970) for stationary cylinders.

4. Torque Transport Extensive torque measurements have been made by Couette (1 890), Taylor (1923, 1936), and Donnelly (1958) in laminar and turbulent flow regimes, considering only smooth rotating surfaces. Eagles' (1974)study was concerned with the torque over the range of just about the critical Taylor number, recognized as the wavy vortex flow. Torque transport accomplished by Reynolds stresses associated with Taylor cells was theoretically predicted and satisfactorily compared with measurements undertaken by Donnelly and Simon (1960). For the case in which both the inner and outer cylinders have smooth surfaces, the torque is transmitted to the outer cylinder and the angular velocity, w, of the inner cylinder is expressed in the Taylor vortex regime by

M

= am-'

+ bw'.36

(25)

where the constants a and b are functions of the clearance ratio (= 2d/(r, + Ti). More extensive measurements of the coefficient of viscous frictional torque, Cf, for rotating rough surfaces were reported by Nakabayashi and Yamada (1982).Compared to smooth surfaces, no effect of surface roughness was found on the critical Taylor number. Moreover, in the range of Taylor numbers of about nine times the critical value, a single curve was found to correlate the measurements with no surface roughness effects on the coefficient of viscous

162

DAVIDMOALEMMARONAND SHIMON COHEN

frictional torque, C , , related to that in laminar flow by C,/Cfo = - 800 Tam + 0.3375 Ta;'*;

d/rm = 0.09

(26)

where rm/K equals 77,151,305 and C , is given by the expression Cf = M/2npr2r:w2

(27)

and the Taylor number, Tam,is that defined in Eq. (12). In a laminar flow, the following relationship holds for C , and Re:

C f 0= 1/Re;

Re

= r,od/v

(28)

However, in the vicinity of Re = 2000-3000, wherein transition to turbulent flow begins, Cf depends on both the Re(CfaRe) and the relative roughness (r,/K) in the case in which the outer cylinder rotates. For a rotating inner cylinder, C, depends only on rm/K,wherein the effect of the surface roughness in this case is more pronounced than that of the outer rotating cylinder.

B. HEATTRANSFER NEARROTATING SURFACES 1. General Heat transfer studies were primarily stimulated by cooling problems in the design of electric motors of high power density. As such, experimental investigations have been mostly concerned with measurements of heat transfer coefficients in a rotating inner cylinder system with and without axial flow. Air serves as a working fluid. The associated thermal characteristics have been studied and expressed in terms of the speed of rotation (Taylor number), the axial velocity (axial Reynolds number), and the radial temperature difference. Also, surface roughness as well as entrance effects have been considered. Although considerable work has been conducted on heat transfer rates with neither cylinder in rotation, few heat transfer data are available for cases in which either cylinder is in rotation. In what follows, these data are reviewed in terms of heat transfer correlations for the various prevailing flow modes.

2. Heat Transfer Correlations Heat transfer rates near rotating surfaces are governed by the following dimensionless parameters:

Nu = f(Re, Ta, d/r,,,, Pr) where the Nusselt number Nu is defined as:

NU = 2UdJk

HEAT/MASSTRANSFER NEARROTATING SURFACES

163

where U is the overall heat transfer coefficient. For a wide gap, the term d/r, can be replaced by the geometrical factor F, of Eq. (1 5). For fixed values of Pr and d/r,, the experimental results can be expressed simply by Nu = f(Re,, Ta)

(30)

a. Zero Axial Flow. As long as laminar Couette flow prevails and free convection is neglected, heat is transferred solely by conduction and the Nusselt number can be shown to be only a function of geometry:

Due to the mixing effect, formation of vortices is expected to increase the rate of heat transfer with increasing rotational speed. Bjorklund and Kaye (1959) have verified this for air in an experimental heat transfer investigation over a temperature range of 40-60°F. Data were obtained for four different values of clearance ratio, d/ri, between 0.054 and 0.246, and for several combination ratios of outer to inner cylinder speed, p, between 2 and 0 to infinity. The heat transfer experimental data were correlated by the relation Nu/NuCond = 0.175Ta1l2;

2.7 x lo3 < Ta < 4 x lo5

(32)

Data of Gazley (1958) for clearance ratios of 0.0068 and 0.095 show a large discrepancy with the above-discussed investigation. This is attributed to the assumptions in deriving the heat and momentum analogy on which the data of Gazley are based. Experimental works on heat transfer with air by Becker and Kaye (1962a,b) and later by Tachibana et al. (1960) are in excellent agreement with the above-stated results of Bjorklund and Kaye (1959). A summary of results of the experimental works on heat transfer in an inner rotating cylinder is presented in Fig. 10. Inspection of the figure reveals that three regimes can be identified. First, it is evident that the horizontal line (Nu = 2) and the correlation line yield the critical Taylor number, between 1700 and 1800, which confirms the aforementioned works in this regime. In the correlation line, two distinct types of heat transfer mechanism characterizing two different flow regimes can be identified and correlated as follows: Laminar plus vortices types of flow: Nu = 0.128(Ta/Fg)E367; 1700 I (TaIF,), I lo4 (334 which agree within 10-150/, with those of Bjorklund and Kaye (1959) but lie about 30% lower than those of Gazley (1958). Turbulent types of flow: Nu = 0.409(Ta/Fg)~241; lo4

-= (Ta/F,),

< lo7

(33b)

164

DAVIDMOALEM MARONAND SHIMON COHEN

(Y’

100 6040-

z

-

2

I

1

1

I

-

0.367

THEORETICAL FOR

LAMINAR PLua vonTicEa O L W F-

Z

I

I

lo*

lo3

I

I

I

10’

10‘

loa

0

MODIFIED TAYLOR NUMBER

(e)(L) F,

FIG. 10. Nusselt number versus Taylor number in a rotating inner cylinder system (Becker and Kaye, 1962a).

A single curve correlating the same data over the whole range of rotations is given by

Nu = 0.409(Ta/F,):241;

1700 < (Ta/F’), c lo7

(334

Note that F,, a geometrical factor introduced to correct for finite gaps, is given in Eq. (15) and (Ta/F,), is the modified Taylor number based on mean radius. Notably, to indicate the same dependency of the Sherwood number on the Taylor number in mass transfer studies, namely, the expression Sh/Sc’I3 (or Nu/Pr”’) gives the same linear relationship with the Taylor number. Ho et al. (1964) obtained an analytical relation that predicted the heat transfer characteristics of the Taylor vortex system for a small range of Taylor numbers above the critical value. The analysis is based on the assumption that disturbances of the conduction temperature field resemble the velocity disturbances determined by Stuart’s (1958) model. It is assumed that the dominant interaction is one between the mean flow and the first harmonic component of the disturbances. Moreover, the finite disturbance in the velocity is similar in shape to that derived by linear theory. The equilibrium amplitude in the supercritical range, Ta > Ta,,, is determined by applying energy balance over the volume bounded by the cell surface of one wavelength. However, the theoretical analysis is only valid for those fluids with a Prandtl number close to unity. It has the further restriction of narrow gap geometry and of a temperature difference, AT, being small. The theoretical prediction

HEAT/MASS TRANSFER NEARROTATING SURFACES

165

takes the form Nu NU,,"*

1708 < Tam < 3 x lo4; Pr

= 0.7

(34)

At larger Taylor numbers there is discrepancy due to the fact that the vortices start to become distorted, and nonlinear analysis has to be used to predict the flow more correctly. Accordingly, the theoretically predicted and measured temperature profiles are increasingly different at high Taylor numbers. A sinusoidal periodicity in the axial direction due to the Taylor vortex motion is clearly observed but still maintains a nearly constant temperature within the vortex with some sinusoidal disturbance. Applying nonlinear theory, Aoki et al. (1967) derived a theoretical prediction of overall heat transfer coefficients limited to small gap widths, to fluids of Pr = 1 and restricted to rotations at the neighborhood of the critical Taylor number. To extend the theoretical predictions to a higher range of Taylor number, a semiempirical modification of the equilibrium amplitude, first derived by Davey (1962) to account for the wide gap width, has been carried out by Kataoka (1975). A satisfactory agreement with measurements is obtained over a wide range of Taylor number (2 IRe/Re,, s 20), wherein the generation of the harmonics and the resultant nonlinear interactions should be considered. As shown in Fig. 1 1, the local Nusselt numbers, experimentally observed, have sinusoidal periodicity in the axial direction measured over the vortical cell. Apart from cycloidal periodicity in a very short supercritical range, the sinusoidal periodicity is quite consistent over the whole range of Taylor numbers.

0

2*

*

X Z/d

FIG. 1 1 . Axial variation of local Nusselt (Sherwood) numbers (Kataoka, 1975a).

166

DAVIDMOALEMMARONAND SHIMON COHEN

Similar behavior has been observed in a subsequent study by Kataoka et al. (1977),concerned with mass transfer near surfaces with and without axial flow. Ho et al. (1964) extended the previous heat transfer results to obtain a generalized empirical correlation for fluids of widely differing properties. Heat transfer results were found to be correlated in the Taylor vortex system by the relation m/2

NUC0"d

=

(&)

0.025 < Pr < 750 ;

(35)

where the exponent m is experimentally found to follow the equation 116

m =

(g)

;

lo7 < P r G r < 40; Ta < 1.7 x 10"

(36)

The Grashof number Gr is the ratio Ra/Pr, where Ra and Pr are defined in Eq. (24). Note that all properties are taken at the mean temperature between the inner and outer cylinder. Very little was done in the study of motion and the associated heat transfer in the case of simultaneous rotation of the inner and outer cylinder. The combined effects of rotation of both cylinders as well as of clearance ratio were correlated by Becker and Kaye (1962a,b) to give the empirical equation Nu -Nbn*

- 1.1

[Tall2 - (Ta;/,$) - Tahi,t]( 1 - 3.5d/ri) C41.1 (Tat!,$) - Ta;/,t](3.5d/ri)

+

(37)

for which values of the quantity in the square brackets range between 2 and 50. Ta,r,rsand Tacr,oare the critical Taylor numbers at velocity ratio ji and at ji = 0, respectively. Theoretical results based on analogy between heat and momentum transfer follow the general trend but are too large. Zmeikov et al. (1970) present an experimental study of the averaged characteristics of turbulent flow for cases of corotation and counterrotation. The stabilizing effect of the outer cyclinder's rotation is shown quite clearly in Fig. 12. In the case of corotation, the Nusselt number first decreases sharply when increasing the outer cylinder's rotation, passes through a minimum, and then increases. The decrease in the Nusselt number is associated with the stabilizing effect of the outer cylinder on the turbulent momentum transfer. Beyond the minimum region, which corresponds to forced vortex flow (mi = mo, b,ro= &,,ri), the Nusselt number increases because the outer cylinder increases the rate of turbulent transfer irrespective of rotation. However, in the case of counterrotation, the stabilizing effect of the outer cylinder retards the rise in the heat transfer rate. When the inner cylinder's rotation is increased while that of the outer cylinder is maintained constant, the Nusselt number increases irrespective of the direction of rotation.

TRANSFER NEARROTATING SURFACES

HEAT/MASS

W i = 2 0 0 0 rpm

167

a

80 -

5s c1

+

6060-

=1200rpm

0 .-

;

-

0


404

W

5

20

I

1

I

I

I

w. /Wi

-4

-2

0 wi /w.

2

4

FIG. 12. Nusselt number versus speeds of rotation ratio. (a) wi/w, (b) wo/wi (w, = 678 rpm).

(0, =

2000 and 1200 rpm);

Actually, natural convection was found to be of minor effect in the aforementioned studies. For cases in which heat transfer by natural convection is significant, the data were correlated by Kraussold (1964): Nu

= 0.22(Gr Pr)0.29;

Gr Pr > lo4

(38)

For air, the Rayleigh number, Ra( = Gr Pr), is small and therefore this effect is negligible. Other correlations of heat transfer rates when natural convection becomes important were established in the theoretical analysis of Leonardi et al. (1982). In a numerical study of convection in a finite annulus (ri/ro = 0.66,L / d = 3) with differentially heated walls and adiabatic ends, heat transfer

168

COHEN DAVIDMOALEMMARONAND SHIMON

rates were correlated by the following relation: Nu = 0.52 Ra'."; Ra/Pr Re2 2 14 (39) For Ra/PrRe2 < 14, rotation makes a significant contribution to the heat transfer and natural convection is extremely small. b. Mixed- Mode Flow. As indicated in Section II,A,2,f, damping of the initial formation of Taylor vortices by increasing the axial flow has been observed for both isothermal and nonisothermal flow. This means that the critical Taylor number has to increase as the axial Reynolds number increases. Accordingly, the Nusselt number decreases with increase of the axial flow, a feature confirmed by Becker and Kaye (1962a,b), Kuzay and Scott (1975, 1976a,b), and Gardiner and Sabersky (1978), when one cylinder is maintained at uniform and constant heat flux and the other is insulated. The inner cylinder is rotating and the outer one is at rest. At a fixed low axial Reynolds number, the Nusselt number increases sharply beyond the conduction value, with an increase of rotation that corresponds to the transition from laminar to Taylor vortex flow. The conduction value and respective transition depend on the axial Reynolds number. However, for very high axial flow rates, rotation has a minor effect on heat transfer rates. Apart from a few experimental studies, no correlations have been proposed to account for the combined effects of rotation and axial flow on heat transfer rates. In a study by Kuzay and Scott (1975), a correlation was established in terms of a physical quantity characterizing the flow helix. With this parameter the correlated form of the Nusselt number is

where Nu, is the Nusselt number in straight axial flow with no rotation, Nu, = 0.038 Re:'

Pr

(41)

and uL, the rotation parameter, is defined as

Note that the heat transfer in this study is from the outer to inner cylinder. Besides the effect of different radii ratio of the annuli, there is substantial agreement with measurements reported in Gazley's (1958) studies, which are for heat transfer from the inner cylinder. Based on theoretical consideration, Nijaguna and Mathiprakasam (1982) adapted use of the ratio, o( = Ta/Ta,,), rather than the Taylor number alone to

HEAT/MASSTRANSFER NEARROTATING SURFACES

169

correlate experimental results, whereby Nu

:=

4 . 2 9 4 c ~ O . ~ ~ ~1 ~<; (Ta/TaCr)< 4.817

(43)

and

N u = 5.08a0.3507; Ta/Ta,, > 4.817

(44)

The two regions of (Ta/Ta,,) cover axial Reynolds numbers of about 2501000 and modified Taylor numbers Ta,/F, up to 3.5 x lo5. Regarding the above studies, which are mainly experimental, no attempts have been made to simulate theoretically the heat transfer rates for the combined effects of rotation superimposed on axial annular flow. A solution, similar to the Reynolds analogy between heat and momentum, was developed by Simmers and Coney (1979) to predict heat transfer characteristics of combined Taylor vortex and axial flow. A three-part velocity profile was assumed, consisting of a laminar sublayer adjacent to the annulus outer surface, a bufler layer, and a core region of constant velocity in the central part of the annular gap. Expressions were obtained for thicknesses of the laminar sublayer and buffer layer as well as the temperature drops across these regions. These were used to give the following form for the Nusselt number:

Nu =

.[

4 Pr Re>5Ta0.3675

A

ri/ro

-]0.5[

( 1 - ri/ro)

(1

-

10.25

ri/ro)

o,6175

Tacr

'

300 < Re, I 1600 Tacr < Ta I 2 x lo6

where A and B are given by

The axial Reynolds number, Re,, is based on twice the mean axial velocity. The analogy solution demonstrates the stabilizing effect of an axial flow and, after criticality, the constant rise in Nusselt number with Taylor number. Also, for a fixed Taylor number the Nusselt number decreases with increasing axial Reynolds number, demonstrating the damping effect of the imposed axial flow. Theoretical prediction based on the heat and momentum analogy were

170

DAVIDMOALEMMARONAND SHIMON COHEN

found to be in accord with experimental results confined to the laminar plus vortices regime, only truly applicable to narrow gap systems. C. MASSTRANSFER NEARROTATING SURFACES 1. General The various flow regimes discussed in Section II,A,2 require different approaches in studying the mass transfer phenomena near a rotating surface. The rotating surface may be reacting or nonreacting. For laminar flow, the velocity profile is readily obtained as a function of the radial position, and the concentration profile can be calculated if the mass transfer coefficient and the transport properties are known. When Taylor vortices appear in either secondary flow or in turbulent vortex flow, the mass transport process becomes complex due to the fluid velocity dependence of more than one independent variable. If fully developed turbulent flow is present, the shear flow characteristics near the inner rotating cylinder are likely to be the same as for fully developed axial flow through the annular passage. Comprehensive theory and supporting experimental data as a means of predicting the increase in mass transfer rates in a rotating system are demonstrated through the use of dissolution and electrolytic reaction systems. As with heat transfer, experimental mass transfer studies are mainly concerned with correlations of the overall transport rates. Film and penetration theories as well as mass and momentum analogies have been proposed to predict the mass transport near rotating surfaces. The recently developing information available on rotating systems with and without chemical reaction reveals the possibility of enhancing the mass transfer rates significantly. In general, the increased mass transfer rates are presented in terms of a reaction factor that expresses the ratio between the mass transfer rate with reaction to that without reaction. These results and other specific mass transfer characteristics are presented in the following section. 2. Mass Transfer Correlations Historically, experimental mass transfer studies were confirmed to the fully developed turbulent flow regime. Mass transfer rates associated with laminar flow plus vortices (Taylor vortex flow) were not considered. Only recently has some information about the specific mass transfer characteristics in this flow regime been evaluated. An experimental study made by Eisenberg et al. (1955) involved solid dissolution and electrolytic redox reactions in fully developed turbulence.

HEAT/

MASSTRANSFER NEARROTATING SURFACES

171

Benzoic acid and cinnamic acid cast into a rotating cylinder were dissolved into water-glycerol solutions. Cathodic reductions of Fe(CN)i3 and anodic oxidation of Fe(CN)i4 on nickel electrodes were used to study mass transfer rates. Mass transfer data in the turbulent region were correlated with a Chilton-Colbourn j factor relation as

lo3 -= Re -= 10’;

835 < Sc < 11490

where k , is mass transfer coefficient and Sc is Schmidt number, defined as SC = v/D

(48)

with D denoting the diffusion coefficient. The effect of the gap vanishes when rotational Reynolds number Re is based on the diameter of the inner rotating cylinder. This mass transfer feature characterizing turbulent flow is remarkably consistent with frictional drag and heat transfer measurements. The data for experimental ruris covering a range of gap to diameter ratio of 0.104 to 3.377 are grouped in the following plot of JI. against Re as shown in Fig. 13 (Eisenberg et al., 1955). The functional dependence of the mass transfer coefficient on the physical properties of the system was studied for Schmidt number values ranging from 835 to 11,490. The study covers a range of Reynolds numbers from 112 to 24 1,000. The results depicted in Fig. 13 lead to the conclusion that

(k,/%,) S C O .af(Re) ~~

(49)

This relation is remarkably close to Chilton and Colburn’s JI., in which the exponent is 213. Also, good agreement is obtained with Roald and Beck’s ( 1951) findings concerning the functional dependence on the rotational speed. In Roald and Beck’s results for a rotating magnesium electrode in hydrochloric acid solutions, it was found that k, = const. V;;’

(50)

which is consistent with Eisenberg’s work. Applying the Chilton-Colburn mass momentum transfer analogy, experimental measured mass transfer rates may be approximated quite closely by

ff2

- 0.0794Re

(51)

where measurements of the frictional drag f/2 on cylinders rotating in various fluids has been obtained by Theodorsen and Regier (1944). Sherwood and Ryan (1959) present an analysis of mass transfer based on

DAVIDMOALEMMARONAND SHIMON COHEN

172

loo

I Re = V o i d

U S C =835-11490

- n

.-

I 100

I I I 1000 10000 100000 ROTATIONAL REYNOLDS NUMBER, Re

FIG. 13. General correlation for mass transfer at rotating cylinders (Eisenberg et a/., 1955).

turbulent layer theory, using empirical relationships between the eddy diffusivity and the distance from the wall, as proposed by Deissler (1955). It is applicable strictly to turbulent flow in tubes, but conditions near the inner rotating cylinder were assumed to be the same as for fully developed flow in a tube. Results for rapid irreversible reactions were presented in terms of the mass transfer enhancement factor, FR( = ki'k,), where k i stands for the mass transfer coefficient with reaction. The agreement between data and theoretical analysis was found to be poor, especially at low values of the Reynolds number. A comparison, made by Marangozis and Johnson (1961), of the recalculated data of Sherwood and Ryan (1959) and Eisenberg et al. (1955) was found to be in excellent agreement with the following Gilliland-Sherwood type of correlation: Sh = 1.97

SC'/~

(52)

where the Sherwood number is defined as Sh = 2 k i ~ , / D (53) The exponent of the Schmidt number in Eq. (52) is in agreement with that of the theoretical value. The Gilliland-Sherwood correlation for the analysis of mass transfer data also yields a satisfactory representation in the Holman and Ashar (1971) experimental dissolution study. Experimental results were obtained for lam, inar vortex and turbulent vortex flow regimes. The experiments involved systems with and without chemical reaction.

HEAT/MASS TRANSFER NEARROTATING SURFACES

173

Mass transfer rates in the presence of Taylor vortices with the inner cylinder rotating were found to differ from those reported for fully developed turbulent flow. The data were analyzed and represented in the following correlations (d/ri = 0.678) (Holman and Ashar, 1971): Sh = 9.30 Re0.425Sc’I3

(54)

Equation (54) is for the nonreacting system, where the Reynolds number is based on the annular gap width. For the reacting system, this is Sh = 8.82

Sc ‘ I 3

(55)

To illustrate the effect of the transition from laminar vortex flow to turbulent flow on mass transfer, data are correlated with the modified Taylor number. An examination of the illustrated data given in Fig. 14 shows that as the modified Taylor number increases, the strength of the vortices increases, and the mass transfer rate with chemical reaction increases in a nearly linear relationship. A hyperbolic relation exists as the trend of the curve near the critical Taylor number for transition. At extremely low rotational flow, the curves for mass transfer with and without chemical reaction approach a diffusion-controlled model. The reaction factor for the system in the work of Holman and Ashar (1971) can be obtained by taking the ratio of Eqs. (54) and ( 5 5 ) to give Re > 300

FR = 9.45Re0.06’;

Taw sh

w Z,Zd’ d 2 :r

z

~

c 10’-

m /

a‘ 3

z

r

o

~

O

REACTING SYSTEM

/

W

m I

(56)

4-

//

Sc a43270

10

- 201

10 lo2 10’ 10‘ MODIFIED TAYLOR NUMBER (T~~,F,J””

FIG. 14. Variation of Sherwood number with modified Taylor number for rotating cylinders.

174

DAVID MOALEM MARONAND SHIMON COHEN

Equation (56) indicates little dependence of FRon the Reynolds number for developed flow. Compared to mass transfer data, presented by Sherwood and Ryan (1959) and Eisenberg et al. (1955) for turbulent flow, Eq. (52), derived by Holman et al., results for the Taylor vortex flow give slightly less dependence on the Reynolds number, 0.492 power in Eq. (55) compared to 0.73 as given in Eq. (53). Presumably, mass transfer rates are affected by an imposed axial flow in a way similar to that described in Section II,B,2,b for heat transfer. Onset of Taylor vortices and the associated enhancement of heat/mass transfer are damped by the imposed axial flow to an extent depending on the relative magnitude between the Taylor number and the Reynolds number. This feature was confirmed in the experimental investigation of the Kataoka et al. (1977) by measuring simultaneously the locally and temporally varying mass transfer coefficients and fluid motion of the axially moving Taylor vortices. Measurements of the mass transfer rates on the internal surface of the outer cylinder were conducted in an annulus of d/ri = 0.62 and L/d = 21, implementing the electrochemical technique. With zero axial flow, the local Sherwood numbers show a sinusoida! periodicity in the axial direction, which is intensified in increasing the speed of rotation (Taylor number). When the Reynolds number of the axial motion is raised gradually, the regular sinusoida! variation of Sherwood numbers is not only distorted, but its mean value and amplitude are greatly reduced. As long as Taylor vortices exist, the average Sherwood number is enhanced with an increasing speed of rotation, but decreases with an increasing axial motion. However, the average Sherwood number increases with higher Reynolds number when the flow is purely laminar. Presumably, internal motions in the Taylor vortices are responsible for the improved mass transfer rates. In further studies undertaken by Kataoka et al. (1975; Kataoka and Takigawa, 1981), the corresponding mixing properties of Taylor vortex flow were examined. Kataoka et al. (1975) showed the possibility of an ideal plug flow if a small constant flow is added to the laminar vortex motion. Also, the rate of the axial mass transfer over the boundaries between Taylor vortices was experimentally determined and presented in terms of Taylor number and Reynolds number. It was found to be almost independent of the speed of rotation, but depends on the axial Reynolds number raised to a power of 1.7 in the range 0 < Re, < 70 and 4.8 x lo4 < Ta < 4.3 x lo5. 3. Wiper Blades (Bafles) EfSect

Practically, diffusion layers in the vicinity of rotating surfaces may be removed by inserting wiper blades into the annular chamber. Moreover, the bulk fluid flow in the annulus will be retarded, increasing the velocity gradient at the inner rotating surface and decreasing the stability of the fluid flow. Two

HEAT/MASS

TRANSFER NEARROTATING SURFACES

175

wiper blades utilized in a rotating cylindrical electrode cell were shown by Nadebaum and Fahidy (1975) to affect the increase in mass transfer rates by a factor of 7-8 with respect to a cell without wiper blades. A theoretical analysis, based on dye experiments, is somewhat similar to the model used to describe flow at the leading edge of a flat plate, although here, the fluid flow is in the opposite direction. By the theoretical model, expressions for the mass transfer rates were derived for multiwiper blade systems. Mass transfer rates can be calculated by ...

/

i

~

?d,/2nri

Re, SC

(57)

The Sherwood number Sh is based on the inner radius, w is the number of wiper blades, d , stands for the wiper blades thickness, and the Reynolds number Re, is defined as Re, = 2biri/V (58) C is a convection correction term for slow rotations; when the flow is predominantly laminar, it takes the form C , = 2 / 3 x SCO.~

(594

At fast rotations, when the flow is turbulent, C is given by - wd,/2xri

C,=O.O164(

)

0.5

(ro/ri)'(r0/ri+ l)1.6 Re:.' sc-0.5 (59b) (ro/ri- 1)0.4

Note that at high Schmidt number (Sc > 450), C , becomes negligible. A further simplification is obtained when wd, << 2xri, that is, when the fraction of the rotating electrode surface covered by wiper blades is small. In cases in which the term C , or C , can be neglected, Eq. (57) reduces to Sh = -

W

Re, SC

where for wd, << 2xri, it becomes Sh = 2x(W Re,

sC)0.5

(61)

which provides an adequate (k15%) estimation of the mass transfer rates over a wide range of operating conditions. Experimental evidence via the reduction of oxygen and ferrocyanide ions indicates that mass transfer rates can be predicted at no more than 5% inaccuracy for Sc > 200. The relative enhancement in the rate of mass transfer due to wiper blades is estimated by comparing Eqs. (47) and (61). When the thickness of the wiper blades is small (wd, << 2xri), the ratio between the respective mass transfer coefficients is

176

DAVIDMOALEMMARONAND SHIMON COHEN

where kgwis the mass transfer coefficient with wiper blades. Note that because mass transfer rates, when the wiper blades are present, do not increase as rapidly with rotation as when wiper blades are not utilized, the enhancement factor in Eq. (62) decreases with increasing rotation rates. At very high rotation, the wiper blades offer negligible improvement. Clearly, increase of mass transfer rates without installing wiper blades can be achieved by resorting to high rotations. The equivalent rotational rates, a;,required to achieve the wiped electrode mass transfer is given by =

1 9 . 7 ~ O . Re-0.286 ' ~ ~R Sco.206

(63)

indicating that the relative increase in rotation rate without wiper blades becomes smaller as the number of wiper blades is increased. The axial contribution to mass transfer can be estimated by comparing the expressions in Eqs. (41) and (61). It can be seen that this contribution is expected to be less than 10% of the total if V, < 8.841j;Y5

(64)

Only relatively high axial flow rates can be expected to soundly affect the mass transfer rates.

111. Closure

Although considerable information has been obtained on stability and flow regimes in a system in which the inner cylinder is rotating and the outer one is at rest, few data for heat/mass transfer are available. The sequence of transitions, as the speed of rotation is increased gradually, is known to involve, at least, steady flow, periodic flow, quasiperiodic flow, and finally turbulent (aperiodic) flow. Demarcation lines between the various flow regimes have been experimentallyestablished for various system geometries (ri/ro, L / d )with and without an imposed axial flow. Theoretical studies mostly deal with stability of the Taylor vortex and wavy vortex regimes. Entended modeling of the higher instabilities to predict the various transition and the corresponding flow characteristics is still lacking. Unlike instability studies, our understanding of the heat transfer characteristics and particularly those of mass transfer is incomplete. Much work remains to be done and the question as to how the Taylor vortex motion controls the boundary layer adjacent to the rotating surface where heat/mass transfer takes place must be answered. Except for a few experimental correlations limited to small ranges, no local and temporal measurements of temperature/concentration have been done in the annular gap, especially in the vicinity of the rotating surface. Practically, the enhancement of heat/mass

HEAT/MASS

TRANSFER NEARROTATING SURFACES

177

transfer rates and the improved mixing in the Taylor vortex flow are very important in chemical processes. NOMENCLATURE constants defined in Eq. ( 2 ) friction coefficient specific heat capacity, cal/g "K convection correction terms for laminar and turbulent flows, respectively laminar mass diffusivity (cm2/sec.) annular gap (r, - r i )(cm) wiper blade width (cm) correction factor defined in Eq. (1 5) reaction factor (k,/k:) friction coefficient [ 2 r i / p ( r i w i ) z 1 Granshof number [fi(T, - 7 J d 3 / v 2 ] gravitational acceleration (cmlsec') roughness height (cm) mass transfer coefficient, without reaction (cm/sec) mass transfer coefficient with reaction (cm/sec) mass transfer coefficient with wiper blades (cm/sec) annulus length (cm) dimensionless length parameter defined in Eq. (23) Nusselt number (2Ud/h-)

Pr Ra r rm Re Re,

Re, S sc Sh T Ta Tam T:

U V

v, Vm W Z

Prandtl number ( C p p / ~ ) Rayleigh number [ P r P ( T , - Il;)d3g/vl radial coordinate average radius [(ri ro)/2](cm) rotational Reynolds number (V,,d/v) rotational Reynolds number (2btrid/v) axial Reynolds number (Vmd/v) correction function defined in Eq. (7) Schmidt number ( v / D ) Sherwood number (2k,r,/D) temperature ("K) Taylor number ( r i w : d 3 / v 2 ) modified Taylor number ( r m w f d 3 / v2, Taylor number ( r : w : d 3 / v 2 r m ) overall heat transfer coefficient (cal/cm' sec "K) velocity (cm/sec) convection velocity of Taylor vortices (cm/sec) mean axial velocity (cm/sec) number of wiper blades Axial coordinate

+

Greek Symbols

dimensionless wavenumber (2nd/1) rotation parameter defined in Eq. (42) bulk coefficient of thermal expansion (l/"K) displacement thickness of the tangential boundary layer eccentricity roughness (cm) wavelength dynamic viscosity (g/cm/"C outer to inner angular velocity (4wi)

v

2 P T LT

0

n R'

kinematic viscosity (cm'/sec) radii ratio ( r i / r o ) density (g/cm3) shear stress (dyne/cm2) Taylor to critical numbers ratio (Ta/Ta,,) Angular velocity (rad/sec) rotational velocity, (rpm) rotational velocity, defined in Eq. (63) (rpm)

Subscripts

inner surface outer surface

r. 0 z cr

radial, tangential, or axial coordinate, respectively critical value

178

DAVIDMOALEM MARONAND SHIMON COHEN REFERENCES

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ADVANCES IN HEAT TRANSFER, VOLUME 21

Modeling the Dynamics of Turbulent Transport Processes

B. A. KOLOVANDIN A . V. Luikoo Heat and Mass Transfer Institute Minsk, 220728 U.S.S.R

I. Introduction Modeling the evolution of turbulence generated by a constant transverse mean velocity gradient has been the concern of a number of fundamental investigations that followed the work of Burgers [l]. The practical interest in this type of turbulence, which, strictly speaking, is inhomogeneous (because the first statistical moment-the mean velocity-is a function of the coordinate), but which is homogeneous for higher order moments, is motivated by the fact that this turbulence is a realistic approach to the turbulence of atmospheric, oceanic, and of a wide class of process fluid flows. Theoretical interest in the nearly homogeneous turbulence is multifarious. First, it is displayed by the “designers” of second-order models, which at present constitute the main tool for calculating inhomogeneous turbulent flows. In fact, being free of the factor of spatial turbulent diffusion, the concept of nearly homogeneous turbulence, compared to inhomogeneous turbulence, admits a more rigorous analysis of the role of a generator-a pure shear-in forming the turbulence anisotropy. Second, some asymptotic solutions for nearly homogeneous turbulence may be used to test the methods of direct numerical simulation of turbulence. Moreover, the nearly homogeneous turbulence admits as many detailed measurements of turbulence characteristics as the isotropic turbulence does; such measurements are required for verifying the adequacy of both the second-order models and the method of direct numerical simulation of turbulence. It seems that the above and related arguments have lately stimulated a number of investigations along the three lines indicated above in the study of nearly homogeneous turbulence. In particular, the construction of secondorder models for modeling isotropic turbulence distortion by a pure shear and 185 English translation copyright 01991 by Academic Press, Inc All nghts of reproduction in any form reserved.

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186

its return to isotropy was the concern of a great number of theoretical works (see, e.g., the recent survey by Savill[2]). Experimental investigation of nearly homogeneous turbulence has also evoked a great number of contributions in the last two decades, from the well-known works of Champagne et al. [ 3 ] to the generalized work of Van Atta et al. [4]. Finally, attempts have been made recently to obtain direct numerical simulations of nearly homogeneous turbulence on the basis of elliptical nonstationary three-dimensional Navier- Stokes equations (see the survey by Ferziger [S] and the works of Deissler [ 6 ] and Schumann et al. [7]). It is impossible to overestimate the importance of this approach for carrying out a detailed investigation into the structure of turbulence, which, apart from a purely scientific interest, ensures receiving statistical information to construct nonasymptotic second-order models, thus presenting an alternative to the physical experiment. Just as nearly homogeneous turbulence is an approach to the actual shear turbulence, the model of the turbulent heat or mass transfer processes is presented by the turbulence of the scalar field, generated by a constant mean scalar gradient transverse to the flow. This problem, which constitutes the basic subject of the present work, has been much less developed than the problem of the nearly homogeneous velocity field. Some recent theoretical, numerical, and experimental investigations were initiated by attempts to construct second-order models of turbulent transfer and have turned out to be not as successful as the corresponding velocity field model. The reason for this seems to be insufficient knowledge about the statistics of the simplest, homogeneous and isotropic types of scalar field, as demonstrated by the fundamentally important experimental work of Warhaft and Lumley [S], who showed that, on the degeneration of the isotropic scalar field, in the decaying isotropic velocity field, the scale ratio parameter €4

= 7 u / z , = (?/EU)/(%

(1)

is a conservative quantity (the numerical value of which is determined by the initial conditions) virtually invariable in time until RA>> 1 and PA>> 1. Thus, the exponents of the turbulence energy degeneration rate and of the mean squared scalar fluctuations t Z in the power laws

42

N

7-m

N

7-"

t2

can be different (because R = n/rn). Based on the condition given by Eq. (l), it was shown by Kolovandin and co-workers [9,lO] and by Newman et al. [111 that, in contrast to the equation for the velocity fluctuation dissipation rate in isotropic turbulence,

MODELING TURBULENT TRANSPORT DYNAMICS

187

where F, = F,, = 1113 for R, >> 1, the equation for the passive scalar fluctuation "smearing" rate in isotropic turbulence has the form

+ Ft1.5JTu+ F , ~ E , / T=, 0

(3)

where F,, = F,,, = F,, - 2 = 513, FI2= FI2, = 2 for R, >> 1, and Pa >> 1. Itwas shown that the solution of the system of Eq. (3) with the equation for t 2 -

t2

+ 2.5, = 0

(4)

together with Eq. (2) and with the equation for the turbulence energy -

q2

has the form

-

+ 2.5, = 0

-

q2 = (q2)0(tO)2/(F".-2)(t

L

t2

R

+ to)-2/(Fu,-2)

= (Fus- 2)(t + 7 0 ) = (,2)0(T0)2R0/(Fu.-2)(T + TO)-2R"/(Fus-2)

T, =

-

(5)

[(F,, - 2)/R0](t

+ to)

(6)

= R o = const.

t 2 ~ : R= " const.

where

is the virtual origin; the values with the superscript zero relate to t = 0. Note that Eqs. (2)-(5) are capable of modeling the degeneration of strong (i.e., for R, >> 1 and Pa >> 1) homogeneous anisotropic velocity and scalar fields, provided that the coefficients F,,, Ftls, and F12sare determined in _ _ - of the anisotropy invariants J2 = aijaji, some way or other as the functions J3 = aijajkaki, where aij = 3uiuj/q2 - 6,, and of the corresponding scalar field anisotropy invariants. The reason for the anisotropy of the velocity and scalar fields in nearly homogeneous turbulence is the spatial gradients of the mean velocities and of the scalars. The anisotropy degree of the fields is characterized by the dimensionless parameters

(7)

188

B. A. KOLOVANDIN

where d&/dXk and dT/dxkare constant, transverse to the flow gradients of the mean velocity and of the scalar. The parameters Pkkand P,,, as production terms in the equations for q 2 and t 2 , respectively, i.e.,

-

+ 2E, = 0 t 2 - PI, + 2E, = 0

q2

- Pkk

-

(8)

(9) in nearly homogeneous turbulence, should be arguments that influence both the production terms in the equations for E, and E,, and the functions F,,, Fils, and F,,,. It seems that the problem of parameterizing the unknown terms in the equations for E, and E , in the parameters P,, P,, J,, J 3 , and R, i.e., the construction of second-order models, received a considerable amount of attention after the appearance of the works by Lumley and co-workers [12-141, who have developed the most general (of those available) method for the invariant modeling of unknown terms in the equations for statistical moments. Attempts to develop less general ad hoc models were undertaken by Elgobashi and Launder [15,16], Elgobashi and La Rue [17], and Jones and Musonge [181. Irrespective of the method of parameterization used, all of the known second-order models of the scalar field are asymptotic: they were developed for strong turbulence ( R , >> 1, PA>> 1). Thus, they are inadequate for calculating the scalar field evolution even in the simplest of possible cases-for a degenerating isotropic velocity field with the scalar field being generated by a constant transverse mean scalar gradient when, at great times of evolution, the final stage of scalar field degeneration exists. The exact solution of this problem was provided by Dunn and Reid [19]. The aim of the present work is, first, to develop the “universal” (with respect to the parameters R,, 0, R, P,, and PI) differential second-order model of nearly homogeneous velocity and scalar fields by the closure technique based on the semiempirical theory of nearly homogeneous two-point correlations, and second, to analyze the process of scalar field transfer in the evolving nearly homogeneous velocity field on the basis of the model suggested. 11. Governing Equations

The initial equations for constructing the second-order model of the nearly homogeneous turbulence are the equations for second statistical moments

-

uiuj - pij + 2Eij - Qij = 0

uit - pi, + Ei, t2

- Pi,

- at, = 0

+ 2Et = 0

(10) (1 1) (12)

MODELING TURBULENT TRANSPORT DYNAMICS

I89

where

and of the mean squared are the production g t e s of the fluxes uiul and scalar fluctuations t 2 due to the mean velocity and scalar gradients. The and dissipation rates of the fluxes due to the molecular viscosity and thermal diffusivitycan be conveniently represented for what follows with the aid of the transformation of coordinates in the form involving two-point correlations

where At = a2fa5,Z is the Laplace operator in the space between two points. Because pressure fluctuations are determined by Poisson’s equation, the components Oij and Oi, in Eqs. (10)and ( 1 1 ) can be presented as follows (in the solution of Poisson’s equation the surface integral is neglected and thus neglected is the region closely adjacent to the wall where the approximation to the nearly homogeneous turbulence is not fulfilled):

B. A. KOLOVANDIN

190

where L~ = a/ag,, L , = a2/agiagj, and L,, = a3/agiagjatkare the differential operators of the first, second, and third order in the space f. Equations (10)-(12) with Eqs. (16)-(20) are the exact equations of nearly homogeneous turbulence. For these equations the terms of Eqs. (16)-(20) are excessive. For these to be found, it is necessary, generally speaking, to know the kinematics of nearly homogeneous turbulence. However, the apparatus of the kinematics of turbulence, the level of the complexity of which exceeds that of axisymmetric turbulence, has not been developed up to now. Therefore, to parameterize the unknown terms of Eqs. (16)-( 18), the phenomenological theory of nearly homogeneous turbulence kinematics for two adjacent points will be used. The adequacy of two-point correlations for closely spaced points in the case of parameterization of unknown terms is attributable to the form of the right-hand sides of Eqs. (16)-(18). As regards the unknown terms of Eqs. (19) and (20), here the application of two-point correlations of the same apparoximation requires that the right-hand sides of these relations can be transformed as

qj= -(L,,,u~u;u;

+L,~~U~U;U;)~A,~,~

where A,, - 4x L A:,

1 =

A,, -

- 4n 1

AUUl

-- 4n

1.fU..T d Vol

d Vol

f.,.,leJ

1. 1.

d Vol

fUJ

d Vol f,.,,

are the squares of the integral length scales related to the corresponding correlation coefficients.

MODELING TURBULENT TRANSPORT DYNAMICS

191

111. Nearly Homogeneous Two-Point Correlations for

Closely Spaced Points On condition that the distance between the points considered is small as compared with the respective integral length scale, i.e., r

= ((:)"'/A

c< 1

. u;

+ Gij...k

(21) it is assumed that the turbulence between the two points differs little from its homogeneous value. The simplest model of two-point correlations is used in this case: UiUj..

= Hij....k

where Hij...,kis the homogeneous (shear-free) correlation for closely spaced points and Gij...kis the correcting tensor that takes into account the effect of the pure shear of the mean velocity. The construction of the homogeneous correlations H i j . , . , k is fulfilled proceeding from the minimum number of necessary conditions that should be satisfied by these correlations: (a) translation invariance, (b) the coincidence of H i j , . , , k in isotropy with the isotropic correlation J i j . . . , k , and (c)the coincidence of H ~ ~ ,at, ,f , = ~ o with the one-point correlation. Proceeding from condition (b) and based on the theory of Robertson's invariants [20], homogeneous correlations can be represented in the form of combinations of the governing tensors, i.e., HI.1 =

( 0 ) 2 Ti HI ( 5 , ) t

H,,i =

+ HY'<,Kli

Hi,j = H\"titj Hij.1

Hij.k

+ H',2'Kij + H\2'(Ki1515j+ K j l t 1 t i )

+ Hi2'Kijt + H\''(Kikttkti + = H\3'
(23)

Kjkt5kti) SjKik)

_ _

where Kij = 3uiuj/q2is the coefficient of the one-point velocity correlation and H',"'(t,2)are scalar coefficients. Under the conditions of Eq. (21), the coefficients H : ) ( t : ) can be expanded into a Taylor series:

Using invariance condition (a) and the condition of incompressibility, which for two-point nearly homogeneous correlations of different orders can

192

B. A. KOLOVANDIN

be presented in the form

a-

=0

-tu; X i

it is not difficultto obtain the relations that connect the coefficients in the series of Eq. (24) and to present homogeneous correlations [Eq. (23)] for closely spaced points in the form

-

-

f4.t

= t 2 - bw:

Ht,i

= Chl‘daij

H i .j

=fw,2[(q2/wU2)Kij

CdK,,

- 3h!2?k%Krnntm
+

0:

+ (-ig - i % ) t i t j

Edaij)r2

+ i\?,trntn+

+

+ ”*I + (+u,”t + Z ~ ~ , ~ , ~+, + f

hd(Kik5ktj

+

Kjktkti)

Hij,,

= (i::)

Hij,k

+ (&\?+ Z\?ntm
where

+ (1 - d)Smn]5,<. + ..*

+

*..)ti<j

Kjk
+

*

(25)

* * . ) ~ i j ~

”’

’ *]

3

= ( - A t a ) , is the mean squared velocity field vorticity and = ( -A$), is the mean squared scalar field (conditional) “vorticity.” The quantities w,’ and wz are the fundamental parameters of turbulence

that are connected with the Taylor macro- and microscales of length by the relations

-A,’ = 5qz/w,Z -Af = 6t2/m,2 L, = 5(~)3’2/cu

--

L, = 6 t 2 ( q 2 ) ” 2 / ~ I

MODELING TURBULENT TRANSPORT DYNAMICS

193

Moreover, the functions YO; = E, and Afw: = E,, together with the quantities and ?,determine the corresponding time scales

2

-

The correcting tensors G i j . . . k in the model [Eq. (22)] for the two-point correlation of nearly homogeneous turbulence will be presented in the simplest form as Gij,,.k

= G'"'UiUj***Uk

where the scalar coefficients G(")can be represented just as H g ) in the form of the series given by Eq. (24). By preserving, for simplicity, only the terms of the series with the exponents at coordinates not involved in the respective homogeneous correlations, and using the obvious condition (uiuj...,;)* = Hij.,,,, (the superscript asterisk means homogeneity) as well as the conditions of incompressibility, the correcting tensors can be presented in the form

and so on. The summation of the corresponding relations, Eqs. (25) and (28), yields approximate expressions of nearly homogeneous two-point correlations for closely spaced points. It is quite obvious that Eqs. (25) and (28) do not yield two-point correlations of the nearly homogeneous turbulence of the most general form. Nevertheless, it will be shown below that the above-given relations make it possible, with sufficient accuracy for the second-order model, to carry out the parameterization of unknown terms in equations for statistical moments.

IV. The Second-Order Model of Nearly Homogeneous Turbulence The parameterization of the unknown terms in the equations for moments leads to the so-called model equations. As indicated above, the present work

194

B. A. KOLOVANDIN

uses for this purpose the above-described two-point nearly homogeneous correlations for closely spaced points. A, THESECOND-ORDER MODELFOR THE NEARLY HOMOGENEOUS VELOCITY FIELD

In Eq. (10) for Reynolds stresses, the terms eij and Qijin the form of Eqs. (16) and (19a) should be parameterized. Using the expressions of two-point correlations qjand u,uiu; [Eq. (25)], it is possible to show that the terms considered are parameterized in the form: ~ i= j $E,[dKij

-

+ (1 - d )

(29)

Sij]

(304

= -u,luuijE,

-

= -b,[a,q2&j

+ (Pub, f

(30b)

rUCij)Pkk]

where 2; = L,/A,, is the ratio of the length scale of energy-carrying vortices to the integral scale of velocity fluctuations, bij = 3 P i ; i / P k k

- 6,

= 3Dij/Pkk

- 6ij

Cij

- A)

a, = 2(&d

=Ad

PU

7, = -&d

and a, and b, are certain unknown constants. In the approximation Eq. (29), the scalar quantity E , is the kinetic energy dissipation rate of nearly homogeneous turbulence, i.e.,

v(2)

-

2*

E , = E,* =

=V

( - A ~ =~ vo,’ ) ~

Note, for the sake of definiteness, that this function in nonhomogeneous turbulence is connected with the well-known quantity E , by the relation

v(g) 2

E,

=

-

+ -41 v ax, y q 2

= vo,”

a2

MODELING TURBULENT TRANSPORT DYNAMICS

195

The coefficient d is a certain function of the turbulent Reynolds number R,, so that for R , >> 1 for R , << 1

d={0 1

In the R , asymptotics, the following relations hold: for R, >> 1

eij = $,(Sij __ #.#.

for R , << 1 q2 these are well-known from the theory of homogeneous turbulence [21]. Equation (30a), which models the return of an anisotropic turbulence to an isotropic one due to pressure fluctuations, indicates that this effect shows up only when R , >> 1. Equation (30b) models the rate of change in the quantity Uiuj under the action of turbulence distortion by the mean shear. In the case of rapid distortion of the preliminary isotropic turbulence, i.e., when the following conditions are fulfilled, E..

=E 2

lJ

-

U.U. = $426, I J

-

p.. = D.. = - 2 39 2,’jij 1J 1J

Pkk = 0 Eq. (30b) has the form

-

@ !112 ) = tb,q2Sij

i.e., it represents the exact solution of Crow [22] at b,

=

2

(33)

To complete the second-ordermodel, it is necessary to derive a differential equation for the function E , = vo,’.The initial equation is the dynamic equation for the two-point correlation of nearly homogeneous turbulence (see, e.g., Hinze [23]):

+ (du.) kj+ (dq) dxk

uiu; -

# #’

~

A

dxk

R

- - L k ( # i # k # ) - #)#;#i) - 2 V AyUiI.4) = O _ .

The augmentation of this equation with the Laplace operator in the = (- ACQj),, space f yields the equation for the vorticity tensor function=

B. A. KOLOVANDIN

196

in the form

+ 2v(A,$&),

=0

the trace of which gives the sought-after equation for the vorticity scalar function

-

-

au, + 2(W'ok + @iok)-

0,'

axk

- - [Lk A&UiUiu; - U i U k U i ) ] o

(35)

+ 2 ~ ( A 5 2 3=) ~0 where W i a k = (-&uJU;)o.

2,taking into account the cor-

In Eq. (35) the gradient-type generator of relational tensor Hi,, [Eq. (25)],is modeled as

With regard to the limiting values [Eq. (3211 of the coefficient d, Eq. (36) shows that for RA>> 1, the gradient-type generator in the equation for 03 should be absent. Despite the fact that this is also true for the equation for E,, which was indicated by LKmley and Khajeh-Nouri [ l 2 ] , for example, this term is present in the popular 42 - E, models [it will be shown in what follows that for RA>> 1, a term of the type given by Eq. (35) should enter into the above equation, but as the model of another factor, which differs from the direct generation of vorticity by mean shear]. In Eq. (35), the factor that reflects one of the fundamental properties of the viscous fluid turbulence distinguishing it from other stochastic processes (sometimes also called turbulence), is of principal value, i.e., the generation of vorticity by stretching. This term can be represented in the following exact form: - CLk A&uiuku; - uiu;ub)lO

MODELING TURBULENT TRANSPORT DYNAMICS

197

where S $ = -S$R,

sg =-6J15 (Lk 7

- A
AC&k.i)O/(

is the skewness of the velocity fluctuation derivatives. In the case of isotropy, this transforms, as can be easily shown, into the well-known parameter (the superscript double asterisk designates isotropy)

s:*

= (aur/axr)3/[(aur/axr)2]3/2

The destruction of vorticity by viscosity in Eq. (35) is given by the last term on the left-hand side; it can be presented in the following exact form:

--I

7

f i SLY-

q2

where S& = S,*R,

6J15

s,*= 7 v(A; Hi,i)o/(-A
s,**= 2 v ( a ~ ~ / a x ~ ) / [ ( a u r / a3/2 xr)z] Thus, formally (but not physically), theterms of Eqs. (37) and (38) can be composed into one (which is done in the q 2 - E, models) with the coefficient 7

F,* ,S= $i(--!---

15fi

- S$)

Taking into account the fact that F,* is at least a function of the parameters R, and P, = Pkk/2&,,it can be modeled in the following form: F:(R,,

F,) = F,**(R,) - 2C:t’(l - d ) P , - 2Cit’dP”

(39) where C:t) and CLt) are some constants to be determined. The isotropic function F,**(R,)can be easily modeled by solving asymptotic (when R, >> 1 and R, << 1) equations of isotropic turbulence with the use of Saffman’s invariant [24] for R, >> 1, -

4%: = const.

B. A. KOLOVANDIN

198

and Loitsyanskiy's invariant [25] for R , << 1,

-

q2A:

= const.

In this case, it is easy to obtain asymptotic estimates of the function F:*:

To approximate the function F:*(R,) at moderate values of R,, use is made of the familiar experimental data on the degeneration of homogeneous and nearly isotropic turbulence behind a grid. These data were generalized, e.g., in the works of Kolovandin and co-workers [9,10]. With regard to the asymptotic values [Eq. (39)], the function F:*(R,) for arbitrary values of R , can be modeled by the relation

F:*(R,) = ?(I

- d ) + yd

= L3L - U d 15

(41)

where 2

d=l1+

Jm

Note that the dependence of the function F: on the parameters R , and puis of fundamental importance for the adequacy of the model as a whole. The importance of this problem is quite equivalent to that of modeling the spectral transfer function in the dynamic equation of the energy spectrum. The present work suggests one of the simplest models of F:(R,, pu).A more general model is considered, e.g., in the work of Chung and Adrian [26]. Thus, the model differential equations for the tensor of Reynolds stresses ujuj and for the kinetic energy dissipation rate E , = vo," in the case of nearly homogeneous turbulence have the form - Pij

+ +[dKij + (1 - d)Sij]~, + a,(l - d)aijEu

+ buCau;;iSij+ (SuBij + ~ u c i j ) ~=, , 0~

(43)

where the following notation is adopted: g i s given by Eq. (13); K, = 3ii&/?; aij = K , - 6,; Sij, bij, cij, a,, p,, and yu are from Eq. (31); is Eq. (36);and F: represents Eqs. (39), (41), and (42). The model given by Eqs. (43) and (44) of a nearly homogeneous velocity field is regarded to be universal with respect to turbulence inertia, i.e., to the values of the parameter R,.

eu

MODELING TURBULENT TRANSPORT DYNAMICS

199

In the case of strong turbulence ( R , >> l), the model has the form

-

-

ej + + E , 6, + U,€,Uij + bua,,q2Sij =0 6 , - 2C!t’Pu~,2/q2+ F , * , * E : / ~=~ 0

UiLj -

It can be easily seen that the above asymptotic model involves all of the basic elements of the familiar second-order (Uiuj - E , ) models. However, the specific feature of the structure of the equation for E, should be noted: the second term characterizes not the direct passage of velocity field vorticity due to the mean shear, but rather an increase in the field vorticity due to its anisotropy; the gradient form of the term considered is attributed to the adopted condition that the field anisotropy is due to the transverse mean shear (it should be noted that in Lumley’s models [12,151, the approximations of the term are used directly in terms of the anisotropy invariants J2 and J3). In concluding this section we note that the proposed model Uiuj - E, is tensor cor*ted via approximation [Eq. (29)] of the tensor dissipative function eij = vw’oj; this approximation is the superposition (with the coefficient d ) of the limiting values of the function E, for R , >> 1 and R , << 1. It would have been more rigorous if the approximation [Eq. (29)] had not beLapplied, but rather the differential [Eq. (34)] had been used directly for aid. However, modeling the tensor functions DIS, and IN, is a more complex problem than the approximation of eij in terms of E , . Nevertheless, such an attempt seems to be worthwhile because this is the direct way to describe the length scale tensors within the framework of the tensor-correlated second-order ii& - eii models.

B. THESECOND-ORDER MODELFOR SCALAR FIELD

THE

NEARLY HOMOGENEOUS

In Eq. (1 1) for the turbulent flux .it of a scalar quantity, the terms to be modeled are ei, and Oi,, presented in the form of Eqs. (17) and (22), respectively. As regards the dissipation rate q , , unfortunately, its parameterization is not as definite as the parameterization of the quantity cij [Eq. (29)], because the coefficient n 1 in the correcting tensor Gi, [Eq. (29)] is unknown. The approximation of can be presented in the form

where T,, is a hybrid time scale differing from the time scales z, and duced previously [Eq. (27)].

T, intro-

B. A. KOLOVANDIN

200

With the notation for the scale T,, allowed for, the correlation first-rank tensor defined by Eqs. (25) and (28), acquires the form

2,

-

uit' = Hi,,

+ 3 - b,,(t~/A~,)$ + ~ b , , ( ~ i ~ s / A , t ) u+, t 2 -

*

9 .

(46) -

Bearing in mind this series and the expression for the correlation tu:u; [Eqs. (25) and (28)], the approximation of the function Oil [see its designation, Eq. (2Oa)l can be presented in the form

With approximations [Eqs. (46) and (47)] taken into account, the model differential equation for the scalar flux in nearly homogeneous turbulence takes on the form

+ b,,("'@dxi

-

Note that the approximation of the functions and 0:;) in strong turbulence is quite equivalent to Monin's relations [27]. For completeness of the second-order scalar field model, it is necessary to derive a differegtial equation for the a t e with which the fluctuations of the scalar E, = Xu: from Eq. (12) for t 2 are smeared and which is the fundamental characteristic of the nearly homogeneous scalar field turbulence [see Eqs. (26) and (27)]. The initial equation is represented by the following dynamic equation of a two-point correlation for the nearly homogeneous turbulence (for homogeneous and isotropic turbulence, such an equation was first obtained by Yaglom [28] and the equation for the homogeneous scalar field turbulence in the isotropic velocity field was reported by Corrsin [29]):

- -

-[L(u,tt' - uit't)]

+ 2X(-A$)

=0

(49)

2

Having taken the Laplace operator of this equation in the space and having thenAssumed that 2 = 0, we obtain the following equation for the function of 0: = (-Ac??),:

MODELING TURBULENT TRANSPORT DYNAMICS

20 1

where the notation is adopted:

a2

-

In this equation, the generating gradient-type term is modeled with the use of two-point correlations for closely spaced points [Eqs. (25) and (28)] in the form

The explanation for the form of Eqs. (51) and (52)regarding the generating terms is quite analogous tothat given previously regarding the term P,, in Eq. (36) for the function 0::in the case of strong turbulence (d = 0) the mean velocity and scalar gradients do not contribute to the "vorticity" of the scalar field. In Eq. (50)the direct generator w;, the inertia term, can be presented in the following form: - [Lk Ag(#ktt' - # ; t ' t ) ] , = 2(Lk A<#,?),

(53)

where STin= S: R,

st" =-2 J G 2(Lk 5

AgHkt,t)O/(

-A c H t , t ) O (

- AcHi,i)h'2

is the dimensionless coefficient,which, in the case of isotropy, converts into the mixed moment

first introduced by Yaglom [30].

B. A. KOLOVANDIN

202

-

The destruction of the quantity o;,because of the finite value of the molecular diffusivity 2,is given by the last term in Eq. (50), which can be

where 1 Pi

= s>- 0

sy

=- J

5

*(A

; ~

R,

A/( , - A
i

is the dimensionless coefficient, which in the case of isotropy converts into the statistical moment known in the theory of isotropic turbulence

- - -

s>*=

*($y/(gy[(zy]1'2

Thus, with allowance for Eqs. (51)-(54), the differential Eq. (50) can be presented in the form

Consequently, the second-order model of the nearly homogeneous scalar field is composed of differential equations: Eq. (12) for 2, Eq. (48) for q,and Eq. ( 5 5 ) for E , . The involved variable coefficients FT1,F 4 , and the hybrid time scale GI are the functions of the parameters R1, 0,R, Pu,and PI. It was shown earlier [9,10] that in the case of the velocity and scalar fields being isotropic, the coefficients F,*, = Fa* and FZ = F&*have the form

where the parameter d is defined as the function R, by Eq. (42). To define the coefficents F:, and F A , the time scale Tu,, and the constants involved in the model, the test problems of the dynamics of nearly homogeneous turbulence are considered below.

V. The Problems of the Evolution of Nearly Homogeneous Turbulence The aim of the analysis that follows is to apply the above second-order nearly homogeneous turbulence model, formed by differential equations [Eqs. (lo), (43), and (4411 for the velocity field and by Eqs. (12), (48), and (55)

I

MODELING TURBULENT TRANSPORT DYNAMICS

203

for the scalar field, to the R1 asymptotic evolving flows with constant transverse velocity and scalar gradients in order to determine the coefficients F: and F ; , the time scale q,, and the involved constants, proceeding from some conditions of the realizability of the flows considered, which were established either analytically or experimentally in Kolovandin et al. [9,10]. A. EVOLUTION OF THE TURBULENT NEARLY HOMOGENEOUS

VELOCITY FIELD The differential equations of the turbulent nearly homogeneous velocity field evolution [Eqs. (43) and (44)]in the presence of the transverse mean velocity gradient dUJdx, are as follows:

-

D”,):(

_ -

-

= {2pu

- 2(pu (57)

-

m2

D Dt

( ) 4

=

-{I- - b,

D DT

-E,

$[-$,,+ &(flu

+

42

yu9]} 2

= 2[6,Ud

- E2 + C!t)(l - d)]P,< -F:*2

4

4,

where the dimensionless parameter of shear anisotropy is pkk p- =-= 2E,

-u,u,dU,ld~2 Ell

Because the above-given model of evolving nearly homogeneous turbulence contains undetermined coefficients, some additional invariance-type conditions are required. Because the a priori solution of the problem is unknown, such conditions are based on qualitative arguments. The simplest

B. A. KOLOVANDIN

204

of these is the hypothesis about the equilibrium of large vortices: because the energy-containing vortices, characterized by the linear dimension L , = 5(;1’)3/2/~,

in conformity to which the following characteristic time scale is set

are produced directly by the mean shear (the only source of.turbulence), which determines the time scale - VT -

Zu-=-

q2

-UIU2/(dU1/dX2) q2

then the time scale ratio t U / Z , should presumably remain invariable during the nearly homogeneous turbulence evolution, i.e.,

where -

Z, = Z,

dUl Jdx2

is the dimensionless time scale and

--

2 2 112 P12 = - u l u 2 / ( U , U 2 )

is the coefficient of one-point correlation. The available experimental data [3,4,31,32] show that Eq. (62) is really fulfilled at any value of the parameter P,. Taking into account the “invariance” of Eq. (62), the differential form of which is

consider the parameter R, asymptotic problems of the nearly homogeneous turbulence evolution in time. 1. Evolution of the Strong Nearly Homogeneous Velocity Field ( R , >> 1)

In the case of a strong nearly homogeneous turbulence (d = 0), Eqs. (57)(61) become

(2)

_ _

= [2Fu

- 2(P, - 1)- - a ,

4 4

: :I;.

(3 2 ) -

1 -- 2

MODELING TURBULENT TRANSPORT DYNAMICS

205

where FZ$ = 1113. Substituting the corresponding differential equations into Eq. (63), we obtain the following relation for the coefficient [Eq. (62)]:

This relation may yield the lower (for p, + 0) estimate of the coefficient-it is necessary to take into account the estimate [Eq. (33)] of the coefficient b,: 2 nus0

1

=15 F,*,*

and

Using Eq. (62), the equations for

DZ,

-= F,*,*

DZ

+ 3a, - 4 E,

(65)

can be presented in the form

- 2 - 2(C'" - l)n,,Z; C"

(67)

where Z = t dUl/dx, is the dimensionless time. When C c ) > 1 (it will be shown below that this condition is fulfilled), Eq. (67) with separating variable becomes -

t, =

CJab

+ bth&Z

+ aCthJabZ

where a = 2(C!:) - l)nus, b = FtZ - 2, and C is the constant of integration. When Z >> 1, the solution obtained has the form

where the coefficients nusand CLt) are unknown as yet.

B. A. KOLOVANDIN

206

Using Eq. (62),it is possible to find the limiting (for Z >> 1) value of the shear parameter 1 F: - 2 P, = r&Z: = 2 C$t' - 1 and the corresponding value of the dimensionless shear stress

Equations (68)-(70) show that in the case of the evolution of a stron (R,>> 1) nearly homogeneous turbulence, the parameters Zu,p,, and U,U,/q tend to limiting values. This conclusion is important in light of the discussions in recent publications on the dynamics of nearly homogeneous turbulence [4,31,33]. It was shown [4] that at great values of dU,/dx2, i.e., at great dimensionless evolution times, p, N 1.67. Using this value, it is possible to obtain an estimate of the constant C ! : ' from Eq. (67)

4

CLi) N 1.5

(71) Taking into account this value of C::) and using the experimental data of Karnik and Tavoularis [31], it is possible to find a rough estimate of the coefficient nu, at pu= F:$ - 2 [see Eq. (70)]:

Bearing h a n d the limiting value of value of ui/q2 from Eq. (64)

u;

4

1

= %bus

+ nu,[2(F:$

p,, it is possible to find the limiting

- 2)

+ (34, - 211 = const.

(73)

Taking tusElation into account, it directly follows from the differential equation for ui/q2 when Z >> l that

Comparison of these relations may give a relationship between the coefficients b, and nus. -Further, noting that the differential equations for u:/q2 and ui/;f" are iiergical for R, >> 1 [see model Eq. (43)], it is possible to obtain the relation for u2

114

2. *

-

-u:

=1

- 2= q2

6(F,*,*- 2) 3 2(F,*,*- 2)

= -1

+ ( 3 ~ -, 2) + ( 3 ~ -, 2)

(75)

MODELING TURBULENT TRANSPORT DYNAMICS

207

_ _

Naturally, this relation is also the solution of the equation for u t / q 2 when ?>> 1. Taking into account the asymptotic relations [Eqs. (68) and (69)] for 7, and p,, it is easy to obtain the asymptotic solution to the kinetic energy equation in the form

Note that this solution was obtained earlier by Tavoularis [33], but it was not considered to be asymptotic; it was assumed that during the turbulence evolution, p, = const. However, as shown above, the condition p, = const. is realized only when 7 >> 1. At arbitrary values of 7, the solution given by Eq. (76) can be conveniently used only to analyze qualitatively the evolution of = ?(?). Depending on the initial gecified value of F,, in a certain range of Z values, the e x ~ n e n t i a l growth of q 2 is not possible, but rather this parameter is constant (at P: ‘v 1, in conformity with Champagne et al. [3] and with one of the experimental findings of Karnik and Tavoularis [31]) or even decreases (when < 1, in accord with the experiment of Rohr et al. [4]). This seems to have been the reason for the discussion about whether the limiting value of is attainable in the flow considered. The above analysis shows that the asymptotic (when 5 >> 1) behavior of is described by the exponential law [Eq. (76)]. In particular, it follows from this relation that to process the experimental data, one should use the dimensionless time 7 as an independent variable, which for the well-known experiments can be presented in the form

2

2

i=r-=-1 dvl dx,

-1 h ( V, h d d xv , )

where h is the transverse dimension of the channel and V, is the center line velocity. Figure 1 presents Eq. (76) at C::) = 3/2 and nus = 1.5 x lo-, as well as a comparison of Eq. (76) with familiar experimental data. It is seen that all the known experimental data are successfully correlated by the exponential relation [Eq. (7611. Further, using rough estimates of the coefficients Cit) and nus, it is possible to find from the experimental data rough estimates of the as-yet undetermined coefficienta, with the aid of the asymptotic relation [Eq. (74)]. Assuming, i& conformity with the experiment of Karnik and Tavoularis [31], that u:/q2 2: 0.55 for Z >> 1, we obtain

a, -N 3 (77) -Moreover, comparing Eqs. (73) and (74) for u:/q2 at the above-determined values of the constants CL:) and nus, it is possible to find the asymptotic value

B. A. KOLOVANDIN

208

G2

=T

9,ef

-

6 -

54-

3.0 2.0

-

1.0 -

-

-

-

0.5 0.4 -

0.3

5

10

I

I

I

15

20

25

f

.,

FIG.1. Evolution in time of the kinetic energy of strong nearly inhomogeneous turbulence: = 12.7, i.e. (x,/&~ = 1 I]; experiments: 0 , the straight line represents theory [Eq. (76) at and A , Ref. [31]; 0, Ref. [32].

of the coefficient bus

bus(Fu = F,*,*

- 2) = 0.71

(78)

Taking into account the lower (for K+O) estimate [Eq. (33)], we shall represent the approximation of this coefficient as the function Fuin the form bus

=

1

+

2 usFu/(F$: - 2)

(79)

where a, = 1. Thus, the asymptotic (for Z >> 1) dynamics of the nearly homogeneous turbulence is governed by experimental law [Eq. (7611 for and by Eqs. (68)(70),(74),and (75) for the remaining functions,which, with the rough estimates

2

MODELING TURBULENT TRANSPORT DYNAMICS

209

[Eqs. (71), (72), and (77)] of the coefficients CLi), nus, and a,, have the form 10 x 102 z, --

-2

9

-

5

P, = 5

--

u i / q 2 = 0.225 --

u:/q2 = 0.548 These asymptotic values correspond to the experimental data of Karnik and Tavoularis [31]. Finally, note that the type of flow considered is characterized by the turbulent Reynolds number increasing in time (or increasing downstream):

i.e., during the evolution of the initially strong nearly homogeneous turbulence, the transition to the final stage is impossible; for this to occur, it is necessary that the turbulence be weak at the start of its evolution. 2. Evolution of the Weak Nearly Homogeneous Velocity Field (R, << 1 ) E,

It is assumed that at the start of evolution (when Z << 1)the values of are such that the turbulent Reynolds number is small,

2 and

In this case d = 1, so that differential equations [Eqs. (57)-(61)] take on the form

210

B. A. KOLOVANDIN

D,-q - -2(1 - PJE, DT

D

-E,,

DT

- E2 E2 = 2d,,,P,,L - F * * L

q2

,,q2

+

where F,*,* = 14/5, d,,, = 7/11 Cl:). The substitution of the corresponding equations into Eq. (63) gives the relation for the coefficient nu, in the case of weak turbulence

This relation may yield the lower (i.e., for P,, -+ 0) estimate of the coefficient considered [in this case it is necessary to take into account the estimate given by Eq. (33) of the coefficient b,,]: nuwo

2 1 1 -15 (F,*,* - 2) - 6

=-

In the absence of experimental data, a semiempirical analysis of this flow for determining the unknown coefficients h,, and S,,,is impossible. Therefore, the hypothetical case, p,, >> I , will be taken, which makes it possible to simplify the equations for q 2 and Zu = ~ , d U , / d x ,by presenting them in the form D -Z, = 2( 1 - d*,,)f,

DZ

DDZ

-92

= 2Fuq2

where, in conformity with Eq. (80), -

f,

= nu,?.'

The solution of these equations has the form

z,- = [ ( 4 2 ) 0 ] ( d , " - 1 ) Z ~ q 2-(d,,-1) = (?)O[l

+ 2n,,(6,"

- l)F3]1'@~"4'

MODELING TURBULENT TRANSPORT DYNAMICS

21 1

i.e.,

F,

= n,,(?,0)2[1

+ 2n,,(6,,

1)?,0~]-~

(83) This relation shows that the condition p,,>> 1 can be realized only in the case of small evolution times by assigning great values of 7:. With regard to Eq. (83), we have

By assuming that the coefficient b,,(F, buw IP,>> 1

-

# 0) is as in Eq. (79), i.e., =0

(85)

Eq. (80) gives

= 2(2 - S,u)n,2,(?,0)2[1

+ 2n,,,(S,u -

-

1)?,0~]-~

On the other hand, the differential equation for u : , which under the condition of Eq. (85) can be presented in the form

=

-2n,,~,0[1

+ ~~,,(S~,,)T,T] 1-0-

-1

has the solution

Comparing this solution with Eq. (86), we obtain an estimate of the unknown coefficient SCu

Seu= 1.5

(87)

-It is also possible to show that when S << 1, the solution of the equation for 14:

is

-

As is seen from Eq. (83), at moderate values of T the condition Fu>> 1 is not fulfilled, so that there is no analytical solution. When ? >> 1, the solution of the problem can be obtained quite analogously to the above-considered case with

B. A. KOLOVANDIN

212

R, >> 1. The asymptotic turbulence parameters are as follows: -2 2,

--(F1 -

::- 2)

nu,

F,

= (F,*,* - 2)

When 7 >> 1, the solution of the turbulence kinetic energy equation has the form

which is formally analogous to Eq. (79) but differs from it in principle by the negative exponent. Thus, in the case of a weak nearly homogeneous turbulence, the kinetic energy of turbulence is an exponentially decreasing function of time (or of position). Because it was assumed, with regard to the universal (e.g., valid for any R,) estimate [Eq. (33)] for F,,+ 0, that the coefficient b,, has the form given by Eq. (79), i.e., -I L

buw

=

1

+ a,F,/(F:,* - 2)

then it is possible to determine the coefficient nu, from Eq. (80)as 1 nu, = -

1

To determine the coefficient a, in the approximation given by Eq. (92), it is possible to use Deissler's spectral analysis results [34] showing specifically that

(-8) ~0.03

fx. 1

With .- this value, Eq. (89)may give an estimate of the coefficient nu, for at P, = F,*,*- 2,

nu,

= 10-3/(~::

- 2)

w2/p (93)

MODELING TURBULENT TRANSPORT DYNAMICS

213

Equation (92)may yield an estimate of the coefficient a, in the approximation given by Eq. (91) of the coefficient b,, a, = 0.5

--

-_

(94)

Then, the asymptotic values of the quantities u:/q2 and uf/q2 [Eq. (8911 will be

-_

(u:/q2)is

N

0.62

N

0.13

-_

(u:/q2),,

These egim_ates are at variance with Deissler’s results [34], according to which ( u i / q 2 )+ 0 when T >> 1, i.e., the turbulence supposedly becomes onedimensional. The results of the present analysis show that the energy of the weak nearly homogeneous turbulence decreases exponentially; with regard to the estimate given by Eq. (94) of the coefficient nuw, the exponent is approximately equal to (- 10-2T). The energy components also diminish at the same rate [see Eq. (89)]; the numerical values of the ratio u2/q2 roughly correspond to the values obtained for RA>> 1. As regards the evolution of the turbulence Reynolds number

[in conformity with Eqs. (89) and (90), when Z >> 11, this parameter decreases at the same rate as the rate of decrease of the velocity fluctuations, i.e., weak nearly homogeneous turbulence has a final stage of degeneration characterized by the asymptotic laws, Eqs. (89) and (90).

3. Formation of the Nearly Homogeneous Velocity Field Model for Arbitrary Values of R,

The analysis carried out in Sections V,A,l and V,A,2 makes it possible to formulate a nearly homogeneous velocity field model at an arbitrary turbulent Reynolds number. With regard for the variable coefficients determined in the above-indicated sections as the functions of the parameters d(R,) and F,, the “universal” (in the sense indicated) second-order tensor model is formulated by Eqs. (43) and (44), with the involved parameters being defined by the following relations: [see Eq. (13)]; aij = 3(U,uj/q2) - 6,; S,, b,, cij, a,,, p,, and yu [see Eq. (3111; and and F,* [see Eqs. (36), (39), (41), and (4211, i.e.,

cj

-2Ey

eu

+ F,* = F,**(R,)- 2F,[G,yd + C!!,’(l

- d)]

214

B. A. KOLOVANDIN

The upper and lower estimates of the coefficient b, are given by Eqs. (78) and (9 l), i.e., 1

+ [~r,(l- d) + awd] F,** - 2

The estimates of the constant coefficients involved in the model are such: 6,” = 3/2 [see Eq. (87)],

a, = 3 [see Eq. (77)], C$i)= 312 [see Eq. (71)], a, = 1 [see Eq. (79)], and aw = 0.5 [see Eq. (94)].

B. EVOLUTION OF THE TURBULENT NEARLY HOMOGENEOUS FIELD SCALAR Differential equations describing the evolution of a nearly homogeneous scalar field in a nearly homogeneouLvelocity field have the form given by Eqs. (95)-(97) [the exact Eq. (12) for t 2 and the model equations Q a n d E, are in the form of Eqs. (48) and (55); to simplify the analysis, the last term, which is analogous in form to the generator pi,, is ignored in Eq. (48)].

-

-dT

uit = -u;-

t2 =

-2(1

dx2

- [d

1+ uuI(1 - d)]-~Zt T,,

- P,)E,

(95) (96)

The variable coefficients F: and FZ can be presented as F$ = F$* - 2C$!’(l - d)F,

FZ = FZ* - 2 C 3 l

- 2D!t’dF,

(98)

- d)F, - 2Di:’dFl

(99)

where

F,

-ii&(dU1/dX,)/Eu, r‘, = -Gt(dT/dx,)/E, The isotropic coefficients FFl* and Ff2* are prescribed in the form of Eq. (57). Note that though the representation of the functions FF, and FZ in the form of Eqs. (98) and (99) is quite arbitrary, nevertheless it will be shown in what follows that the coefficients C;;),Cp), D $ t ) ,and DLf)can be found quite definitely within the frameworks of the hypothesis adopted, as functions of the anisotropy parameters p, and The additional condition, which makes it possible to parameterize the =

e.

MODELING TURBULENT TRANSPORT DYNAMICS

215

unknown coefficients of the model, is the hypothesis concerning the equilibrium of large scalar “vortices”: large “energy-carrying’’ scalar vortices characterized by the linear dimension -2 2 112 L , = 6t (4 1 1% in conformity to which the time scale is set -

-

T, = t 2 / E , = L,/6(q2)1’2

are produced directly by the mean scalar gradient d T / d x , (this is the sole source of turbulent fluctuations of the scalar). This determines the time scale

-

TT

= ZT/q2 = [ - Q / ( d T / d x 2 ) ] / ?

and this time scale ratio 7 T / q should presumably remain invariable in the process of evolution of the nearly homogeneous scalar field in time, i.e.,

-

where p2, = - s / ( u $ t 2 ) ’ I 2 is the coefficient of the one-point correlation of transverse velocity and scalar fluctuations; 7, = z , d U l / d x 2 is the dimensionless time scale. The differential form of the hypothetical invariant [Eq. (loo)] is as follows:

Adopting Eq. (100) and its differential form [Eq. (lOOa)] as a hypothesis, we shall consider the R, asymptotic problems of the nearly homogeneous scalar field evolution in time. In view of the fact that the number of governing parameters in scalar turbulence considerably exceeds the number in the velocity field turbulence (0, R = 7 , / 7 , , PI are supplemented to the parameters R, and p,), Eq. (100) is insufficient for determining the hybrid time scale T,, and the unknown coefficients of the scalar field model. Therefore, in addition to Eq. (loo), one more condition will be used as a minimum-an assumption about the structural equilibrium of the velocity and scalar fields at large evolution times, i.e.,

R,,,

= R, = const.

The differential form of this equation is as follows:

(101)

B. A. KOLOVANDIN

216

1. Evolution of a Strong Nearly Homogeneous Scalar Field in a Strong Evolving Homogeneous Turbulence ( R , >> 1) In the case of strong turbulence, Eqs. (95)-(97) have the form

-

u;t =

- dT -24;-

t2 =

-2(1

dx2

1 -

- a,,-u,t

TItS

- Fl)€,

where F?,? = F,*,* - 2 = 513 F:,: = 2 Substituting Eq. (61) for and Eq. (103) for G,?, and E, into differential conditions [Eq. (100a)], it is possible to obtain the following relation for the time scale:

The symptotic condition, Eq. (102),at p,, = 0 yields an estimate of the coefficient CLf' cg)= 1 (104) At p,, # 0, this very condition determines the second coefficient

cg)= 4

(105) Taking into account these estimates and designations [Eqs. (98) and (99)], the mixed time scale is defined as

whereas differential Eq. (103) takes on the forms

MODELING TURBULENT TRANSPORT DYNAMICS +I

-=

- 1

(F,*,*- 2 - Pu)-

( 109)

7,

TI

217

It follows from comparing the last equation with Eq. (107) for t, that the scale ratio parameter R = ~ , / qis an invariable quantity during the evolution of a strong nearly homogeneous turbulence, i.e.,

R = R o = const.

(1 10)

Recall that Eq. (101),used for determining the coefficients CLt) and C::), is asymptotic, i.e., it is less powerful than the hypothesis about the equilibrium of large vortices. a. Evolution of a Strong Nearly Homogeneous Scalar Field in a Strong Degenerating Isotropic Velocity Field. In the simplest of the possible cases, when there is no gradient-type generator of velocity fluctuations (F,, = 0), the velocity field is isotropic and degenerating; the laws of its evolution are given by Eq. (6). With the use of Eq. (loo),i.e.,

where the subscript zero stands for in the form

t2

+ 2R?/tu

= 0, the equation for p c a n

be presented

=2n,,(dT/d~,)~&,/R

= 2nl,o(dT/dx2)2[(F,*,* - 2 ) / R ] ( z ) o ( ~ o ) 2 / ( F- t ' (T . + to)1-2/(P"'.-2)

It can be easily shown that the solution of this equation is

t2

= [ ( F ) O - ( 0 2 ) 0 1 ( ~ *+ +( 0 2 ) O ( T *

1)-2R/(Ft%-2)

+ 1 ) 2 ( F t f - 3MFt'.

- 2)

( 1 12)

where

(v2)0 = n,,,[(F,*s*- 2)/R(F,*,*- 3

+R ) ] ( d T / d ~ ~ ) ~ ( ~ ) ~ ( z ~ ) ~

t* = t / T o

If the initial conditions are such that ( 0 2 ) 0 << (p)',the well-known isotropic solution [Eq. (6)] results, i.e.,

-

_

t 2 = (t2)0(T* + 1)-2R/(Ftf-2)

When t* >> 1, Eq. ( 1 1 1 ) takes on the form (tZ).>>, = (02)O(t*)2(Ftf-3)/(FL'.-2)

= (0 2 10 ( T * )4/5

(113)

218

B. A. KOLOVANDIN

Thus, at small times of the scalar field evolution, when the linear gradient dT/dx2 has not yet had time to develop (or at small values of this parameter), t decays according to the isotropic “law” [Eq. (6)]. At large values of z*, when the gradient dT/dx2 is well developed, the evolution of ? follows the power law [Eq. (1 13)]. Taking into account the solution, Eq. (113), it is possible to show that the asymptotic solution Eq. (11 l), for the parameter is

E,

F, = 1 + (F,*,* - 3)/R0

(114)

i.e., in the case of strong turbulence the ratio of the scalar field turbulence generation rate to the rate of “dissipation” has an asymptotic value that depends on the scale ratio R, which, according to Eq. (1lo), is determined by the initial conditions. Asymptotic relations for other characteristics of the strong nearly homogeneous scalar field turbulence in a strong, evolving isotropic velocity field are given by Eqs. (115a)-(l16). To compare the results obtained with the experimental data, the characteristics given below have been normalized by the scales: Urnis the velocity of the flow incident on an agitating grid and M is the size of the grid cell.

To verify the validity of the hypothesis [Eq. (loo)], the asymptotic relations [Eqs. (114)and (1 15)], and of the solution [Eq. (112)] for t“, let us refer to the only known experiment of Sirivat and Warhaft [35], in which the problem considered was studied. First of all, note should be made of the validity of Eq. (loo), both for the “mandolin” (for all of the geometries considered) and for the “toaster,” where

ntsO1: lo-’

(117)

MODELING TURBULENT TRANSPORT DYNAMICS

219

As regards Eq. ( 1 lo), it is fulfilled at any values of x,/M for the “toaster” and when x,/M >> 1 for the “mandolin.” Asymptotic evolution laws [Eqs. (115)] are fulfilled for both types of experiment (“mandolin” and “toaster”), with the numerical values of the parameters and p:, at ntsO= lo-’ and R o = 3/2 (determined in the experiment for the “toaster”) fitting quite closely to the experimental values.

F, = 13/9 = 1.44;

p;, = 13/30 = 0.43

The validity of Eq. (1 12), written in terms of the experiment, i.e., z * = (x,/M)/(x?/M) = x / P

where x o is the virtual origin, was verified for both realizations of the experiment. It was found that 3.28 x lo-’

(q2/U;)0= XO =

-2.9

(q2/U;)0 = XO

=

i i

mandolin

1.3 x lo-’ -6.4

toaster

For each of the configurations of the “mandolin,” the parameter R was assigned to be equal to the corresponding averaged (over x 1/ M ) value. The results of calculations [by Eq. ( 1 12)] of the evolution of the parameter t 2 over the coordinate i= x,/M for the “mandolin” show (Fig. 2) quite a satisfactory

10

20

30

40

50

XX0

FIG.2. Evolution of the mean squared temperature fluctuations in the wake behind the “mandolin.”Theory [Eq. (1 12)]: Solid line, at R = 1.2; dashed line, at R = 1.75; dot-dash line, at R = 1.8. For experimental designations, see Ref. [35].

B. A. KOLOVANDIN

220

agreement with the experiment at least for geometries (2,l)and (2,2),for which the parameter R varies insignificantly. The comparison is less satisfactory for geometry (10, l), where the parameter R falls appreciably; this is probably due to the not very large values of X at which the temperature field parameters were measured. The data for the “toaster” (Fig. 3) indicate that the curves predicted for R N 1.55 correlate quite satisfactorily with the experiment. As is seen from Figs. 2 and 3, the basic difference between the evolutions of ? in the two versions of the nearly homogeneous tempgature field organization is due to the qualitatively different behavior of t 2 at a small distance behindlhe grid: in the ‘‘mandolin’’-producedtemperature field, the initial decay of t 2 is observed analogous to the isotropic scalar field degeneration. This situation is predicted by Eq. (112) for (u2)0 << (t’)’. The processed experimental data indicate that this condition is realized for the “mandolin.” For the “toaster,” the asymptotic solution [Eq. (1 13)] is realized virtually at all values of X. b. Evolution of a Strong Nearly Homogeneous Scalar Field in a Strong Evolv# 0, the nearly ing Nearly Homogeneous Velocity Field. In the case when homogeneous velocity field evolves following the laws considered in Section V,A,l. It w_as shown that with Z >> 1, the dynamics of the kinetic energy of turbulence q 2 obeys an exponential law [Eq. (76)] and that the parameters

72

( dT/dx.J2M2

0.4

-

0.3

-

0.2

-

20

60

40

ao

1

0.1 R

0.2

0.3

100 I 0.4

120 I

0.5

140 I

XIM I

I

7[SI

FIG.3. Evolution of ? in the wake behind the “toaster.” Theory [Eq. (112)]: Solid line, at = 1.5; dash line, at R = 1.6. For experimental designations, see Ref. [35].

MODELING TURBULENT TRANSPORT DYNAMICS

22 1

Tu,u,Uj/z, and tend to the asymptotic values defined by Eqs. (68)-(70), (74),and (75).If in such a velocity field the scalar field is generated by a transverse gradient dT/dx, = const., then its evolution is given by Eqs. (107)-( 109), where, according to the equilibrium hypothesis [Eq. (loo)], the parameter has the form

and the scalar field time scale t;is expressed, according to Eq. (1 lo), in terms of the time scale Tu -

Z, = Fu/Ro

(119)

Substituting these relations into Eq. (108) for solution of the latter in the form t2

=-

c

a+b

exp(bT)

t',it is possible to obtain the

+ Cexp(--a?)

where a = 2[nu,/(F:,* - 2)] 112 b = 2(F:: c = 2n,,

- 2)[n,,/(F:,*

dT/dx, (dUl/dx,)

-

1

- 2)] 'I2

F:,* - 2

s( nus

)

'I2qfef

where C is a constant of integration. Because the above solution was obtained with the use of the asymptotic parameters of the velocity field, it holds only for i >> 1, i.e., -

(t2),>>

1

= a,(q2)C+1

(120)

where 1

With this solution taken into account, Eq. ( 1 18) shows that there is a limiting value of the parameter defined by the relation PI = 1

+ (F:,*

- 3)/R0

i.e., the limiting relation [Eq. (1 14)] is peculiar to both the case = 0 and the case P, z 0. Asymptotic relations for the other characteristics of the nearly homogeneous scalar field, taking into account the corresponding asymptotic relations that describe the nearly homogeneous velocity field characteristics, have

B. A, KOLOVANDIN

222

the following form (here the designations from Tavoularis and Corrsin [32] are employed for the convenience of comparison with the experiment):

(A,/h)’ = 6- - (r R, Ro

(121b)

To verify the validity of the above-described asymptotic “laws” of the nearly homogeneous scalar field dynamics, the results were used of the only known experiment, that of Tavoularis and Corrsin [32], for which n,,N 2.5 x Ro ‘v 2.22 With these estimates, the following asymptotic values of the parameters given by Eqs. (114),(121b), (l22), and (123) = p2,

(A,/h)2= 1.35 x

1.3; = (2.5 x

10-2)(5)(1.3) = 0.16

are obtained for the corresponding velocity field parameters: 0, =

(1.5/2.5)(2.22)= 1.33

These calculated values are seen to be very close to experimental data. The mean square of the temperature fluctuations was calculated from the asymptotic “law” [Eq. (120)] in the form of

_ _ t2/&

--

= cr,(42/4,2e‘)

MODELING TURBULENT TRANSPORT DYNAMICS

0.3

I

I 20

I 15

I

I

5

10

223

f

FIG. 4. Evolution of t 2 at d U , / d x 2 = const. and d T / d x , = const.; straight line, theory [Eqs. (120)and (76)]; experiment ( 0 )Ref. , [32].

zef

-

The values for the characteristic parameters and tfef were selected at the end of the test sections of the setup of Tavoularis and Corrsin [32], i.e., at Tref =

p)(--) h

h dU, U, d x ,

=

12.7

A comparison of the predicted and experimental data given in Fig. 4 shows that the experiment [32] was carried out at values of Tinsufficient for attaining the asymptotic “law” [Eq. (120)] that governs the growth of temperature fluctuations. Judging from the fact that in this experiment the asymptotic values were attained for (see Fig. l), it is possible to assume that the asymptotic values of the parameter ? develop at greater times (i.e., distances). Figure 4 presents the theoretical curve that should be followed by the experiment at great values of T. In concluding this section, we note that the hybrid time scale defined by Eq. (106) involves the empirical coefficient nts, the numerical value of which depends (as is shown by the analysis carried out in the latter two subsections) - parameter the lower estimate (at = 0) and the upper estion the shear mate (at P, = F,*,* - 2) of this coefficient, Eqs. (1 17) and (124), were determined on the basis of an assumption about the structural equilibrium of the

4‘

E;

B. A. KOLOVANDIN

224

fields. For nonequilibrium situations, when the hypothesis [Eq. (loo)] is not fulfilled, the following simple approximation, can be recommended: which takes into account estimates given by Eqs. (1 17) and (124), 1

1

n1s

ntso

1+3

Ft:

I',- 2

)

2. Evolution of a Weak Nearly Homogeneous Scalar Field in a Weak Evolving Homogeneous Turbulence If the turbulence is weak, the equations of the scalar field dynamics, Eqs. (95)-(97), become -

- dT

u,t = -u;

-

1 -

-- d-uzt

dx2

TI,,

t2 = -2(1 -

where

+ d(l - 2D!:')Pu F,*,, = F,*;C, - 2Pt + 2d(l - DrT))Pt FZW= Ff&

- p,

(127)

Ff2$ = 1013;

FflC = 0 Substituting Eq. (61) for E , and Eqs. (126) into the differential condition of equilibrium Eq. (100a), one may obtain the following relation for the mixed time scale: -

{;[

u2 1

+

3)

- 2(1 - PI)- F&,]R + 2 - Fflw}/( 1 - l + O (128) 4 ntw Equation (102) makes it possible to find the asymptotic value of the mixed scale in the form

=

T,,W

(")

TUIW

= F W l

{

[FTZw - 2(1 - Pta)]Ra

+ Fflw - (F,*,*- 2) + P,,

where the subscript a means 7 >> 1. Comparing these relations for .C >> 1, one may obtain the following relation for the unknown coefficients:

MODELING TURBULENT TRANSPORT DYNAMICS

[FZw - 2(1 - P,,)]Ra

-

+ (FZ*, - 2) +

(

1 -f:0)(4

- F:

- Pus)

=(%)(s) 1+a

q2

225

( 129)

a n!-Ra tw

Because the coefficient Ftr,, is independent of [see Eq. (127)], then at the latter relation may yield the following functional dependence:

=0

where the subscript zero stands, as before, for Pu = 0 and the asymptotic values of the parameters RaOand Ptaoare determined by Dunn and Reid's analytical solution [19] in the form of the following functions of the molecular Prandtl number:

[ ( + .)[ (&J/[

Rao=1 - 2 - 2o )3'2 + .3/2]/[ 50 l+0

pta0=

-

1 - 2 ( L ) 1 1 2 + .1/2] l+0

-

2(&y/2

(130)

+

Substituting the expression of FZ, [Eq. (12711 into the left-hand side of the above relation gives an expression for the coefficient 0;:)in the form

Taking this relation into account, the general (for any R,) approximation, Eq. (99),for Ff2 becomes

T

Then, substituting Eq. (131) into the asymptotic relation, Eq. (129), for >> 1, it is possible to find the asymptotic value of the coefficient Fflwa:

B. A. KOLOVANDIN

226

The further substitution of Eq. (127) into the left-hand side of the above relation for .S >> 1 yields an expression for the coefficient DL:)

+ RO[(:)

a

n,,

] , / ( F : : - 2) - I}(%) I f f 7 3nlw0

-L

With this relation being substituted into the approximation given by Eq. (98), it is possible to find the general expression (for any R,) of the coefficient Flr, in the form

+

+

( )[ 2Fta 1+0

-

'" ( 1 - Ra/Rao)

-

J-)

F::-2

3n,,,

F:$ Ra- 2

-

l]}P,,

Taking into account Eq. (128) for the mixed time scale T',,,, Eqs. (131) and (132), for the coefficients Fir,, and FZ,, we shall consider the problems of the weak nearly homogeneous scalar field evolution in a weak homogeneous turbulence.

a. Evolution of a Weak Nearly Homogeneous Scalar Field in a Weak Degenerating isotropic Velocity Field. In the absence of a transverse velocity gradient, i.e., at F,, = 0, the velocity field is isotropic and degenerating. It obeys the laws of the final stage of degeneration [see Eq. (6) with F:$ = 14/51. The differential equations of the nearly homogeneous scalar field (is, # 0) are of the form of Eq. ( 1 26), in which, according to Eqs. (128), (13 l), (1 32), and (57), the hybrid timescale T,,,and thecoefficients F:lw and F&, have the form

MODELING TURBULENT TRANSPORT DYNAMICS 1

+ 2(1 - PI) - FT2,,]R + 2 - FTlwo}/(

1-

227

3) l + O

FT2t = 1013 F:,* = 1415 With the use of these designations, the differential equation for takes on the form

4

-=

71 =

P/E,

{(FF2t - 2)(1 - Pl/Plao)R

TI

(133)

From this equation it follows that, unlike the case of strong turbulence, when the scale ratio parameter R is constant at any value of time [see Eq. ( 1 lo)], in the present situation considered, this condition is fulfilled only asymptotically when Eq. (133) acquires the form

With the condition given by Eq. (loo), i.e., at

the differential Eq. (126) for 2 at great evolution times has the form of Eq. ( 1 13), in which the coefficient F:: is replaced by F::, i.e.,

As is seen, this solution corresponds to Dunn and Reid’s exact solution [191.

B. A. KOLOVANDIN

228

With regard for this solution, Eq. (135) for -

E, can be presented in the form

Ptao = 1 - (3 - FE)/Rao

where the asymptotic value of the scale ratio parameter RaO is given by Eq. (130). It can be easily demonstrated that at this value of Ra, Eq. (137) converts into the exact asymptotic solution [Eq. (130)] for ptao. Using Dunn and Reid's exact solution for the correlation coefficient, 1+0

1+0

from Eq. (100) one may obtain the value of the coefficient n,, in the form

ntw0

-[

0 1 - (-)'I']Ra0 2O 91-CJ l+O

=5

The asymptotic relations derived above make it possible to determine the evolution of the characteristic length scales

2-(3-F%)/(F%.-2)

T-1/4

In addition to the characteristics considered above, an important characteristic of the nearly homogeneous scalar field turbulence is the turbulence Prandtl number:

Taking into account Eq. (8 1) for nuwO,Eq. (1 37) for n,,,, and Eq. (1 30) for RaO, the previous relation takes on the form =25 6 (F,*,* - 2)

k 0/ [ l

-(&r2]

i.e., the parameters ntwOand nTaoare interrelated as

(139)

MODELING TURBULENT TRANSPORT DYNAMICS

229

Note that the numerical values of the coefficient in Eq. (139) differ somewhat from those obtained by Deissler [36]. Unfortunately, the work does not contain the detailed numerical estimates of the above-mentioned coefficient and this does not allow one to judge its validity. b. Evolution of a Weak Nearly Homogeneous Scalar Field in a Weak Nearly Homogeneous Velocity Field. If, apart from d T / d x 2 = const., the transverse velocity gradient dU,/dx2 is also constant, the scalar field dynamics is described by a system of equations [Eq. (126)] with the coefficients FZ and F Z , specified by Eqs. (131) and (132),and by the hybrid time scale in the form of Eq. (128). In the case considered, the velocity field dynamics .S >> 1 is described by Eq. (90), where the coefficients b,, and nuware determined in the form of Eqs. (92) and (93). With these relations taken into account, it is possible to show that in the asymptotics .S >> 1, the sojution of the differential equation for 7, has the form .St = TJR, = const.

Based on Eq. (loo), the parameter

p, can be presented in the form

2

so that the solution of the differential equation for when Z >> 1, obtained in the fashion analogous to Eq. (120) for R A >> 1, has the form

2

where the asymptotic “law” for is defined by Eq. (91). Thus, in the case o h weak nearly homogeneous turbulence the quantity t 2 ,just as the quantity q2, is an exponentially decreasing function of time. Taking into account Eq. (139), the asymptotic value of the parameter has the form (141) P,, = 1 - (3 - F,*,*)/R, analogous to the relation for the case = 0. Just as in the case of Eq. (138), the turbulent Prandtl number can be presented as OTa = ( n u w / n t w ) R a ( 142) where the coefficient nuwis determined in the form of Eq. (93). The absence of an analytical solution for the case considered, which would have been analogous to that of Deissler [34], does not allow the determination of the parameters n,, and R,, i.e., the coefficient F:, in the form of Eq. (132) remains undetermined. Therefore, let us consider in more detail one of the

230

B. A. KOLOVANDIN

approximate methods for determining these coefficients. By presenting the coefficient n,, on the basis of Eq. (137) as

it is possible to obtain Eq. (142),taking into account Eq. (93),for the coefficient nu, in the form

which shows that in the asymptotics 7 >> 1, the turbulent Prandtl number at P, # 0 differs little from aTaO (this result is also at variance with Deissler's analysis, according to which aT + 1 at any values of 0). For lack of more rigorous information about the scale ratio parameter, we shall assume that Ra = Ra0 (145) and, consequently, in accordance with Eq. (141), C a

= iiao

(146)

With Eqs. (140) and (143)-(146) taken into account, Eqs. (131) and (132) for the coefficients FZ and F$ take on the following, more simple, form F:, = F':

- p,, + dfl(0)Fu

F& = F&* - 2p, + d f 2 ( ol )+ (Ox )

(147)

Equations (143) and (147)determine the hybrid time scale [Eq. (128)] for a weak, nearly homogeneous turbulence and other asymptotic characteristics of the scalar field turbulence:

and so on.

MODELING TURBULENT TRANSPORT DYNAMICS

23 1

The estimates of the numerical values of the above-given characteristics at the limiting values of the molecular Prandtl number are as follows: when a << 1:

when a = 1:

when a >> 1:

The proposed parametrization technique for unknown terms in equations for one-point statistical velocity and transporting scalar moments can be used for constructing the closed second-order model of inhomogeneous turbulence. The use of the Burgers variables (xi)AB
= *[(xi)A

+ (xi)Bl

= (xi)B - (xi)A

of the unknown terms in the equations of transfer in a form that involves the operators of two-point correlations in the space t taken at the point 4 = 0, and the corresponding differential operators in the Cartesian system xi, e.g.,

yields a transformation that makes it possible to fulfill the parameterization of the first term on the right-hand side, i.e., v( - d2ii&/a(f)o = eij,within the framework of semiempirical kinematics of nearly homogeneous turbulence (two-point correlations for closely located points). The second term on the right-hand side of the relation for cij (or any other unknown gradient-type term) does not introduce new unknowns into the model equations. It is quite clear that the proposed parameterization of the “new” unknown terms is not complete, because the resulting two-point correlation tensors are not the most general for the nearly homogeneous turbulence. Unfortunately, the absence of a rigorous apparatus for the kinematics of the nearly homogeneous turbulence restricts the possibilities of the approach to the parameterization of unknown terms in nonclosed differential equations of transfer for statistical

232

B. A. KOLOVANDIN

moments. One of the advantages of such an approach is its universality (irrespective of the term being modeled). Thus, the second-order model of nearly homogeneous turbulence transfer is a specific case of the second-order model of inhomogeneous velocity and scalar fields. On the other hand, the proposed model of nearly homogeneous turbulence at = 0 and = 0 goes over into the second-order correlation model of homogeneous and isotropic turbulence (in this case R,, = 315). One of the specific features of the model considered-the universality with respect to the turbulence inertia, i.e., to the turbulent Reynolds number and to the anisotropy parameters and as well as to the molecular Prandtl number-is fundamental for modeling decaying turbulence. This condition is a prerequisite for inhomogeneous turbulence to model any types of flows, both free and wall flows. The testing of the model of nearly homogeneous turbulence using the results from flows well-detailed experimentally (when R, >> 1)or on problems that have analytical solutions (when R, << 1) had two aims: to show the adequacy of the model and to determine a number of unknown - - coefficients and functions, in particular, the hybrid time scale T,, = T,,(P,, P,, R A ,cr, R). The proposed second-order model of nearly homogeneous turbulence can be used for solving the problems of oceanography and meteorology, for which the pure shear in the mean velocity and the linear distribution of the mean scalar are typical, i.e., dU,/dx, = const., dTldx, = const. The final formulation of the model is given in Section VII.

E,

VII. Appendix A. SECOND-ORDER MODELOF

THE NEARLY HOMOGENEOUS VELOCITY FIELD[EQS. (43) AND (44)]

'I

+ (1 - d)-6ij 3 +

- E2

E,

+ ~ , ( -l d)aijru

iu(F,** - 3 P J Z = 0

where

4,

MODELING TURBULENT TRANSPORT DYNAMICS 1 dQ s..=-+-

dQ dxi)

2(dxj

'I

b.. =3 ?I

P.. 2 - 6.. pkk

"

c..= -3' D..

6.. JJ

pkk

'I

D..=-

(m - + f i I

kd4

dxi

p- = - - pkk 2.5, d = 1 - 2/(1

11 F:* = - ( I 3

a,

N

3

6,

N

2800

a,

N

1

N

0.5

a,

dxi

+ Jm)

14 1 1 13 d) + -d = - - -d 5 3 15

B. SECOND-ORDER MODELOF THE NEARLY HOMOGENEOUS SCALAR FIELD

? + 2(1 - F1)EI = 0

233

234

B. A. KOLOVANDIN

where

NOMENCLATURE UY

h,

K,J - 15,~,deviator of Reynolds

stress tensor 3P,/Pkk- S,,, deviator of the rate of Reynolds stress production tensor

c,,

D,J

f"." G~~ k

3Dij/Pkk- S, deviator of the D, tensor

-(WiULmk/axj + qk suk/ax,), u , u : / q * , two-point velocity

correlation coefficient correcting tensor [Eq. (22)]

MODELING TURBULENT TRANSPORT DYNAMICS two-point correlation tensor in homogeneous shearless turbulence coefficient of molecular conductivity 3 G j / q 2 ,second-order onep i n t correlation coefficient 5(q2)3'2/c,,Taylor macroscale of length related to velocity field 6(7)'/'?/~~, Taylor macroscal of length related to scalar field a3/agiagjagk,differential third-order operator in 6 space -2u,U, aU,/axk, rate of doubled kinetic energy of turbulence generation Pky/k,,ratio of doubled kinetic energy of turbulence generation to rate of its dissipation - 2 Q d T / d x k , rate of scalar variance generation PIl/2q, ratio of scalar substance variance generation rate to rate of its destruction

-

-

€1

+

aTaxk u ~aui/axk), i rate of turbulent scalar substance flux generation pLessure fluctuations P (q2)1'2At/X, turbulent Peclet PA -number u f , doubled kinetic energy of turbulenceR L/T, = ( q * / ~ u ) / ( t 2 / ~ ltime ). -scale ratio (q2)'/2A,/v,turbulent RA Reynolds number t ( a u i / a x j aq axi), Sij tensor of mean shear mixed velocity-scalar T"I substance time scale mean value of scalar substance T scalar substance fluctuations t scalar substance variance t2 vector of mean velocity 'Ji velocity fluctuations ui Reynolds stress tensor U1uj velocity-scalar substance uit correlation vector two-point velocity correlation uiu; tensor two-point velocity-scalar Uit' substance correlation vector xi(i = 1,2,3) Cartesian coordinates pit

Greek Svmbols A: Kronecker symbols Laplace operator in ( space v(du,/ax,)', rate of kinetic energy nonhomogeneous V turbulence dissipation .@(atlax,)', rate of scalar substance variance destructi tion in nonhomogeneous 0 turbulence YO:, rate of kinetic energy 0, homogeneous turbulence dissipation 4 Ho:, rate of scalar substance variance destruction in komogenesuaurbulence 0: 5 v q 2 / ~= , 5qL/w3,squared Taylor microscale of.length related to velocity field

a2/ag,

235

-(@k

+

+

--

6.@2/cT= 6t 2/ o: ,squared Taylor microscale of length related to scalar field integral length scalar of turbulence coefficient of molecular viscosity (xi)a- ( x i ) A .vector of distance between two points v/X: molecular Prandtl number turbulent Prandtl number ( - Ac%hs o, mean squared vorticity of homogeneous velocity field ( mean squared conditional vorticity of homogeneous scalar field

B. A. KOLOVANDIN

236

Superscripts

* **

homogeneity isotropy

0

initial value

Subscripts

a ref t U

asymptotic value reference value notation of scalar field notation of velocity field

S W

0

strong turbulence weak turbulence shearless turbulence

REFERENCES 1. Burgers,J. M. Someconsideration on turbulent flow with shear. Proc. K . Ned. Akad. Wet.. Ser. B 56, 125-147 (1953). 2. Savill, A. M. Recent developments in rapid distortion theory. Annu Rev. Fluid Mech. 19, 531-575 (1987). 3. Champagne, F. H., Harris, Y. G., and Corrsin, S. Experiments on nearly homogeneous turbulent shear flow. J. Fluid Mech. 41,81-139 (1970). 4. Rohr, J. J., Itsweire, E. C., Helland, K. N., and Van Atta, C. W. An investigation of the growth of turbulence in a uniform-mean-shear flow. J. Fluid Mech. 187, 1-33 (1988). 5. Ferziger, J. H. High-level simulation of turbulent flows. “Computational Methods for Turbulent, Transonic and Viscous Flows.” Hemisphere, Washington, D. C., 1983. 6. Deissler, R. G. Turbulent solution of the equations of fluid motion. Reo. Mod. Phys. 56, No. 2, pt. 1,223-254(1984). 7. Schumann, U., Elgobashi, S. E., and Gerz, T. Direct simulation of stable stratified turbulent homogeneous shear flows. Notes Numer. Fluid Mech. 15,245-264 (1986). 8. Warhaft, J., and Lumley, J. L. An experimental study of the decay of temperature fluctuations in grid-generated turbulence. J. Fluid Mech. 88,659-684 (1978). 9. Kolovandin, B. A., Luchko, N. N., and Martynenko, 0. G. Modelling of homogeneous turbulent scalar dynamics. Proc. Int. Symp. Turbul. Shear Flows 3rd, Davis, Calif. pp. 10171025 (1981). 10. Kolovandin, B. A., and Luchko, N. N. Asymptotic modelling of turbulent passive scalar field dynamics. Lett. Heat Mass Transfer 8,321-328 (1981). 11. Newman, G. R., Launder, B. E., and Lumley, J. L. Modeling the behaviour of homogeneous scalar turbulence. J. Fluid Mech. 111,217-232 (1981). 12. Lumley, J. L., and Khajeh-Nouri, B. Computational modelling of turbulent transport. Adv. Geophys. MA, 169-192 (1974). 13. Lumley, J. L. Computational modelling of turbulent flows. Ado. Appl. Mech. 18, 123-176 (1978). 14. Zeman, 0.Progress in the modelling of planetary boundary layers. Annu. Rev. Fluid Mech. 13, 253-272 (1981). 15. Elgobashi, S., and Launder, B. E. Modelling the dissipation rate of temperature variance in a thermal mixing layer. Proc. Int. Symp. Turbul. Shear Flows 3rd Davis, Calif. pp. 1517-1547 (1981). 16. Elgobashi, S., and Launder, B. E. Turbulent time scale and the dissipation rate of temperature variance in the thermal mixing layer. Phys. Fluids,26,2415-2419 (1983). 17. Elgobashi, S., and La Rue, J. C. The effect of mechanical strain on the dissipation rate of a scalar variance. Proc. Int. Symp. Turbul. Shear Flows, 4th. Karlsruhe additional paper (1983).

MODELING TURBULENT TRANSPORT DYNAMICS

237

18. Jones, W. P., and Musonge, P. Modelling of scalar transport in homogeneous turbulent flows. Proc. lnt. Symp. Turbul. Shear Flows, 4th. Karlsruhe pp. 17.18-17.24 (1983). 19. Dunn, D. W., and Reid, W. H. Heat transfer in isotropic turbulence during the final period of decay. Natl. Aduis. Comm. Aeronaut. TN-4186 (1958). 20. Robertson, H. P. The invariant theory of isotropic turbulence. Proc. Cambridge Philos. SOC. 36,209-223 (1940). 21. Batchelor, G . K. “The Theory of Homogeneous Turbulence.” Cambridge Univ. Press, London, 1953. 22. Crow, S. C. The viscoelastic properties of fine grained turbulence. J. Fluid Mech. 33, 1-13 (1968). 23. Hinze, J. 0. “Turbulence: An Introduction to Its Mechanism and Theory.” McGraw-Hill, New York, 1959. 24. Saffman, P. G. Note on decay of homogeneous turbulence. Phys. Fluids 10, 13-49 (1967). 25. Loitsyanskiy, L. G. Some basic laws governing the isotropic turbulence. T r . R A G 1 No. 440 (1939). 26. Chung, M. K., and Adrian, R.J. Evaluation of variable coefficients in second-order turbulence models. Proc. lnt. Symp. Turbub. Shear Flows, 2nd. London pp. 10.43-10.48 (1979). 27. Monin, A. S. About the properties of turbulence symmetry in the near-ground layer of air. lzu. Akad. Nauk SSSR. Fiz. Atmos. Okeana 1,45-53 (1965). 28. Yaglom, A. M. Concerning the local structure of temperature field in a turbulent flow. Dokl. Akad. Nauk SSSR 49,743-746 (1949). 29. Corrsin, S . Heat transfer in isotropic turbulence. J . Appl. Phys. 23, 113-118 (1952). 30. Yaglom, A. M. The inequality for the mixed moment of velocity and temperature derivatives in locally isotropic turbulence. Izu. Akad. Nauk SSSR, Fiz. Atmos. Okeana 3, 1014-1017 (1967). 31. Karnik, U., and Tavoularis, S. The asymptotic development of nearly homogeneous turbulent shear flow. Proc. lnt. Symp. Turbul. Shear Flow, 4th. Karlsruhe pp. 14.18-14.23 (1983). 32. Tavoularis, S., and Corrsin, S. Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient, Part 1. J. Fluid Mech. 104, 311-347 (1981). 33. Tavoularis, S. Asymptotic laws for transversely homogeneous turbulent shear flows. Phys. Fluids 23,999-1001 (1985). 34. Deissler, R. Effect of inhomogeneity and of shear flow in weak turbulent fields. Phys. Fluids 4, 1187-1198 (1961). 35. Sirivat, A., and Warhaft, Z. The effect of a passive cross-steam temperature gradient on the evolution of temperature variance and heat flux in grid turbulence. J. Fluid Mech. 128,323346 (1983). 36. Deissler, R. Turbulent heat transfer and temperature fluctuations in a field with uniform velocity and temperature gradients. Int J. Heat Mass Transfer 6,257-270 (1963).

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ADVANCES IN HEAT TRANSFER, VOLUME 21

Radiative Entropy ProductionHeat Lost to Entropy

VEDAT S. ARPACI Department of Mechanical Engineering and Applied Mechanics The University of Michigan Ann Arbor, Michigan 48109

I. Introduction The foundations of knowledge of entropy production go back to Clausius and Kelvin’s studies on the irreversible aspects of the Second Law of Thermodynamics. Since then, the theories based on these foundations have rapidly developed, first by the efforts of natural philosophers, followed by astrophysicists, and later through the endeavors of applied scientists and engineers. However, the entropy production resulting from temperature differences has remained untreated by classical thermodynamics and is the motivation for this study. This thermal (conduction and/or radiation) energy is the lost heat that goes into entropy production. Clearly, all forms of entropy production result from dissipative processes (involving mass, species, momentum, and/or heat transfer, and electromagnetic or nuclear transport). Less known is the fact that the dissipation may have a diffusive or hysteretic origin, the diffusion being directional and the hysteresis being cyclic. However, except for a few cases (such as strain hardening and electromagnetic saturation), the majority of dissipative processes, including the dissipation of radiation, is diffusive in nature. Accordingly, the present study is primarily devoted to diffusive dissipation, except for Section XI, in which hysteretic as well as diffusive electromagnetic dissipation are identified as lost work. The study consists of twelve sections: Section I1 explores the thermodynamic foundations of entropy production, Section 111 briefly reviews radiative stress, Section IV develops the local entropy production in terms of this stress, Section V illustrates entropy production in a stagnant gas (plasma), 239 Copyright 01991 by Academic Press, Inc. All rights of reproduction in any form reserved.

240

VEDATS. ARPACI

Section VI develops a qualitative understanding of radiative heat transfer, Sections VII and VIII discuss the relation between entropy production and heat transfer, and the maximum entropy production at flame quenching, respectively, Section IX briefly reviews the microscales of turbulence, Section X deals with radiation-affected turbulence, Section XI distinguishes the diffusive and hysteretic aspects of lost electromagneticwork, and Section XI1 concludes the study. 11. Thermodynamic Foundations

The concept of entropy production is now well understood (see,e.g., Arpaci [1,2] and Bird et al. [3]). A renewed interest in the concept is toward its utilization for engineering problems. Because of the abundance of publications on entropy, no attempt is made here to review the literature (see Bejan [4-61 for a review on the application to problems of fluid mechanics and heat transfer). However, an inspection of the entropy literature reveals that, for thermomechanical processes, the concept of lost heat, as opposed to that of lost work, remains largely untreated except for the recent studies by Arpaci [7-91, Arpaci and Selamet [10,11], and Selamet and Arpaci [l2]. The purpose of this section is to introduce the concept of lost heat, to show the relation between this concept and entropy production, and to identify the effects of conduction and radiation arising from this production. Consider a reciprocating engine. For each cycle of this engine, the First Law of Thermodynamics states (Fig. la) that Qi - Q2 =

W

FIG.1. Two laws of thermodynamics. (a), First law; (b), second law (proposed).

(1)

RADIATIVE ENTROPY PRODUCTION

24 1

The usual approach to the efficiency of this engine operating under a reversible cycle leads to a scale for the absolute temperature (Kelvin) QdQ2

= T1IT2

(2)

and to the definition of entropy flow, Y = Q/T

(3)

The classical Second Law of Thermodynamics for an engine states that (4) or (5)

the equality being for a reversible cycle and the inequality being for an irreversible cycle. This inequality suggests, from the view of a balance principle, an entropy production ll,which is a measure for the lost (irreversibly dissipated) energy. In terms of this production, the Second Law for any (reversible or irreversible) cycle may be expressed as (Fig. lb)

Y2- Yl

=

n

(6)

For a reversible cycle

n=0

(7)

and

Yl= Y2 = const. which is identical to Eq. (2). To illustrate the concept of lost energy further, consider a thermomechanical process through an infinitesimal control volume. The First Law applied to this control volume gives (Fig. 2a) where HO

=

u + p v + u, + u p

(9)

is the stagnation enthalpy, V is the volume, and U , and Upare the kinetic and potential energies. Now, recall the definition of entropy flow [Eq. (3)] and express heat flow in terms of entropy flow,

SQ

= S(TY)

or, explicitly,

SQ = T S Y

+YST

VEDATS. ARPACI

242

FIG.2. Two laws for a differential control volume.

where 6Qs is the entropical part of heat explicit in the Second Law and 8QL is the heat dissipated or lost to entropy. After a sign change, let 6 W be split, in a manner similar to SQ, as' 6 w = -(6wM

+ 6wL)

(13) where SW, is the portion of mechanical work balanced in the mechanical energy and 6 WL is the work dissipated or lost to heat. Then Eq. (8) becomes

4 U + UK + U p ) = 6(Qs + QL)

~ P +U6(WM + WL)

(14) [with the sign change introduced in Eq. (13), the work terms now reflect the opposite sign convention for pressure and shear stress]. The mechanical energy balance, obtained either by eliminating thermal effects from Eq. (14) or from the mechanical energy associated with Newton's Law, is d(UK

-

+ U p ) = - V d p + 6wM

(15)

Note that 6 WL,being dissipated into heat, is a thermal term. Also, p dV reversibly affects the internal (thermal) energy. As is well known, Eq. (1 5 ) is reduced to the Bernoulli equation for steady, incompressible, and inviscid flow. The Second Law (proposed) for the control volume (Fig. 2b) is dS = 6Y

+ 6l-I

* The explicit tensorial form of 6W is given in Section IV.

(16)

RADIATIVE ENTROPY PRODUCTION

243

For a reversible process, 6l-I = 0, T = const., and Eq. (16) is reduced to the usual form of the Second Law, dS = 6Q/T

(17)

Now, for a thermomechanical process, consider the energy difference (Fig. 3) First Law - Mechanical Energy - T(Second Law)

or (Total - Mechanical - Thermal) Energy

First Law = Thermomechanical Energy

Law =

Temperature x Second Law Thermal Energy

FIG.3. First law - velocity x Newton's law

- temperature x

3

second law.

VEDATS. ARPACI

244

which, in terms of Eqs. (1 4), (15 ) , and (16), yields dU + p d l ' - T d S = d Q L + ~ W L - T 6 l l

(18)

or dG = BQL

+ SW, - T6l-I

Under local thermodynamic equilibrium,

G=U

+ pV - TS

defines the Gibbs function, and (21)

dG = 0

or, because of locally uniform p and T, d U = TdS - pdV

(22)

Then 6l-I = (l/T)(SQL

+ 6 WL)

(23)

Now, the Second Law given by Eq. (16) may be given alternative forms in terms of the lost heat and work as dS = 6"

+ (l/T)(SQL + 6WL)

(24)

or, in view of SQs = T 6 Y , as (Fig. 4a)

dS = 6Qs/T

+ (l/T)(aQL + ~ W L )

or, in view of Eq. (12), as (Fig. 4b) dS = SQ/T+ 6WLfT

FIG. 4.

Two alternatives of the second law for a differential control volume.

(25)

RADIATIVE ENTROPY PRODUCTION

245

The explicit forms of bQL and SWL will be given in the next section, which deals with the rate of the First and Second Laws of Thermodynamics. When the First Law includes all forms of energy (heat, work, radiative, electromagnetic, chemical, and nuclear), Eq. (23) is generalized to Entropy production = (l/T)(AIl forms of lost energy) and Eq. (24) becomes dS

=

bY

+ (l/T)(All forms of lost energy)

or, with the definition of Lost energy (except for heat) = Energy dissipated to heat becomes

dS = bQ/T + (l/T)(Energy dissipated to heat)

(28)

As already mentioned in the introduction, the two sources of dissipation are diffusion and hysteresis, which are elaborated in the following sections. Finally, let the internal energy, heat, and work associated with the optical limit of electromagnetics or gas radiation be UR*Q" and W", respectively. As is well known,

U R<< u,

Q"

N

QK,

W R<<

w

provided the characteristic transport velocity remains much less than the velocity of light. Then, under the influence of radiation, Q = QK

+ Q"

(29)

QKbeing the heat flow by conduction. For an understanding of Q", the next section is devoted to a brief review on gas radiation. 111. Radiative Stress

The spectrally averaged definitions of the radiative internal energy, heat flux, and stress in terms of the intensity I are

qy

=

Ili dQ

246

VEDATS. ARPACI

where the J scalar and the nijtensor are introduced for notational convenience, c is the velocity of light, Ii is the unit vector in the direction of intensity, and R is the solid angle. In terms of these definitions, the first three specular moments of the transfer equations are

a'' axi

-- - up(B - J )

(33)

with Mijpq...= 411 1 j j i I j l p l , . . .) dR

(36)

Here B = 4&,, Eb = 0T4 is the Stefan-Boltzmann law for the black-body emissive power, upand K~ are the Planck and Rosseland means of the absorption coefficient, respectively, and KM = ( K ~ K ~ )is~ the ' ~ geometric mean of these coefficients. The incorporation of up and uRinto the foregoing equations is discussed by Traugott [13] and Cogley et al. [14], and their use in a variety of problems is discussed by Arpaci and co-workers [8,12,15-221. Clearly, Eq. (33) denotes the thermal balance, Eq. (34) denotes the momentum balance associated with radiation, and Eq. (35) gives the llij tensor in terms of a series based on the specular moments defined by Eq. (36). Note that the radiative heat flux given by Eq. (34), rearranged as

can be interpreted as a generalized diffusion process for any optical thickness. A procedure for the evaluation of Eq. (36) in terms of the Wallis integrals is described in Unno and SpiegelC231. After lengthy manipulations, this procedure leads to

where ai = a/ax, and 8, are used for notational convenience. The same result may be found also in earlier works (see,e.g., Milne [24]). The formal similarity of Eq. (38) to the Hookean constitution for elastic solids should be noted (see Arpaci [25]). An alternate form for this stress may be given in terms of the isotropic radiative pressure. First, invoking the assumption of isotropy, Eqs. (30) and

RADIATIVE ENTROPY PRODUCTION

247

(32) are related as which implies

where is the (isotropic) pressure of radiation. Then, from the trace of lklk = 1,

n,, noting that

Now, in a manner similar to the incorporation of the isotropic pressure into the development of viscous stress from elastic stress (see, e.g., Arpaci and Larsen [2]), adding the identity to Eq. (38), the TIij tensor may be rearranged in terms of the radiation pressure,

The formal similarity of Eq. (44) to the viscous (Stokesean) stress and the electromagnetic (Maxwell) stress should be noted (see Arpaci [25]). This similarity is to be expected in view of the assumed isotropy for the elastic, viscous, and electromagnetic continua (see, e.g., Stratton [26] and Prager [27]). The use of the first term of Eq. (44)in place of Eq. (38) is the well-known Eddington approximation, which leads to a diffusive heat flux,

for any optical thickness. The maximum deviation of this flux from the exact flux given by Eq. (37) is about 29% at T = l / f i (see Arpaci [25]). The next section is devoted to an expression for the rate of entropy production, which includes a radiative part in terms of Eq. (37) or (45).

IV. Local Entropy Production The entropy production discussed in Section I1 is extended here to moving media, which requires as well the consideration of the momentum balance. For the Stokesean fluid, this balance in terms of the usual nomenclature is

248

VEDATS . ARPACI

ap axi

~v~

p-=--+-+ph

Dt

azij

(46)

axj

In terms of the entropy flux, (471

Yi = qi/T

the entropy balance (the rate of the Second Law balanced by the rate of local entropy production) is

where s"' denotes the local entropy production. Also, the conservation of total (thermomechanical) power (or the rate of the First Law) including the heat flux expressed in terms of the entropy flux,

is

D P-(U

Dt

+ 40:)

=

a

----(YiT) axi

a

- -(PU,)

axi

a + axj

-((~ij~i)

+ p f , ~ +i u"'

(50)

Now, the fundamental difference of power, Rate of Total Energy - (Momentum)iui- (Rate of Entropy)T

or Rate of (Total - Mechanical - Therma1)Energy

(51)

lead, in terms of Eqs. (46), (48), and (50)and the conservation of mass,

+

D ~ / D C p(aui/axi)= o

(52)

to

where sij is the rate of deformation. For a reversible process, all forms of dissipation vanish, and

Du/Dt - T(Ds/Dt) + p(Du/Dt) = 0

(54)

which is the Gibbs thermodynamic relation. For an irreversible process, Eq. (54) continues to hold provided the process can be assumed in local equilibrium. Then, Eq. (53)gives the rate of local entropy production SI((

="

T

( y )+

- Yi -

+

zijsij u"'

]

(55)

RADIATIVE ENTROPY PRODUCTION

249

where the first term in brackets denotes the dissipation of thermal energy to entropy (lost heat), the second term denotes the dissipation of mechanical energy to heat (lost work), and the third term denotes the dissipation of any (except for thermomechanical) energy to heat. When radiation is appreciable, qi denotes the total flux involving the sum of conductive flux and radiative flux, q, = q r 4; (56)

+

In terms of the usual conductive constitution and the radiative constitution given by Eq. ( 3 7 ) ,the rate of local entropy production is found to be s"'

= -1[ - ( k1 -

T T

.)(")

dT + 1 dll.. tiR dxj

axi

dxi

+ rijsij + u"'

whose radiative part needs to be related to temperature through Eq. ( 3 8 ) or (44). Also, consideration of only the first term of Eq. (44) yields

s"' =-[-(A1 1 T T

dT + 1 dJ(dT) dxi 3tiR dxi axi ~

+ T..s.. + u"' "

'

whose radiative part is an Eddington approximation and needs to be coupled with (v2- 3 K i ) J = - 12tiiEb (59) (see, for example, Arpaci and Goziim [ 153). The next section is devoted to radiative entropy production in a stagnant gas.

V. Stagnant Gas Consider a stagnant gas between two parallel plates. The emissive power and emissivity of and the distance between the plates are E,, and Eb,, and e2, and L , respectively. The rate of uniform energy per unit volume, u"', is steadily generated in the gas. For negligible conduction, distributions of the gas emissive power and of the entropy production between the plates are desired. The thermal balance for a one-dimensional differential system is The radiative balance for the same system [recall Eq. (33)] is

4: = tip(4Eb - J )

~

dX

Also, the one-dimensional form of Eq. (45) is

VEDATS . ARPACI

250

Now, inserting Eq. (60) into Eq. (61) gives J = 4Eb - U"'/Kp

(63)

and this result into Eq. (62) yields 4: = -(4/3KR)(dEb/dx)

(64)

Note that Eq. (64), valid usually for a thick gas, now applies to a gas of any optical thickness because of the neglected conduction. The combination of Eqs. (60) and (64) gives the governing equation, d2Eb/dx2+ 3KRU1''/4 = 0

(65)

subject to the gas emissive power on the plate walls, Eb(0) = Ebo

and

Eb(L) = EbL

(66)

Because of neglected conduction, these emissive powers are different than those imposed on the walls (the radiative jump). The solution of Eq. (65), subject to Eq. (66), is Eb - Eb,

=

-(Ebo - EbL)(X/L) + (3/8)KRu"'(xL - x2)

(67)

which gives, in terms of Eq. (64),

However, Eqs. (67) and (68) need to be expressed in terms of the wall emissive powers rather than the wall values of the gas emissive power. To accomplish this, consider the radiative boundary conditions (see, e.g., Goody [28]),

which may be rearranged in terms of Eq. (60) to give

The difference between Eqs. (71) and (72) yields

RADIATIVE ENTROPYPRODUCTION

25 1

where, employing Eq. (68),

Note that

+ q!(L)

-&O)

= u”‘L

(76)

as expected. Now, from Eqs. (73), (74), and (79, after some manipulations,

7 M = aML being the optical thickness. Then, q!, in terms of the imposed wall emissive powers, obtained by inserting Eq. (77) into Eq. (68), is

(Eb1 - E b 2 )

4:

Also, for x

=

+?($-;)

1 37, -+-+--1 49 El

u”’ + -(2x 2

1

- L)

(78)

€2

= 0, (Ebl

qfl(0) =

- Eb2)

+ e(L -

37, 1 -+-+--I 49 €1

2

El

1

i)

u

~

~

t

--

2

~

(79)

€2

For u”’ = 0 and 7, = 0, Eq. (79) is reduced to the enclosure radiation between two closely located parallel large plates, as expected. To express Eb of Eq. (67) in terms of the wall emissive powers, rearrange Eq. (71) as u“‘

(80)

Then, in terms of Eqs. (74) and (80),

Eb(x)- Ebl = -

--1 € (:

where q t ( 0 )is given by (79).

2

+37Mx) q!(O) + U”’L ( 1- - 3i;t2) 4qL 49 TM

(81)

VEDATS. ARPACI

252

Proceeding to the radiative entropy production, first express the local production [recall Eqs. (47) and (%)I,

in terms of Eb rather than T. Employing E, = gT4, Eq. (82) may be readily rearranged as

This production is evaluated for the two special cases that follow.

A. HEATEDPLASMA Let the stagnant gas be an electrically conducting fluid (plasma). For E,, = E,, and E , = = E,, Eqs. (67) and (68) yield, in terms of Eqs. (73), (74), and (76), Ebz =

U"'L [ L - - + -1+ - ( l 1 Eb(X) - Eb, = 2 Ew 2 2q2,

q: = (u"'/2)(2x - L), &O) Then, Eq. (83) gives, in terms of Eq. (84),

32Mx 4qL

I):-

= -u"'L/2

(84) (85)

Figure 5 shows (for q = 1, E , = 1, and u"'L/2Eb, = 1) the distribution of the entropy production against the optical thickness.

B. AERODYNAMIC HEATING

Let one of the plates move steadily with velocity U relative to the other plate. In the absence of any pressure gradient, the energy generated by viscous dissipation in an incompressible gas between these plates is u"' = (3/4)PKR(u/L)'

(87)

Let E,, = Ebw, Eb2 = Ebm,El = E,, and € 2 = E , = 1. The adiabatic wall emissive power Eb, is the specific value of Ebw satisfying &(O) = 0. This

RADIATIVE ENTROPYPRODUCTION

253

XlL

FIG.5. Entropy production in a stagnant plasma between two parallel plates.

Note that the emissivity of the insulated surface does not play any role, as expected. The entropy production associated with Eq. (90) is

where

254

VEDATS . ARPACI

S"'

u"'IT,

0.0

0.2

0.6

0.4

0.8

1.0

XlL

FIG.6. Entropy production in a plasma between two parallel plates in relative motion.

Figure 6 shows (for q = 1 and u"'L/2Eb, = 1) the distribution of the entropy production against the optical thickness. More complicated examples, which will be carried out by dimensional arguments, need to be examined regarding qualitative radiation.

V1. Qualitative Radiation Reconsider the thermal part of Eq. (58), (93) Introduce an entropy production number,

ll, = s"'l2k

(94)

I being a characteristic length, and a heat transfer number, H = (aJ/axi)/3~-,k(aT/8xi) qr

(95)

RADIATIVE ENTROPYPRODUCTION

255

In terms of these numbers, Eq. (93) becomes

Also, from Eq. (49, on dimensional grounds,

- (Jw

6 (97) where 6 is the thickness of thermal boundary layer and Jwand J, are the wall and ambient values of J, respectively. To relate J to temperature, consider the radiative constitution given by Eq. (59).By the help of Fourier transforms, for example, exp(ikjxj), kj being the wavenumber vector, qR

or, in view of ko

-

- Jm)/3KR

V2 = - k i ,

h-', V 2

kg = k :

+ k: + k$

- - K 2 , and Eq. (59) yields

(6-2 + 3Ki)J

-

12K;Eb

(98)

Then, in terms of the optical thickness 2

J

-KM6

- (x)& + 3t2

(99)

1

which, together with Eq. (97), leads to the radiative heat flux, qR

'v

4q[2/(1

+ 322)1(Ebw

-

Ebm)

(101)

which is valid for any optical thickness but excludes any boundary effect. For q = 1, Fig. 7 shows the radiative heat flux versus emission and absorption, which, respectively, are the measures for the hotness and thickness of the gas. Equation (101) excludes any effect of scattering. To include this effect, consider the thick gas limit with isotropic scattering,

where PR = xR+ o,, PR and 0, being the Rosseland mean of extinction and scattering coefficients, respectively. On dimensional grounds, and in terms of the albedo of isotropic scattering, w =4

and Eq. (102) yields

P

R

VEDATS . ARPACI

256

\

/

Hot

/

Thick Gas

ABSORPTION TION

FIG.7. Radiative heat flux versus emission and absorption.

Also, noting that the thin gas limit is independent of any scattering, Eq. (101) may be rearranged to include for z 00 the effectof isotropic scattering. Then, --f

qR

"[

z

1

1

+ 3z2/(1 - o) ( E b w - E b m )

Figure 8 shows the radiative heat flux versus absorption and scattering. To include any wall effect in Eq. (105),first consider the boundary-affected thick gas and thin gas approximations with negligible scattering. For the thick gas, from Arpaci and Larsen [171,

RADIATIVE ENTROPYPRODUCTION

257

‘a0 FIG.8. Radiative heat flux versus absorption and scattering.

where pw is the wall reflectivity and E , and E , are the usual exponential integrals of order three and four. On boundaries,

or, dimensionally,

where q = ( K ~ / K ~ ) ” ’ and E, is the wall emissivity. For the thin gas, from Lord and Arpaci [lS] and Arpaci and Troy [22], aqF/ay = 4KPC(Eb - Ebm) - (Ew/2)(Ebw

- Ebm)E21

(109)

-

where E2 is the exponential integral of order two. Outside of a thermal boundary layer, E b E b m , and Eq. (109) is reduced to aqF/ay = -4KP(Ew/2)(Ebw

- Ebm)E2

(1 10)

258

VEDAT

s. ARPACI

or, near boundaries,

which, on dimensional grounds, yields R

4qz(Ew/2)(Ebw - Ebm)

4w

(1 12)

The comparison of Eqs. (108) and (1 12) with the thick gas and thin gas limits of Eq. (101)identifies the wall effect near a boundary layer or a cold wall next to this layer by the factor ~ , / 2 .This is a heat source effect. It can also be shown that the wall effect of a hot boundary is 1 - eW/2,which is a heat sink effect. Accordingly, the radiative heat flux, including a hot wall as well as the emission, absorption, and scattering effects, is found to be

Furthermore, introducing the Planck number, gw=

Emission Conduction

'v

Ebw - Ebm k(Tw- Tm)/6

(1 14)

Eq. (95) may be rearranged as R

z

( :.>[

H w = -4 -wK4 q l-1 + 3z2/(1 - w,]gw 4w Finally, Eq. (96) yields, in terms of Eq. ( 1 15),

n!i

[(TW

or, explicity,

n,

-(

Tw

Tm)2{

1

-

+ Hw),

T,)/TI2(1

+ 4q( 1 - ?)[

(1 16)

z

1

+ 3r2/(1 -

0)

The smallest value of this production is on the hot boundary, and its radiative part becomes, after some rearrangement,

For a proportionality constant of unity [chosen arbitrarily for a graphical representation of Eq. (1 l8)], E , = 1 and w = 0, Fig. 9 shows the boundary production of radiative entropy versus the optical thickness and the temperature ratio. The foregoing dimensional considerations will prove useful in the next three sections on the evaluation of entropy production within a boundary layer, and across a discontinuity (such as flames).

RADIATIVE ENTROPY PRODUCTION

259

0

FIG.9. Radiative entropy production.

VII. Heat Transfer

Consider a thermal boundary layer of local thickness 6 next to a wall (Fig. 10). On dimensional grounds, the local thermal entropy production [recall Eqs. (47), (55), and (56)] on the wall is

0

x

Tw

FIG. 10. Wall entropy production

260

VEDATS. ARPACI

where the subscripts w and Eq. ( 1 19) as

GO

indicate the wall and ambience. Rearrange

or, in terms of the convective heat flux,

as

or, in terms of the definition of the local Nusselt number,

as -s;’x2 --(I k

-$r(l+$)Nu;

Introducing a local entropy production number,

IT, = s:‘X2/k and, for a hot wall and a small T, further rearranged as

-

(125)

T,, noting T,/Tw << 1, Eq. (124) may be

or, in terms of Eqs. (1 13), and (1 14), and (121), as

nx--[1+49(1-’iz)[

1

+ 3t2/(1 z -

O )]Bw]Nu:

(126)

Clearly, the wall entropy production being proportional to Nu: provides no new information. However, if a value Nu, were to be obtained from some approximate temperature profiles, the principle of the “least entropy production” provides the Nu, closest to the actual Nu,. These variational considerations are not the concern of this study. The next section deals with another example illustrating the application of entropy production to flames.

RADIATIVE ENTROPYPRODUCTION

26 1

VIII. Flame Quenching Consider the entropy production in a steady flame stablized on a porous flat-flame burner as depicted in Fig. 11. On dimensional grounds, and in a manner similar to Eq. ( 1 19),

where A is the quench distance (the thickness of reaction zone is d, and d << A). Rearrange Eq. (127) as s"'

- $( + $)(T) 1

Tb

-

Tu

or, in terms of

as s"'

-(;

1

+

$)(T)

T, - Tu

-

Reaction

FIG. I 1. Quenched laminar flame.

VEDATS. ARPACI

262

In view of the fact that most of the reaction occurs close to the highest temperature, use Tb for the characteristic temperature in Eq. (130). Accordingly, s'"

- $!( + $)$ (1 -

1

or, in terms of a characteristic length 1 = a / S t , where a is the thermal diffusivity and St is the laminar flame speed at the adiabatic flame temperature, assuming Tu/Tb << 1, s'"12/k

-

+ qR/qK)Pe-2

(1

(132)

where Pe = All

=

S: A / a

(133)

is the flame Peclet number. Separately, when based on the characteristic length I = a/St, the Planck number given by Eq. (1 14) yields (in terms of temperatures & and Tu)

which may be arrranged as the flame Boltzmann number, Ebb

Emission Flame enthalpy flow

- Ebu

B - pc,st(Tb - T,)

(135)

Then, in view of the relation

Eqs. (132), (1 33), and (135) lead to

n,

--

1 T Pe + 4 ' ( 1 - ? ) [ 1 + 3 ~ 2 / ( l - o )

(137)

The linearized B, is independent of the flame temperature, or, with the approximation (Eb Bb

- Eu)/(&

- <)

h,

(EE

- Eu)/(Tbo - <)

becomes independent of this temperature. Thus

n, = f(v,,T, B b r Pel where Pe = f(A)

and

A = f(&)

RADIATIVE ENTROPYPRODUCTION

263

and Il, depends on the flame temperature only through the Peclet number (dimensionless quench distance). The U-shaped nature of A = f(Tb) is well documented in the literature (see Ferguson and Keck [29,30] for the case excluding radiation, and Arpaci and Tabaczynski [20,21] for the case with radiation; see also Kooker [31] and Sohrab and Law [32] for the importance of radiation on the quenching process, and Lee and Tien [33] for the effect of condensed fuels on this process). The foregoing Arpaci- Tabaczynski and Ferguson-Keck studies follow the usual practice and evaluate the minimum quench distance from the tangency condition, a/dTb(Pe) = 0

(139)

Actually, an extremum in the entropy production,

is shown by Arpaci [S] and Arpaci and Selamet [113 to provide the physical justification for this condition (note that the terms in brackets are positive). The next section briefly deals with the microscales of turbulence, which are needed for entropy production in radiation-affected turbulent flows.

IX. Microscales of Heat Transfer A. KINETICSCALES

Following the usual practice, decompose the instantaneous velocity into a temporal mean and fluctuations vi

+ ui

(141)

and let Ui be statistically steady. The balance of the mean kinetic energy of velocity fluctuations is (see, e.g., Tennekes and Lumley [34]) q(aK/axj) = -(d.9j/j/dxj)

+ 9- E

(142)

where K = $ G

(143)

is the mean kinetic energy,

gj = $puj

+ 3uiUiuj - 2vu,sij

(144)

VEDATS. ARPACI

264

is the mean transport (flux),

9 = -uiujsij

(145)

= 2vs,jsij

(146)

is the inertial production, and €

is the dissipation of turbulent energy. Here sijdenotes the rate of fluctuating strain and Sij denotes the rate of mean strain. For a homogeneous pure shear flow (in which all averaged quantities except Q are independent of position and Sij is a constant), Eq. (142) reduces to

Y’=E On dimensional grounds, Eq. (147) gives 9

-211 -

v(u2/n2)

(147)

-

E

( 148)

where 1 is an integral (or geometry) scale and R. is a dissipation (and diffusion) scale. The velocity from Eq. (148) is, in terms of the integral scale, (41/3

(149)

and, in terms of the dissipation scale, I4

N

A(€/V)’/2

Elimination of the velocity between Eqs. (149) and (150) yields the dissipation scale in terms of the integral scale , I ~/3(~3/41/6

(151)

This scale was originally proposed by Taylor, and hereafter will be called the Taylor (micro)scale. It describes the isotropic dissipation of a homogeneous (also, on dimensional grounds, a nonhomogeneous) flow (Fig. 12). Consider now a thin layer of turbulent flow (sublayer next to a wall, or a vortex tube or sheet) in which the Taylor scale and the integral scale approach the same asymptotic limit and become indistinguishable. Let this limit be u,

Then, Eq. (151) is reduced to the length scale,

In terms of this scale, Eq. (149) or (150) gives the velocity scale

RADIATIVE ENTROPYPRODUCTION

a

265

b

FIG. 12. (a) Homogeneous flow (isotropic dissipation); (b) homogeneous, isotropic flow (isotropic production, isotropic dissipation).

and the ratio of Eqs. (153) and (1 54) yields the time scale ‘f

- q/u

(155)

(V/€)”2

These smallest (or isotropic flow) scales of turbulence are called the Kolmogorov microscales [35 3. B. THERMAL SCALES

Let the instantaneous temperature of turbulent forced convection be decomposed into a temporal mean and fluctuations, 0 + 8

(156)

and let 0 be statistically steady. The balance of the root mean square of thermal fluctuations, Ke -1282 is (see, e.g., Tennekes and Lumley [34]) L$(aK,/axj) = - ( a / a ~ j () 9 e ) j

where

+

9 0

- €0

-

(9e)j = (1/2)o2uj - a(a/a~j)~(1/2)e21 is the mean thermal transport (or turbulent thermal flux),

9 = -u,e(ao/axj)

VEDATS. ARPACI

266

is the thermal production, and eg = a(ae/axj)(aO/axj)

is the thermal dissipation. Here a denotes the thermal diffusivity. Equation (158) is related to an entropy balance that can be demonstrated in terms of Eq. (161). The thermal part of entropy production given by Eq. (55), s”’ = -(Yi/T)(8T/aXi)

( 162)

may be rearranged in terms of Fourier’s law of conduction to yield

(s”’/pc,)T2 = a(a T/axj)(aT/8Xj)

(163)

For a homogeneous flow (in which all averaged quantities except Ui and 0 are independent of position and in which Sij and d @ / a x j are constants), Eq. (158) becomes

Y g = €0

(164)

On dimensional grounds, Eq. (164) gives 90

-

-

-

ug(~2/i,) a(e2/A,2)

€0

(165)

where ug is the velocity and lo is the integral scale associated with enthalpy flow. Further development on thermal microscales is carried out in terms of two limits corresponding to Pr + 0 and Pr co,and for the range Pr 2 1. --+

1. Pr + 0 (Liquid Metals) Assuming ug and lo to be proportional to those of momentum flow, lg

-

1,

ug

-

u

(167)

an inertial estimate, obtained from Eq. (148), gives ug

-

(~lg)”~

(168)

This assumption implies a kinetic dissipation layer much thinner than the thermal dissipation layer; it applies to fluids with small Prandtl numbers. Elimination of Ug between Eqs. (166) and (168) yields

At

-

1i/3(a3/~)’/6, Pr + 0

a thermal Taylor scale for liquid metals. In the isotropic limit

( 169)

RADIATIVE ENTROPYPRODUCTION

Eq. (1 69) gives

-

267

Pr + o (171) the thermal scale independently discovered by Oboukhov [36] and Corrsin P71. qi

(c~~/E)’/~,

2. Pr + 00 (Viscous Oils) A viscous (rather than inertial) estimate for u, obtained from Eq. (148), is u

‘v

A(€/V)”2

(172)

The viscous estimate for u and ue implies a kinetic dissipation layer much thicker than the thermal dissipation layer; it applies to fluids with large Prandtl numbers. Elimination of ug between Eqs. (166) and (173) yields

A,B 1 i / 3 ( v a 2 / ~ ) 1 / 6 ,Pr + co a thermal Taylor scale. The isotropic limit, ‘v

of this scale is q,B

-

(1 74)

( v a ’ / ~ ) ’ / ~ , Pr + co

the thermal Kolmogorov scale discovered by Batchelor [38]. Although proposed for forced convection, the same scale is recently found to apply for natural convection as Pr + 00 [39]. 3. Pr 2 1 (Gases, Liquids, and Viscous Oils) The foundations of thermal scales for Pr 2 1 rest on the similarity between the homogeneous balances of kinetic energy and entropy. In terms of A scales, this similarity implies

At

is a proposed thermal Taylor scale. Note that the production of where kinetic energy and entropy is now based on the same integral scale. From Eq. (177)

(u/ue)(A/At)’ = Pr

(178)

VEDATS . ARPACI

268

U

FIG. 13.

Velocity for enthalpy flow (Pr 2 1).

which may be rearranged in terms of Fig. 13 and

Then, in terms of Eq. (15 l),

n,A

-

Pr 2 1

1'/3(vcr2/~)'/6,

(181)

which is a thermal Taylor scale. Note the difference between the integral scales of Eqs. (174) and (181). In the limit of isotropic flow, Eq. (180) leads to

n/n,A -+

q/q,A

-

Pr ' I 3

(182)

or, in terms of Eq. (153), to a thermal Kolmogorov scale [40], q,A

-

Pr 2 1

(183)

The foregoing considerations on microscales are extended in the next section to radiation-affected microscales of and entropy production in turbulent flows.

X. Radiation-Affected Turbulence Under the influence of radiation, the homogeneous entropy balance for thermal fluctuations given by Eq. (164) becomes

%=E t

+

€?

(184)

RADIATIVE ENTROPY PRODUCTION

269

or, explicity, -p~,i&m/ax,)

=

-(s:

+ qp)(ae/ax,)

(185)

or, in terms of entropy production, -pc,u,e(ao/ax,)

=

rqs;;,+ SR')

(186)

Equation (185) is the starting point for the following development of radiation-affected thermal microscales of turbulence. With an integral scale I and a thermal Taylor scale &, Eq. (185) dimensionally yields Pcu9(e2/1) Iv q K ( l + 4R/4K)(e/43

(187) or, in terms of Fourier's law of conduction and that of Eq. (185) for o = 0,

Under the assumption of isotropy, ug

-+

1 --*I],

vg,

l.9 - 4 9

(1 89)

where u, 4, and 4, are Kolmogorov scales, Eq. (1 88) becomes.

vB vl

ly

[+ 1

44( 1 - fiz>(*).@w]&2

Separately, from the Kolmogorov scales given by Eqs. (153) and (1 54), v

-

v/4

(191)

Then, noting the isotropic limit of Eq. (1 79), v/v9

= 4/49

(192)

and eliminating v and ug among Eqs. (190), (191), and (192) yields

where Pr is the Prandtl number. In terms of the Kolmogorov scale,

4

-

(V3/€)"4

Eq. ( 193) yields the radiation-affected thermal Kolmogorov scale

( 194)

270

VEDATS. ARPACI

The diminishing radiation reduces Eq. (195) to the thermal Kolmogorov scale given by Eq. (183). Clearly, when 6 of Section VII is replaced with q e , the dimensional arguments apply to turbulent heat transfer. Increasing radiation in the thin gas range and decreasing radiation in the thick gas range increase qo and decrease the entropy production in a turbulent thermal sublayer characterized by q’. In the preceding sections, the irreversibility of gas (or optical limit of electromagnetic) radiation has been identified by the concept of lost heat. However, the irreversibility of electromagnetic radiation, in general, is lost electromagnetic work rather than heat. The nature of this work is elaborated in the following section.

XI. Lost ElectromagneticWork Consider the electromagnetics of media at rest.’ Let the electromagnetics of these media be described by the four Maxwell equations

v x H = J + aD/at

(196)

pe = V * D

(197)

v x E = -aB/at

(198)

O=V*B (199) where the first and third relations are, respectively, related to Ampere’s Law and Faraday’s Law, and the second and fourth relations are their divergence. Here, E and H are the electric and magnetic field intensities, D and B are the flux densities, J is the current density, and pe is the charge density. As is well-known, the dot product of H and Eq. (198) subtracted from that of E and Eq. (196) yields, after some vector manipulations, the balance of electromagnetic power/volume (the Poynting theorem) aD aB E . -at+ H . - = -atv . N - E . J where N=ExH is the electromagnetic flux. For an infinitesimal process in local equilibrium over a time interval dt, Eq. (200) becomes

* The theories on moving media appear to remain incomplete and will not be concerned here.

RADIATIVE ENTROPY PRODUCTION

27 1

+

or, in terms of the electromagnetic constitutions D = eOE P and B p o w + M) duE

+ E*dP+ pOH*dM= - 6 N

- E.SQ*

=

(203)

Here

+p o w )

UE = + ( E 0 E 2

(204)

is the energy, P is the polarization, M is the magnetization, Q* is the charge, and e0 and p o , respectively, are the permittivity and permeability of the empty space (vacuum). Now, rearrange Eq. (203)for a system of volume V fixed in stagnant media, dUE

+ VdN = - V[E

*

-

dQ*

+ (E

*

dP + poH ~ M ) R + ~ ](205)

Here, UE= uEV,E SQ* is completely irreversible, and the subscripts R and L introduced into

+ poH

(E * dP

*

dM)R+L

(206)

indicate a reversible as well as an irreversible part. Now, the fundamental difference, First Law - Temperature(Second Law) yields, in terms of Fig. 14,

dU

-

T d S - V(E * dP + poH * dM)R

+ VE

= SQL

*

SQ* + V(E SP + p0H * 6M)L - TSII

or

dG

= SQL

+ V[E .S Q * + (E SP + poH +

*

6M)J

For an irreversible process in local equilibrium,

G

=

U - TS - V(E * P

+ poH. M)R

defines the Gibbs function, and dG = 0

or, because of locally uniform T, E, and H,

dU = T d S + V(E .dP

+ p o H . dM)R

which is the Gibbs thermodynamic relation [41]. Then, Eq. (207) gives 1

SII = -(SQL T

+ V[E

*

SQ* + (E * SP + poH * 6M)J)

(212)

VEDATS . ARPACI

272

vs

n

FIG.14. Thermoelectromagnetics of a system fixed in stagnant media. (a) Electromagnetic energy. (b) First Law. (c) Temperature x Second Law.

where the terms in brackets denote the electromagneticwork lost to heat. That is, E SQ* is the Joulean dissipation of energy by diffusion and

-

(Em 6P + poH * 6M)L is the hysteretic dissipation of energy by Foucault currents. Note that hysteresis is a cyclic process involving a finite energy loss/cycle that corresponds to a finite area on E, P and/or H, M plane(s). A differential representation of this process actually implies

@-6P+p,H*6M)

RADIATIVE ENTROPYPRODUCTION

27 3

XII. Concluding Remarks The concept of lost heat has been proposed, as opposed to that of lost work. It has been shown that a part of entropy production is related to the irreversible (lost) part of all forms of energy except for heat, and that the remaining part of entropy production is related to the irreversible (lost) part of heat energy. The lost heat energy includes a part of the radiative as well a part of the conductive modes of heat transfer. A dimensionless number for entropy production was introduced. This number is evaluated in terms of some illustrative cases. In the case involving a thermal boundary layer, the entropy production is found to be proportional to the square of the Nusselt number. Unless it is tied to a variational problem that selects the physically meaningful solution among all mathematically possible solutions, the entropy production provides no new information for this case. In the case involving a quenched flame, the entropy production is found to be inversely proportional to the Peclet number. The tangency condition, usually considered in the literature to determine the minimum quench distance, is related to an extremum in entropy production. The relation between the turbulent energy dissipation and the entropy production has been demonstrated. In terms of this relation, a radiationaffected thermal Kolmogorov scale qe was introduced. The entropy production in a laminar thermal boundary layer of thickness 6 and in a turbulent thermal sublayer of thickness qo is related to the Nusselt number based on thickness and scale. The increasing radiation in the thin gas range and decreasing radiation in the thick gas range increase 6 and qe, increase the Nusselt number, and decrease the entropy production within 6 and q e . The lost electromagnetic energy outside the optical and infrared spectra is identified as lost work. The diffusive part of this work is the Joulean heat dissipation whereas the hysteretic part is the heat dissipation due to Foucault currents. The present review is based on the assumption of local thermodynamic equilibrium. For problems associated with nuclear explosions and shock tubes, this assumption is known not to hold any longer. One then needs to assume (translational, rotational, vibrational, and electronic) nonequilibrium for the internal structure of matter affected by (photon) radiation. These kinetic and statistical considerations, however, are beyond the scope of the present study, but may be found in the work of Arpaci and co-workers [19,42-441. NOMENCLATURE B B

Boltzmann number, or equilibrium intensity magnetic flux density

c

cp

velocity of light specific heat at constant pressure

VEDATS . ARPACI

274

P

d

thickness of reaction zone or change in a property D electric flux density black-body emissive power Eb integro-exponential function of E" order n = 2,3, or 4 E electric field intensity body force fi H heat transfer number H magnetic field intensity complex unit i I intensity J averaged intensity J current density thermal conductivity k wavenumber k0 wavenumber in x i ki K mean energy of fluctuations 1 a characteristic length unit vector in x i 4 M magnetization operator defined by Eq. (36) M i j p q ... N electromagnetic flux Nu Nusselt number pressure P

Pe Pr 9i

Q

Q* S S"' Sij

S SO

t T U U1I)

U J'i

V

Vi

V

W Xi

polarization Peclet number Prandtl number heat flux in x i heat charge entropy/mass rate of entropy generation/volume rate of deformation entropy laminar flame speed at adiabatic flame temperature time temperature internal energy/mass or volume rate of energy generation/volume internal energy mean velocity in x i specific volume or Kolmogorov velocity velocity in x i volume work coordinate axis

Greek Symbols a

6

A V2

E €0

rl

0 0 K

I

thermal diffusivity thickness of boundary layer or net flow of energy quench distance Laplacian emissivity or turbulent dissipation permittivity of vacuum weighted nongrayness or Kolmogorov scale turbulent temperature mean turbulent temperature absorption coefficient Taylor scale

PO V n i j

n P

Pe

U

T

Tij

4 Y

R

permeability of vacuum kinematic viscosity radiative tensor entropy flow or entropy number density or reflectivity charge density Stefan- Boltzmann constant optical thickness or Kolmogorov time stress entropy flux entropy flow solid angle

Script Symbols

53

9

turbulent diffusion turbulent production

9

Planck number

Subscripts

b g

burned generation

M

P

mean Planck mean

RADIATIVE ENTROPY PRODUCTION Rosseland mean entropy unburned wall local

0 \m

275

thermal ambient

Superscripts

A B C K

proposed scale Batchelor scale convection or Corrsin scale conduction

R %

-

radiation instantaneous value mean value

ACKNOWLEDGMENT This study is dedicated to the memory of Professor Joseph H. Keenan, our thermodynamics teacher, whose high standards I hope 1 have to some degree achieved.

REFERENCES I . V. S. Arpaci, “Conduction Heat Transfer.” Addison-Wesley, Reading, Massachusetts, 1966. 2. V. S. Arpaci and P.S. Larsen, “Convection Heat Transfer.” Prentice-Hall, Englewood Cliffs, New Jersey, 1984. 3. R. 8. Bird, W. E. Stewart, and E. N. Lightfoot, “Transport Phenomena,” Wiley, New York, 1960. 4. A. Bejan, “Entropy Generation through Heat and Fluid Flow.” Wiley, New York, 1982. 5. A. Bejan, Second Law analysis in heat transfer and thermal design. Adu. Heat Transfer 15, 1-58 (1982). 6. A. Bejan, “Advanced Engineering Thermodynamics.” Wiley (Interscience),New York, 1988. 7. V. S. Arpaci, Radiative entropy production. A l A A J. 24, 1859-1860 (1986). 8. V. S. Arpdci, Radiative entropy production-lost heat into entropy. Int. J. Heat Mass Transfer 30.21 15-2123 (1987). 9. V. S. Arpaci, Foundations of entropy production. In “Advances in Thermodynamics” (S. Sieniutycz and P. Salomon, eds.). Taylor & Francis, New York, 1990. To appear. 10. V. S. Arpaci and A. Selamet, Radiative entropy production. Proc. Int. Heat Transfer Conf.,8th 2,729-734 (1986). I I. V. S. Arpaci and A. Selamet, Entropy production in flames. Combust. Flame 73, 251-259 (1988). 12. A. Selamet and V. S. Arpaci, Entropy production in boundary layers. J . Thermophys. Heat Transfer to appear (1990). 13. S. C. Traugott, Radiative heat-flux potential for a nongrey gas AIAA J . 4, 541-542 (1966). 14. A. C .Cogley, W. G. Vincenti, and S. E. Gilles, Differential approximation for radiative transfer in a nongrey gas near equilibrium. A l A A J. 6, 551-553 (1968). 15. V. S. Arpaci and D. Gozum, Thermal stability of radiation fluids: The Benard problem. Phys. Fluids 16, 581-588 (1973). 16. V. S . Arpaci and Y. Bayazitoglu, Thermal stability of radiating fluids: Asymmetric slot problem. Phys. Fluids 16, 589-593 (1973). 17. V. S. Arpaci and P. S. Larsen, A thick gas model near boundaries. A l A A J. 7,602-606 (1969). 18. H. Lord and V. S. Arpaci, Effect of nongray thermal radiation on laminar forced convection over a heated horizontal plate Int. J. Heat Mass Transfer 13, 1737-1751 (1970).

276

VEDATS. ARPACI

19. W. F. Phillips and V. S. Arpaci, Monatomic plasma-thermal radiation interaction: A weakly ionized kinetic model. J. Plasma Phys. 13,523-537 (1975). 20. V. S. Arpaci and R. J. Tabaczynski, Radiation-affected flame propagation. Combust. Flame 46, 315-322 (1982). 21. V. S. Arpaci and R. J. Tabaczynski, Radiation-affected laminar flame quenching. Combust. Flame 57, 169-178 (1984). 22. V. S . Arpaci and J. S . Troy, On the attenuating thin gas. J . Thermophys. Heat Transfer to appear ( 1990). 23. W. Unno and E. A. Spiegel, The Eddington approximation in the radiative heat equation. Publ. Astron. Sac. Jn. 18,85-95 (1966). 24. E. A. Milne, Thermodynamics of stars. Handb. Astrophys. 3,65-255 (1930). 25. V. S. Arpaa, Hookean and Stokesean implications of the radiative stress. Fundam. T h e m Radiat. Heat Transfer, ASME, HTD 40, 1-5 (1984). 26. J. A. Stratton, “Electromagnetic Theory.” McGraw-Hill, New York, 1941. 27. W. Prager, “Introduction to Mechanics of Continua,” pp. 87-92. Ginn, Boston, Massachusetts, 1961. 28. R. M. Goody, “Atmospheric Radiation.” Oxford Univ. Press, London, 1964. 29. C. R. Ferguson and J. C. Keck, On laminar flame quenching and its application to spark ignition engines. Combust. Flame 28, 197-205 (1977). 30. C. R. Ferguson and J. C. Keck, Stand-off distances on a flat flame burner. Combust. Flame 34, 85-98 (1979). 31. D. E. Kooker, Numerical study of a confined premixed laminar flame: Oscillatory propagation and wall quenching. Combust. Flame 49, 141-149 (1983). 32. S. H. Sohrab and C. K. Law, Extinction of premixed flames by stretch and radiativeloss. Int. J. Heat Mass Transfer 27,291-300 (1984). 33. K. Y. Lee and C. L. Tien, Flame wall-quenching by radiation and conduction in combustion of condensed fuels. Combust. Sci. Technol.43, 167-182 (1985). 34. H. Tennekes and 0. L. Lumley, “A First Course in Turbulence.” MIT Press, Cambridge, Massachusetts, 1972. 35. A. N. Kolmogorov, Local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30,299-303 (1941). 36. A. M. Oboukhov, Structure of the temperature field in turbulent flows. Izv. Akad. Nauk SSSR, Ser. Geogr. GeoJiz. 13,58-69 (1949). 37. S . Corrsin, O n the spectrum of isotropic temperature fluctuations in isotropic turbulence. J. Appl. Phys. 22,469-473 (1951). 38. G. K.Batchelor, Small-scale variation of convected quantities like temperature in a turbulent fluid. J. Fluid Mech. 5, 113 - 133 ( 1959). 39. V. S. Arpaci, Microscales of turbulence and heat transfer correlations. Int. J. Heat Mass Transfer 29, 1071-1078 (1986). 40. V. S. Arpaci, Microscales of turbulence and heat transfer correlations. Annu. Rev. Heat Transfer 3,195-231 (1990). 41. G .N. Hatsopoulos and J. H. Keenan, “Principles of General Thermodynamics.” Wiley, New York, 1965. 42. W. F. Phillips. V. S. Arpaci, and P. S. Larsen, Diatomic gas-thermal radiation interaction between parallel plates. J. Plasma Phys. 4,429-439 (1970). 43. W. F. Phillips and V. S. Arpaci, Diatomic gas-thermal radiation interaction: A model equation for the internal fluid. J. Plasma Phys. 7,235-246 (1972). 44. J. C. Haas, V. S. Arpaci, and G. S . Springer, Mass and heat transfer in a diatomic gas. J. Plasma Phys. 6,547-560 (1971).

ADVANCES IN HEAT TRANSFER, VOLUME 21

Heat, Mass, and Momentum Transfer in a Multifluid Bubbling Pool*

GEORGE ALANSON GREENE Department of Nuclear Energy Brookhaven National Laboratory Upton, New York 11973

I. Introduction Reassessments of the public risk due to unterminated severe accidents in nuclear power reactors, as well as the recognition of the need for improvements in the process and metallurgical industries, have pointed to the need for a better understanding of the heat, mass, and momentum transfer processes in gas-sparged multifluid bubbling pools. In the case of severe core meltdown accidents in nuclear power reactors, the public risk is a direct function of the physical threat to the reactor containment and the release of radionuclides from the molten core to the environment during the stage of the accident termed the molten core-concrete interaction (MCCI) stage. The MCCI presupposes that the core meltdown has proceeded uninterrupted, that the high-temperature core melt has penetrated the reactor pressure vessel, and that the core has been deposited on the concrete floor below. Heat transfer from the molten core debris is sufficient to melt the concrete, which will then release H,O and C 0 2 gases due to the decomposition of hydroxides and carbonates in the concrete as well as to the vaporization of water. As these gases rise through the molten pool of core debris, they can react exothermically with molten metals in the pool, releasing more energy into the core debris, converting H 2 0 and COz into H, and CO, and chemically transforming some condensed-phase radionuclides into intermediate volatile species by reactive vaporization. Because the core debris consists of molten oxides (UO,, ZrO,, FeO, etc.) and molten metals (Zr, Fe, Ni, Cr, etc.), which are immiscible,

* Work performed under the auspices of the U.S. Nuclear Regulatory Commission. 277 Copyright 01991 by Academic Press, Inc. All rights of reproduction in any form reserved.

278

GEORGE ALANSON GREENE

the molten pool would be vertically stratified in the absence of the gas bubbling action of the concrete decomposition gases. It is the action of the rising gas bubbles that enhances the boundary surface (bottom and side) and interfacial heat transfer rates and that drives the entrainment mass transport across the interface between the stratified layers of oxides and metals. If the rising gas bubbles are not able to entrain the lower liquid into the upper layer, the pool will remain stratified and heat transfer between the liquids will be across a well-defined interface. Heat transfer across this interface will be enhanced by surface renewal processes, as would be heat transfer at the lower bubbling surface and along the vertical walls of the cavity or container. Under some circumstances, the rising bubbles may drive mass transport across the interface. If the rate of mass entrainment can be balanced by the rate of settling of the entrained droplets, the upper layer will become a heterogeneous mixture layer; if the rate of entrainment exceeds the rate of settling of the droplets, the pool will eventually become completely mixed and the liquid-liquid interface will disappear. In either case, heat transfer from one layer to the other would be augmented even more than for the stratified case. In the last decade, there has been a significant increase in research activity in the area of heat, mass, and momentum transfer in gas-sparged multifluid bubbling pools. Some studies have looked at the heat transfer processes across a liquid-liquid interface due to the bubble action across the interface. Other studies have examined the bubbling-enhanced heat transfer at solid boundaries. Attention has been focused on both horizontal and vertical surfaces; in addition, some investigations have considered surfaces with bubbling along the boundary layer whereas others have considered surfaces with gas injected through the surface. There has been little activity in the study of the criterion for onset of entrainment between liquid layers due to gas bubbles rising across their interface, and, similarly, little activity in the investigation of the rate of mass entrainment by the rising bubbles and the subsequent rate of settling of the entrained droplets. The objectives of this paper are to review the available literature that is applicable to heat, mass, and momentum transfer in multifluid bubbling pools. In the course of this review, a series of seven separate experimental investigations will be discussed, which the author has conducted in order to develop a comprehensive set of models for incorporation in the CORCON computer code, which models molten core-concrete interactions. The results of these seven separate investigations will be discussed and the mathematical models for each of the processes that are described will be presented. Finally, the major remaining uncertainties and areas in need of further investigation and model development will be presented.

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

279

11. Enhancement to Heat Transfer between Stratified Immiscible

Liquids by Gas Bubbling across the Interface In early nuclear safety analyses, the core debris was assumed segregated into overlying immiscible layers of core oxides, core metals, and melted concrete slag in a vertical, cylindrical geometry. When the core oxide layer became diluted by concrete decomposition products to the extent that it became less dense than the metallic layer, these two layers would physically invert. The core oxide layer and the overlying slag layer would then combine into one layer. These layers would be continuously sparged by concrete decomposition gases from below, keeping the individual layers well mixed and isothermal. It is this gas flux that would drive the heat and mass transfer processes between the overlying layers. Separate analyses and experimental evidence have revealed that under a wide range of conditions, bubble-induced entrainment may dominate the interlayer transport processes (Greene et a[., 1988;Greene and Schwarz, 1982).However, under conditions in which the rising bubbles are unable to support entrainment, transport processes between liquid layers would be controlled by bubble agitation and turbulence at the liquid-liquid interface. In the stratified state, fission products and their decay heat would be concentrated in the oxide layer while chemically reactive metals, principally zirconium, and their heats of reaction would be concentrated in the metallic layer. Interfacial heat transfer between layers would determine not only the temperature of each layer but also the upward-to-downward heat transfer split. This is turn would control the downward concrete erosion versus upward radiative or boiling heat flux. In the absence of models for mixing and entrainment, early assessments of the consequences of molten core-concrete interactions were based solely upon analyses that assumed this stratified configuration. Modeling of this stratified heat transfer mode will be discussed in this section. A. BACKGROUND

The earliest stratified interlayer heat transfer model that was developed was a version of the Konsetov model (1966)for heat transfer to a surface with bubble agitation, modified by Blottner (1979)to include two effects: (1)a lower asymptotic limit for natural convection in the absence of bubbling and (2)an arbitrary increase in the coefficient of the void fraction term from 0.4 to 50.0 to shift the model into better agreement with some limited experimental data. Blottner’s interfacial relationship, applicable for each fluid, is as follows:

hi = k(Prg/~~)’/~(O.O0274B AT+ 5 0 ~ r ~ ) ” ~

280

GEORGE ALANSON GREENE

This relationship can be nondimensionalized as follows: Nu = (0.00274 Gr,

+ 50 Gr,)'l3 Pr '13

(2)

where Gr, is the Grashof number based upon the thermal buoyancy and Gr, is the Grashof number based upon void fraction-inducedbuoyancy. Gr, is not unlike that derived by Greene et al. (1980) except that the void fraction dependence is arbitrarily quadratic instead of linear. In nearly all cases, the term involving Gr, is much less than the term involving Gr, and can be neglected. Another model for heat transfer across liquid-liquid interfaces agitated by bubbles was developed by Szekely (1963) using surface renewal principles that assume periodic destruction of temperature gradients at the liquid-liquid interfacesby the arrival of bubbles. His result was further modified by Blottner (1979), resulting in the following interfacial heat transfer coefficient for either fluid, hi = 1.69(p~kj,/r,)'~~ (3) This relationship can be nondimensionalized as follows: Nu = 1.69Re'', Pr 1/2

(4)

where Re is the Reynolds number based on the superficial gas velocity and the volume-equivalent bubble radius. A third model for heat transfer across stratified liquid-liquid interfaces agitated by bubbles was developed by Lee and Kazimi (1984). Using Szekely's original result, as modified by Blottner, further adjustments in the modeling assumptions were made to cause the bubble arrival period to be inversely proportional to the bubble frequency. This was done to force the heat transfer coefficient to be proportional to the superficial gas velocity, a behavior apparently supported by some available experimental data. Introducing the bubble rise velocity as a parameter, Lee arrived at the following relationship for the overall heat transfer coefficient across the two layers:

where subscripts 1 and 2 in this equation refer to the lower and upper fluids, respectively. If the heat transfer resistance for one layer is neglected, several terms may be omitted, resulting in an interfacial heat transfer relationship similar to Eq. (3). The resulting relationship may then be made dimensionless as follows: Nu = 8.8(Re Pr J:)1/2

(6)

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

28 1

where J i = jg/[ga(pl - p,)/p:]1/4. In this form, it becomes apparent that the in only change from Eq. (4) is the inclusion of an additional term involving an effort to force Nu to be a linear function of j,. These three models were subsequently compared to some early experimental data of Werle (1982)and Greene and Schwarz (1982)with surprisingly mixed results. Equations (l), (3), and (5) all were found to compare favorably with the data of Werle for the 10 cS oil/Wood’s metal system (cS = centistokes). None were able to predict the water/mercury data of Greene with any success with the exception of Eq. (3,which was essentially forced to do so. An attempt was made to reproduce the data of Werle to try to resolve this apparent discrepancy. It was found that Werle’s data could not be reproduced and, in fact, were lower than Greene’s data for water/mercury by almost an order of magnitude. On this basis, any appearance of favorable comparison between the three models just discussed and the data base for interlayer heat transfer between overlying, stratified liquid layers were dispelled. As a result of this comparison, the experimental investigation to be described was initiated.

Ji

APPARATUS AND PROCEDURE B. EXPERIMENTAL An experimental apparatus with which to investigate heat transfer between immiscible liquid layers with gas agitation was constructed. The apparatus (see Fig. 1) was a Pyrex glass cylinder, 10 cm in diameter and 1 m deep. The apparatus was insulated to minimize radial heat losses by a 2-inch thick cylindrical blanket of Fibrefrax insulation. A porous stainless-steel frit was mounted in the base to provide a spatially uniform gas flux from below. Internal heating of the lower fluid layer (mercury) was accomplished by a cartridge heating assembly which penetrated the side of the cylindrical test vessel into the fluid chamber. Power was measured by a precision watt meter. The flow through the porous frit was measured by a bank of air rotameters. Bubble size was characterized photographically as a function of the pressure and flow rate. Over the range of pressure and flow rate covered by these experiments, the volume-equivalent bubble radius was correlated to the superficial gas velocity as rb (cm) = 0.233

+ 0.109jg

(cm/sec)

(7)

This relationship was developed only to determine the bubble size in this study and not as a general formula. A vertical traversable thermocouple assembly was installed along the centerline axis of the pool for determination of the temperature distribution in each layer and the temperature difference across

282

GEORGE ALANSON GREENE

FIG. 1. Schematic of experimental apparatus.

layers. Ten thermocouples were aligned at a nominal separation of 1 inch. Fluid pairs chosen for these nonentrainment interlayer heat transfer tests were mercury-water and mercury-silicone oil. Mercury was chosen as the lower fluid to suppress entrainment and to have a minimal resistance to heat transfer. With respect to stratification versus entrainment, a necessary condition for entrainment by bubbles rising across a liquid-liquid interface was derived by Greene et al. (1988). Application of that criterion to the water (or oil) and mercury system indicates that rising gas bubbles are unable to

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

283

support liquid entrainment for the fluid pairs used in the present study. This behavior has been verified experimentally. With respect to resistance to heat transfer, one can assume for the time being that the interfacial heat transfer process is a transient surface renewal process on both sides of the interface. Examination of the physical properties, in particular the thermal diffusivities of each liquid layer, suggests that the resistance to heat transfer on the mercury side of the interface would be, for the worst case, at least a factor of 10 less than on the water side. Neglecting this resistance would result in an uncertainty in the heat transfer coefficient of, at most, 10% for the water/mercury tests and negligible uncertainty for the two oils over mercury. In this fashion, the measured overall heat transfer coefficient would be approximately equal to the heat transfer coefficient on the oil or water side of the liquid-liquid interface. Fluid density and viscosity were measured in the laboratory. Specific heat and thermal conductivity values were taken from the literature or from vendorsupplied data. The apparatus was charged with the test fluids and power was supplied by the immersed cartridge heating assembly. The temperature distribution in the liquids was monitored until steady-state conditions were achieved at a prescribed heat flux. The overall heat transfer coefficient was calculated as the net power input divided by the interfacial temperature difference and the cross-sectional area. The superficial gas velocity was calculated as the volumetric gas flux divided by the cross-sectional area. Each data point to be presented represents an average value of from 5 to as many as 13 tests, with the individual data variations being in most cases less than 6%.

C. EXPERIMENTAL RESULTS The experiments were performed in three series as listed in Table I. The number in parentheses next to each heat transfer coefficient listed is the standard deviation of the value presented. The superficial gas velocity covered the range 0.30-8.35 cm/sec and the measured heat transfer coefficient varied from 2986 to 50,216 W/m2 K. The dimensional experimental data for the three fluid pairs are shown in Fig. 2. The experimental data greatly exceed the data from Werle (1982) for the same fluids by as much as a factor of 10. With one exception, the data exceed the model predictions of Eqs. (1) and (3) by as much as a factor of 7-14 at the highest superficial gas velocities investigated. The only exception is the comparison of the water/mercury heat transfer data to Eq. (l), which lies a factor of 2 below the data. It was found that Eq. ( 5 ) deviates from the oil/mercury data as badly as Eqs. (1) and (3). It agrees with the water/mercury data only in the lower range of j , = 1 cm/sec; the comparison rapidly deteriorates at higher gas velocity. Because the lower range of gas velocity is not the range of greatest interest,

TABLE I LISTINGOF EXPERIMENTAL DATA

h"

JB

Run

W/m2K)

(cm/s)

1 2 3 4

8823 (1023) 14424(4317) 14309(1687) 15842(1837) 2083 l(2174) 27907(4222) 3 1039(8117) 27766(1767) 32458(3079) 38 158(5766) 38946(4910) 42805(1364) 50216(4800)

0.30 0.59 0.88 1.17 1.47 2.05 2.65 3.23 4.38 5.63 6.28 7.21 7.75

T ("C)

Nu

Re

37.9 67.6 74.6 90.8 128.7 203.8 254.6 259.6 367.5 516.5 571.5 698.6 873.3

11.4 28.5 43.2 59.5 83.6 134.7 208.8 277.8 434.9 675.9 808.1 993.8 1101.3

Pr

Waterlmercury

5

6 7 8 9 10 11 12 13

37 45 40 37 39 38 41 39 36 37 36 35 33

4.60 4.00 4.32 4.60 4.41 4.51 4.23 4.41 4.71 4.60 4.71 4.81 5.07

10 cS oil/mercury

14 15 16 17 18 19 20 21 22 23 24 25 26

5177 (118) 5796 (68) 7196 (128) 8146 (164) 9609 (127) 10784 (40) 14395 (250) 14347 (415) 16870 (290) 20674 (416) 24980 (631) 3 1604(1239) 29105 (842)

0.30 0.60 0.90 1.22 1.52 2.13 2.8 1 3.41 4.64 6.06 6.74 7.74 8.35

56 56 56 63 63 63 77 77 77 82 82 82 77

104.3 129.8 177.2 224.9 286.8 378.2 580.1 642.4 931.6 1373.0 1808.0 2547.0 2476.0

1.18 2.63 4.34 7.17 9.65 15.9 32.8 44.2 74.2 136.2 165.1 211.1 205.6

79.4 79.4 79.4 72.6 72.6 72.6 52.8 52.8 52.8 44.9 44.9 44.9 52.8

52.4 73.1 93.0 115.1 157.2 215.9 276.5 371.6 525.2 813.8 1142.0 1429.0 1831.0

0.19 0.35 0.63 0.85 1.27 2.09 3.51 4.76 7.79 12.5 14.9 19.1 21.6

501.0 540.0 501.0 540.0 50 1.O 501.0 445.0 457.0 457.0 445.0 445.0 445.0 457.0

100 cS oil/mercury

27 28 29 30 31 32 33 34 35 36 37 38 39

2986 3750 4338 4924 6052 7073 7885 9382 10930 14082 18328 20566 24737

(32) (42) (47) (66) (25) (54) (133) (169) (193) (175) (845) (577) (480)

0.33 0.61 0.9 1 1.21 1.52 2.13 2.80 3.43 4.63 6.03 6.70 7.7 1 8.34

61 56 61 56 61 61 78 72 72 78 78 78 72

Numbers in parentheses are standard deviations

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

285

LEGEND WATER / MERCURY DATA A 1OcS OIL/MERCURY DATA 100 cS OIL / MERCURY DATA 0

0

0

0 0

0

A A A A

'0

I

2

A

3

rn

4 5 6 2 Jg(m/s x I 0 )

7

8

9

FIG.2. Interfacial heat transfer coefficients versus superficial gas velocity.

it was concluded that the overall comparison of Eq. ( 5 ) with the present experimental data is at least as poor as Eqs. (1) and (3). The heat transfer coefficient was cast as a Nusselt number and the superficial gas velocity as a Reynolds number based upon the volume-equivalent bubble radius and the properties of the water or oil layer. The Prandtl number was similarly chosen as that of the water or oil layer. The correlation that was developed to predict interlayer heat transfer between overlying immiscible liquid layers agitated by rising bubbles is shown in Fig. 3. The form of the correlation is

1.95 and is the relationship recommended for interlayer heat transfer. Nu

=

(8)

GEORGE ALANSON GREENE

286 lo3

Nu

1o2 N

2

.

& 10 2

X

z

X

.6

1

lo-' lo-'

1

10

X

10-cS Oil / Hg

0

100-cS Oil I Hg

lo2

1o3

Re

FIG.3. Dimensionless correlation of interfacial heat transfer data.

D. SUMMARY The data cover a range of Prandtl numbers from 4.0 to 540, as listed in Table I. Therefore, the correlation given by Eq. (8) would be expected to be reasonably accurate for situations in which one fluid layer has a Prandtl number greater than unity (core oxides), and the other fluid layer has a negligible resistance to heat transfer (core metals). This is precisely the situation in the case of a molten core-concrete interaction. In practice it is recommended that Eq. (8) be applied to the fluids on each side of the interface and then the overall heat transfer resistance can be calculated across the interface from the bulk temperature of one fluid layer to the bulk temperature of the second fluid layer.

111. Onset of Entrainment between Immiscible Liquid Layers

due to Rising Gas Bubbles In the previous section, the enhancement to heat transfer across a stratified liquid-liquid interface due to rising gas bubbles was treated. It was explained that this situation is found in such diverse applications as metallurgical processing and nuclear reactor safety. In all these occurrences, one is concerned with the possibility of entraining the denser liquid from the bottom layer into the top layer of lighter liquid by the rising gas bubbles. Such

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

287

entrainment can significantly increase both heat and mass transfer between the two immiscible liquid layers, a phenomenon which could be either detrimental or desirable depending upon the specific application. In the absence of such a criterion for the onset of entrainment, the molten layers are required to remain stratified in spite of the action of the rising gas bubbles. In this section we describe the investigations of entrainment onset and the criteria developed that allow relaxation of the stratification constraint. A. BACKGROUND

Previous investigations reported in the literature indicate that entrainment between the liquid layers is clearly observed in some cases but apparently is not present in other cases. Szekely(1963) deals with the case of a bubble-stirred interface between the two immiscible liquids without entrainment. Porter et al. (1966) reported observations of aqueous solutions overlying liquid mercury with no apparent entrainment of the mercury into the aqueous phase. Poggi et al. (1969)report experimental indications of entrainment occurring between pools of glycerine-water over mercury, fused salt over lead, and slag over molten copper. More recently, Werle (1982) and Greene and Schwarz (1982) investigated the fluid mechanics and heat transfer behavior of silicone oil and water over liquid metal pools of Wood’s metal and mercury, and silicone oil over water pools, both cases with gas bubbling from below. It was found in both investigations that the rates of heat transfer between the two overlying immiscible liquid layers driven by gas bubbles rising through the interface were strongly dependent upon whether the system exhibited entrainment or stratification. The gas bubbling was able to support entrainment for the oil-over-water system, however, the systems of water/oil over Wood’s metal/mercury remained stratified with no mass transfer between phases. As reported by the authors, the fluid systems that underwent entrainment exhibited rates of interlayer heat transfer that greatly exceeded those measured for the fluid systems that remained stratified. Epstein et al. (1981) tested stratified pools of organic liquid and water with nitrogen as the bubbling gas. These authors reported that a well-mixed emulsion of the two liquid phases was formed upon exceeding some critical value of gas flux. Gonzalez and Corradini (1987) and Suter and Yadigaroglu (1988) also reported clear indications of mixing between stratified layers of liquids under certain operating conditions. The accumulated evidence indicates that entrainment between the immiscible liquid layers could occur or not occur depending on the specific liquid system and operating conditions. The objective of this section is to develop a criterion for predicting the onset of interfacial entrainment between the liquid layers. A conceptual limiting criterion is developed and then compared with experimental observations.

288

GEORGE ALANSON GREENE

B. PHYSICAL PHENOMENA Consider a pool consisting of stratified layers of two immiscible liquids. If gas is injected at the bottom of the pool and flows upward through the liquid layers, entrainment and mixing of the heavier liquid (density pz) into the lighter liquid (density p l ) can occur, as reported above. Our objective is to develop a criterion that can predict the minimum gas flow needed to cause such entrainment. Clearly, the phenomenon is affected by the two-phase flow regime of gas in the lower liquid pool. As gas flux increases, the two-phase flow regime changes from bubbly flow to churn-turbulent flow, annular flow, and eventually to dispersed flow. Because our concern is with the minimum gas flow that would induce entrainment, attention is centered on the regime of discrete bubbly flow. In the first part of this study, a series of photographic records were obtained to aid in qualitative understanding of the physical phenomenon. Stratified layers of silicone oil over colored aqueous solutions of copper sulfate in a transparent 10-cm-diameter column were used for these experiments. Single bubbles of air were injected at the bottom of the column and photographed as they penetrated the interface between the two liquids. Examples of the photographic records of the phenomenon are shown in Fig. 4. It is seen in Figs. 4a and 4b that as the gas bubble penetrates the liquid interface, it begins to drag a column of the denser fluid in its wake into the lighter fluid. As the bubble continues to rise (Figs. 4c and 4d), this “wake” column elongates and then necks down and finally snaps to free a glob of the denser fluid, which is then successfully entrained into the upper fluid by the rising gas bubble. In these experiments, depending on the size of the gas bubble, globules of the

FIG.4. Photograph of entrainment phenomenon.

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289

lower liquid would be so entrained or not entrained into the upper pool. In the latter case, a raised column (such as in Fig. 4a) could still occur, but then the column would be pulled back into the lower pool by surface tension without necking down and breaking off as an entrained globule. It was also observed that small gas bubbles would sometimes be trapped at the two-fluid interface and be prevented from penetrating into the upper layer of lighter liquid. The question to be addressed is, “Under what conditions does a rising bubble penetrate the interface between the two liquid pools and entrain some volume of the denser liquid into the lighter liquid?” Our interpretation of the photographic records indicated that the entrainment process was caused by the rising gas bubble having sufficient buoyancy to penetrate the liquid-liquid interface and to drag in its wake a globule of the denser fluid into the upper pool, with sufficient force to overcome interfacial tension between the two liquids as well as the negative buoyancy of the denser fluid. In the following discussion, a criterion is developed based on this hypothesized mechanism to estimate the minimum size of gas bubble able to cause entrainment for a given pair of immiscible liquids.

c. FORMULATION OF CRITERION Consider a gas bubble of volume V, as it rises through the lower liquid layer and approaches the interface. To successfully entrain fluid of density pz into the fluid of density p I , the gas bubble must necessarily meet both of the following requirements: 1. Successfully penetrate the liquid-liquid interface and rise into the upper liquid layer. 2. Successfully drag a globule of the p2 fluid in its wake, into the upper liquid layer, a sufficient distance so as to detach the globule from the liquidliquid interface. We will formulate the necessary criterion for each of these two requirements. The following simplifying assumptions are taken as reasonable hypotheses: 1. Consider only individual bubbles, i.e., no interaction between successive or neighboring bubbles. 2. Cylindrical symmetry about the vertical axis of the rising bubble. 3. One-dimensional motion in the vertical direction only. 4. Limit to low velocities so that inertial and viscous shear forces are negligible compared to buoyancy and surface tension forces.

GEORGE ALANSON GREENE

290 1. Penetration

First consider a bubble of volume V, as it passes the liquid-liquid interface. Visual observations in this investigation and elsewhere (Suter and Yadigaroglu, 1988)indicate the tendency for a small “film” of the p2 liquid to surround the bubble as it attempts to penetrate the interface. Neglecting dynamic effects, the forces acting on the bubble-film assembly are upward buoyancy and downward surface tension. The effective buoyancy force decreases as the bubble passes from the heavier p2 fluid into the lighter p1 fluid. At the same time, the downward component of the surface tension force increases as the angle 0 changes from 0 to 71/2. Therefore, the critical moment is when all of V, is in the upper layer and the angle 0 is equal to n/2. For successful penetration by the gas bubble, this minimum net upward force must be greater than zero. Neglecting the mass of the thin film around the bubble, and for the limiting case of a spherical bubble, the penetration criterion can be written as

That is, for the gas bubble to penetrate the interface, its volume Vg must exceed a critical volume V,* given by Q(P1- PJ where a12is the interfacial tension between the two liquids. 2. Entrainment Next consider a gas bubble of volume V, after it penetrates the liquid interface, as shown in Fig. 5. Under buoyancy-driven motion, the gas volume rises through the upper lighter liquid and attempts to pull in its wake a column of the heavier lower liquid. Let V2 indicate the volume of the denser fluid pulled above the liquid interface and R, denote the column radius, with interfacial contact angle p as indicated in Fig. 5a. A vertical force balance on the combined V2 configuration indicates an upward buoyancy force of

<+

Buoyancy (UP) = C<(Pl - P g ) - V2(P2 - Pl)lg and a downward force due to surface tension of Surface tension (down) = 271R,aI2 sin

For the entrainment process to continue successfully, the upward force must exceed the downward force, hence

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

29 1

FIG.5. Geometrical system for analysis.

The surface tension force is maximum when sin fl approaches unity, corresponding to the situation shown in Fig. 5b. In this state, assuming a hemispherical volume, the wake volume Vz corresponds to V, 2 $ r R :

at

p =4 2

(12)

The photographic records indicate that radius of the wake column (Rz) is approximately equal to the projection radius of the leading gas bubble. Hence one can write the approximate relationship between R, and V, as follows: R2 = [ ( ~ / ~ I T ) V J ” ~ S

(13)

where S is a shape function for the gas bubble; measuring its projection radius versus the radius of a sphere, S = 1 for spherical bubbles and S > 1 for spherical-cap bubbles. For the minimum condition, take S = 1, resulting in the following necessary criterion for entrainment of p, fluid into the p1 layer:

We can define a dimensionless bubble volume by normalizing with respect to the minimum-penetration gas volume (V:), as given by Eq. (lo), w 3

V,/v:

GEORGE ALANSON GREENE

292

The criterion for interface penetration by the rising gas bubble becomes o>l (16) and the criterion for possible entrainment of the heavier liquid into the lighter liquid is

For most cases, the gas density is negligible compared to the liquid densities, and Eq. (17) simplifies to w > [2/(3 - p * ) 1 3 / z

(18)

where p* = p2/p1. The results are plotted in Fig. 6,showing the regimes where (a) gas bubbles do not penetrate the liquid-liquid interface, (b) bubbles penetrate the inter-

I

I

I 1

12 -

10 -

’

1 1

--I

(d) Entrainment NotPossible By Single Bubbles

w

-

4-

-

4

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

293

face, but cannot entrain the heavier liquid into the upper layer of lighter liquid, (c) entrainment is possible, and (d)entrainment is not possible by single bubbles by this mechanism. Equation (18) indicates that entrainment by this single-bubble process is not possible when the ratio of liquid densities exceeds the limit p* > 3

This is consistent with the observations of Porter er al. (1966), Werle (1982), and Greene and Schwarz (1982) that no entrainment occurred for water over mercury (p* ‘Y 13) or for water over Wood’s metal ( p * 2: 9.5). It should be emphasized that this formulation treats only entrainment by single bubbles, giving a necessary criterion for entrainment. Thus, this deals primarily with applications at the lower range of gas flow rates, where bubbly flow exists.

D. EXPERIMENT In order to test the criterion for bubble penetration and entrainment derived above, experiments were carried out with eight different pairs of immiscible liquids. For each pair of fluids, gas bubbles of varying sizes were injected to determine the regimes of entrainment and nonentrainment. As illustrated in Fig. 7, the experimental apparatus consisted of a vertical glass column having two sections and provided with devices for injecting and measuring gas bubble volumes and for collection of entrained liquid globules. The denser and lighter liquids were contained in the lower (6.0-cm i.d. x 23.5cm length) and upper (16-cm i.d. x 14.5-cm length) sections of the glass column, respectively. In operation, the liquid-liquid interface would be adjusted to lie just below the junction plane of the two sections. A mechanical shutter was mounted just above the junction plane to collect any volume of the denser liquid that is entrained into the lighter liquid. Air bubbles of varying sizes could be injected at the bottom of the glass column by means of a micrometer syringe and holding-cup mechanism. After initial purging, a metered volume of gas would be injected by the syringe into the inverted holding cup. When ready, the cup would be quickly turned upward, allowing the injected gas to rise through the liquid layers as a single bubble. An inverted funnel, with attached burette mounted at the top of the test section, permitted trapping of the rising gas bubble and measuring its volume with a precision of 0.002 cm3. The liquid properties were measured as a part of the test program. Densities were obtained by gravimetric measurements to a precision of 0.01 g/cm3 and surface tensions were measured with 8 Fisher Surface Tensionmat (Model 21) to a precision of 0.05 dyn/cm. The tabulated values are averages obtained from 10 measurements for each fluid. It should be noted that surface tension is sensitive to small amounts of impurities or additives (e.g., copper

GEORGE ALANSON GREENE

294

SECTION

0

HOLDING-CUP MECH A N ISM

p2

-LOWER

SECTION

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

295

sulfate used to color the water) and direct measurements had to be obtained with the actual test fluids. In the experiments, a pair of liquids would be charged into the apparatus and adjusted to set the liquid-liquid interface at the desired height. A specified volume of air would be injected into the holding cup, then permitted to rise as a single bubble through the two liquid layers. Visual observation and video recordings would be made during the time of bubble transit. The morphology observed was as qualitatively described above and as illustrated in Fig. 4. Entrainment of the denser fluid into the upper liquid pool was discernible by eye and from the video films whenever it occurred. All experiments, with the exception of those using R11 refrigerant, were performed at room temperature. Due to the low boiling point, the R11 fluid had to be slightly refrigerated to prevent its vaporization during the tests.

E. RESULTS Sample results are shown in Figs. 8a and 8b for.the eight pairs of test fluids. The numerical results are listed in Table 11. Each point represents a test condition wherein multiple observations were made. Due to second-order perturbations, there was some stochastic variation regarding whether a specific bubble of a given size would induce entrainment. If bubbles of a given size were observed to cause entrainment in 75% (or more) of the repetitions, that condition was considered to be an “entrainment” point. Usually, the bubble volume that satisfied this 75% threshold was also close to the minimum bubble size with observable entrainment. In Figs. 8a and 8b, the cases with and without entrainment are indicated by dark and open symbols, respectively. The threshold criterion for entrainment, given by Eq. (14), is indicated by the horizontal line for each pair of test fluids. There is a 500-fold range in the predicted bubble volume for onset of entrainment for these widely different fluids. The results of Figs. 8a and 8b show generally good agreement between experiment and theory, with the onset of significant entrainment occurring close to the proposed thresholds. The acetone/glycerine fluid pair exhibited an interesting but unexplained quirk, that is, an onset-of-entrainment transition followed by a subsequent return to conditions not supporting entrainment. This anomalous behavior has not been investigated further at present, and for the purposes of this paper, the first onset-of-entrainment volume was chosen for analysis. According to the theory proposed above, the onset criterion for entrainment is affected primarily by just two physical property groups: ratio of liquid densities, p*, and ratio of interfacial tension/liquid density, olz/pl. The eight pairs of liquids used in these experiments covered a 20-fold range in alz/pl and almost a 3-fold range in p*. These data were examined to test the effects of

ENTRAINMENT NO ENTRAINMENT -PREDICTED THRESHOLD 0

o NO ENTRAINMENT -PREDICTED THRESHOLD

0

GLYCERINE

WATER BROMOFORM

BROMOFOPM

FIG.8. Sample entrainment-onsetresults.

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

297

TABLE I1 COMPARISON OF MEASURED AND CALCULATED MINIMUM BUBBLEVOLUME FOR ENTRAINMENT ONSET Minimum bubble volume for entrainment onset" Fluid pair 10 cS silicone oil/water (colored with CuSO,) 100 cS silicone oil/water (colored with CuSO,) Water/Rll (refrigerant) Water/bromoform Hexane/water Glycerine/bromoform Acetone/glycerine 100 cS silicone oil/glycerine ~~~~~

Measured

Measured range

Calculated

0.015

0.010-0.020

0.011

0.030

0.020-0.035 0.030-0.050 0.55 -1.40 0.15 -0.21 0.03 -0.09 0.001-0.003 0.05 -0.09

0.048 0.046 1.02 0.22 0.043 0.002 0.058

0.050 0.90 0.17 0.04

0.002 0.07 ~

~

Bubble volume in cm3. ' Entrainment occurred in 75%of the cases at this volume.

physical properties on the entrainment onset criterion. Figure 9a plots the conditions corresponding to onset of entrainment determined experimentally for the eight pairs of test fluids as a function of the density ratio, p * . The gas volume for onset, V,, is normalized with respect to the penetration volume, V:, to give the dimensionless onset volume, o,as defined by Eq. (1 5). The points in Fig. 9a represent the smallest o that caused significant entrainment for each pair of fluids. It is seen that the 3-fold range in p * caused two orders-ofmagnitude change in the onset w. The curve in the figure represents the theoretical criterion for onset of entrainment, as given by Eq. (18). Good agreement between experiment and theory was obtained; the theory not only predicted the parametric effect of p * but also gave good estimates of the magnitude of o. Figure 9b examines the parametric effect of the property group, a12/p1. According to the proposed theory, the group Vg[g(3 - ~ * ) / 7 . 8 ] should ~ / ~ be equal to (a12/p1)3'2,where 5 is the threshold volume of the gas bubble to cause entrainment. The experimental measurements for the test fluids are plotted in Fig. 9b and confirm this prediction over a 20-fold range of o12/p1.The theoretical prediction, Eq. (14), is also plotted and shows very good agreement with the experimental results over the 100-fold range of the threshold volume. The axes in this figure are in CGS units. The proposed theory, based on first principles, obtained this good agreement with measurements without the use of any empirical parameters.

r-

-

ONSET

OF ENTRAINMENT

-

EXPERIMENT PROPOSED THEORY

-

.-

N R

aD

b’

\

*P

--

100

--

-

-

-

I

0

m

---

-

I

p2 ’PI

I

1

I

I

I

I

10 Q12’PI

FIG.9. Effect of (a) liquid densities and (b) interfacial tension-liquid density ratio on entrainmentonset.

1

1 1 1 1 1

I00

HEAT,MASS,A N D MOMENTUM TRANSFER IN FLUIDS

299

F. SUMMARY The problem of entrainment between stratified layers of immiscible liquids caused by rising gas bubbles was examined to develop a criterion for onset of entrainment. Visualization experiments led to a hypothesis that entrainment by single bubbles is caused by the levitation of a small column of the denser fluid in the wake of the bubble as the bubble passes across the liquid-liquid interface. A first-principle analysis led to a theoretical criterion for the threshold volume of gas bubble necessary to cause entrainment, indicating a strong dependence on the liquid-liquid density ratio ( p * ) and the interfacial tension-liquid density ratio (q2/pl). Experiments were conducted to measure actual onset of entrainment for eight different pairs of fluids. The experiments covered a 3-fold range in p *, a 20-fold range in qi2/p1,and a 2000-fold range in the gas bubble volume. The proposed theoretical criterion, with no empirical parameters, was able to predict the experimental measurements with good agreement over the entire test range.

IV. Bubble-Induced Entrainment between Initially Stratified Liquid Layers As illustrated in Fig. 4,it is known that gas bubbles of sufficient size rising through stratified layers can entrain some volume of the denser liquid from the lower layer into the upper layer of lighter liquid. This entrainment phenomenon has significant effects on both heat and mass transfer between the two liquid layers (Szekely, 1963; Greene and Irvine, 1988). This situation is encountered and is of concern in a number of applications. Some examples are chemical processing, assessments of severe nuclear reactor accidents, and metallurgical processing. In the previous section, an entrainment onset criterion was developed that enabled relaxation of the limiting constraint requiring the two liquid layers to be stratified under the action of gas bubbles rising across their interface. In this section, a model is developed and presented that will enable calculation of the amount of the lower liquid that is entrained by the rising gas bubbles.

A. BACKGROUND Greene et al. (1988) showed that discrete gas bubbles must exceed a certain minimum volume to cause entrainment of the denser liquid into the upper pool; a theoretical criterion was developed to predict this onset condition. The objective of the present section is to determine the volume of the denser fluid that will be entrained by a discrete bubble, once this onset criterion is exceeded. The study is limited to that regime of gas flux that results in bubbly

300

GEORGEALANSON GREENE

flow. Higher gas fluxes would lead to alternate flow regimes (e.g., churnturbulent flow), which may have very different mechanisms for fluid mixing between the two immiscible liquids. Previous investigations reported in the literature indicate that entrainment between overlying immiscible liquid layers by bubbling gas is clearly observed for some fluid systems but apparently is not observed for other fluids. Szekely (1963) and Greene et al. (1988) report on studies of liquid systems that did not support entrainment, but rather remained stratified. Szekely presents a theoretical analysis of transient heat transfer at the interface by surface-renewal principles, whereas Greene et al. present experimental results for interfacial heat transfer and a dimensionless correlation of their data. Mercier et al. (1974) report on visual observations in which a minimum bdbble volume threshold was observed (0.020 cm’) for onset of entrainment between water and a variety of mineral oils. A similar observation is reported by Mori et al. (1977),who observed a minimum bubble volume for penetration of an aqueous glycerol-R113 interface; for bubbles larger than this penetration threshold value (0.020-0.045 an’),entrainment of the lower fluid always occurred and increased with increasing bubble volume. Epstein et al. (1981) report on the onset of mixing and stratification in bubbling systems; they suggest that it is only necessary to know the mixture density to predict the pool configuration. In a similar study, however, Suter and Yadigaroglu (1988) develop an entrainment criterion on the basis of stability considerations that include density and interfacial and surface tensions. Greene et al. (1988) present a theoretical and experimental study of entrainment in which they report both a bubble penetration threshold and an entrainment onset threshold. The model is a function of the densities of the gas and two liquids, the interfacial tension between the liquids, and the bubble volume; entrainment onset data for eight separate fluid pairs were found to be in good agreement with the entrainment onset criterion. The bubble penetration criterion was in agreement with the experimental observations of Mori et al. (1977). That the rate of entrainment between immiscible liquid layers increased with increasing bubble volume for a particular pair of fluids is reported by Poggi et al. (1969),Cheung et al. (1986),Veeraburus and Philbrook (1959),and Mori et al. (1977). Mori et al. present data for glycerol-R113 that demonstrate an increase in entrained volume with an increase in bubble volume and a decrease in the viscosity of the light liquid. Entrainment mechanisms of film and bubble transport were observed by Poggi et al. (1969), Veeraburus and Philbrook (1959), and Mori et al. (1977). Cheung et al. (1986) observed that entrainment could be affected by nonuniform gas bubbling. Although no model for the rate of entrainment is presented, Gonzalez and Corradini (1987) report experimental results for entrainment onset and emulsification for several fluid pairs as a function of the superficial gas velocity.

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

301

Heat transfer between overlying liquids both with and without entrainment is reported by Greene and Schwarz (1982), Greene and Irvine (1988), and Werle (1982). Each observed that pools could be stratified, mixed, or homogenized depending upon the fluid properties and volumetric gas flux. The component of heat flux due to entrainment was observed to substantially increase the interlayer heat transfer over the stratified situation. Greene and Schwarz (1982) present a model for calculating the entrainment heat transfer as a combination of the stratified component and that due to direct mass transfer. The analysis requires a model for the rate of mass entrainment that, until now, has not been available. It should be noted that Suo-Anttila (1988) recently proposed a framework for calculating entrainment and entrainment heat transfer between overlying liquids with gas bubbling. However, models for the various processes involved are not offered.

B. EXPERIMENT In order to investigate the phenomenon of bubble-induced entrainment between stratified liquids, the experimental investigation to be described was initiated. In this section we will discuss in detail the experimental apparatus and the experimental procedure. It became necessary during this investigation to measure the physical and transport properties of the fluids that were used, as well as the bubble rise velocities for each of the fluids. These measurements will be discussed as well. The experimental apparatus was previously illustrated in Fig. 7. It consisted of a vertical glass column having two cylindrical sections, fitted with devices for injecting and measuring gas bubbles of precise volumes and for collection of the entrained liquid volume. The denser and lighter fluids were contained in the lower and upper sections of the glass column, respectively. A mechanical shutter was mounted just above the junction plane to collect the volume of the denser, lower liquid that would be entrained into the lighter, upper liquid by a rising bubble. In the experiments, a pair of immiscible liquids would be charged into the apparatus and adjusted to set the liquid-liquid interface at the junction of the two cylindrical sections. A specified volume of air would be injected into the holding cup, then allowed to rise as a single bubble through the two layers. Visual observations and video recordings would be made during the time of bubble transit and entrainment. The volume of the lower, heavy fluid that was entrained in the wake of the rising bubble would be intercepted by the mechanical shutter device. The shutter would be simply rotated in a horizontal, sweeping motion to cover the junction at the interface of the two liquids, capturing the entrained liquid volume as it once again settled. The bubble would continue to rise until trapped in the burette, at which point an

302

GEORGE ALANSON GREENE

accurate determination of the actual bubble volume could be made. The entrained droplets were recovered from the shutter plate, separated from the lighter liquid, and measured. All experiments, with the exception of those using R11 refrigerant, were performed at room temperature. Due to its low boiling point, the R11 fluid had to be refrigerated to 40-50°F to prevent vaporization during the tests. However, the properties of the R11 that were used in the data analysis were measured at this refrigerated temperature. C. BUBBLE RISEVELOCITY

It was necessary to know the bubble rise velocities for these tests in order to calculate the bubble Reynolds number. No theory was found that would accurately and reliably predict the bubble rise velocity in any fluid other than water. Therefore, the bubble terminal rise velocities were measured for air bubbles in each of the test fluids as part of the experimental program. A separate cylindrical test column was constructed, into which the bubble holding-cup mechanism could be installed. Bubbles, which covered the size range used in the entrainment experiments, were released and their time of flight was timed over two separate courses during their ascent. It was required that the velocities over both courses be identical in order for them to be considered at terminal velocity. The terminal velocities indicated in Fig. 10 represent the average of 20 runs for each bubble size tested. D. RESULTS In this section we present sample experimental results and discuss their implications.

1. Observations of the Phenomenon Figure 4 shows a series of frames from a high-speed photographic record of a single, discrete bubble rising through the stratified liquid layers. During this series of frames, the gas bubble has passed into the upper pool and a volume of the denser liquid is clearly seen to be entrained in its wake, having been pulled through the interface between the two liquid layers. Studies of the high-speed movies showed that entrainment starts when a gas bubble of sufficient volume penetrates the surface and pulls a column of the lower liquid in its wake into the upper pool. If the bubble volume is of insufficient size, the levitated column falls back to the lower pool and the gas bubble continues to rise through the upper pool without having caused any entrainment. For sufficiently large gas bubbles, the levitated column rises to a sufficient height such that it becomes hydrodynamically unstable; as it elongates, it necks down to snap free a glob

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

303

25

23

15

-.21 5 2 19 v)

13

.0

W > W

-

17

11

15

9

13

.3 n

m

7

5 0.4

0.5

08 0.7 0.8 Bubble Radius (cm)

0.9

1.0

FIG. 10. Measured bubble rise velocities: (+) water, ( )acetone, ( A ) RI I, (u) hexane, ( x ) 10 cS silicone oil, (0) bromoform, ( c ) 100 CS silicone oil, and (m) glycerine.

of the denser fluid, which is then considered to be successfully entrained into the upper fluid layer. Our experiment permitted the capture and measurement of the volume of such entrained globules. Observations of the phenomena from the visual recordings indicated that the volume of entrainment would be affected not only by the size of the gas bubble but also by the densities of the two liquid fluids, their viscosities, and the interfacial tension between the two liquids. Experimental indications of these parametric effects are described below. 2. Egect of Heavy-Liquid Density Figure 1 l a shows the experimental data indicating volumes of entrained liquid for individual bubbles of various sizes (volumes). Results are plotted for two pairs of fluids, water overlying Refrigerant-1 1 (R 11) and water overlying

2.0

a 0

1.6 Water / R11

4"I 0.'

01

10

20 Gas Bubble Volume. Vp ( Cm3 )

30

40

0 Gas Bubble Volume, V, ( cm 3 )

20

0.24

C

d 0.20

1.6

P)

5

0

-5

1.2

>-

i

-5

-5

d

d

g P-

9 0

W

8

0.16

1

-

;.

0.12

-e

0.t

c

UI

w c

0.08

01

0.04

01

0 00 Gas Bubble Volume. V o ( cm3 )

J

1

10

20

30

40

Gas Bubble Volume. Vg ( Cm3 1

FIG. I 1 . Parametric effects of (a) heavy-liquid density, (b) light-liquid density, (c) heavy-liquid viscosity, and (d) light-liquid viscosity upon entrainment.

306

GEORGE ALANSON GREENE

bromoform. The major parametric difference between these two pairs is the density of the heavier liquid, bromoform being 2.9 times more dense than water. It is seen that the entrainment volume for a given bubble volume decreased very significantly with increasing density of the heavy liquid, the water/bromoform entrainment volumes being almost an order of magnitude lower than the corresponding entrainment volumes for water/Rl 1. 3. EfSect of Light-Liquid Density

The effect of the light-liquid density is just the opposite. Figure 1l b shows data for two pairs of fluids, silicone oil over water and hexane over water. The major parametric difference between these two pairs is the density of the upper-layer light fluid, hexane, with a specific density of 0.7 as compared to silicone oil, with a specific density of 0.9. These results clearly show that entrainment volumes increased with increasing density of the light liquid. For the fluids indicated, a density ratio of 1.5 in the upper-layer fluid resulted in a factor of approximately four in the entrainment volumes, for the same bubble volume. Combined with the observation of Fig. 1la, the experimental results demonstrate the importance of the buoyancy difference between the two liquids.

4. Effect of Interfacial Tension Measurements obtained with the two pairs of fluids, water over R11 and hexane over water, can be compared to evaluate the effect of interfacial tension on entrainment. Both of these pairs have a relative density difference of approximately 0.4 g/cm3 between the heavier and lighter liqujds, so that relative buoyancy should not be a factor. The major parametric difference between these two pairs is the interfacial tension (qJ, which differed by almost a factor of 10. The results showed that there is little discernible difference in the entrainment volumes between these two pairs of fluids. However,tthe minimum gas bubble volumes for which measurable entrainment could be obtained was found to be 0.03 cm3 for water over R11 and 0.002 cm3 for hexane over water. The conclusions from this observation is that the interfacial tension affected the onset of entrainment but had a relatively small effect on the volume of entrainment. The first effect is consistent with the results reported in Greene et al. (1988). 5. EfSect of Heavy-Liquid Viscosity

Although the liquid viscosities do not affect static force balances, they would be expected to affect bubble rise velocities, which in turn influence the

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

307

characteristics of the bubble wake. Figure 1 l c examines the potential effect of the liquid viscosity of the heavier liquid in the lower pool. The two pairs of fluids illustrated in this figure have essentially identical viscosities in the lighter fluid and also similar density differences; the major parametric dependence is in the viscosities of the heavy liquids, which differ by a factor of over 1000 for water and glycerine. The experimental results show a strong influence on the entrainment volumes. The low-viscosity case (hexane/water) had entrainment volumes greater than 10 times that of the high-viscosity case (acetone/glycerine). 6. EfSect of Light-Liquid Viscosity

Figure 1Id illustrates the effect of different viscosities for the light liquids in the upper pool. Data are plotted for water over bromoform and glycerine over bromoform. These two pairs of test fluids had the same heavier liquid (bromoform) and similar buoyancy density differences. Thus this figure illustrates the parametric effect of the upper fluid (light-liquid) viscosity. It is seen that while there is a discernible increase in entrainment volume for the less viscous case, the magnitude of the effect for changing light-liquid viscosity is much less than that for changing viscosity of the heavy liquid.

E. CORRELATION OF DATA The experimental results clearly indicate that the volume of entrainment from the lower pool into the upper pool by a discrete gas bubble is a function of the gas bubble volume, the densities of the two liquids, the viscosity of the two liquids, and the interfacial tension between the two liquids. Over the range of experimental tests for eight different fluid pairs, the resultant entrainment volume per bubble varied over two orders of magnitude. The development of a correlation to represent this experimental data base is presented here. It was first noted that, for a given pair of liquids, entrainment would be possible by discrete gas bubbles only if the bubble volume exceeded the criterion for onset presented by Greene et al. (1988),

V,, was theoretically derived by Greene et al. (1988),and is given by

A static force balance was performed (assuming negligible inertial forces) to determine the maximum theoretical liquid volume that could be entrained across the liquid -liquid interface by a rising gas bubble that exceeds this

308

GEORGE ALANSON GREENE

entrainment onset volume. As the bubble (in this analysis the bubble is assumed to be a spherical cap) rises across the liquid-liquid interface, it is observed that a column of the lower, heavy fluid rises with it due to buoyancy and wake effects associated with the bubble. We will consider a control volume consisting of the rising gas bubble and the levitated column of the lower, heavy liquid. By considering the upward lifting force due to the buoyancy of the bubble on the column of entraining fluid, the restoring force on the fluid column due to interfacial tension with the upper fluid, and negative buoyancy of the entrained column in the upper layer, the following force balance was derived:

The first term is due to bubble buoyancy, the second term represents the effect of interfacial tension as the levitated column attempts to tear free from the lower layer, and the third term is the restoring buoyancy of the entraining column. This can be rearranged to solve for the maximum volume of lower liqiud (Vm,Jthat can be levitated into the lighter upper liquid as follows:

One can consider an “efficiency” of entrainment (E) by taking the ratio of actual entrainment volume (V,) to the maximum entrainable volume ( Vm,J, E = V,/Vm,e

(23)

Figure 12 shows plots of the experimentally obtained entrainment efficiencies versus the gas bubble volume, for two sample liquid pairs. It is seen that for a given pair of fluids, E initially increases rapidly when Vgexceeds V,,, but then seems to approach asymptotic limits at higher V, values. For the cases tested, the entrainment efficiency varied over two orders of magnitude but did not exceed 0.3. Other pairs of test fluids presented parametric curves similar to those plotted in Fig. 12. A normalized driving force for entrainment can be defined, based on the excess bubble volume beyond that required to cause onset of entrainment, as

4 = (Vg - ~ g o ) / & o

(24)

When the entrainment efficiencies are plotted versus 4, approximately linear line segments are obtained on log-log coordinates. Parametric variations in the magnitude of E are obviously related to the fluid properties of the stratified liquid layers.

HEAT,MASS,AND MOMENTUM TRANSFER I N FLUIDS 0 50

309

I

I

,

I

c

0 0)

'6

030

n u U ~~~

0

Water / Bromoforrn

Yl

10

20 3.0 Gas Bubble Volume. V, (cm3)

4.0

FIG.12. Examples of entrainment efficiencies.

As indicated above, photographic observations had indicated that entrainment occurs by capturing a volume of the dense liquid in the wake, as the gas bubble passes through the liquid-liquid interface. It seems reasonable to expect that any inefficiency in the entrainment process would be associated with the size and character of the bubble wakes. The bubble wakes are in turn affected by bubble diameter and velocity, the fluid viscosity, and density. These parameters are conveniently grouped in the bubble Reynolds number,

Re, = PIUbldb/p(,

for lighter liquid

Re, = PZUb2db/&

for denser liquid

(25)

where dbis taken to be the bubble diameter for an equivalent spherical volume of gas. In this investigation, the terminal rise velocities (Ubl and U b 2 ) were experimentally determined as a function of the gas volume (V,) for each fluid. In situations when experimental determination is not possible, published correlations can be used (Wallis, 1969). Following this reasoning, a functional relationship of the form E

=

.mJ, Re,, Re,)

was sought. The final correlation was represented as (d/4(Rel)"(Rez)b= K d C

(26)

where the constants were empirically determined to be a = 0.119, b = 0.380, c = 0.848, and K = 806.

GEORGE ALANSON GREENE

310 1o6

lo2'

I

lo-'

11111111

1

I

1 1 1 1 1 1 ' 1

I

1 1 1 ' 1 1 ' 1

I

1 1 1 1 1 1 1 1

10 1o2 Normalized Bubble Volume, cp

lo3

IIIIIIJ

1o4

FIG.13. Correlation of data: ( 0 ) water/bromoform, (0)100 CS silicone oil/brornoform, (v) 10 CSsilicone oil/water, (*) water/Rll, (+) 100 cS silicone oil/water, (x)glycerine/bromoform, (a) acetone/glycerine, and ( x ) hexane/water.

Figure 13 shows a graphical comparison of this correlation with all the experimental data obtained in this investigation. The statistical fit corresponds to a mean deviation of 0.35 in the ratio of measured to calculated entrainment efficiency.

F. SUMMARY The process of liquid entrainment between stratified liquid layers by rising gas bubbles was experimentally investigated for eight different pairs of liquids, covering a wide range of physical properties. In addition to photographic observations, quantitative measurements of the entrained liquid volumes were obtained for a range of gas bubble sizes. The results indicated that once the gas bubble exceeded the necessary minimum size to cause onset of entrainment, measurable entrainment occurred, which increased with excess gas volume (above the required onset volume). The efficiency of entrainment, defined as the ratio of entrained volume to maximum entrainable volume as derived from a limiting static force balance, was found to range over two orders of magnitude, but did not exceed 0.3 for any of the test cases. A dimensionless correlation was obtained to represent the entrainment efficiency as a function of the excess gas volume ($), and the gas bubble Reynolds numbers in each

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

311

liquid (Re,, Re,). Agreement between the proposed correlation and the 189 experimental data points indicated a mean deviation of 0.35.

V. Heat Transfer between Stratified Liquids with Entrainment across the Interface Consider the case of two immiscible liquids in initially stratified layers and agitated by rising gas bubbles as illustrated in Fig. 14a-c. As gas bubbles rise upward through the liquid-liquid interface, they not only disturb the temperature gradients on both sides of the interface, but may entrain the lower, heavy liquid into the upper, lighter layer, further increasing the heat transfer from liquid to liquid. If the rising bubbles are not able to support entrainment, the stratified state will prevail, and heat transfer will be across a well-defined interface between two well-mixed pools at different temperatures. This is the configuration presented in Fig. 14a and discussed in Section 11. Under some circumstances, the rising gas bubbles not only agitate the liquidliquid interface but also drive a mass transfer process by entraining the lower heavy liquid upward in the wakes of the rising gas bubbles. If the rate of entrainment of the lower phase can be balanced by the rate of droplet settling (or deentrainment), a partially mixed configuration will result in which the liquid -liquid interface will be preserved between a homogeneous lower layer and a heterogeneous upper mixture layer. This is the pool configuration presented in Fig. 14b. Criteria for onset of entrainment and rate of entrainment

FIG.14. Configurations of rnultifluid bubbling pools.

312

GEORGE ALANSON GREENE

were discussed in Sections I11 and IV, respectively. Deentrainment will be discussed in Section VIII. If the rate of entrainment is greater than the rate of settling of the entrained droplets, the two liquids will form a pool that is fully mixed, or fully entrained; there will only be one heterogeneous liquid layer and the concept of interlayer heat transfer will not be applicable. This is the configuration that is presented in Fig. 14c. This configuration will not be addressed in this discussion. IN STRATIFIED CONFIGURATION A. HEATTRANSFER

If conditions are such that entrainment of the lower fluid into the upper fluid cannot be supported by the rising gas bubbles, the liquid layers will remain stratified as illustrated in Fig. 14a. In this stratified configuration, the rate of heat transfer between the two layers is controlled by the rate of bubble agitation at the liquid-liquid interface. Szekely developed an analytical solution for the surface renewal heat transfer coefficent for each side of the interface (representing the heat flux divided by the average bulk layer-tointerface temperature difference) by integrating the time-dependent heat flux on each side of the interface over the period of arrival of successive bubbles. He arrived at the following result for the heat transfer coefficient on each side of the interface,

hi = 1.69(p~,kj,/r,)'/~

(3)

which was cast in dimensionless form as Nu, = 1.69 Re,!/'

PrfiZ

(4)

where Rei = j g r b / vand i = 1 or 2, representing the upper or lower liquid layer, respectively. The overall heat transfer would be evaluated by summing the two series resistances on both sides of the interface. Due to the absence of data with which to evaluate this model, Greene and Irvine (1988) performed an experimental investigation into heat transfer between overlying, stratified immiscible liquids with bubble agitation. The fluids that were tested were water and 10 and 100 cS silicone oils over a layer of mercury. Three sets of experiments were performed. The experimental data were nondimensionalized as in Eq. (4)and an empirical correlation was developed. The resulting correlation is given by Nu, = 1.95

Pr?72

(8)

This is the model that is recommended for surface renewal heat transfer driven by gas bubbling at a liquid-liquid interface in the absence of mass entrainment.

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

313

B. ONSETOF BUBBLE-INDUCED ENTRAINMENT Experiments with liquids such as water or light oils over a pool of mercury (and other heavy metals as well) have shown that the stratified configuration is possible, even under the influence of gas bubbling across the interface. However, there is evidence from other studies indicating that with some fluid pairs, the gas bubbling can drive mass transfer across the interface if some threshold entrainment condition is satisfied. Greene et al. (1988) performed a systematic experimental and analytical investigation of the conditions for the onset of mass entrainment across a stratified liquid-liquid interface due to single rising gas bubbles. Experiments were performed for eight fluid pairs with single air bubbles. The bubble volumes in the entrainment-onset experiments covered over three orders of magnitude. For each of the fluid pairs, a distinct bubble volume at which entrainment onset occurred was observed. The onset volume was observed, however, to be strongly dependent upon the fluid pair, ranging from 0.002 cm3 for acetone/glycerine to 1.0 cm3 for water/bromoform. An analytical criterion for the prediction of entrainment onset was developed, based upon a static force balance on the bubble/liquid system as it attempts to levitate a column of the lower, heavy liquid, and interfacial tension, which acts to restrain the entrainment. The mathematical criterion for onset of entrainment that was developed is given by the dimensionless inequality

where o is the dimensionless bubble volume equal to V,/V:, V, is the bubble volume, V z is the necessary bubble volume to penetrate the interface, and is the entrainment onset bubble volume. These are defined as

KO

7.80,~ v,o

=

(3P,

-

P2 - 2 P J S

and

I”’

The proposed theoretical criterion [Eqs. (lo), (14), and (17)], with no empirical parameters, was found to predict the experimental measurements with good agreement over the entire test range.

c. RATEOF BUBBLE-INDUCED ENTRAINMENT There is clear evidence in the literature that entrainment between overlying immiscible liquid layers by gas bubbling from below occurs for some fluid

314

GEORGE ALANSON GREENE

pairs, and is a function of the bubble volume as well as the physical and transport properties of the liquids and the gas. This is the configuration shown schematically in Fig. 14b. Greene et al. (1990b) performed a systematic experimental investigation of the parameters that contribute to the volume of heavy liquid that can be entrained by a gas bubble rising across a liquid-liquid interface to develop a general entrainment rate model. Experiments were performed with eight separate fluid pairs and single air bubbles as previously described for the entrainment onset studies (Greene et al. 1988). The experimental results clearly indicate that the volume of entrainment from the lower pool into the upper pool by a discreet gas bubble is a complicated function of the gas bubble volume, densities, and viscosities of both liquids, and the interfacial tension between the two liquids. An analysis based upon a static force balance was developed to predict the maximum volume that can be entrained, neglecting inertial forces as

One can consider an entrainment efficiency, E, by taking the ratio of the actual measured entrained volume, V,, to the maximum entrainable volume, V,,,,

(23) It is reasonable to expect that any inefficiencies in the entrainment process would be associated with the size and character of the bubble wakes. The bubble wakes are affected by Re, and Re,, the bubble Reynolds numbers in the upper (I) and lower (2) liquids, respectively. A functional relationship for the entrainment efficiency was sought of the form E = f(4, Re,, Re,), where 4 is the dimensionless excess bubble volume beyond that required for onset of entrainment, E = V,/Vm,e

The final relationship was found to be (t$/~)Ret"~ = 806t$0.848 (26) This correlation and the experimental data were found to be in good agreement. D. OVERALL INTERLAYER HEATTRANSFER WITH ENTRAINMENT In order to evaluate the magnitude of the heat transfer between overlying immiscible liquid layers with bubbling-induced entrainment across the interface, the configuration shown in Fig. 14b, it is necessary to calculate the component due to surface renewal as well as that due to the entrained mass.

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

315

The total heat transferred between the two liquid layers is assumed to be the sum of the heat transfer across the interface and the heat transfer from the entrained drops of the lower, heavy fluid while suspended in the upper layer. The overall surface renewal heat transfer coefficient, hSR,is constructed by summing the series resistances to heat transfer on both sides of the interface, h , and h 2 , as follows:

hi;

= [l/h,

+ l/h2]

Both hl and h2 can be computed from the surface renewal heat transfer correlation, Eq. (8), by simply substituting the fluid properties appropriate to the continuous fluid on each side of the interface, i = 1 and 2. The heat transferred by the entrained droplets while suspended in the upper, light liquid layer is modeled as a fraction of total excess (or deficit) enthalpy transferred from the drops to the continuous liquid. This can be represented in terms of an entrainment heat transfer coefficient, he, as follows:

where j , is the superficial (volumetric) entrainment flux, analogous to the superfical gas velocity, j , ( 3 volumetric gas flux/interfacial cross-sectional area). The superficial entrainment flux, j , , is defined as the product of the superficial gas velocity and the ratio of entrained volume per bubble to the bubble volume: j e = j,(K/%)

(29) Both Eqs. (27) and (29) illustrate the importance of not only the physical and transport properties of both liquids, but the sizes of the rising bubbles as well, in evaluating the heat transfer under entraining conditions. The sizes of the bubbles were directly measured in the course of these tests. The parameter K in Eq. (28) is an efficiency factor for the droplet-liquid heat transfer, which represents the actual fraction of excess enthalpy transferred from the drop to the surrounding liquid. For the experimental data to be presented, K was found to be almost always nearly equal to one, and will be assumed equal to one for this model. The total interlayer heat transfer coefficient can now be represented by summing Eq. (27) and (28) as follows: where the first term on the right-hand side is the contribution due to surface renewal effects and the second term is due to mass transport effects. This heat transfer coefficient is multiplied by the overall temperature difference from bulk layer to bulk layer to calculate the overall heat flux. It is possible to nondimensionalize Eq. (30) by multiplying both sides by the quantity (db/k2), resulting in the following dimensionless form of the total interlayer heat

GEORGE ALANSON GREENE

316 transfer model:

NUT = Nu,,

+ Re, Pr,

(31) where NUT = hTdb/k2,Nu,, = h,,d,/k,, Re, = jedb/v2,and Pr, is the Prandtl number of the lower, heavy liquid layer. This form of the model may be useful for evaluating data for additional fluid pairs as they become available. In order to evaluate the model for total interlayer heat transfer with entrainment [Fig. 14b; Eq. (30)], a series of two heat transfer experiments were performed (Greene and Schwarz, 1982). These entrainment heat transfer experiments differed from the stratified heat transfer experiments, which were discussed in Section 11, only in the choice of the immiscible fluid pairs tested. For the stratified heat transfer studies, one of the fluids chosen was mercury; this ensured that the liquids would remain stratified because the fluid pairs did not satisfy the entrainment-onset criterion. In addition, the resistance to heat transfer on the mercury side of the interface was negligible, allowing direct comparison of the data to models that were applicable for only one side of the interface. In this manner, the uncertainty in the data and models would not be compounded by having two comparable heat transfer resistances in series. In the heat transfer experiments with entrainment across the interface, the fluid pair was chosen to admit entrainment across the interface by the rising gas bubbles. Therefore, the overall heat transfer measured would be the sum of the contribution due to surface renewal processes in series on both sides of the TABLE I11 ENTRAINMENT HEATTRANSFER DATA FOR 10 CS SILICONE OIL/WATER

101 102 103 104 105 106 107 108 110 200 210 220 230 240 250

0.17 0.17 0.25 0.25 0.38 0.38 0.63 0.63 1.25 0.12 0.19 0.25 0.38 0.63 1 .00

2578 2490 3819 3953 7115 7490 27,344 28,646 75,300 2136 4143 7382 12,058 23,670 52,513

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

317

interface (stratified contribution) and the contribution due to mass transport of the lower, heavy liquid into the upper, light layer by the rising gas bubbles (entrainment contribution). The experiments utilized 10 CS silicone oil over water with gas bubbling from a porous frit installed in the test section base. The test apparatus was previously discussed in Section 11. The heat transfer coefficient that was measured in these tests was the net overall heat transfer coefficient, including both the surface renewal and entrainment components. The resulting experimental heat transfer data for the two series of tests are listed in Table 111. The heat transfer coefficient was then calculated by Eq. (30) and compared to the measured heat transfer coefficient, as shown in Fig. 15. The measured overall heat transfer coefficient is defined as the overall heat flux divided by the difference in bulk temperature between the two fluid layers. Equation (30) was found to be in good agreement with the experimental data over two orders of magnitude in measured heat transfer coefficients. I

I

I

I

I

I

l

l

]

0

10 cS Oil /Water 0 Series 100 0

Series 200

103 0.1

1

Superficial Gas Velocity ( crn/s)

FIG. 15. Measured overall heat transfer versus gas velocity.

GEORGE ALANSON GREENE

318

E. SUMMARY A framework for evaluating the total rate of heat transfer between overlying immiscible liquid layers with bubble-induced entrainment was presented. It was found to be a synthesis of surface renewal heat transfer and that due to direct heat transfer from the entrained drops. Models for surface renewal heat transfer, entrainment onset, and entrainment rate were presented and compared to available experimental data. The models and data were found to be in very good agreement. These models were then combined to construct the overall heat transfer model, Eq. (30). When the total interlayer heat transfer model was compared to available heat transfer data for bubbling pools of 10 CS silicone oil over water, the model and data, which both included the effects of entrainment, were found to be in very good agreement over two orders of magnitude in the measured total heat transfer coefficient.

VI. Heat Transfer from a Liquid Pool in Bubbly Flow to a Vertical Wall In a number of cases of bubbling of a noncondensable gas through an overlying liquid pool, it is desirable to know not only the rate of heat transfer between overlying immiscible liquid layers, but also heat transfer rates to the boundaries of the vessel. These structural boundaries include the base as well as the vertical side walls of the vessel. In this section, we shall consider the bubbling enhancement to the heat transfer from the bubbling liquid pool to the vertical side walls of the vessel. In the subsequent section, we shall consider bubbling-enhanced heat transfer to the bottom surface. A. PHYSICAL PHENOMENA Consider a liquid pool with an internal heat source that is held in a container. In the absence of gas bubbling, the heat transfer from the heated pool to its surroundings would be controlled by free convective processes driven by the temperature differences between the pool and its walls. In the free convection configuration, the magnitude of the boundary heat transfer coefficient is typically on the order of 100-500 W/m2 K. Boundary layers are stable under these conditions and may, in fact, be laminar over a considerable portion of the boundary surface. If a noncondensable gas is injected into the pool from below, a two-phase flow environment would be created. The rising gas bubbles would create severe circulation patterns and turbulence. Depending upon the rate of gas injection, the flow regime may be bubbly or churn-turbulent. The length and velocity scales of the turbulence in the pool would be functions of the

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

319

characteristic bubble size and gas injection velocity, respectively, particularly in the bubbly flow regime, which is characterized by distinct, rising bubbles. As the bubbles rose in the pool, they would interact with the wall boundary layer. The kinetic energy of the turbulent eddies would be orders of magnitude greater than that of the boundary layer flow per unit volume, resulting in destruction of the boundary layer processes. This could occur by two mechanisms: direct interaction of the bubbles with the wall boundary layer, and convection of turbulence from the bubbling core of the pool to the boundaries. It is expected that the heat flux from the bubbling pool to the boundaries would be considerably enhanced over the free convection case. The heat flux along the wall may be spatially uniform and cannot be predicted by free convection or single-phase flow considerations. This situation just described is what would be encountered in gas-sparged contactors, metallurgical furnaces, and molten core-concrete interactions.

B. PREVIOUS STUDIES There have been several previous studies concerning bubbling heat transfer to vertical surfaces. The first was an experimental study by Kolbel et al. (1958). They investigated bubbling heat transfer in a vertical cylinder to a cylindrical heat transfer element. They investigated eight fluids ranging in viscosity from 0.85 to 946.5 cS. They observed that if the viscosity was increased, the heat transfer coefficient was decreased. For water, in particular, they observed that for superficial gas velocities in the range 3-10 cm/sec, the heat transfer coefficient was in the range 4000-5000 W/m2 K. They accounted for the viscosity effect in a Reynolds number but neglected modeling the heat transfer as a function of a Prandtl number. They offered the following relations for heat transfer in terms of a Nusselt number:

Nu

= 43.7

Re > 150

Nu

= 22.4

Re < 150

for j , I10 cm/sec

(32)

The Reynolds and Nusselt numbers in Eq. (32) employ the diameter of the heating surface and the superficial gas velocity. Note, however, the absence of any Prandtl number dependence. Subsequently, Fairer al. (1962)reported on their study of heat transfer and gas holdup in a gas sparger. The technique they used was somewhat different than Kolbel’s. Air and water were injected into a cylindrical test section simultaneously. A heater was submerged in the flowing air-water pool and the enthalpy rise of the overflowing water was used to calibrate the electrical power input. Using the temperature difference between the electrical heating surface and the water, and the area of the heating element, they calculated a

320

GEORGE ALANSONGREENE

heat transfer coefficient.They found their data best correlated, in dimensional terms, as

h (W/m2 K) = 3300j:.22 (cm/sec)

(33) This dimensional relationship was found to be in good agreement with Kolbel’s water data, for superficial gas velocities in the range 0.3 to 15 cm/sec. Hart (1976) reported that the available literature on heat transfer from bubble-agitated systems was inadequate either due to the dimensional nature of the correlations developed or because the dimensionless representation neglected important parameters and variables. Hart used dimensional arguments to identify the dominant variables (superficial gas velocity, gravity, density, viscosity, specificheat, and thermal conductivity) that determine the heat transfer coefficient, and to reject the container dimensions as the significant length scale. He then proceeded by dimensional analysis to develop a general dimensionless structure for the heat transfer coefficient, St Pra = a(Re Fr)Y

(34) where St = h/pcJg and Re Fr = pji/pg. Experiments similar to those of Fair were performed with water and ethylene glycol. The temperature differences between the heater and the flowing water, along with the power and heater area, were used to calculate the heat transfer coefficient. Once again, dimensional trends in the data were used to determine the coefficients and exponents in Eq. (34). It was found that tl = 0.125, = 0.6, and y = -0.25. This formulation predicts h j:.25, which is in reasonable agreement with the trends reported by Kolbel et al. (1958) and Fair et al. (1962). Konsetov (1966) developed an analytical solution to the problem of heat transfer to a boundary during bubbling of gas through the pool. In his approach, Konsetov assumed that the pool was turbulent and that heat transfer was determined by the size and velocity of these turbulent eddies. He estimated the size of the turbulent eddies to be one-sixth the apparatus diameter, in accordance with some existing theories. Using the data of Kolbel, he arrived at the following dimensional relationship for the bubbling heat transfer coefficient to an apparatus wall:

-

s

h =0 . 2 5 k ( P r a g / ~ ~ ) ” ~ ( ~ / p , ) ~ ~ ’ ~ (35) where the term (p/pW)O.l4can be neglected for most practical applications. This relationship was later modified by Blottner (1979) to tailor it in a more general form for inclusion in CORCON. Blottner included a free convective term that would extend the applicability of Eq. (35) to the case of the free convective asymptote when the superficial gas velocity, and therefore the void fraction, was equal to zero. After changing the coefficient to improve comparison to the data of Hart (1976), Blottner arrived at the following di-

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

32 1

mensional relationship:

h = k(Pr g/~~)'/~(O.O0274P AT + 0.05~r)"~ However, it can be easily shown that for almost all cases with bubbling, the free convective term is negligible and Eq. (36) can be simplified to

h = 0.37k(Pr ug/v2)'I3

(37)

It is clear that the only difference between Eqs. (35) and (37) is in the magnitude of the coefficient, which reflects the data base preference of the authors. Neither Eq. (35) nor (37) has an explicit length scale in the formulation. However, the development of Konsetov did involve consideration of turbulent eddy size, so a length scale is implicitly considered. In the Konsetov and Blottner formulations the heat transfer coefficient is proportional to Because in bubbly flow, the void fraction is approximately linearly dependent upon j g , this is equivalent to j:l3. This compares relatively well with the observations of Kolbel et al. (1958), Fair et al. (1962), and Hart (1976). Significant questions remained unanswered concerning bubbling heat transfer from a liquid pool to a vertical boundary. Dependence upon bubble size has not been established. The dependence of the heat transfer coefficient upon the gas superficial velocity has been reported to vary from jyzto j:33. Some studies suggest a forced convective behavior (Reynolds number) whereas others suggest a buoyancy-type (Grashof number) or a mixedconvection dependence. Finally, the dependence upon the Prandtl number has yet to be established on a firm technical basis. In order to address these concerns and to develop a reliable model of bubbling heat transfer to a vertical boundary, the following experiment was performed.

C. EXPERIMENT A rectangular test vessel was fabricated out of 1-inch plexiglass stock and had the following inside dimensions: length = 16.9 cm, width = 15.3 cm, and depth = 41 cm, resulting in a horizontal planform cross-sectional area = 258 cm2. In the base was installed a porous plenum assembly through which the gas was injected into the pool. Submerged in the test pool was a horizontal, coiled calrod heater capable of dissipating nominally 2.5 kW into the pool fluid. It was installed in such a manner as to reside directly over the gas injection plenum. The gas was metered through a bank of rotameters and preheated to the pool temperature prior to injection. The calrod heater power was measured by a precision watt meter as well as by root-meansquare voltage and amperage. Discrepancies in calculated power were always negligible.

322

GEORGE ALANSON GREENE

A vertically oriented instrumented test wall was submerged in the pool. It was constructed out of micarta in a window-frame design. A 0.25-inch-thick copper plate was installed in such a manner that one surface was exposed to the bubbling pool, flush with the micarta frame, and the backside was exposed to a serpentine coolant passage through which water was circulated to cool the test plate. Two thermocouples were installed in the coolant inlet and outlet ports to measure the coolant enthalpy rise. A vertical rake of 10 thermocouples was installed along the pool centerline to measure the average pool temperature. Five microthermocouples were installed in the copper plate to perform precise local surface temperature measurements along the vertical axis of the test wall. The water pool depth would be set to cover the test wall (approximately 30 cm). The gas flux, pool heater, and test plate coolant would be set to specifications. The gas heater was energized and monitored in order to keep the pool temperature and gas temperature as equal as practical. The pool temperature was targeted to 41°C in order to keep the coolant temperature rise around 5°C. The gas flux was varied to cover a range of superficial gas velocities from 2 to 25 mm/sec. The size of the bubbles generated by the gas plenum was approximately 3 mm in equivalent spherical radius and was essentially constant over the range of the experiments. The test fluid was water and the water properties were evaluated at 41°C for all test runs. All tests were executed under steady-state conditions and each data point to be presented represents the average of five separate tests under nominally identical conditions.

D. RESULTS The modeling approach adopted by Konsetov (1966) and Blottner (1979) requires knowledge of the void fraction in order to correlate the heat transfer data, and a model to predict the void fraction in order to apply the heat transfer model. As a result, the void fraction was measured in the course of these tests. 1. Void Fraction Measurements

The pool-average void fraction was evaluated as a function of the superficial gas velocity as part of this investigation.The data were visually observed to all reside in the bubbly flow regime, an observation that is in accord with observations of transition to churn-turbulent flow in a pool with bottom gas injection at a superficial gas velocities of approximately 10 cm/sec. A variety of models have been developed to predict this pool-average void fraction. The one that will be compared to the data was reported by Blottner (1979). The

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

323

form of the void model developed by Blottner, assuming that the two-phase distribution parameter is equal to unity, is shown below:

+ 4)

a = 4/(1

(38)

where 4 is the dimensionless superficial gas velocity. The void data were compared to Eq. (38). The results of the comparison, shown in Fig. 16, were quite favorable. 2. Heat Transfer Measurements The measured temperature throughout the bubbling liquid pool was found to be spatially uniform. The variations in surface temperature along the plate were found to be small compared to the temperature difference from the pool to the plate, generally less than 0.1 K, within the range of the calibration uncertainty of the measurement system. The pool-to-plate temperature difference was generally on the order of 5 K. The wall heat transfer coefficient was calculated according to the following formula:

t Adiabatic air-water x Series 200 oil / watei

%,-

Equation 38

m egl

0

m

I

I

I IIIIII

I

I

I

I1111l

I

I

U

Superficial Gas Velocity ( cm / s )

FIG. 16. Average void fraction in bubbling pools with bottom gas injection.

324

GEORGE ALANSON GREENE

where the numerator represents the enthalpy rise of the test wall coolant and the temperature difference in the denominator represents the average temperature difference across the wall boundary layer. The measured heat transfer coefficientsfrom two series of tests are shown in Fig. 17 as a function of the superficial gas velocity. It was found that the heat transfer coefficient was a monotonically increasing function of the superficial gas velocity in the bubbly flow regime. The lowest superficial gas velocity at which data were taken was approximately 2 mm/sec. The wall heat transfer coefficient at this lowest superficial gas velocity was measured to be nearly 2000 W/m2 K, a factor of five or more greater than calculated for free convection on a vertical plate. This suggests that the bubbles very readily caused the pool to become turbulent, essentially at the onset of any bubbling, causing a discontinous large increase in the heat transfer coefficient, quite similar to the approach assumed by Konsetov (1966). The heat transfer coefficient was found to increase with increasing superficial gas velocity, and was measured to be approximately 5000 W/mZ K at a superficial gas velocity of only 2.36 cm/sec.

6

I

I

I

I

I

Series 1 0 Series 2

0

C

- 5 Y

cu

. E

2-

* c) .-0u

4

0

-

u L w

3

r

u)

e

I-

* m

2 2

1

0

0.5

1

1.5

2

2.5

3

Superficial Gas Velocity ( crn / s )

FIG.17. Bubbling heat transfer coefficients versus superficial gas velocity for a water pool to a vertical wall.

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

325

The trend of the heat transfer coefficient data was observed to be similar to that reported by Kolbel et al. (1958), Fair et al. (1962), and Hart (1976). Whereas Kolbel et al. and Fair et al. found the heat transfer coefficient proportional to j i . 2 2 ,and Hart found proportionality to jF5,the present data on the average were found best approximated as proportional to j F 3 3 , in agreement with the results of Konsetov (1966) and Blottner (1979).

E. MODELDEVELOPMENT A composite of all the heat transfer data from a bubbling pool to a vertical instrument is displayed in Fig. 18. Included in the figure are the data of Greene (1989), Kolbel et al. (1958), Hart (1976), and Fair et al. (1962). These data all demonstrate the trend of increasing heat transfer coefficient with increasing superficial gas velocity as previously observed. The large spread in the data for the heat transfer coefficient (200-5400 W/m2 K) is due to the large range

Kolbel ( 63% sucrose )

Greene ( water ) 15% sucrose ) 0 Kolbel ( 30% sucrose ) X Kolbel ( 41% sucrose ) A Kolbel ( 51% sucrose ) X Kolbel ( 56% sucrose ) A

x Kolbel ( 73% sucrose )

% Kolbel (

6000

I

I

A

5000

0 Hart ( water )

t Kolbel ( water ) X Fair ( water )

o

Hart ( glycol ) I

I

t

++

I

+

.

N

E

-

4000

c

c 0)

.-

3000 0 0 L

W

E

2000

2 + c

2 I

1000 0

0

0.02 0.04 0.06 0.08 0.1 Superficial Gas Velocity ( m / s )

0.12

FIG.18. Composite data for bubbling heat transfer coefficients versus superficial gas velocity: Prandtl number range 2.5-7500.

326

GEORGE ALANSON GREENE

of Prandtl number reflected by this data base, from 2.5 to 7500. It is desirable to develop a model that would be applicable over this range of Prandtl number. The present data will be compared to the analytical model of Konsetov (1966) (because it is the only model that was presented that had a theoretical basis) and the modified version reported by Blottner (1979). Neither the Konsetov nor the Blottner model is in dimensionless form. Introducing the Nusselt number and Grashof number as defined in the Nomenclature at the end of this article, it is straightforward to transform Eqs. (35) and (37) to dimensionless forms: Nu = 0.25 Gr:13 Pr 1/3

(354

and Nu = 0.37 Gr:13 Pr 1/3 In the application of either of these equations, the void fraction may be estimated from Eq. (38). All the heat transfer data displayed in Fig. 18 were nondimensionalized and correlated according to the same format just described for the Konsetov and Blottner models. The dimensionless model that was developed covers a range of Grashof number (Gr3) of seven orders of magnitude and is shown in Fig. 19. The dimensionless correlation that was developed is Nu = 0.37 Gr:13 Pr 'I3

lo2

FIG. 19. Dimensionless correlation of bubbling heat transfer data to a vertical wall.

(374

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

327

This equation is identical to the dimensionless form of Blottner’s (modified Konsetov) equation, and is applicable for fluids with Prandtl numbers greater than unity. F. SUMMARY Data were presented for heat transfer from a bubbling pool to a vertical wall, covering a range of Prandtl numbers from 2.5 to 7500. The data were nondimensionalized in a manner suggested by the analysis of Konsetov (1966). The resulting correlation was found to be a function of the Grashof number defined on the basis of void buoyancy (Gr3), and was identical in structure to the model developed by Blottner (1979), a modified form of the Konsetov model. The resulting dimensionless model was found to cover a range of modified Grashof numbers from to lo5.

VII. Heat Transfer from a Horizontal Bubbling Surface to an Overlying Pool In addition to the enhanced lateral heat transfer from a bubbling pool to vertical side walls, it is of interest to examine the effect of gas bubbling on the downward heat transfer from the bubbling pool to the base, when the gas bubbling is through the base. This is primarily of interest when considering hypothetical core melt accidents in nuclear reactors. If a reactor core becomes molten and penetrates the reactor vessel, it may attack the concrete basemat of the containment. The ablating concrete would release noncondensable gases that rise through the molten pool. It is important to know the heat transfer rate from the molten pool to the bubbling concrete surface in order to accurately calculate the rate of ablation of the concrete. A. BACKGROUND

Heat transfer to a gas-evolving surface from an overlying pool was initially modeled as if there were a stable gas film between the surface and the pool. This situation is similar to film boiling as shown by Benjamin (1979). In film boiling on a horizontal flat plate or heat transfer through a stable gas film, as in the sublimation of ice under a pool of water (Dhir et al., 1977; Reimann and Alsmeyer, 1982), the gas is released in a fixed geometrical pattern, where the relative bubble locations are governed by the Taylor instability theory. Recently, Bradley (1988) suggested that heat transfer from an overlying pool to an outgassing surface is distinctly different from simple gas film models. He found that the gas release from a concrete surface is less than required to form

328

GEORGE ALANSON GREENE

a gas film, and intermittent pool-concrete contact would occur. In this case, the heat transfer would be strongly dependent upon the liquid- solid contacts and the increased movement of liquid past the heat transfer surface caused by the bubbling action. Sims and Duffield (1971) attempted to correlate porous and drilled surface bubbling heat transfer with nucleate boiling to explain the hydrodynamic aspects of the heat transfer mechanisms in nucleate boiling. They found that when the latent heat transport is small, porous surfaces do imitate nucleate boiling, but that all their drilled surface data poorly modeled nucleate boiling, because “the number of bubble generating sites does not vary with the amount of gas leaving the heat transfer surface.” Kutateladze and Malenkov (1976) used several drilled heat transfer surfaces with varying hole densities, 4-400 sites/cm2. They showed that as the number of holes increased, the heat transfer coefficient of a drilled plate approached that of a porous surface, which in turn could be equated to nucleate boiling under a limited range of superficial gas velocities. A complicating factor in trying to understand the heat transfer mechanisms in bubbling is the interaction between bubbles as they leave the surface. When the bubbling sites are far enough away, the interbubble effects can be neglected. Without the interbubble interactions, only the effect of the bubbles jetting into the overlying pool on the surface heat transfer needs to be studied. An important question is how far away the bubbling sites have to be from each other in order to neglect interbubble effects. Bard and Leonard (1967) showed that the effect of bubbling from a single orifice on the local heat transfer decreases inversely with the distance from the orifice. They noted that for distances along a surface greater than 10 mm from the center of an orifice of 1 mm or less in diameter, the heat transfer was not affected by the presence of the bubble. This was confirmed by several nucleate boiling studies dealing with nucleation site densities on horizontal surfaces (Tien, 1962; Lienhard, 1963; Kurihara and Myers, 1960). Lienhard found that when the nucleation site density is less than approximately 0.32 sites/cm2, the sites cease to influence one another. This site density leads to a bubble influence area on the heat transfer surface, which, if assumed circular, has a radius of 10 mm, corroborating the observations of Bard and Leonard (1967). Therefore, if the bubbling sites are located far enough away from each other, the main contribution to the surface heat transfer is due to the movement of the fluid on that surface by the stirring action caused by the bubbles. Konsetov (1966) proposed that heat transfer from a surface in a bubbling pool is due primarily to the action of turbulent eddies. He stated that approximately one-third of the heat was removed by flow normal to the surface and the remaining heat by the parallel movement of the eddies. Interestingly, Konsetov (1966) made no mention about the orientation of the heat transfer

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

329

surface in the pool, whether the bubbles were emanating from the heat transfer surface or from another source, or about the specific shape of the surface. He developed a correlation using the data from two bubbling pool studies by Fair et al. (1962) and Kolbel et al. (1958). Both studies measured the heat transfer from vertical cylinders in deep pools and the bubbling came from a different source lower in the pool. The phrase “deep” means that the pool height is equal to or greater than its width. Using their data, Konsetov (1966) developed a correlation that accounted for the void fraction in the bubbling pool as a function of the superficial gas velocity and a range of sizes for the turbulent eddies. From data that covered a range of dynamic viscosity of over three orders of magnitude, he developed the following heat transfer relationship: h/k(~’/g)’/~ Pr-’/3(pW/p)0.’4= 0 . 2 5 ~ r ” ~

where ci is the void fraction. The void fraction was empirically determined by Kutateladze, as shown in Konsetov (1 966), to be = o.4[ji/3(sdC~l - P,])-”~(P~/P,)~~’~]

(41)

Konsetov evaluated Eq. (41) for an unspecified gas-liquid system and, upon substitution into Eq. (40), arrived at the following equation for the bubbling heat transfer coefficient: h (W/m’ K) = 0.2O[kj~.’’ Pr 1/3(P/~w)0.14(91v2)1~31

(42)

It should be pointed out that this formula is quite insensitive to variations in the properties in Eq. (41) and is applicable to a wide range of fluids. The correlation coefficient in Eq. (42) is not dimensionless; it has units of (m/sec)-0.22and requires the other variables to be of consistent units. In this correlation all properties are calculated at the pool temperature unless indicated otherwise. Duignan et al. (1990) examined the heat transfer rate from a horizontal bubbling surface when the bubble sites are “sparsely” located. A bubble site density of 0.1 1 sites/cm2 was chosen to approximate the spacing of bubbles generated by a Taylor instability mechanism. The interdependence of the bubbles should be minimized and the predominant heat transfer mechanism should be the stirring action created by the bubbling jets into the pool. This study is described in some detail below.

B. EXPERIMENT The apparatus consisted of two 6.35-mm-thick stainless-steel plates that sandwiched a flexible electrical heater in a horizontal position. A vertical

3 30

GEORGE ALANSON GREENE

quartz tube of 101.6 mm inside diameter was mounted over the top plate and was filled to varying heights of water. The pool height was adjusted by changing the height of a makeup reservoir, which was maintained at the pool temperature. The two steel plates had nine 1-mm holes that were drilled in a square geometry with the closest interhole spacing being 27.2 mm, to conform with previously assumed bubble locations (Benjamin, 1979). These drilled holes were fed by a gas-filled plenum. The entire apparatus was insulated so that heat flow radially outward and downward was negligible in comparison to the upward flow through the bubbling pool. The upper heat transfer plate had twelve type-K thermocouples. They were positioned and installed in order to determine the average heat flux through the plate and the average surface temperature. After the gas flow was established through the apparatus, it was filled with water to a predetermined height. The heating coil maintained a constant heat flux through the heat transfer plate of approximately 26 kW/m2 for all the experimental runs.

C. RESULTS Figure 20 shows the deep-pool data of Duignan et al. (1990) over the entire range of superficial gas velocities. Over that range, the heat transfer coefficient varied only about 20%, demonstrating only a small dependence of the heat transfer coefficient on the superficial gas velocity. These data are for pool heights greater than the pool diameter, which was 10.2 cm. Included in Fig. 20

I

-

I

I

I

I

Experimental Data 0 Kolbel __ Konsetov ( modified ) ._____ Fair

I

0

Y

.8 -

"E

2

-

Z 6-

-

01 .-c 0

0

.-

i=,

25 4 L

01

c

r E c

I*

-

--

c 9 '

-

2-

01 m

1

0

I

I

I

I

Superficial Gas Velocity ( cm / s )

F~G. 20. Heat transfer coefficient from a horizontal bubbling surface versus superficial gas velocity.

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

33 1

are the water pool data of Kolbel et al. (1958) and the correlation of Fair et a/. (1962). Even though these studies involved heat transfer surfaces that were neither flat, horizontal, nor the source of the bubbling, as was the case in the Duignan’s study, their data agree reasonably well with Duignan’s measurements in both magnitude and trend. Konsetov’s (1966) turbulent heat transfer model seems to predict Duignan’s heat transfer data. The correlation coefficient in Eq. (42) was recalculated by Duignan for his data and was found to be 0.28. As previously mentioned, the constant that Konsetov obtained was 0.20. This observation is consistent with the observation of Greene (1989),who found that, on the average, the measured bubbling heat transfer coefficient from a water pool to a vertical boundary exceeded the Konsetov model prediction by a similar amount. An interesting result shown in Fig. 20 is that for superficial gas velocities greater than 0.6 cm/sec, the heat transfer from a solid surface to a liquid pool is neither dependent on the orientation of the heat transfer surface (horizontal or vertical) nor on the location of the bubbling source; that is, it is unimportant whether the bubbles pass through the heat transfer surface or come from some other, lower, surface. These results support the following observations: the bubbles that are leaving the heat transfer surface do not contribute to the local heat transfer when they are “sparsely” located, the role of the superficial gas velocity is to maintain the liquid pool turbulence, and the pool is deep enough to not affect the turbulent structure in the pool. The first observation is substantiated by a criterion developed by Kutateladze and Styrikovich, as found in Wallis (1969). They showed that when the gas leaving an orifice exceeds a critical velocity, it no longer forms bubbles at the orifice but forms a jet that breaks into bubbles later in the pool. Applying the Kutateladze and Styrikovich criterion to the present tests predicts a superficial gas velocity of approximately 2.8 cm/sec in the present apparatus for transition from discrete bubbles to jet flow from the drilled orifices. Because there was no discernible change in the heat transfer coefficient at that gas velocity, this suggests that the bubbles that form and break off from an orifice do not control the heat transfer from that surface when they are located far enough away from each other. The second observation is supported by the result that the heat transfer coefficient is not strongly dependent upon the superficial gas velocity. It is observed that for j , greater than 2 cm/sec the data are bounded by Konsetov’s turbulent pool model, which would suggest that the liquid pool was turbulent. The third observation was investigated by varying the pool height. The Konsetov model assumes that the characteristic length scale, the average size of a turbulent eddy, is a constant; he assumed it only to be a function of the inside diameter of the pool. Figure 21 shows that the heat transfer coefficient was not affected by changing the pool heights as long as the height was at least

GEORGE ALANSON GREENE

332 1

I

1

1

I

I

I

2

4

6

8

10

12

14

16

Pool Height ( ern ) FIG.21. The effect of pool height on the heat transfer coefficient from a horizontal bubbling surface at a constant superficial gas velocity.

60% of the diameter. Below that level, the heat transfer coefficient appeared to be directly proportional to the pool height. This decrease in heat transfer may have been caused by a reduction in the average turbulent eddy size, a reduction in eddy motion due to the smaller pool, or a combination of both effects. Heat transfer measurements could not be made accurately for pool heights lower than approximately 2.5 cm because of the difficulty in maintaining a constant liquid height and the violent splashing of the liquid from the bubbling pool. The dependence of the heat transfer coefficient on the pool height, as shown in Fig. 21, was measured at a constant superficial gas velocity of 6.4 cm/sec. This dependence was also investigated at lower gas velocities. Although the data base for the lower superficial gas runs was much smaller and had considerably more scatter than the data in Fig. 21, the heat transfer coefficient was found to behave similarly. D. SUMMARY For a bubble site density of 0.11 sites/cm2, the overall heat transfer mechanism in pool bubbling was not affected by the formation of the bubbles at the surface. The heat transfer appeared to be a function of the turbulence generated by the bubble action in the pool. The Konsetov turbulent heat transfer model, developed for a heat transfer surface in a bubbling pool in which the bubbles come from below the surface, predicted the heat transfer with bubbling through the surface, as long as the bubbling sites were sparsely

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

333

located. For the present data, changing the coefficient in Eq. (42) to 0.28 gave a best fit to the data. Heat transfer in a bubbling pool has been found to be a function of the pool height if the pool height is lower than 60% of the pool diameter. This is rigorously true for the study of Duignan et al. (1990), which utilized a pool 10.2 cm in diameter with a bubbling site density of 0.1 1 site/cm2. For pool heights greater than 60% of the diameter, the heat transfer coefficient (at a fixed superficial gas velocity) is a constant. For a deep pool (pool height greater than 60% of the vessel diameter), the heat transfer coefficient is a weak function of the superficial gas velocity, for j , greater than 0.6 cm/sec.

VIII. Drag and Instability of Liquid Droplets Settling in a Continuous Fluid To determine the configuration of multifluid pools with bubbling-induced mass entrainment across the liquid-liquid interface, it is necessary to address the motion of the entrained liquid drops as they resettle downward under the action of gravity. In the absence of entrainment, the pool will remain stratified as seen in Fig. 14a. If the rate of settling of the entrained drops can balance the rate of entrainment driven by the rising gas bubbles, the pool can take on an equilibrium mixture, as seen in Fig. 14b. If the rate of settling cannot keep pace with the rate of entrainment, the lower fluid will continue to be pumped into the upper layer until the pool would become fully mixed and the concept of layers becomes irrelevant (Fig. 14c). In this section we will examine the drag characteristics of solid spheres and liquid droplets. Studies of the relationship between the terminal velocity and drag coefficient of spheres falling in a semiinfinite medium have been numerous. The broad scope of applications for such research, from the analysis of particle motion in fluids to the study of phenomena such as rainfall and boiling, is a clear indication as to the importance of this field of study. A review of the literature indicates that there exists a substantial body of work devoted to the study of both solid spheres and liquid droplets. We will discuss the existing data and the data from our studies over a wide range of Reynolds numbers, and will compare the data with the predictions of existing drag correlations. In addition, the onset of droplet deformation and oscillation will be discussed and the effect upon the droplet drag will be demonstrated. A. BACKGROUND

The construction of a force balance on a sphere moving at a constant terminal velocity, u, results in the following expression for the drag

GEORGE ALANSON GREENE

334 coefficient:

which is usually presented as a function of the Reynolds number, defined as,

(44)

Re = udd/v, These relationships will be needed in the following discussion. 1. Solid Spheres

A number of investigators have undertaken to develop a general correlation between the drag coefficient and Reynolds number for solid spheres. Barnea and Mizrahi (1973) studied single-particle and multiparticle fluidization and sedimentation systems, and synthesized data from 12 different sources. They proposed the following simplified correlation: cd

= (0.63

+ 4.8/fie)2

10-3 c Re c 2 x 105

(45)

Zanker (1980) used their work as the basis for the development of nomographs for determining particle settling velocities. Turton and Levenspiel (1986), in order to simplify a 10-polynomial regression correlation that had been proposed by Clift et al. (1978), developed the following relationship: cd

24 Re

= -[1

+ 0.173

0.413 + 1 + 16300Re-’.09’

10 c Re c 2 x lo5

(46)

which was found to correlate the existing data with better accuracy than the cumbersome set of equations suggested by Clift. In addition, Flemmer and Banks (1986), modifying the traditional Oseen (1927) approximation using a regression technique, proposed the following empirical correlation for a settling sphere: where E = 0.261

- 0.105

- 0.124/[1

+ (log Re)2]

(48)

2. Liquid Droplets A number of authors have examined the motion of liquid droplets falling in liquid-filled test columns. Hu and Kintner (1955) studied droplets of 10 organic fluids falling in a square, water-filled column whose temperature was

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

335

maintained to within +0.5"C. The study covered a Reynolds number range of 10-2000. The authors observed that there was little evidence of surface flow or internal circulation effects below a Reynolds number of 300,since the c d vs. Re curve in this range was nearly identical to that for solid spheres. Beyond this Reynolds number range, droplet behavior departed from the predictions of solid-sphere theory. A peak diameter corresponding to the maximum droplet velocity was observed in each system studied, and it was found that droplet drag increased rapidly if the Reynolds number were to exceed the critical Reynolds number based on this peak diameter and velocity. The authors calculated the Weber number based on these values for peak diameter and velocity for each system, and presented the following average critical value, Wecri,= 3.58, where the Weber number is defined as, We = p,u:d/o

(49)

and o is the interfacial tension between the droplet and the continuous liquid. They concluded that droplet oscillation induced at this peak velocity was the primary cause for this rapid increase in droplet drag, and that droplet deformation was secondary. The authors proposed an empirical dimensionless physical property group, p , defined as 3 Re4 p=and found that the data from all but one of their fluid systems collapsed to one curve when presented in the following format: (CdWe PO"')

vs.

(Relp'.'')

(51)

Winnikow and Chao (1986) performed a similar study examining droplets of four organic liquids over a Reynolds number range of 100-1000. The experiments were performed at room temperature in a 4-in.-square waterfilled test column. Similar to Hu and Kintner (1959, the authors identified three distinct regions in the drag coefficient vs. Reynolds number relationship. For Reynolds numbers less than 1.0, the liquid droplets were observed to maintain their spherical shape, and the data for each system were found to be in excellent agreement with the Stokes solution for solid spheres. A transition region was indicated for 80 < Re < 300, in which droplet deformation became apparent and the measured drag coefficients were higher or lower (depending on the particular fluid involved) than those for solid spheres of equivalent diameter. A third region, beginning at Reynolds numbers between 700 and 900, marked the onset of droplet oscillation and a sharp increase in droplet drag with a slight increase in the Reynolds numbers. The authors noted that the minimum drag and maximum droplet terminal velocity coincided with this onset of oscillation, as had been found by Hu and Kintner.

336

GEORGE ALANSON GREENE

They hypothesized that this increase in droplet drag was due to increased pressure drag resulting from droplet deformation and a shift in the position of boundary layer separation. The continued increase in droplet drag was attributed to the combined effect of deformation-induced form drag and droplet oscillation. They proposed an average critical Weber number of 4.08 as an indication of the onset of liquid droplet oscillation. This compared favorably with the value of 3.58 proposed by Hu and Kintner. Beard and Pruppacher (1969) studied water droplets injected into a wind tunnel with a hypodermic needle. The air speed in the tunnel was adjusted until droplets of a certain size became suspended, at which point the droplet terminal velocity and tunnel air speed were equal. Droplet diameters were determined by photographic analysis. The authors' study covered a Reynolds number range of 0.2-200, and they proposed the following correlations for determining droplet drag as a function of Reynolds number: 1 + 0.102Re0.955,

DID, = 1 + 0.1 15 1

+ 0.189

0.2 < Re < 2.0 2.0 < Re < 21.0

(52)

21.0 < Re < 200.0

where D is the measured droplet drag and Ds is the Stokes drag for a solid sphere at the same Reynolds number. The results agreed quite well with those of prior studies involving both solid spheres and liquid droplets over a comparable Reynolds number range. For Reynolds numbers greater than 200, the droplet drag was observed to increase due to distortions from the spherical shape. As a result these equations were not recommended for Reynolds numbers greater than 200 nor for the case of droplets that are experiencing distortion or oscillation, whichever comes first. Klee and Treybal (1956) examined both rising and falling droplets for 11 fluids in a square-test column. The study covered a Reynolds number range from 1 to 2000. The authors identified two distinct regions in plotting the droplet velocity vs. diameter, and noted that the maximum point in each curve corresponded to the point at which droplet oscillation developed. Droplet drag coefficients were consistently found to be smaller than the values predicted for solid spheres, a phenomenon which the authors attributed to the oscillation and internal circulation present in a droplet. The interfacial tension between the dispersed and continuous phase liquids was found to be a controlling parameter for droplet stability. The authors presented the following equations relating Re, C,, and We: Re = 22.2C65.'8We-0.'69 Re = 0.00418C,2.91We-'.''

(Region I) (Region 11)

(53) (54)

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

337

where I and I1 refer to the regions of observed droplet stability and oscillation, respectively. The critical Weber number at which the droplets deviated from solid-sphere drag was found to be, on the average, approximately equal to 8.21. This result was in good agreement with the observations of Hu and Kintner and Winnikow and Chao, who reported average values of the critical Weber number of 3.58 and 4.08, respectively (for organic droplets settling in water). Once again, however, the range of the critical Reynolds number over which the droplet drag coefficient was observed to deviate from solid sphere data was small, in the range 300-600. Similar studies were performed by Licht and Narasimhamurty (1955) and by Krishna et al. (1959), who studied liquid droplets of 6 fluids and 31 fluids, respectively. The droplets were of various organic liquids and the continuous fluid was once again water, just as in the studies of Klee, Winnikow, and Hu that were discussed previously. (Recall that in the study of Beard, the droplet was water and the continuous fluid was air.) In both these studies, the measured droplet drag coefficient was observed to conform closely to the established theory for solid spheres at low Reynolds numbers. Departure from solid-sphere drag behavior once again occurred in a narrow range of the droplet Reynolds number; the critical Reynolds number reported for the drag deviation was in the range of 600-1000 for Licht and 500-1000 for Krishna. For Krishna’s data, the critical Weber number for the departure of drag from solid-sphere drag (an average of 31 data points) was found to be 4.04, in remarkable agreement with the result of Winnikow and Chao (1966) of 4.08. Closer examination, however, suggests that this good agreement, particularly between the results of Hu and Kintner (1955), Winnikow and Chao (1966), and Krishna et al. (1959),may be a result of the narrow range of variation in physical properties of the fluids used in these three studies, i.e., all three used very similar droplet fluids settling in a water column. As a result of this limitation, it is suspected that the critical Reynolds and Weber numbers observed, and the correlations developed on the basic of these data, may be applicable only to the narrow category of organic liquid drops settling in water. B. EXPERIMENT

Droplet settling studies were conducted with eight different pairs of fluids in order to further examine the droplet drag- Reynolds number functional relationship. Liquids utilized for the droplet phase included water, water/ CuSO,, bromoform, and mercury; liquids utilized for the continuous phase included 10 and 100 CSsilicone oils, hexane, and water. The droplet volume was varied over the range 1 pl to 6 ml; combined with the variations in liquid

338

GEORGE ALANSON GREENE

properties, this resulted in a data base for the droplet drag coefficient covering seven orders of magnitude in Reynolds number, from to lo4. In order to measure the terminal velocities of the droplets, droplets of precise volumes were placed in a Teflon cup at the top of the liquid-filled test column. The descent of the droplet was electronically timed as it traversed a fixed course (1 m) inscribed on the column's outer surface. The time of flight for 20 separate trials for each droplet volume was then averaged, and this average time was used to calculate an average terminal velocity. Precise measurements were made of the physical properties for each of the fluids studied over a temperature range appropriate to the droplet settling data (20-40°C). Temperature-dependent values of density, interfacial tension, and viscosity were used to calculate the Reynolds number, Weber number, and the dropletdrag coefficient.

C. EXPERIMENTAL RESULTS The experimentally measured drag coefficient data for four of the droplet systems are illustrated in Fig. 22. These droplets were observed to remain spherical during the fall period, and the measured drag coefficients were found to agree well with established models for solid sphere drag; shown in Fig. 22 are the Stokes flow model for Reynolds numbers less than 0.1 and the correlations of Beard and Pruppacher (1969) for Reynolds numbers in the range 0.2-200 [Eq. (52)], shown here extrapolated to Re = 1OOO. These data

10 0

lo4

100 cS Oil / CuS04

A 10 cSOil/ H20

c W

0 100cSOil/H20

c

:3 l o 3

5, 0 0

0 lo2 e

n

1 1o-'t-u"1"1I

10.~

I 1"1111'

lo-*

I i"lld

10.'

I 1111'111

11111111 I

1 10 Reynolds Number

lo2

I

lo3

ulud lo4

FIG.22. Drag coefficients versus Reynolds number for stable liquid drops settling in a liquid column.

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

339

are in excellent agreement with the droplet drag data of Hu and Kintner (1955), Licht and Narasimhamurty (1955), and Krishna et al. (1959) for Reynolds numbers below the values of the critical Reynolds number for droplet oscillation observed in each of these studies. Previous studies have shown that, at the onset of droplet oscillation, the droplet drag suddenly departs from the solid-sphere drag curve (this will be called the critical drag coefficient)and increases rapidly with a further increase in the Reynolds number. This Reynolds number has been called the critical Reynolds number, and was measured by previous investigators to lie in the range 500- 1000. For nearly all these data, the corresponding critical Weber number for the onset of this phenomenon was measured to be approximately equal to four. The data for seven series of experiments from the present study are shown in Fig. 23. These data were intended to cover the widest possible range of Reynolds number and Weber number, in order to test the observations of other investigators concerning the critical Reynolds number and Weber number for the onset of oscillation and drag enhancement. As is shown in the figure, the data for five of the seven fluid pairs followed the stable drag curve of Beard and Pruppacher (1969) over a range of Reynolds number 0.2- 1000. However, depending upon the fluid pair tested, deviation from the Beard model was observed to suddenly begin at Reynolds numbers as low as eight and as high as 1000. As shown in Fig. 23, six of the seven fluid pairs tested experienced this departure from stable drag. Interestingly, no

10" F

---

I I 111111'

I I 1111111

2

10 E

-

P

0 % I

0 0

g

0

OO

-

-

n

e

D

1:

-

-

-

-

10.'

--

-

Beard 10 = -

c a l

'

100cS011/H~O + 100 CS Oil / Bromoforrn HLO/Brornoforrn = Y Mercury I10 cS Oil 5 x Hexane / H 2 0 0

c

I I li'll1' I I 1111111 I A 10 cSOil / H 2 0

I 1111111'

I 11111111

I 111'111'

I 11111111

GEORGE ALANSON GREENE

340

droplet system tested in the present study, nor from any of the previous studies cited, were able to exceed a Reynolds number of 1000 without experiencing drag departure from the stable drag curve. In previous studies, it has been reported that the point of droplet oscillation and drag enhancement coincided with a value of the droplet Weber number approximately equal to four. However, in the present study, the critical Weber number was found to cover a much wider range than previously reported, a range of 2- 50. The relationship between the critical droplet Reynolds number and critical Weber number for the onset of droplet oscillation and departure from stable drag is shown in Fig. 24. It is apparent from this figure that nearly all the previous data experienced drag enhancement at a Weber number of four and in a range of Reynolds number of 400-1500. The present data, however, demonstrated this behavior for Weber numbers over the range 2- 50 and Reynolds numbers from less than 10 to greater than 1000. From the present data, a functional relationship between the Reynolds number and Weber number at this point of departure from stable drag was calculated as shown below:

Wecri,Re:!: = 165

(55)

It is clear that for values of WeRe0.65 less than 165, a particular dropletcontinuous liquid pair would be stable, and the droplet drag would be well characterized by any of the solid-sphere drag models. As the critical value of We is approached, the droplet would begin to experience oscillation and deformation. For values greater than 165, extensive droplet oscillation

J

A

x +

1

PRESENT DATA Hu & Kintner Winnikow & Chao Krishna et al I

1

1

I

I

~3 I

, , , I

I

,

I

10 Weber Nurnbei

FIG.24. Critical droplet Reynolds number versus critical Weber number for onset of droplet oscillation and deviation from stable drag.

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

34 1

and deformation would result in an increase in the droplet drag far above that for an equivalent spherical drop as the Reynolds number were to increase. Liquid droplets for which the value of WeRe0.65 exceeds 165 will have deviated from the stable drag curve and the trend of the droplet drag coefficient will be to increase with increasing Reynolds number. The data from the present study that evidenced this behavior as shown in Fig. 23 are those that depart from the Beard line and move up and to the right. As shown in the figure, this behavior was found to occur for Reynolds numbers as low as eight. It is evident that many droplet systems may experience this drag enhancement and that for practical applications, the unstable drag regime, described by We greater than 165, may be the dominant regime. In order to develop an empirical model for the data in this regime (for WeRe0.65 > 165), the droplet drag coefficients and droplet Reynolds numbers for these data were normalized by the corresponding critical values for departure from stable drag which were previously determined and shown in Fig. 24. From the normalized data, a functional empirical relationship between the droplet drag coefficients and Reynolds numbers was calculated by least squares regression analysis and is given by

G/G,crit = CRe/RecritI

(56)

The normalized data are shown along with Eq. (56) in Fig. 25. This is the drag relationship that is appropriate for droplet systems for which We exceeds 165.

H20 I Bromoform 10

+ 100 CSOil I Bromoform A

= u

10 CSoll/ H20 10 cS011I Mercury

. d

0

D

0

X

FIG.25. Correlation of unstable drag data.

342

GEORGE ALANSON GREENE

D. SUMMARY Date for droplet drag have been found to agree well with established solidsphere or spherical-droplet models and correlations as long as the droplets maintained their spherical shape while settling under the influence of gravity. At the Reynolds and Weber numbers indicative of the onset of droplet oscillation and deformation, the drag coefficients for the droplets were observed to suddenly depart from the solid-sphere drag curve and increase with increasing Reynolds number. This sudden departure from classical drag has been found to occur for Reynolds and Weber numbers over the ranges 10-loo0 and 2-50, respectively. No stable drag data were found for Reynolds numbers greater than approximately 1000. It was found that the parameter, We at this critical point was approximately 165. For values less than this criterion, the droplet drag was stable and agreed well with spherical drop models; for values greater than this criterion, the droplet drag departed from stable drag models and the drag coefficient increased with increasing Reynolds number. When the droplet drag coefficients and Reynolds numbers in the unstable drag regime were normalized by the corresponding critical values for the departure from stable drag, the normalized droplet drag coefficient was found to correlate well to the normalized droplet Reynolds number.

IX. Concluding Remarks This article has been prepared with the intention of presenting an introduction to the problems posed by heat, mass, and momentum transfer in multifluid bubbling pools. A survey of contributions in each of the seven technical areas that were discussed has been presented, along with a more indepth discussion of the investigations undertaken by the author and coworkers. It is clear from this review that this field of heat, mass, and momentum transfer in multifluid bubbling pools has not been the subject of extensive research, and may represent a fertile research area for those who find its remaining problems interesting. Nevertheless, the existing contributions have been significant, and a comprehensive set of models has been developed that, when applied to the analysis of multifluid bubbling pools, yields reliable results. At this time, the remaining uncertainties present an immediate opportunity for research by investigators who are so inclined. One issue that needs to be explored is the effect of widespread bubbling upon the entrainment onset and entrainment rate models. You recall these models were based upon data and analyses of single-bubble systems. A similar situation exists for the study of droplet drag and settling and the onset of droplet instability; the work

HEAT,MASS,AND MOMENTUM TRANSFER IN FLUIDS

343

that was reported here was similarly based upon single-droplet studies. The study of bubbling heat transfer to vertical boundaries has an impressive data base, covering a range of Prandtl numbers from 2.5 to 7500. However, there are no data for bubbling heat transfer from liquid metal pools (low Prandtl numbers). This same situation exists for horizontal bubbling surfaces as well; all the data are for water. The author intends to investigate liquid metal pools in the near future. Finally, nearly all the studies presented represented singlebubble or droplet systems or pools in the bubbly flow regime. Little is known concerning the applicability of these models at higher gas injection rates, rates at which the bubbling pools could be in churn-turbulent flow. NOMENCLATURE A CP

Cd

d db

Fr 9

G Gr, Gr2 Gr3 h JC

j, k

K Nu Pr 4 rb

radius Reynolds number (= ud/v) superficial Reynolds number

test plate surface area specific heat drag coefficient equivalent spherical diameter spherical equivalent bubble diameter Froude number gravitational acceleration coolant mass flux Grashof number ( = y / l A T r i / v 2 ) Grashof number ( = y a 2 r i / v 2 ) Grashof number ( = g a r z / v 2 ) heat transfer coefficient superficial entrainment flux superficial gas velocity thermal conductivity droplet heat transfer efficiency Nusselt number (= hr,/k) Prandtl number heat flux spherical equivalent bubble radius

(=Jk!rb/V)

entrainment Reynolds number (=jJ,/v2)

bubble Reynolds number in layer 1 or 2 (= ubd,/v) Stanton number shape factor for gas bubble temperature velocity bubble rise velocity bubble terminal rise velocity volume entrainment onset bubble volume minimum bubble penetration volume maximum theoretical entrainment volume Weber number ( = pu2d/cr)

Greek Symbols a

B B E

P

P* P

void fraction, thermal diffusivity coefficient of thermal expansion column contact angle entrainment efficiency density density ratio dynamic viscosity

CT

0 12

V

w

Q Q

surface tension interfacial tension kinematic viscosity dimensionless bubble volume dimensionless excess bubble volume dimensionless superficial gas velocity (=j,/Urn)

Subscripts

b crit

bubble critical value

d e

droplet entrainment

344 g i

in I m out plate

GEORGE ALANSON GREENE gas side of interface (i = 1 or 2) coolant inlet liquid maximum coolant outlet test plate average

pool SR SURF T W

1 2

liquid pool average surface renewal surface total wall upper light liquid lower heavy liquid

REFERENCES Bard, Y.,and Leonard, E. F. (1967).Heat transfer in simulated boiling. Int. J. Heat Mass Transfer 10, 1727. Barnea, E., and Mizrahi, J. (1973).General correlation for fluidization and sedimentation. Chem. Eng. J . 5, 171. Beard, K. V., and Pruppacher, H. R. (1969).A determination of the terminal velocity and drag of small water drops by means of a wind tunnel. J. Atmos. Sci. 26, 1066. Benjamin, A. S. (1979).Core-concrete interfacial heat transfer and molten pool dynamics. Trans. Am. Nucl. SOC.33,527. Blottner, F. G. (1979). Hydrodynamics and heat transfer characteristics of liquid pools with bubble agitation. U.S. Nucl. Regul. Comm. NUREG/CR-0944. Bradley, D. R. (1988). Modelling of heat transfer between core debris and concrete. Natl. Heat Transfer Conf., ANS Proc. 3, 37. Cheung, F. B., Leinweber, G., and Pedersen, D. R. (1986). Bubble-induced mixing of two horizontal liquid layers with non-uniform gas injection at the bottom. Proc. I f . Exchange Meet. Debris Coolab., 6th EPRI NP-4455. Clift, R., Grace, J. R., and Weber, M. E. (1978). “Bubbles, Drops, and Particles.” Academic Press, New York. Dhir, V. K., Castle, J. N., and Catton, I. (1977). Role of Taylor instability on sublimation of a horizontal slab of dry ice. J. Heat Transfer 99,411. Duignan, M. R., Greene, G. A., and Irvine, T. F., Jr. (1990). Heat transfer from a horizontal bubbling surface to an overlying water pool. Chem. Eng. Commun. 87, 185. Epstein, M., Petrie, D. J., Linehan, J. H., Lambert,G. A., and Cho, D. M. (1981).Incipient stratification and mixing in aerated liquid-liquid or liquid-solid mixtures. Chem. Eng. Sci. 36,784. Fair, J. R., Lambright, A. J., and Andersen, J. W. (1962).Heat transfer and gas holdup in a sparged contactor. Ind. Eng. Chem., Process Des. Dev. 1, 33. Flemmer, R. L. C., and Banks, C. L. (1986). On the drag coefficient of a sphere. Power Technol. 48,217. Gonzalez, F., and Corradini, M. (1987). Experimental study of pool entrainment and mixing between two immiscible liquids with gas injection. Proc. CSNI Specialists’ Meet. Core Debris-Concr. Interact. EPRI NP-5054-SR. Greene, G. A. (1989). Heat transfer from a liquid pool in the bubbly flow regime to a vertical boundary. AIChE Symp. Ser. No. 269,223. Greene, G. A., and Irvine, T. F., Jr. (1988).Heat transfer between stratified immiscibleliquid layers driven by gas bubbling across the interface. Natl. Heat Transfw Conf., ANS Proc. 3, 31. Greene, G. A., and Schwarz, C. E. (1982). An approximate model for calculating overall heat transfer between overlying immiscible liquid layers with bubble induced liquid entrainment. Proc. Post Accid. Heat Removal lnf. Exch. Meet., Sth, Karlsruhe, F.R.G. I, 251. Greene, G. A., Jones, 0.C., Jr., and Abuaf, N. (1980).Correlation of local heat flux from inclined volume-heated pools in bubbly flow. Proc. Natl. Heat Transfer Conf., ASME Paper NO. 80-HT-91.

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Greene, G . A,, Chen, J. C., and Conlin, M. T. (1988). Onset of entrainment between immiscible liquid layers due to rising gas bubbles. Int. J . Heat Mass Transfer 31, 1309. Greene, G. A,, Chen, J. C., and Irvine, T. F., Jr. (1990a).Heat transfer between overlying immiscible liquids due to bubbling-induced mass entrainment across the interface. Proc. Int. Heat Transfer Conf., 9th, Jerusalem 3,467. Greene, G. A., Chen, J. C., and Conlin, M. T. (1990b).Bubble-induced entrainment between stratified liquid layers. Inr. J. Hear Mass Transfer 34, 149. Hart, W. F. (1976). Heat transfer in bubble-agitated systems: A general correlation. Jnd. Eng. Chem., Process Des. Den 15, 109. Hu, S., and Kintner, R. C. (1955). The fall of single liquid drops through water. AIChE J . 1,42. Klee, A., and Treybal, R. E. (1956).Rate of rise or fall of liquid drops. AIChE J . 2,444. Kolbel, H., Siemes,W., Maas, R., and Muller, K. (1958).Warmeubergang and Blasensaulen. Chem. Ing. Tech. 30,400. Konsetov. V. V. (1966). Heat transfer during bubbling of gas through liquid. Int. J . Heat Mass Transfer 9, 1103. Krishna, P. M., Venkateswarlu, D., and Narasimhamurty, G. S. R. (1959).Fall of liquid drops in water: Drag coefficients, peak velocities, and maximum drop sizes. J . Chem. Eng. Data 4,340. Kurihara, H. M., and Myers, J. E. (1960). The effects of superheat and surface roughness on boiling coefficients. AIChE J . 6, 83. Kutateladze, S. S., and Malenkov, 1. G. (1976).Fluid- and gas dynamical aspects of heat transfer in the injection bubbling and boiling of liquids. High Temp. (Engl. Transl.) 14,703. Lee, M., and Kazimi, M. S. (1984). Interfacial heat transfer between bubble agitated immiscible liquid layers. Proc. I f . Exch. Meet. Debris Coolab.. 6th. EPRI NP-4455. Licht, W., and Narasimhamurty, G. S. R. (1955). Rate of fall of single liquid droplets. AIChE J. 1, 366. Lienhard, J. H. (1963). A semi-rational nucleate boiling heat flux correlation. Int. J . Heat Mass Transfer 6,215. Mercier, J. L., da Cunha, F. M., Teixeira, J. C., and Scofield, M. P. (1 974). Influence of enveloping water layer on the rise of air bubbles in Newtonian fluids. J . Appl. Mech. %, 29. Mori, Y. H., Komotori, K., Higeta, K., and Inada, J. (1977).Rising behavior of air bubbles in superposed liquid layers. Can. J. Chem. Eng. J5,9. Oseen, C. W. (1927). “Hydrodynamik.” Akad. Verlagsges., Leipzig. Poggi, D., Minto, R., and Davenport, W. G . (1969). Mechanisms of metal entrapment in slag. J . Met. 21, 40. Porter, W. F., Richardson, F. E., and Subramanian, K. N. (1966). Some studies of mass transfer across interfaces agitated by bubbles. In “Heat and Mass Transfer in Process Metallurgy” (A. W. D. Hills, ed.), Chap. 3. Inst. Min. Metall., London. Reimann, M., and Alsmeyer, H. (1982). Hydrodynamics and heat transfer processes of dry ice slabs sublimating in liquid pools. Proc. Int. Heat Transfer Conf., 7th. Munich 4, 167. Sims, G. E., and Duffield, P. L. (1971).Comparison of heat-transfer coefficients in pool barbotage and saturated pool boiling. Trans. Can. Soc. Mech. Eng. 14, 1. Suo-Anttila, A. J. (1988).The mixing of immiscible liquid layers by gas bubbling. US.Nucl. Regul. Comm. NUREG/CR-5219. Suter, A,, and Yadigaroglu, G. (1988). Bubble-driven mixing of the oxidic and metallic phases during MCCI. Trans. Am. Nucl. Soc. 56,401. Szekely, J. (1963). Mathematical model for heat or mass transfer at the bubble-stirred interface of two immiscible liquids. Int. J . Hear Mass Transfer 6, 41 7. Tien, C. L. (1962). A hydrodynamic model for nucleate pool boiling. Int. J . Heat Mass Transfer 5, 533. Turton, R., and Levenspiel, 0. (1986). A short note on the drag correlation for spheres. Power Technol. 41, 83.

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Veeraburus, M., and Philbrook, W. 0.(1959). Observations on liquid-liquid mass transfer with bubble stirring. In “Physical Chemistry of Process Metallurgy” (G. St. Pierre, ed.), p. 559. Interscience, New York. Wallis, G. B. (1969). “One-Dimensional Two-Phase Flow,” McGraw-Hill, New York. Werle, H. (1982). Enhancement of heat transfer between two horizontal liquid layers by gas injection at the bottom. Nucl. Technol. 59, 160. Winnikow, S., and Chao, B. T. (1966). Droplet motion in purified systems. Phys. Fluids 9, 50. Zanker, A. (1980). Nomographs determine settling velocities for solid-liquid systems. Chem. Eng. p. 147 (May 19,1980).

Subject Index

A

Adhesive force, 63-64 Aerodynamic heating, entropy production, 252-254 Angular velocity, critical, induced rotational ROW,148-149 Axial flow effect, induced rotational flow, 156-159

rate, stratified liquids, 313-314 Bubble radius, volume-equivalent, 281 Bubble rise velocity, 302-303 Bubble volume entrainment onset, 3 13 maximum entrainable, 3 14 minimum, entrainment onset, 295, 297 Burgers variables, 231

C

B Baffles effect, induced rotational flow, 174- I76 Boltzmann number, flame, 262 Bond number, 1 17 Boussinesq relation, 7 Bubble-induced entrainment efficiency, 314 initially stratified liquid layers, 299-31 1 background, 299-300 bubble Reynolds number, 309 bubble rise velocity, 302-303 correlation of data, 307-310 driving force, 308 efficiency, 308-309 experiment, 301-302 force balance, 308 heavy-liquid viscosity effect, 305-307 heavy-liquid density effect, 303-304 interfacial tension effect, 306 light-liquid density effect, 304, 306 light-liquid viscosity effect, 307 phenomenon observations, 302-303 interlayer heat transfer, 3 14-3 17 onset, stratified liquids, 313

Capillary tube, wettability, 61-62 Carbon dioxide, physical properties, 2-3 Chilton-Colbournjfactor relation, 17 1 Clausius-Clapeyron relation, 70 Coefficient of nonstationary heat transfer, 45 Coefficient of turbulent viscosity, 23 Cohesive force, 64 Condensation adhesive force, 63-64 balance of interfacial forces, 62-63 cohesive force, 64 contact angle, 62-63 types, 57-59 wettability, 61-65 Condensation coefficient dependence on system pressure, 73-75 experimental determination, 73-80 metal vapor, 73-75 polyatomic fluid, 79-80 Condensation curves, heat transfer coefficient, 85-86 Condensation heat transfer, 55- 132, see also Dropwise condensation; Film condensation electric field enhancement of, 125- 131 341

SUBJECT INDEX

348

Condensation and heat transfer (continued) EHD instability of vapor-liquid interfaces, 126-131 principle, 125- 126 noncondensable gas effect, 107-108 surface tension force enhancement of, 115-125 fin geometry for vertical condenser tube, 117-119 horizontal low-fin tube, 120-123 principle, 115- 117 wire wound around tube, 124- 125 theory, 55 transport at vapor-liquid interfaces, 65-60 condensation coefficient determination, 73-80 interphase mass transfer theory, 67-72 modification of Schrage’s theory, 72-73 Nusselt equation, 66-67 Contact angle, 62-63 Convective heat flux, 260 Cooling, turbulent heat transfer under, 34-37 Couette flow inducedrotational flow, 144-147, 150-151 stability studies, 150-151

D Degraded heat transfer, supercritical pressures, 47-48 Drag coefficient, 333-334 liquid droplets, 334-337 oscillation and instability effect, 339 versus Reynolds number, 338-339 solid spheres, 334 Dropwise condensation, 59-61,80- 104 drop departure, 82 drop growth, 82 drop size distribution, 96-97 timewise change, 99-100 heat transfer coefficient measurement, 82-98 condensation curves, 85-86 drop size effect, 83-84 high-pressure steam, 90 low-pressure steam, 83 material thermal properties effect, 86-90 metal vapors, 92

organic vapors, 91-92 small vapor-to-surface temperature differences, 9 1 steam at near-atmospheric pressure, 82-83 surface thermal conductivity, 88-89 thermal resistance, 87-88 heat transfer theory, 92- 100 Le Fevre-Rose theory, 92-96 Tanaka’s theory, 96- 100 high-performance condensing surfaces, 101 initial droplet formation, 80 maintaining for long periods, 100- 104 microscopic mechanisms, 80-82 polymer coating, 103-104 surface heat transfer coefficient, 94 surface subcooling, 93 things not known about mechanism, 81 Duhamel integral, 33

E Eccentricity effect, induced rotational flow, 157-160 Eddington approximation, 247,249 Eddy viscosity, 13 Electric charges, redistribution, 126-127 Electric field, condensation heat transfer enhancement, 125- 131 Electrohydrodynamic instability, vaporliquid interface, condensation enhancement, 126-1 3 1 Electromagnetics, lost work, entropy production, 270-272 Energy, Joulean dissipation by diffusion, 272 Entrainment bubble-induced, see Bubble-induced entrainment immiscible liquid layers, 286-299 background, 287 criterion formulation, 289-293 experiment, 293 experimental results, 295-298 penetration criterion, 290 physical phenomena, 288-289 threshold criterion, 295-296

SUBJECT INDEX wake column, 291 wake volume, 29 1 Entropy balance, 248 Entropy flow, 241 Entropy production, 239-240 flame quenching, 261-263 fundamental difference of power, 248 heat transfer, 259-260 microscales, 263-268 local, 247-249 low electromagnetic work, 270-272 number, 254-255 Nusselt number, 260 qualitative radiative, 254-259 entropy production number, 254-255 isotropic scattering, 255-256 wall effect, 256-259 radiation-affected turbulence, 268-270 radiative stress, 245-247 stagnant gas, 249-254 aerodynamic heating, 252-254 gas emissive power, 250-25 1 heated plasma, 252-253 radiative balance, 249 radiative boundary conditions, 250-251 thermal balance, 249 thermodynamic foundations, 240-245 control volume, 242-244 heat flow, 24 1-242 laws, 240-243 mechanical energy balance, 242 wall, 259-260 Evaporation, microscopic description, 57 Evaporation coefficient, 73-74

349

thermodynamic cycle, 110-1 11 Nusselt number, 105-106 single component vapor, 105-107 Finite annulus length effect, induced rotational flow, 165 First Law of Thermodynamics, 240-241 Flame quenching, entropy production, 261 -263 Fluid, thermophysical properties, 2-3 Forced convection condensation, 1 13-1 14 Free convection horizontal wires, 42-44 vertical surfaces, 37-42 laminar flow, 37-39 turbulent flow, 39-42 Friction factor, 11-12, 15 turbulent heat transfer under cooling, 36 G Gas heat transfer microscales, 267-268 incompressible, energy generated by viscous dissipation, 252-253 noncondensable, condensation heat transfer effect, 107-108 stagnant, entropy production, 249-254 Gas bubble, entrainment requirements, 289 Gas-liquid interface, Maxwell stress, 127-1 28 Gibbs function, 244 Gibbs thermodynamic relation, 248,27 1 Gold-plated surfaces, dropwise condensation, 103 Grashof number, 280

F Film condensation, 59-61, 104-1 15 heat transfer coefficient, 105 multicomponent vapor, 107-1 15 forced convection condensation, 113-1 14 heat-transfer coefficient, 108- 109 KBrmBn-Pohlhausen method, 108- 1 10 Minkowycz-Sparrow theory, 109-1 10 natural convection condensation, 114-1 15 nonazeotropic mixture, 110-1 12 noncondensable gas effect, 107-108

H Heated plasma, entropy production, 252-253 Heat flow, 24 1-242 Heat flux, 248 convective, 260 dropwise condensation, 87 radiative, see Radiative heat flux Heat transfer entropy production, 259-260 versus gas velocity, 3 17 microscales, 263-268 kinetic scales, 263-265

SUBJECT INDEX

350

Heat transfer (conrinued) Kolmogorov microscales, 265 mean transport, 264-265 Pr-00.267 Pr-.O,266-267 PrZ 1,267-268 Taylor scale, 264 thermal scales, 265-268 Heat transfer coefficient, see nlso Dropwise condensation bubbling, 320-321, 329 versus superficial gas velocity, 324-325 condensation, 108- 109 entrainment, 315 film condensation, 105 general dimensionless structure, 320 horizontal bubbling surface, versus superficial gas velocity, 330-33 1 interfacial, 70-71, 280 versus superficial gas velocity, 283,285 interlayer, 3 15 pool height effect, 331-332 Prandtl number effect, 66-67 stratified configuration, 312 surface, 94 surface renewal, 3 15 turbulent free convection, 40-41 vapor-liquid interface, 70 wall, 323 Heat transfer theory, dropwise condensation, 92-100 Heavy-liquid density, bubble-induced entrainment effect, 303-304 Heavy-liquid viscosity, bubble-induced entrainment effect, 305-307 Helical electrodes, 130-1 3 1 Helium, relative change in heat transfer coefficient, 16 Horizontal low-fin tube, 120-123 Horizontal wires, free convection, 42-44

I Incompressible gas, energy generated by viscous dissipation, 252-253 Induced rotational flow, 141-142 boundary layers, 142 heat transfer, 162-163 mixed-mode flow, 168- 170 natural convection, 167- 168

zero axial flow, 162-168 isothermal flow secondary flow, 146-147 stability limit, 149 mass transfer, 170- 174 wiper blades effect, 174-176 stability, transition, and flow regimes, 143- 144 axial flow effect, 156-159 Couette flow, 144-147, 150-151 eccentricity effect, 159-160 finite annulus length effect, 156 isothermal flow, 144-160 nonisothermal flow, 160- 161 Taylor vortex flow, 147-152 torque transport, 161- 162 turbulent flow, 153-155 wavy vortex flow, 152-153 thermal stability, 142 transition to turbulence, 154-155 transport characteristics, 142 Inertia factor, 28-29 Interfacial forces, balance between, 62-63 Interfacial heat transfer, dimensionless correlation of data, 285-286 Interfacial tension, bubble-induced entrainment effect, 306 Interfacial tension-liquid density ratio, entrainment onset, 297-298 Interphase mass transfer theory, 67-72 Maxwell velocity distribution function, 68 net condensation rate, 68-70 Isothermal flow, induced rotational flow, 144-160 axial flow effect, 156-159 Couette flow, 144-147, 150-151 eccentricity effect, 159-160 finite annulus length effect, 155-156 Taylor vortex flow, 147-152 turbulent flow, 153-155 wavy vortex flow, 152-153 Isotropic pressure of radiation, 247 Isotropic scattering, 255-256 Isotropic turbulence, velocity fluctuation dissipation rate, 186

J

Joulean dissipation of energy by diffusion, 272

SUBJECT INDEX K KArmAn-Pohlhausen method, 109-1 10 Kinematic energy dissipation rate, nearly homogeneous turbulence, 194 Kinetic energy, strong nearly homogeneous turbulence, 207-208 Knudsen number, 72-73 Kolmogorov microscales, 265 Kolmogorov scale, 268, 269 radiation-affected thermal, 269-270, 273 viscous oils, 267 Konsetov model, 279 Konsetov turbulent heat transfer model, 331-332

L Labuntzov-Kryukov theory, 72-73.79 Laminar flame, quenched, 261 -262 Laminar flow forced flow in round pipes, 6 vertical surface, supercritical pressures, 37-39 Laplace equation, pressure difference, 1 16 Laws of thermodynamics, 240-243 Le Fevre-Rose theory, 92-96 Light-liquid density, bubble-induced entrainment effect, 304, 306 Light-liquid viscosity, bubble-induced entrainment effect, 305, 307 Liquid droplets drag and stability, 333-342 drag coefficient, 334-337 Liquid layers immiscible, entrainment onset, see Entrainment initially stratified, see Bubble-induced entrainment Liquid-liquid interface models for heat transfer, 279-28 1 Nusselt number, 280, 285 Liquid metals, heat transfer microscales, 266-267 Liquids heat transfer microscales, 267-268 stratified immiscible, heat transfer by gas bubbling across interface, 279-286 Liquid-vapor interface, see Vapor-liquid interface

351

Loitsyanskiy’s invariant, 198 Lost heat, 240 Low electromagnetic work, entropy production, 270-272 Low-yield processes, 141

M Mass entrainment, rate, 278 Mass transfer, induced rotational flow, 170-174 Material thermal properties, heat transfer coefficient effect, 86-90 Maxwell equations, electromagnetics, 270 Maxwell stress, gas-liquid interface, 127-1 28 Maxwell velocity distribution function, 68 Mechanical energy balance, 242 Metal vapor condensation coefficient, 73-75 dropwise condensation, 92 Minkowycz-Sparrow theory, 109- 1 10 Mixed mode flow, induced rotational flow, 168-170 Molten core-concrete interaction, 277 Multifluid bubbling pool, 277-343 bubble-induced entrainment, 299-3 1 1 configurations, 3 1 1 drag and instability of liquid droplets settling in continuous fluid, 333-342 background, 333-337 experiment, 337-338 liquid droplets, 334-337 results, 338-341 solid spheres, 334 entrainment onset, see Entrainment gas-sparged, 278 heat transfer by gas bubbling across interface, 279-286 apparatus and procedure, 281 -283 Blottner’s interfacial relationship, 279-280 experimental data, 283-286 models, 279-28 1 heat transfer from horizontal bubbling surface to overlying pool, 327-333 background, 321-329 experiment, 329-330 results, 330-332

352

SUBJECT INDEX

Multifluid bubbling pool (continued) heat transfer from liquid pool to vertical wall, 318-327 experiment, 321-322 heat transfer measurements, 323-325 model development, 325-327 physical phenomena, 318-319 previous studies, 319-321 void fraction measurements, 322-323 heat transfer with entrainment across interface, 3 I 1-3 18 rising gas bubbles, 278 stratified configuration, heat transfer, 312

total interlayer heat transfer model, 3 15-3 16 zero axial flow, 163- 164 Nusselt theory, 105

0 Organic vapors, dropwise condensation, 9 1-92

P

N Natural convection condensation, 114-1 15 Navier-Stokes equations, induced rotational flow, 144-146 Nearly homogeneous scalar field, secondorder model, 199-202,233-234 Nearly homogeneous turbulence, secondorder model, 193-202 Nearly homogeneous velocity field model formation, 213-214 second-order model, 194-199,232-233 Noble metal-plated surfaces, dropwise condensation, 103 Nonazeotropic multicomponent mixture, 110-112 Nonisothermal flow, induced rotational flow, 160-161 Nonstationary heat transfer determined by external conditions, 44-46 thermoacoustic perturbations, 46-47 Nuclear power reactors, accidents, 277 Nusselt equation, condensation heat transfer, 66-67 Nusselt number, 22, 260,273 axial variation, 165 film condensation, 105- 106 free convection, 40,43-44 heat transfer relations, 319 induced rotational flow, 162-163 liquid-liquid interface, 280,285 mixed-mode flow, 168-169 multifluid bubbling pool, 326 normalized, 33-34,45 versus speeds of rotation ratio, 166- 167 versus Taylor number, 163-164

Peclet number, flame, 262 Planck number, 258,262 Plasma, heated, entropy production, 252-253 Poisson’s equation, 189 Polyatomic fluid, condensation coefficient, 79-80 Polymer coating, dropwise condensation, 103- 104 Poynting theorem, 270 Prandtl number heat transfer coefficient effect, 66-67 molecular, 225 multifluid bubbling pool, 325-326 turbulence, 228 zero axial flow, 164-165 Pseudodropwise condensation, 130

R Radiation-affected turbulence, entropy production, 268-270 Radiative heat flux, 246 versus absorption and scattering, 256-257 versus emission and absorption, 255-256 Radiative internal energy, 245 Radiative stress, 245-247 Rayleigh inviscid criterion for rotational instability, 157 Rayleigh number, turbulent flow, 39 Reynolds number axial, 169 based on superficial gas velocity, 280 bubble, 309 critical, 337, 339 versus critical Weber number, 340-341

SUBJECT INDEX versus drag coefficient, 338-339 droplet drag as function of, 336 induced rotational flow, 153-154 liquid droplets, 335 rotational, 151 solid spheres, 334 turbulent, 209 Rotational flow, induced, see Induced rotational flow Round pipes, see Supercritical pressures

S

Saffman's invariant, 197 Scale ratio parameter, 186 Schmidt number, mass transfer, 17 1 Schrage's theory, modification, 72-73 Second Law of Thermodynamics, 240-243 Second-order model nearly homogeneous scalar field, 199-202, 233-234 nearly homogeneous turbulence, 193-202 equation for vorticity tensor function, 195- 196 vorticity scalar function, 196 nearly homogeneous velocity field, 194-199,232-233 Shear anisotropy, dimensionless parameter, 203 Sherwood number mass transfer, 172- 174 wiper blades effect, 175 Silicone oillwater, entrainment heat transfer data, 316 Solid spheres, drag coefficient, 334 Solid surface, wettability, 61 Stagnant gas, entropy production, 249-254 Stanton number, 14-15 versus buoyancy parameter, 17- 18 thermogravity effect, 19-20 Steam high-pressure, heat transfer coefficient, 90 low-pressure, heat transfer coefficient, 83 near-atmospheric pressure, heat transfer coefficient, 82-83 Stefan-Boltzmann law, 246 Stokesean fluid, momentum balance, 247-248 Stratified liquids, heat transfer with entrainment. 311-318

353

bubble-induced entrainment onset, 313 rate, 313-314 heat transfer, versus gas velocity, 3 17 overall interlayer heat transfer with entrainment, 314-317 Strong nearly homogeneous scalar field, evolution in strong degenerating isotropic velocity field, 217-220 strong evolving homogeneous turbulence, 216-224 strong evolving nearly homogeneous velocity field, 220-224 Strong nearly homogeneous velocity field, evolution, 204-209 Supercritical pressures, 1, see also Free convection boundary-layer approximation, 3 degraded heat transfer, 47-48 forced flow in round pipes hydrodynamic entry region, 5-6 laminar flow, 5-6 turbulent flow, 6-17 turbulent mixed convection, 17-32 turbulent transfer at varying heat flux, 32-34 turbulent transfer under cooling, 34-37 general premises and approaches to problem solution, 3-5 heat transfer augmentation, 47-48 nonstationary heat transfer, 44-47 thermophysical properties of fluid, 2-3 turbulent energy balance equation, 4 Superficial entrainment flux, 315 Superposition principle, 33 Surface condensation, 58 Surface condenser, 55 Surface heat transfer coefficient, 94 Surface tension force, condensation heat transfer enhancement, see Condensation heat transfer

T Taylor number critical appearance of vortices, 150- 15 1 dependence on axial Reynolds number, 158

354

SUBJECT INDEX

Taylor number (continued) turbulent momentum transport induced rotational flow, 148, 151-152 coefficient, 8 nonisothermal induced rotational flow, upstream flow history effects, 13-14 160 vertical surface, supercritical pressures, variation with eccentricity ratio, 159 39-42 wavy vortex flow, 153 Turbulent heat transfer induced rotational flow, 157-158 at heat flux varying lengthwise, 32-34 versus Nusselt number, 163- 165 under cooling, 34-37 Taylor scale, 264,267-269 friction factor, 36 liquid metals, 266 velocity profiles, 35 viscous oils, 267 Turbulent mixed convection Taylor vortex balance equation of turbulent stresses, 24 critical speeds for onset, 156 buoyancy parameters, 24-25 induced rotational flow, 146-1 52 hydraulic drag, 28 Thermal conductivity inertia factor, 28-29 free convection, 39-40 supercritical pressures, 17-32 near pseudocritical temperature, 43 coefficient of turbulent viscosity, 23 surface, dropwise condensation, 88-89 density fluctuations, 23 Thermal resistance, 87-88 horizontal pipes, 30-32 Thermal stability, induced rotational flow, 142 Nusselt number, 22 Thermoacoustic perturbations, nonstationary parameter I?, 20 heat transfer, 46-47 Stanton number versus buoyancy Thermoelectromagnetics, system fixed in parameters, 17- 18 stagnant media, 27 1-272 thermogravity effect, 19-20 Thermophysical properties, fluid, 2-3 velocity profiles, 22 Thin-film surface thermometer, 89 vertical pipes, 17-30 Torque transport, induced rotational flow, wall temperature distributions, 21 16 1- 162 threshold acceleration parameter, 29 Total interlayer heat transfer model, 315-3 16 upward water flow, 25-26 Transfer equations, specular moments, 246 velocity profiles, 28 Turbulence Turbulent nearly homogeneous scalar field, parameters, asymptotic, 212 evolution, 214-231 radiation-affected, entropy production, Turbulent stresses, balance equation, 24 268-270 Turbulent transport process modeling, Turbulent flow 185-232 induced rotational flow, 153-155 anisotropy degree, 187 supercritical pressures, 6-17 correcting tensors, 193 boundaries in absence of Archimedes fundamental parameters of turbulence, 192 forces effects, 10 governing equations, 188-190 Boussinesq relation, 7 nearly homogeneous two-point correlations degraded heat transfer regime, 11- 12 for closely spaced points, 191-193 eddy viscosity, 13 problem of evolution, 202-23 1 friction factor, 11-12, 15 “mandolin” and “toaster”, 218-220 models, 12- 13 nearly homogeneous velocity field model Nusselt number, 16 formation, 213-214 relations for turbulent transport strong nearly homogeneous scalar field, coefficients, 7-8 see Strong nearly homogeneous relative change in heat transfer scalar field coefficient, 16 strong nearly homogeneous velocity Stanton number, 14-15 field, 204-209

355

SUBJECT INDEX turbulent nearly homogeneous scalar field, 214-231 turbulent nearly homogeneous velocity field, 203-2 14 weak nearly homogeneous scalar field, see Weak nearly homogeneous scalar field weak nearly homogeneous velocity field, 209-2 13 second-order model, nearly homogeneous turbulence, 193-202 Two-point correlations, closely spaced points, 191-193

U Unstable wavelength, dependence on applied electric field intensity, 128- 129

V Vapor multicomponent film condensation, 107-1 15 single component film condensation, 105-107 Vapor-liquid interface, see also Condensation heat transfer electrohydrodynamic instability, condensation enhancement, 126- 13 I heat transfer coefficient, 70 shape, 1 17 temperature at, 66-67, 74-75 transport at, 65-60 condensation coefficient determination, 73-80 interphase mass transfer theory, 67-72 modification of Schrage’s theory, 72-73 Nusselt equation, 66-67 Vapor-to-surface temperature differences, dropwise condensation, 91 Velocity fluctuation dissipation rate, isotropic turbulence, 186 Vertical condenser tube, optimum fin geometry, 117-119 Viscous oils, heat transfer microscales, 267-268 Void fraction, measurements, 322-323

Volume condensation, 58 Vortex flow, wavy, 152-153 Vorticity scalar function, equation, 196

W Wake column, 291 Wake volume, 291 Wall temperature distribution, 33-34 turbulent mixed convection, 2 1 standard deviations in determination, 76 Water condensation/evaporation coefficient, 73-74 physical properties, 2-3 Watt, James, 55 Wavy vortex critical speeds for onset, 156 induced rotational flow, 152- 153 Weak nearly homogeneous scalar field evolution in weak degenerating isotropic velocity field, 226-229 evolution in weak evolving homogeneous turbulence, 224-23 1 evolution in weak nearly homogeneous velocity field, 229-23 1 Weak nearly homogeneous velocity field. evolution, 209-2 13 Weber number critical, 336-337, 339-340 versus critical Reynolds number, 340-341 liquid droplets, 335 Wettability, condensation, 6 1-65 Wiper blades effect, induced rotational flow, 174-176

Y Young’s equation, 62

2 Zero axial flow, induced rotational flow, 163- I68

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