Steam Boilers of Thermal Power Stations

the inlet (iE i 1,. 1 = i') and x = 1 at the outle t (i[ i 0 u t = i"), (Fig . 9.2). On the other ha nd, t he volume steam content ~ and the actual steam content rp, which both have zero initial values, increase sharply in the initial portion of a tube. Further in the tube, they increase less intens ively to the maximum value oE unity at the tube end. Wi th an increase in press ure, rp and ~ approach x . The pressure steam content q> is an i mpo rtant flow characteristic, since it describes the actual distribution of water and steam in a mixture and their individual velocities in the combined £low. The actual velocity of steam is

w,

=

w;trp

(9.37)

and the actual velocity of water is

~

If t~ere is no difference in velocity, that IS w, = Ww = Wm, c = 1 and (9.36)

Fig . 9.1. Relation between mass steam conlent z and volu.me steam content ~ at vunous pressures

f .O

cp<~

For a descending flow

w,

103

9.3. R egimes of Steam-water Mtztare Floro

The proportionality fac tor c characterizes the ratio of the velocity of the steam-water mixture Wm to the ~ctual velocity of steam w., and t hus 1t characterizes the relative velocity of steam w,. With a n increase in pressure, the relative velocity of steam decreases and, as pressure approac~es the critical value, c tends to uruty while rp tends to ~The mass steam content in a uniformly heated evaporating tube varies linearly along the tube length between its extreme values: x = 0 at

Ww

= w~/(1 -

rp)

(9.38)

The_apparent density of a steam-water m ixture, Pm• kg/m3 , is the density

4h

G, w'Qf

+ Gw

IDo/

(9.39)

The actual density of the steam water mixture, Pa• kg/m3 , i.e. the density at the actual v el ocities of s team and wa ter, can be found from the Eollowing consideration. Let us s eparate a tube element of height !J.h filled with a steam-water mixture (Fig. 9.3). L et steam and water i n the element be represented by corresponding elementary volumes. The fractio ns of the cross sections occupied by steam and water may be denoted as f , 1 and !w a nd thei r sum, as f. The masses of t he two components may be written as /, 16.hpw and fw!J.hp' and their s um is the mass of the sepa rate d volume of the steam-wa ter mixture, f!J.hp 0 • This gives us an expression for the actual density of steam-water mixture:

Pa = p' -

!p

(p' - P *) = (1 - cp) p' rpp. (9.40)

+

In an ascending tube, the steam has a relative velocity W ro which results in a decrease in the fractio n of tho cross section occupied by the steam, rp, and an increase in the fraction occupied by water, (1 - rp), s o t hat Pa > Pm· With increasing pressure, Pa tends to Pm· The circulation ratio K is an inverse of the mass steam content and is essentially the ratio of the quantity of circulati ng wa ter to the quantity of steam produced for the same time interval (see Sec. 1.2}: K = 1/x (9.41)

9.3. Regimes of Steam-water Mixture Flow f

Fig. 9.3. To derivation of the formula for the density or steam-water mixture

The intensity of heat removal from a heating surface substantially depends on the con ditions, or regimes, of the motion of the steam-water mixture on t hat surface. Under identical conditions, the s t ructure and

I

104

Ch. 9. Parameters an¢ Motion Equations of Working Fluid

·-

-•,_

--. ·a

_ _o _

-a

0

.

• . ..-·-

.... ..

--~~'­

'- -' _ G_ a_

-r--·

.

!~..

. . . .

••t

0

!..~

o__o_

•• •

--• -

-.-

(a)

{b)

••

-.!.-

.• '

•

••• •• •

.

••~ : •• • •0 • •

•• • •

• • • ••

•

•••

--~

••

0

.!...!..... 0

•

•

-'-·

...

.-.• ••

.,_qr_.

•

0 0

0

.

.

-·-

•

-

-

'-

(c)

•• • ••

·-. -

(d)

(e)

=

Fig. 9.4. Steam-water mixture flow modes in vertical tubes (a) bubble !low; (b) emu ls ion flow; (c) slug now; (d) dispersed annulnr flow; (c) inverse dispersed annular f low

I

regime of flow are determined by the spatial orientation of the heating tubes, which is used to organize the motion of th e working fluid in steam-generating tubes. Vertical tubes. An adiabatic flow of a steam-water mixture at a low steam content and slow velocity is essentially liquid with rare fine vapour bubbles distributed in it. This is what is called bubble flow (Fig. 9.4a). As the steam content increases and if wp is high, the moving mixture contains a larger number of fine vapour bubbles; this is the emulsion flow, or frothy flow (Fig. 9.4b). At low values of wp, an increase in steam content may result in the coalescence of fine bubbles into larger formations whose size may be comparable with the tube diameter and whose l ength may be many times the diameter. These formations are called 'slugs' and, correspondingly the flow regime is the slug regime (Fig. 9.4c). Behind a slug there is a thin liquid bridge containing fine vapour bubbles. With a further increase in the steam content, water bridges between the slugs disappear and the slugs merge into a continuous vapour column with atomized water droplets in it. This

column moves aloog the tube core and i~ ·surrounded hy a continuous annular water film which moves aloog the tube wall. The water film intensively cools the internal surface of the tube. This is what is called the disperse-annular (wet-wall) flow (Fig. 9.4d). The thickness of the water film depends on the ratio between the flow rates of water and steam. At a high pressure and high steam velociLy, the major mass of the film is broken off and carried as droplets by the steam flow, leaving only a very thin water film on tho wall, which soon evaporates. In heated channels, some specific flow regimes may take place. In film boiling, for instance, a vapour film may separate the liquid from the heat-exchange surface, while liquid fills in the core of the channel; this is called the inverse disperse-annular flow (dry-wall flow), Fig. 9.4e. The flow regimes described above are rather conditional since they gradually change from one typo to another with no distinct boundaries between them. Horizontal tubes. The flow of the steam-water mixture in a horizontal tube is characterized by a non-uniform distribution of structural flow components over the tube cross section. Since steam has a lower density, it moves primarily along the upper tube wall, while the main mass of water is concentrated at the lower wall. The asymmetry of flow relative to the horizontal depends on the velocity of the steam-water mixture and tube diameter. At higher velocities, the flow is less asymmetrical. In steam-generating tubes 30-40 mm in diameter and at relatively high inlet velocities (w > 1 m/s), vapour bubbles form in the initial tube section. They are detached from the tube surface and move along with the liquid (Fig. 9.5a). The vapour bubbles increase in number in the direction of the flow and begin to merge into larger · formations. An ever increasing quantity of steam is involved in the combined motion of

10!)

9.4. H ydraullc Resis.t.ances

•

..

-

(a)

•

(6)

Fig. 9.5. Two-phase flow modes in horizontal tubes at (a) high and (b) low inlet velocity

the two phases, so that at a high steam content the flow in a tube becomes almost axisymmetrical and resembles the; disperse-annular regime in vertical tubes. \Vith a low velocity of water at the inlet to a steam-generating tube (w < 0.5 m/s), asymmetrical motion of water and steam may result in the exposure of substantial portions of the tube surface (Fig. 9.5b). The flow then becomes essentially asymmetrical along the whole length of the tube, with the steam moving along the upper tube wall. In other words, the two-phase flow is separated, as it were, into two individual flows. This flow regime is unstable. With an increase in the flow velocity, waves may form on the separating surface with their tops periodically touching the superheated upper wall. At supercritical pressures, the working fluid is a homogeneous medium . Nonetheless, even with a directed motion of the fluid at supercritical pressure in a horizontal channel, free convection may take place in a transverse direction, resulting in density variation along the height of the flow. This inhomogeneity can be characterized by the motion of a lighter (less dense) medium along the upper surface of a tube and of a heavier (denser) medium, along the lower surface, with no distinct boun-

dary between them. The difference in densities increases with an increase in the vertical size of a channel or of the tube diameter. In tube bends, some portions of the tube surface may be washed less intensively by the fluid than others, which is due to the centrifugal effect by which water is thrown towards the outside surface of a bend, while the tube wall at the inside surface is insufficiently cooled by water. 9.4. Hydraulic Resistances Since tubes offer resistance to the motion of water, steam-water mixture or steam, a pressure gradient forms between any two sections along the length of a tube. The general equation of the total pressure gradient l~ns been given in Sec. 9.1. For practical calculations, it can be written in a more convenient form:

t:..p = t:..p/r

+

t:..p,

+

t:..Poc ±

t:..ph (9.42)-

i.e. the total pressure gradient between any two sections of a heated tube is the sum of pressure gradients due to· fdction, t:..p1,, local resistances t:..p, acceleration t:..Pac• and hydrostatic head t:..p h· The resistance due to friction is caused by the viscosity of the moving:

.• 1 06

Cit. 9. Parameters and Motion Equations of Working Fluid

fluid. For a single-phase isothermal flow in a straight channel of constant -cross section, this term is found from the formula: l

t.p,, =

Ik

whore t.p1, is the resistance due to friction, Pa, A. 0 = A.ld is the resolved friction coefficient, 1/m, l is the length of the channel , m, w is the velocity of the fluid, m/s, and p is the density of the fluid, kg/m 3 • Tho friction resistance for a twophase flow can be principally determined by tho formula for single-phase flows, by replacing the s ingle-phase Jlow velocity by the velocity o.E steamwater mixture wm, provided that tho two-phaseflowcan be considered homogeneous, i.e. obeying the following relationship: (9.44) Noting the law of mass conservation {soe Sec. 9.1), we can write: PmWm

=

pfwo

(9.45)

Noting equation (9.25), formula (9.44) can be re-written as foll ows: t.p,, = "-ol

The pressure loss due to friction for a flow of variable steam content can be determined from the formula: w~ [_1+XI!J - - ( rr p' l!.pfr = "-ol--fp'

(9.43)

• 0

~0

p' [

w+ w; (1 - ~:)] 0

noting equation (9.29):

I!.Ptr = Aol

4- pf [ 1+X(f.- -1) J (9.47)

I 11 most flow regimes, the structure {)[ a two-phase flow differs noticeably from a homogeneous structure (see F ig. 9.4), for which reason correction factor 'iJ is introduced into the last ~qu ation to account for the effect {){ the flow structure. With a constant steam content, we then have: t.p,, = A0 l

4-;P' [1+ x'IJ (f.- -1 )J (9.48)

J

where x is the average steam content in the channel and \~jXJ-IJ>t.:Z:j

"' = --'-....;___ Xf -

Xj

_

where 'IJ 1 and 'i'J are correction factors to account for the effect of the initial and final steam content, x 1 and x 1 , on tho flow structure. Pressure losses due to local resistances are explained by the energy consumption for detachment of the boundary layer from the tube wall and the formation of whirls in the flow. Local r esistances appear in places where the shape or direction of a channel changes; conditionally, they are considered to be localized in a particular section and do not include fr iction resistance. The pressure loss in local resistances to a single-phase flow is determined by a formula similar to (9.43) in which the resolved friction coefficient A.0 = Aid is replaced by the coefficient of local resistance ~~ (it may be found in reference books):

(9.46) {)f,

1)

(9.49)

-

(9.50) For a two-phase flow: [t.p 1 = 1:£{

L4p' [1+x ( :~ -1) J (9.51)

where !;! is a conditional coefficient of local resistance for a steam-water mixture, usually !;; > £1• For a s ingle-phase flow across a tube bundle, the hydraulic resistance is t.p,b =stb

w2

2

1.07

9.5. Thermoph ytlcal Properties of Working Fluid

p

(9.52)

The coefficient !; 1 for a flow across a tube bundle depends on the design of the bundle.

For a two-phase flow:

ilPtb =

s;b ~~ p'[ 1+ x( f.--1)](9.53)

The pressure loss due to acceleration is caused by a change in the volume, and therefore, in the velocity of a flow. It can appear as the steam -content of a fl ow increases due to heating, as the fluid passes through a reduced cross section, or both. The pressure loss due to fl9w accel eration can be found by the· formula: D.Pac

=tot wp dw

(9.54)

wl

For a steady-state flow:

wp = constant Therefore, at p >Per llPac = wp (w, - w 1) = (wp) 2 (v1 - v 1) (9.55) For a two-phase flow, p < p cr· Expressing v1 and v 1 through the mass steam contents: v1 = v' (1 - x1) u·x, u1 = u' (1 -xi) v •x 1 and substituting them into formula (9 .55), we finally have: llPoc = (wp) 2 (u • - u') X \xl - Xt) (9.56) The hydrostatic component of pressure loss for single- and two-phase v ertical· flows is: WtPt

= w 1 p, =

9.5. Tbermophysical Properties of Working Fluid in the Path of a Monobloc Unit Va1·iations of the parameters and physical properties of the working fluid in various sections of the watersteam path of a supercritical-pressure monobloc unit are shown in Fig. 9.6. T he highest pressure of the fluid is at the inlet to the high-pressure water heater, downstream of the feed water pump, Pw 1, and tho lowest pressure is in the turbine condenser, p c· For supercritical-pressure pl ants, this pressure range is from 32 MPa to 0.003 MPa and for high-pressure power plants, from 17 MPa to 0.003 MPa. In the condensate path, which includes a condensate pump, and in the feed water path with a feed water pump, the

t (A I I

+ +

h

t.p,. = g ) p dh =

pgh

---l'- I

''I .

(9.57)

0

For a single-phase flow, p is taken as the average density of the fluid in section h, and for two-phase flows is determined by formula (9.40). When determining the total pressure gradient, l!.p h is assumed to be positive for ascending flows since it increases the total resistance of a tube, and negative for descending flows where it ceases to be a resistance and, {)n the contrary, acts in the direction {)f the flow.

Mtrln

1/iglr· :;;

S11por- , prt.·~-sll ti

!JMior :ylintfe

I

II

"ill

IY

l ow·

Pl'l!~·svriJ

~ -ylri ufpr

..

Y

1!1.

~ ~

"l:l

.§

Y11

Fig. 9.6. Variations of parameters and thermophysical properties of the working flu id in the water-steam path of a supercriticalpressurc monobloc unit

i08

Ch. 9. Parameters and Motion Equations of Working Flulct

pressure of the turbine condensate is raised to a value equal to tho pressure at the inlet. to the high-pressure water heater, which ensures the motion of the working fluid through the entire water-steam path of the boiler. In tho turbine, steam moves due to a pressure gradient between tho inlet, where the pressure is equal to that downstream of tho boiler, and the outlet, where it is equal to the pressure in the condenser. The highest temperature of the working fluid is the temperature of superheated steam and the lowest is that in the condenser. In high-pressure and supercritical-pressure plants, this temperature interval is from 545° to 30°C. Turbine condensate is preheated in the low-pressure water heaters and the deaorator of the condensate path from 30° to 105-165°C aud further in the higl1-pressure water heaters of the feed water path to the feed water temperature t1w (usually 145-270°C). Water pressure in this path varies from p = 0.3-1.0 MPa to 15-17 MPa in high-pressure plants or to 30-32 MPa in supercritical-pressure pl ants. The total pressure gradient in the path, flp = flP ec flPdca flp31, constitutes 20-30% of the pressure of superheated steam. On the other hand, the temperature of the working fluid is raised substantially, from t1w to superheated-steam tomperature t ••. Superheated steam is fed into the turbine with the parameters t,. and p ••· As steam expands adiabatically in tho turbine, its pressure and temperature decrease. In power plants of high and supercritical parameters, the total efficiency is increased by taking off steam from an intermediate stage of t.he turbine (at a pressure of 3-4 MPa and temperature of 290-320°C) and recirculating it back into the boiler where it is reheated, usually to the initial steam temperature, 54SC, in an intermed iate steam superbeater (rehoater). Since t,he working fluid changes its pressure and temperatm·e in the wa-

+

+

ter-steam path or a supercritical monobloc power unit, this involves certain changes in its thermophysical properties. As shown in Fig. 9.6, the water-steam path can be divided in to seven zones in which typical changes in the th ermophysical properties of the working fluid take place. The first zone includes the condensate and feed water path of the power unit with convective and radiationheated economizers. In this zone, feed water pumps create tho highest pressure in the water-steam path needed to produce steam of the required pressure. The working fluid in this zone remains liquid and therefore, its thermophysical characteristics are changed relatively weakly, notwithstanding the high rise in pressure; its dielectric permittivity e, however, decreases to a smoll rraction of its initial value. The most t.yp ical is the second (near critical) zone which is usually located in the boiler furnace. In this zone, the change from water to steam takes place, so that even a small rise in the temperature of the flow causes sharp changes in the thermophysical characteristics: near tho temperature of the highest heat capacity, the enthalpy of tho flow increases sharpl y, while the viscosity ll• heat conductivity A., and, what is especially important, the density p and dielectric permittivity e decrease substantially, though smoothly, not jumpwise. In the third zone, which iocl udes the main superheater, the temperature and enthalpy of the superheated steam increase further, but all other parameters, including density and dielectric permittivity, change less than in the previous zones. In the fourth zone (high-pressure cylinder of turbine), the thermal energy of the superheated steam is converted into mechanical work and its pressure and temperature decrease. Accordingly, the enthalpy and density of the steam decrease, the dielectric permittivity increases somewhat (due to temperature reduction), and tho

109

9.5. Thermophyslcal Properties of W orking Fluid

other properties (heat conductivity, heat capacity' and dynamic. viscosity) continue to decrease smoothly. [n tho fifth zone (reheater), steam temperature is raised again at an almost constant pressure roughly 3.5 MPa, resulting in an appreciable in crease in its enthalpy and slight increase in viscosity and beat conductivity. The dielectric permittivity of steam decreases by 67-80%, whereas the density and isobaric heat ca'pacity diminish only slightly. In the sixth zone (low-pressure cylinder of turbine), the physical parameters of steam are changed in a manner similar to the fourth zone, the only difference being that the pressure changes much more substantially t han in the fourth zone. For that reason, the density of steam decreases roughly to 1/300 or 1/400 o[ its init ial value. Its enthalpy also decreases Sllbstantially, wh ile changes in the other properties are rather s mooth. The seventh zone includes the turbine condenser. The condensing steam hero has constant parameters: pressure 0.003 MPa and a temperature of around 30°C. Only the enthalpy and steam content of the flow are diminished. As steam is condensed fully , the density, heat conductivity, heat capacity, viscosity and dielectric permittivity increase sharply. The specifics of subcritical pressure units are most pronounced in the second zone of the water-steam path, which is within the boiler furnace (Fig. 9.7). This zone is characterized by the conversion of water to steam (boiling); steam generation takes place at a constant temperature t' with two phases: water and steam, existing simultaneousl y (curve 1). For comparison, the figure shows temperature variations in the working flu id at supercritical pressure (curve 2). As may be seen from Figs. 9.6 and 9 .7, the thermo physical characteristics of the working fluid in the zone of high heat capacities at suporcritical pressures vary smoothly (though rapidly), whereas at subcritical pres-

p

Pu

I~ 2'::::,.·..... / / ~ /

/--x-:o

/://

:r:~t

t'

.

~

.....

\

~

'

~ r\Pm \

\

~ r-....

~ ._ '\em

\

....

.I!

-~

"" c::S"l::"

"'til~

:

' :>..,

I

~

._..,. -

f11rnaco ·~ ~ wafer walls ~\,.

•

•"' ~~~ c.a";-

Fig. 9.7. Variations of density p and dielectric permittivity e in evaporating zone

1- lcmperaturc or the working !luid at subcr!tical pressure; B- d!tto, at sul?ercri tical pressure (for compartson)

sures the change from the proper ties of water to those of steam suddenly takes place at the saturation temperature. As seen in Fig . 9.7, as the water to steam transition takes place, tho density of the water, Pw• abruptly changes to that of steam, p., which results in an increase in the enthalpy of the working fluid. Similarly, the dielectric permittivity of water changes abruptly to that of steam. In the zone of constant temperature at the saturation line and constant pressure, bot;h phases (water and steam) co-exist; in the whole range of steam contents of the two-phase flow, 0 < < x < 1. The dot-and-dash lines in tho figure show variations in the density· of the steam-water mixture, Pm, and in its dielectric permittiv ity em along the flow. The thermophysical properties of water and steam are determined by

i10

Ch. 10. Temperature Conditions on Heating Surfaces

\

,1/r

J11alt?r r t?!IID/1

e~

, Pcr,tcr

Fig. 9.8. Effect of pressure on the density of

wn ter aod steam, p, and dielectric pcrmi Uivity e at the saturation li ne

pressure. Figure 9.8 shows v ariations in the density and dielectric permitti-

vity of water and steam; as may be seen, these characteristics are substantially different in a wide range of subcritical pressures, but come closer and closer together in the critical point. The above relationships between the parame~ers and properties of the working £lu id determine the hydrodynamic and heat- transFer processes on heating surfaces and the carry-off by steam of impurities present in water; they also pl ay a v ital part in the formation of deposits on boiler and turbine elemen ls.

TEMPERATURE CONDITIONS ON HEATING SURFACES 10.1. Classification of Heating and Cooling Modes

U A knowledge of temperature distribution in the metal of heating surfaces, which operate under complicated and heavy conditions, is essential for estimating the reliability of a steam boiler. In boilers operat ing on organic fu el , there are three differen t regions of heat transfer. The first region includes the heating surfaces (water walls) arranged in the furnace; they receive heat mainly by radiation. The most important factor of heat transfer in this region is the pattern of distribution of the heat flow along tho height of the furnace (see Fig. 20.2) and over the cross-sectional periphery of waterwall tubes (see Sec. 10.4). The second heat-transfer region comprises the semiradiant heaLing surfaces which are arranged in the boiler zone where radiation from gaseous vo-

lumes is still subst,anLial and the gas temperature is quite high (1 2008000C). This group of heating surfaces includes primarily platens and water walls in the horizontal duct of the furnace. The heating surfaces in the third reg ion receive heat mostly by convection. T his r egion has a r elatively low temperature of combustion products (800-900°C at tho inlet, to the convective shalt and up to 100-150°C at i Ls • outlet) and accommodates convective heating surfaces: an economizer, air heat er, and some superheater banks . These heating surfaces have a small Lube pitch, i .e. small spaces between tubes for Lhe passage of hot gases. Under such cond itions, the fraction of radiant heat transfer from combustion products onto t ubes is not high . In the steam generators of nuclear power st ations , Lwo r egions of heat transfer are usually distinguished. The first r egion includes the heating sur-

1U

10.1. Claulflcatlon of Heating and Cooling .Modes

faces arranged in the reactor core. The heat in the reactor is liberated due to nuclear fission in ·a very restricted volume around the pl ace where nuclear fission occurs. This makes it. possible to ass ume that practically all the heat released by tho core is concentrated in tho volume of fuel elements. The maximum amount of heat released, qmax• through the s1.1rface of fuel elements is in mid-height and the zero value (in the absence of reflectors), a t the end faces. Reflect ors can substantially equalize the intensity of heat released. In the second region, heat transfer takes place by convection at a moderate temperature of the heat-transfer agent (water, liquid metal or gas) . For instance, in a water-healed primary circuit, water Lem perature does not exceed 330-350°C (at a pressure of 13-15 MPa); in ci rcuits heated by liquid metal or gas, this temperature may be higher, up to 700-800°C. Under such conditions, ther e is no immediate danger of tube burn-out should the heat-Lransfer conditions worsen, but the durability of the heating surfaces' metal can be impaired and cause emergency situations. Present-day engineering has made possible the release of enormous quantities of heat in the boiler furnace or the core of a nuclear power reactor. Irrespective of t,he type of power plant, the beat released must be actively r emoved through heating surfaces. This is ensured by the motion of the working fluid at a definite velocity. The motion of the steam-water mixture, and therefore, cooling of the evaporating tubes of steam boilers and steam generators can be organized in a different manner (Fig. 10.1). In free-·c irculation plants , the steamwater mixture moves under the freecirculation head that appears when the tubes heat up. With an increase in load, tho mass fl ow rate at the inlet, to the evaporating tu bes first

c

c'

D/Dr

Fig. 10.i. Effect or boiler load on mass. velocity J - gravlly clrculntlon; t-mulliple forced c irculati on: J-c lrculotlon In once-through boiler; 4-comblned circulation

increases sharply to a maximum value and is then almost stabilized or oven decreases somewhat , since the increasing steam generation at. a high unit volume of steam results in an increased resistance of the Lubes (curve 1 in Fig. 10.1). In mul t iple forced circulat-ion plants, the circulation of the water and steam-water mixture in evaporating tubes is effected b y a circulation pump , so that the mass flow rate of the working fluid is almost constant irrespective of boil er load (curve 2). In once-through boiler plants, tl1e mass flow rate is proporLional to t he boiler load (curve 3). At a low load, the mass fl ow rate may turn out t<> be inadmissibly low and cause burnout of tubes. For this reason , load shedding is restricted to no less than 30% of the rated load. Curve 4 in Fig. 10.1 depicts the characteristic of a boiler plant in which tho motion of the working fluid is based on the principle of mulLiple forced circulation at low loads and on the once-through principle at high loads. When operating on the principle of mult-iple forced circulation, the mass flow rate is expressed by the sum of the ordinates ab- the mass flow rate corresponding to recircul ation through the steam-generating surfaces, and be- the mass flow rate in the economizer and superheater in once-through operation. With a chan-

Ch. 10. T empera ture Conditfons on Heating Surfaces

112

ge to the once-through operation, the cross section of a tube (noting the mass flow rate is the same in all that heat supply or heat r emov al may the heaLing surfaces of the water-steam b e uneven around the periphery of a path, the ordinate a' c'. Thus, the tube}. comb ined circulation system ensures adequate cooling of all the heaLing 10.2. Heat-transfer Crisis surfaces irrespective of the load. Acin Evaporating T ubes cordingly, fo rced-circul ation singleFor a g iven heat release intens ity, pass motion of water and steam takes place in the economizers and super- the temperature conditions on an evaporating tube are m ainly determiheaters. The thermal conditions in the Lubes ned by the steam content, mass flow of tho steam-water path of a steam rate and pressure of the flow . Various inadequate temperature conboiler or steam generator are established depending on the ratio between ditions which may appear in evapothe quantity of heat s upplied by the rating tubes can resu lt in a substanheat-trans[er agent \.o the outside tial rise in tho temperature of the surface and tho quantity of heat metallic tube wall which carl someremoved by tho working fluid from times cause tube burn-out. One should the inside surface. T he si multaneous differentiate between tho inad equate processes of heat rel ease and heat temperature condition s in bubble El ow removal ensure that any point of a and in disperse-annular flow . With bubble flow of a s team-water heating surface is in the necessa ry t herma l s tate and has a specific mixture having a moderate steam t emperature. The temperatures in va- content, a thick layer of liquid coveL"S rious points of a heating surface form the tube wall (Fig. 10.2a) and the what is called the temperature field . temp erature of the tube is maintained One can distinguish between a tem- at a substantially low level (section perature field along the motion of 1-2). Even with intensive heating, it t he working fluid, i.e. along the length exceeds the saturat ion temperature of tu bes (assuming that t hey are hea- only by a few tens of degrees. As th e working fluid moves and is ted uniformly around their periphery) and a temperatu re field across heated in t he tube, more and more J I

-,

f

f'

2

t - - - -- -

t'- - -

q,

gUll.LLLI.Lj.J.lll.UJJ

. .......... . ... .. . .' . . : . .. ' . .' .. . . .

•

.

,.

(a)

•

.

• t

q2rrrrrn""T"T"i-rrhrTTTTTT1rTTTTT"M u~~~~~~~~

• •

I

(h)

Fig. 10.2_ General pattern of temperatures in a steam-generating lube at (a) bubble Oow and (b) dispersed nnnular Oow of steam-water mixture

10.2. Heat-tran•fer Crl1ll In" Euapor4tlng Tabu

vapour bubbles appear on tho tube walls; they coal esce with .o ne another and may eventually form a continuous vapour film· which will separate the flowing liquid from the heated tube wall and thus sharply impair the conditions of heat transfer. T he effect of a sharp worsening of heat transfer on a change fro m bubble boiling to film boiling is called tho boiling crisis (burn-out conditton.~) and t he corresponding heating · ·t oad is called the critical load qcr· On occurrence of the boiling crisis (point 2), the layer of superheated steam at the heated wall becomes thicker, the heat-transfer coefficient drops down sh arply, resulting in a sharp rise in the wall temperature and often in burn-out of the t ube wall (point 3). These temperature conditions can occur in bubble flow if the heating rate of one of the evaporating tubes is t oo high and a thick layer of steam accumulates oo the superheated tube wall. In disperse-annular (wet-wall) flow, saturated steam with susp!:nded liquid droplets moves in the core of a tube and a water film flows along the tube walls (Fig. 10.2b). Without heating, the thickness of th e liquid film depends on the ratio between the flow rates of water and steam, the quantity of moisture t hat is separated from the flow core and wets the wall, and . the. quantity of moisture r emoved from the wall due to break-off and mechanical carry-off by the steam flow. In heated tubes, the intensity of heating has a strong ,additional effect on the thickness of•'"the water film. A continuous water fil m can still ensure proper heat removal from the heated surface and the wall tempeL·ature can be maintained at an allowable lovel (line 1 ' -2' ). In further heating, the water film becomes thinner owing to ev aporation and break-off, so that only a very thin fil m (micro-film) re mai_ns on the wall. Under a particular heating load, t he microfilm is destroyed and separated into individual islets and streaks whose Iiumbor 8 -01524

113

and size diminish in the course of further vaporization. T hus, a continuous steam layer forms at the wall, while the core of the flow contains ii slightly superheated steam with wator droplets whose concentration decreases gradually in the direction of the flow. Under such condit ions , water dro plets do not roach the tube walls. Besides, on entering a hotter zone of the tube they evaporate. Since vaporization is more intensive at the walls, the water droplots are repulsed from tube walls back into the steam flow. T:Jnder such conditions, heat transfer occurs through a continuous steam layer, i.e. the heat-transfer · c.oefficient decreases sharply and the wall temperature increases (poin t 2' in F ig. 10.2b}, resulti ng in a boiling crisis . In contrasl. to , the previously mentioned case of a boiling crisis at a mode1·ate steam content during a cbange· from bubble to "film boiling, this type of boiling crisis is caused by a · change fro m disperse-annular to disperse flow with a complet e drying out of the liquid film at a high steam content of the £low. This type of boiling crisis can even · take place at a · low h eating load and rel atively high .heattransfer coefficien t as th e liquid phase evaporates almost completely. Since the unit volume of steam is much higher than that of water, the linear velocity of the flow increases substantially. In a boiling crisis when the liquid fil m dries out, the temperature of the tube wall rises less significantly (line 2' -3'), as in tho former case. Boiling cris is in the one-sided heating of evaporating tUbes. We have discussed the mechanism of boiling crisis in an ascending flow of working fluid in vertical tubes heated u niform ly all over their periphery. l n modern powerful steam boilers, th~ main steam-generating surfaces nrc formed by furnace water walls which are unevenly h eated over their periphery. The distribution pattern of the heating load over the periphery of a water-wall tube is shown in

114

Ch. 10. Temperature CondLttons on Heating Surfaces

Fig. 10.3. Heat distribution in a smooth tube heated from one side

10.9: Temperature Conditions Along the Channel

ce-through boilers are shown in Fig. 10.4. In boilers of any type, the temperature of feed water in the economizer, t 1w, is raised to the saturation point (in the limit), so that the heat absorbed by evaporating tubes is almost fully spent on vaporizatioi;I, and the temperature of the steam-water mixture t' remains almost constant. In the superheater. the temperature of steam is further r aised to the rated value t ••. In the economizer section I, the temperature of the flow t1 and that of the tube wall, tt. are below the saturation temperature: t 1 < t' and· t 1 < t'. The thermo physical properties of water in the layer near the walL vary only slightly with temperature. Under such conditions, heat transfer is mainly governed by the laws of convection:

Fig. 10.3. The front side of the tube, which faces the furnace, is heated most intensively and the rear side, (10.1) least intensively. Uneven heating causes transverse circulation of the work- Where ing fluid from the less heated to the o:d Re - wp1d Pr - cptf.Lr u N 1,_'J..!' ! ~lf. ,,, more heated portions of the tube periphery; some heat is also transThe subscript 'f' implies that the· ferred by conduction in the metal wall. These two circumstances en- flow temperature is taken as the decihance the cooling effect of the working sive parameter. For straight channels,. fluid, thus retarding the appearance c = 0.023 . Heat exchange occurs at a temperaof a boiling crisis. For this reason, burn-out conditions in tubes that ture gradient M = t 1 - t1 = qlrzzare unevenly heated across their pe. riphery can appear at substantially /! 3 / J :... higher values of qcr than in those 1>:1 8,8'-.../ / ,-J -c >Sl < which are heated uniformly from all i \ sides. This means that one-sided heat1:>. . ing surfaces can be designed for more ~ . forced heating, q~~ > q~;· (see Fig. l>:t -c >I f-.rcr ' z 10.3). The allowable heating load 2~ J 'B.o' ·C AI to avoid a boiling crisis can also be • increased by artificial turbulization "== "== ,_ of the flow with helical inserts, screw ...., ..... f 1 threading of the internal surface of tss trw t' tubes, etc. (b) (a)


~~~

-

//

i

10.3. Temperature Conditions Along the Length of a Channel

Fig. i0.4. Temperature variations in the water-steam path of (a) drum-type and (b)· once-through subcritical-pressure boilers

Straight channels. Curves of temperatures in the water-steam path of subcritical-pressure drum-type and on-

1- economizer: 2-evaporating tubes ; J-superbeater· A-temperature or the working !luld ~ B-wall temperature or a drum-type boil er; B'wall t empe rature or a once-through boile r; C-ailowa ble metal tempera ture

In the economizer section, heat exchange between the wall and the water takes place at a low heating intensity (q ~ 5-10 kW/m2 ) and high values o[ the heat-transfer coefficient [a. = = 2-5 kW/(m 2 K)}, so that the wall temperature is only a few degrees higher than that of the water. In section II, the temperature of the flow t 1 is lower than t', while the wall temperature is higner than t'. This means that vaporizatfon (surface boiling) has already started on the wall. As steam bubbles are formed, they pass into the water flow and are condensed. In this section, the water flow is heated in the temperature range up to t'. This section too, belongs to the economizer, but the heat transfer in it is more intensive than in section I. As heat is further supplied, stable bubble boiling begins in the tube: t1 = t' and t 1 > t'. In this boiling zone (section I II), heat transfer takes place irrespective o[ the flow velocity (at velocities typical for steam boilers), but is determined by the heating intensity q and the thermophysical properties of the liquid and steam at the saturation line, which depend uniquely on pressure p. In the pressure range 0.4-16 .MPa, the heat-transfer coefficient in the zone of intensive boiling can be approximated by the empirical formula ~ . - 0 · 31•po.4sqo.7 (10.2) . '-"bo II • Heat transfer in this section takes place at a substantially high heattransfer coefficient [rz 2 = 50-100 2 kW/(m K)J, and therefore, the temperature of the tube wall only slightly exceeds the temperature of the flow even with very intensive heating, as is possible in high-forced furnaces of steam boilers without running the risk of scale deposition (see Sec. 10.4). Such conditions exist along the whole length o[ evaporating tubes in drumtype boilers (section III) where the mass steam content x at the outlet 1s rather low (less than 20%) due to the high circulation ratio. In once-

H5

through boil ers, the steam content varies along the length of the evaporating tubes in the range 0 < x < 1. In tube portions where x is relatively low due to extensive boiling, a 2 is rather high and can be calculated by formula (10.2). Under such conditions, t 1 is close to t'. ·· Beginning hom a certain steam content Xcr. which depends on pressure and heating intensity 1 th_e temperature of the wall rises st~;bstantial­ ly (boiling crisis associated with ' water film drying, see Sec. 10.2),, which is indicative of deterioration of the heat exchange conditions (section. IV). For this section, i.e , in the zone x > > X cr. or upon occurr~nce of a. .boiling crisis, it may be assumed with a good approximation that the coefficient of heat transfer from the wall to steamwater mixture (rz1 ) varies . roughly proportionally to the linear velocity of the steam-water mixture to the power of 0.8; a similar relationship may be assumed fr9~ the. coefficient of heat transfer t.o · dry saturated steam a*. Therefore:

'wm

•

•

t ••

(10.3)

'. Assuming that the steam-water mixture in the zone of deterioration of heat transfer moves as a homogeneous medium, the coefficient of heat transfer can be determined as for dry saturated steam whose average velocity is equal to the velocity of the mixture: '

Wm

-. = x Ul

'

'

p• +-,.. (1-x) p

(10.4)

The formula for a 1 in the . zones of deterioration of beat transfer has been proposed by Z. L. Miropolsky [61]. I t includes a correction factor y to account for inhomogeneity of the fluid in the zone at a steam content that is slightly less than unity: Nu1 = c (Re")o. spro · 8 t' '

• ]0 8 +f,(1-x) ·. y [

X _x

(10.5)

·H6

Ch. 10. Temp erature Conditions on Heating Surfaces

~vh ere

Nu1 =

i/

y :: 1 -0.1 (

I •

I

I

where Nu 0 is the Nussel t nu mber under isothermal conditions:

kW/(m21()

and Re" = (w:~ d

'The correction factor y, w hich de-pe•~ds on the steam content and p' /p • rat1o, can be found ily the empirical form ula:

~: - 1

r·"

oc 15 +-~ - ---

10

·u nder impairment 'of heat transfer conditions, the coefficient a. 2 is substantially lower than in intensive boiling, but is still sufficiently high to ensure reliable operation of the m etal of the heating surfaces at a properly selected mass lvelocity of the ilow (see Sec. 10.2). vVilh an increase in pressure, surface tension decreases , so that impairment of heat transfer occurs at a lower value of Xcr· T he same effect on Xcr has an increase in tho heating load (due to quicker ovaporaLion of moisture in the tubes). The boundary of the transition to .deteriorated heat-transfer conditions ·can move along the length of an eva-

~

5

(1 - x("

On atlaining a maximum, the te m·perat ure of the heated tubes decreases. 'T his is due to the fact that heat "transfer is mor~ in tensive with an :increase in l.he linear velocity of :steam in the zone where the liquid pl1ase is evaporated completely (section V). A t the boundary between sections IIf. and IV for drum-typo boilers or lf and VI for onco-tllrough. boilers, tile• mass steam con tent be-comes equal to unity (x = 1) and the •enthal py is i •, so that any further :supply of hoat results i11 steam super~ .heatiug. '· In all types of boilers, in tho super:hoater region (section VI), heat translfer to s uperheated s team again wor:sens and the temperature of the tube metal increases. In this region, the laws of convective heat exchange for a s ingle-pllase medium become valid .again - but for superheated steam we .have: (10.6) N u ,, = cR e.0 · 6p rl0·8

.

20

-......

' 0

t

Fig. 10.5. Wall and flow temperatures in the water-steam path of a supercriti calp ressure boiler 1- tl ow temperature; 2- wnll temperature al low q;

HT

10.3. ·r emperatarc Conditions Along .the Channel

3-Wnll temperature at high q

porating tube d epending on certa in operating factors. T he metal at that boundat·y is subjected to temperature ch anges which may cause thermal fatigue. The amplitude of tempet·ature variations can be decreased by I imiting the temperature g•:adient between the in ternal wall Of a t ubo and the flow in the transition region (6.t :s;;;; ,s;;;; 80°C}. This is attai ned by maintaining a sufficientl y high flow velocity. Curves of the temperature variatio ns of the flow and tube wall in a vertical once-through circuit o[ supercritical pressure at various heating intensities are shown in Fig. 10.5. The entire region of steam generation cart be divided into three h eat-transfer regions: I or I' -water preheating; II or [['-pseudo-boiling; and IIIsteam Superheating. [n l'egiOil f , tho temperature oE the fl ow and woll at a given pressure is below the temperature of conditional phase transition, tph· The thermophysical prope.·ties of the working fluid in t he boundary layer on the wall vary only slightly. UndeL· such cond itions, h eal, Lransfer is governed by convection and can be calculated by formula (10.1). The wall temperature in r egion I increases slowly, following v adations in l,he flow temperature.

.....-::

/,.... ......

!;

v

""""~ t'<

J

\ 4

8

Rc,Prt N u 0 = ------=~-=----

I

2

1.07 -l- 12.7

!'...

5 2000

Other quan liLies needed fo1· the cal cul ation: Gr is the :G t·ashof number:

JrJ/Irg

I

Fig. 10.6. Effect of the working fluid enthalpy and of heat flux on the }teat-transfer coefficient at supercritical pressure P - 24 MPa, wp = 700 kg(m' s); q, k\V/m': 350; 2-520; 3-64 0; 4-7 50; S- 870

J-

ln region II, the temperature of the flow remains lower than tp 1., but the temperature of the wall is higher th an tp 1, . For this reason, the tbermophysical properties in the boundary layer and in the main fl ow may differ substantially (see Sec. 9.5) resulting in the features s pecific to the heat transfer in that region : heat transfer is intensified at low values of q (line 2 in Fig. 10.5) and, on the con trary, is impaired at high values of q (line 3). Figure 10.6 shows the effect of q on a. 2 in the phase transition region. As may be seen , a. 2 has a maximum at a low q and decreases to a minimum at higher values of q. With a decrease in the flow velocity or an increase in the heating load, the region of unit enthalpies typical of the im paired beat tr ansfer becomes larger. Ats'upercritical pressures, heat transfer in this region is caused by two m ain factors: tho effect of varying properties of the fl ow on the processes of turbulent mass exchange and the appearance of noticeable thcrmogrnv itational forces (free convection). Th errnogravilutional forces can be characterized by the ratio Gr/Re 2 . At Gr/Re2 < 10 - 2 , the effect of free convection ca n be neglected and boat transfer in a vertical ascending flow can be calcul ated by the formula proposed by V.S. Protopopov:

Nu, =- Nu 0

(.2_)n (_E_I_ )"' c ,t PJ . 1

(Pr2f'-1)

(10.8) l

IJOO

V{

(10.7)

Cr -

g

(Pt -r.~!l'!:. . VJPt

~ is the coefficien t of hydraulic resi-

s

stance, = (1.82 log Re - 1.64) - 2 ; cp is the mean specific henl.: -c

It - it - _..:.......__,_ p -

It -

It

is the specific heat of the flow~. and Pt and p 1 arc the How density at. the flow temperature and wall temperature, respectively. The exponents' n and m can be foun d by empiricaL formulae. At Gr/Re 2 = 1 0 -~-0.4, the contribution of gravitational forces is large· and heat transfer deteriora tcs owing: to a partial degeneration of the turbu-lence in an ascending flo w. For such• condi tions , the calculation should be· corrected for the effect of free convection, i.e. for Lhe ratio Gr/ Re 2 • The conditions described by curve 3. (see Fig. 10.5) can ap pear at q > · qt,. at a given value wp or at qlwp >> lq (wp)Jb 122). In region III, the temperature of' the flow exceeds t 1, 11 and heat transfer is governed b y the laws for superheated steam [formula (10.6)]. Bent channels. T he tubing of a boiler· or steam generator has straight tubular portions and numerous tube bends , such as bent tu bcs in cylindrical cyclone primary furna ces , coi led tubebanks in nucleat· steam generators, bends in multi-p ass sections of water wall, bent tubes around burner ports m a nholes, etc. ' Tlto motiou of flui ds in bent channels can be characterized by the appenl'ance of centrifugrll forces directed towards the outsid e of a beud. The· cp 1

i20

Ch. 10. Tempuatu.rc ·Conditions on Heating Surface.s

at that point should t)le heating . be uniform with a heat flow qmnx· . Non-uniformity of the temperature field is most essential for horizontal tubes of subcritical-pressure plants and also for tubes of any orientation of supercritical plants in zones where the working fluid has a high heat capacity. At supercritical pressures, the thermophysical properties of the \\rorking fluid may substantially change in a rather narrow temperature range (Fig. 9.6). In vertical heated tubes, this may result in a density gradient along the radius and the formation of a zone of lower density and lower conductivity near the heating surface whose temperature is naturally higher. This can lower the intensity .of heat transfer and worsen the temperature conditions on the heating surface. In horizontal tubes of a diameter more than '15-20 rom,. even if h eated uniformly over the periphery, in Lernal heat transfer in the upper portion is much worse than in the lower, which is due to asymmetry of the mixture flow under the effect of gravitational forces. F or this reason, the range of deteriorated heat transfer is larger in horizontal tubes than in vertical. In inclined tubes, even when heated uniformly over the periphery, unsymmetrical two-phase flow can also appear under certain conditions, resulting. in poor beat transfer in the top portion of a tube. Thus, the region of deteriorated heat transfer is l arger in inclined tubes than in vertical, but smaller than in horizontal tubes. At subcritical pressures and separated flow of the working fluid, heat transfer is unsymmetrical and the temperatures on the upper and lower surfaces of tubes are different (Fig. 10.9). Heat transfer at the upper tube surface, where the wall temperature is close to the saturation point, is worse than at the lower surface. This is associated with particular conditions of two-phase flow in horizontal tubes where the flow can be separated

10.4. Temperature Conditions around Periphery of Channel

For a ,platen wall, the. t.emperatur& of a fin root 'is found froui the formula

180" 0.9 0.8 0.7 0.6

I

2

......... qfurl' =0.2!! /7 ~~ 0.42 ./ /I ,\ VI

~

0.5,2{10

(1! ~) J

\ "--._0.53./ I "0.84 / 1600

v

'(10.16)

2fl01l t.lrJ/Ir!!

3 o 9o tou 27fl Jofl" An!lle from lower· line of tube

Fig. 10.9. Overheating of the upper surface of a horizontal tube with a separated flow inside, relative to its lower surface J-p

= 11 MPa;

Z-p =

1 B MPa; 3-p = 22.4 MPa'

by gravitational forces (see Sec. 9.3). Waves which form on flow separation can periodically splash onto the over., heated wall. Repeated water splashes cause a sharp cooling of the tube wall. Temperature variations in Ute metal can cause fat igue phenomena. For this reason, in free-circulat ion boilers, which are characterized by low velocities of flow in evaporating tubes, horizontal tubes are not heated. In once-through boilers, flow velocities are substantially higher, and therefo-. re, flow separation does not occur within a v,rider range of heating loads. The superheating /:;.t of the upper portion of a tube relative to i ts lower portion (upon flow separation) can be somewhat diminished by increasing, the thickness of the tube wall and • the heat conductivity of the tube metal, which causes heat to spread thr~>Ugh the tube metal. At supercritical pressures, the wor-. king fluid can separate in horizontal tubes by density in the vertical direction. For th is reason, under identical conditions, Lhe temperature of the upper portion of a horizontal tube turns out to be higher than the wall temperature of a vertical tube. The ratio a.h";,~/a ocr depends on the enthalpy of the flow and the parameter q/wp (Fig. 10.10). F inned tubes. As has been given earlier, under id en tical conditions the

Fig. 10.10. Ratio a~~Javer at supercritical pressUJ'e (p = constant)

heating load near the front wall on a smooth and a finned tube is roughly the same. This makes it possible to calculate the wall temperature on the front surface of finned tubes of water walls (equal one-sided heating of parallel tubes) as for smooth tubes [see formula (10 ..12)1. To estimate the reliability of the tube metal, it is essential to know the temperature of metal at the tip of fins . According to (14], with a symmetrical temperature . field (the same diameter of welded tubes and the same temperatures of working fluid and h eat -transfer coefficients a 2 in the tubes), the temperature at the tips of the fins can he found from the formula: (10.13) The coefficient lew considers the effect of welded seams; this can be taken from reference data. The temperature gradient between the root and tip of a fin can he found as: (10.14)

where A is the coefficient of the fin shape depending on the geometrical characteristics hlbr and b11br of fins (to be taken from reference data) . The t emperature in the root of the fin s of a furnace water wall can be expressed by the formula: 1.

tr = t}l + ~J.Lrqmax [ a 2

+ Am (1~~) fit

121

J

(10.15)

When it is essential to obtain the· mass velocity of the flow, use is often made of multi-pass hydraulic systems in t.he form of individual banks or sections connected in series so that, as the working fluid [lows through them, its temperature· and . enthalpy gradually increase. 011 the qther hand,. these sections are arranged in parallel relative to the heat-transfer· agent (combustion products), so that each of them absorbs roughly the sam& quantity of heat. Under such conditions, an unsymmetrical tern perature field can appear in the extreme tubes of adjacent sections. If these tubes are welded together, as is done in gas-tight water walls, the temperature difference between connected fins can give rise to appreciable temperature stresses, sometimes causing fracturing in the welded sections. Temperature asymmetry in gas-tight (membrane) walls creates the problem of proper temperature distribution in such walls. The problem is solved by applying the superposition principle: with a number of external actions. it is assumed that their effects can be added together. The quantity of heat absorbed by a finned tube c,:onsists of two components: the heat absorbed in the cylindrical portion (from the front point to the fin root) and the heat absorbed in the fin proper. Each of these portions has a particular tempera Lure field which can be added together to obtain the total temperature field of a finned tube. P roblems with temperature symmetry (Fig. 10.11a) are solved by assuming that the heat absorbed by the· connecting piece between two tubes is distributed evenly between the tubes, and that the maximum of temperature is in the middle of the piece.

1.22

Ch. 10. Temperature Condition s on Heating Surfaces

0

Tubef

i!t =t,.,-Lr

• Tube2 t'

10.5. Heat Exchange In Steam Generators

0

H (a)

(a)

'L( l

s(IJ/tf

1

\ j· }r..

mH ( -m H II l~ig.

' {h)

10.11. Temperature distribution in n

connecting piece

t< 1l - t< 2l; (b) nsymmclricnl cond itions. t< 1 l > t< 1 >

h -......:...,

I

Y \

s(2)/lf

.

//Jjd

-<~ "

.

0

"'1 '/(

{<>) lbcrmnl symmetry ,

With Lam perature asymmetry (Fig. 10.11b), Lhe temperature maximum is shifted towards the tube with a lower temperature level, so that the distribution of heat between the tubes will be uneven: it will be divided in proportion to tho length of the piece from the point of the highest temperature to a particular tube, i.e. mH for the tube at a higher temper ature and (1 - m) H for that at a lower temperature, where m is the coefficient of displacement of the temperature maximum; its calculations can be found in reference book~ [141. Temperature conditions in a finned tube arc determined by its thermal and geometrical parameters . In Fig. 10.12, typical temperature fields are shown in terms of excessive temperatures, i.e. tempet·ature differences between t he metal and the working fluid, for a developed half-tube. For all tho curves shown, it is typical that tho highest temperature is in the front zone <>f a tu ho a nd at the tip of fins. Betw een these points, the tomperat1.1re decreases monotonically from the front point to tho fin root, then rises sub-

Fig. 10.12. Temperature distribution over the outer surface of a finned tube of water wall (a) at different values of internal hentt~ansfer coefficient a 2 (cd < ai < al); (b) at different relative pitch s/d (s 1 > s~ > s3) stantially c1t the fin tip and again decreases from the fin tip to the rear side of the tube. The effect of deposits on temperature conditions on heating surfaces. We h ave discllSSed temperature conditions on clean (deposit free) heating surfaces. S1.1ch conditions can be en- • sured by proper organization of the processes of steam generation. On the other hand, in boiler operation there is almost always a potential danger that deposits will form from impurities whi ch are presen L, in a d isso I ved or susp ended state, in water (see Chs. 14 and 15). When deposits are present on the inner surface of tubes, th e wall temperature is found from the formula: 1 1 t~n=tfz +qin(...!...+ ~' " ') as r!cp

(10.17)

With a larger thickness Odcp of the deposited layer and with a lower hea t conductivity 'J..dcP• the thermal re-

123

which always takes place over the cross section of the core (especially with intensive heating which is typical o£ nuclear reactors), the differences in the flow rates of the working fl1.1id in the cells and over the periphery of rods may result in different unit heaL absorp tion dq/d (wp) (per unit flow rate of the coolant) and the appearance of temperature gradients between the rods. Since there is no active turbulent mass exchange of the fluid through the narrow spaces between the rods in t he cells, the velocity field of the working fluid may be distorted oven more by deformation (bonding) of the rods undel' the action o[ temperature gradients. The critical heat flow at which detodorated conditions of heat transfer can appear on heating surfaces deponds on the following principal factors : steam content x , mass velocity wp, and pressure p. For a system of parall el channels in a boiling reactor, these parameters are average if each of the channels receives the same quantity of the working fluid, so that Lhe 10.5. Heat Exchange in Steam heat transfer crisis in rod bundles Generators of Nuclear Power Stations is qualitatively the same as in evapoThe unit heating load on the surface rating tubes, but the quantitative <>f fuel elements is quite high (1 000 relationships turn out to be substankW/m 2 or even more). This circum- tially different. Considering all the st ance makes the appearance of poor geometry specifics of rod assemblies cooling conditions in fuel elements and Lhe condi tions of their operation, differences in unit heat release or heat very probable. The steam-genet·ating channels of absorption can result in differences in nuclear reactors have an intricate sha- the flow rate wp and steam content x. pe. In reactor construction, wide uso Since qcr is a decreasing function of is made of heating surfaces in the form x, a boiling crisis is more likely to of bundles of heat-releasing rods bet- occur in the space between tubes at ween which the heat-transfer agent a lower limiting steam content than in tubes. flows in a longitudinal direction. Various types of flow whirlers are The geometry of chan nels formed b etween the rods may have a signi- mounted in tho outlet section of a reacficant effect on the hydrodynamics of tor channel to increase its power ; t he working flu id: the flow rates thro- they are often made integral with ugh the cells of a rod bundle may turn spacers. Under the effect of whirlers, out to be different or the velocity of water droplots are thrown from the the working fluid may vary over tho flo\v core onto the walls and replecross section of a cell; this may result nish the moving liquid film on the in the formation of diffori ng cooling walls . Thus, a boiling crisis caused by conditions over the periphery of Lhe Liquid film drying is delayed and at rods. With uneven heat absorption, the outlet the steam-generating reac-

sistance across the wall to the working fluid is higher, and thus, .the wall will have a higher temperature. The conduct ivity of deposits depends on the composition of impurities in the water and may vary within very wide limits-from 0.1 to 0.5 W /(m K) for mineral deposits and u p to 3-5 WI /(m K) for ferric oxides. Since the conductivity of deposits is substantially lower than that of the metal, even slight deposits, in a · layer of <>nly a few fractions of a millimetre, in the tubes of water walls, which are heated quite intensively, can raise the wall temperature to a value that is inadmissible for reliable operation <>f the t ube metal. Since d eposi t.s grow gr adually during boiler operation, t h is circumst ance limits the continuous operation of a plant. For this reason, one of t he most essential problems in boiler operation is to prevent or restrict the formation of deposits on the steam-water side of heating surfaces (for more detail see Ch. 15).

I

124

tor can operate at a Jaigh er s team content. I n steam generators operating in combination with pressurized water-cooled reactors or sodium-cooled fast-neutron reactors, during evapora tion the working fl uid passes through the same stages as in organic-fuel steam boilers . The only point of significance is that deteriorated heat tra nsfer in the steam generators of nuclear reactors does not involve failure of th e apparatus, since the temperature of its elem ents cannot exceed that of th e heat- tr~n s[er agent. However, . an increase m the length of the boiling cris is zone, where th e i nlcnsity of heat transfer is low (sec Sec. 10.2), r esults either in an increase in the steam generator size or in a loss o[ its capacity .

Moreover, the wall in the zone of liquid film drying is a lternately washed with liquid and steam, causingtemperatu re pulsations in the wall (which can be quite substantia l in. liqu id-metal heating) a nd varying stresses in the m etal, which can resu lt ifr tube cracking. All these phenomena arc probablein steam generators where the workingfluid mo ves in the narrow cells of a channel or in chann els of a substAntial length. - On the other hand, boiling crisis in the reactor core is extremely dangerous. To prevent i ls occunencc iu wat er-moderated water-cooled power reactors, th e heal-transfer agent of th o- . primary circuit. has a tom pornl.u re a few tens of degrees below th o hoi ling point (subcooling margin) .

125

11:1. Classification of Open Hydraulic Sydem1

Ch. 11. Hydrodynamics of Op en Hydrazlllc _S ystems

•

•

Heating surfaces '

. ..

..•

.

plalen-lgpe

ra diant

convective

.

.. I • I I

v . v· I/ \_i ._~ ~

.

B .

\

'

\

\

.

~

'(b)

"

•

I

l

F F

' 'I •

--

(ti}

(C)

'

' n

(h)

(g )

~~

I

l

~

••

(a) I

-

'-

\_ j

6

'

I

(i)

F

I

. (l)

F

t

'

.

•

(m}

{j)

{k)

(f)

feJ

Fig. 11.1. Principal types of hydrau lic circuits with forced motion o[ the working fluid ( a) horizon tally wound water wall tubes; (b) vertica l nrrnnl!Cment of water wa ll tubes; (c) U-sbaped water wa ll : (d) N-shaped water wall;(<) multl-pa'l.• water wnll witll vcrli cul tubes; (f) mulli-pa_ss wa ter wall with hor izontal tubes; (g) L-shaped platen: (h) double L-sllaped platen: (i) horizontal platen: (J) U-sbaped platen; ( k ) multi-pass vertica l platen; (I) vertica l convection bank; (m) horizontal con'l'eclion bank

HYDRODYNAMICS OF OPEN HYDRAULIC SYSTEMS 11.1. Classification of Open Hydraulic Systems The principal cliagrams of open hydraulic circu its in heating surfaces are sh own in Fig. 11.1. In any circuit the r eliabl e oporaliou of the steam~ generating channels depends substantially on stability of mot ion, i. e. a s table flow rate of the \vorld ug flu id through parall el tubes anct channels . Under parti cula r condi lions (pressure, mass velocity, cuthalpy of the wo rking flu id a t th e inl et heating intensity) a nd depending o~ the design of th o s tcam-generati ug channels. uns table motio n of the work ing fluid (i.e. with a variable flow rate) can appear in them. One must distinguish, b etween static a nd dynamic instabi-

liLy. S ince instability, per so, is a dynamic process, tho term 'static inst ability' is rather cond i tiona!. Under statica lly unstable conditions, the fl ow . rates through various tubes (channels) are different and may vary in limo with an apprecialJle hequ ancy. Th o flow rate in individual tubes may turn out to beinsufficient to ensure proper healtransfer conditions. Varying conditions of cooling of t he t.uLes aud variations of the wall temperature may cause ther mal fatigue of the metal. Jn som e cases, especially in operntion at, variable or off-design loads, dynamic (or oscillating) insta bility of motion appears in lh e for m of excessive flow pulsations which lead to variable flow r ate of the worki ng fluid .

.

.

.

d ecreased cri tical heating load, cyc- 11..1a), the length L of a single tube may be as high as a few hundred melic temperatu..re variatio ns in the heatres and tho tube has a large number t ed walls a nd, as a fi nal result, to of bends and therefore, a high hydrauemergency situations. lic resistance D.p hr· The hyd rostatic Out of all the causes or hydraulic component D.p,. of the total pressure i nstability, and therefore, of maldigradient in such a circuit is not large s trib ution of heat , special attention in view of the small heigh t of the s hould be g iven to the effect o[ a n circuit compared with the length u nsteady hydrodynamic cltaractorisof the developed tube, i.e. H « L. t ic, flow pulsat ions and hoadet'S on T ho resistance due to flow accelera; he distribution oE the flow between tion 11Pac is also not large, especially [) arallel tubes. The pressure grad ient at a high pressure. Therefore, t.he n _heated tubes can be represented as total pressure gradient in a circuit the sum D.p = 11Ptr D.p, 'D.P ac± ± D.ph (see Sec. 9.4). Combining t ho with horizontal or slightly inclined tubes is mainly determined by the h yd raulic resistances in a si nglo torrn, hydraulic resistance: 11p1, 'D.p 1 = flp 1.,, we obtain: D.p = 11Phr -1- 11Pac ± flp,, (1.1.2) . (11.1) B oiler circuits w ith ver tical ascenIn o nce-through boilers with hori- ding or ascending-descending motion zontal or slightl y inclined lubes (Fig. of tho working fluid (Fig . 11 .tb to e)

+

+

+'

•

126

Ch . 11. H ydrodynamics of Open Hydraulic Systems

.dp

t!p 0

2' I' 2

11

Fig. H .2. Hydraulic characteristics: stable, (2) unstable

(1)

are charac terized by a s mall number of passes, and therefore, by a relatively short tubes and low number of bends. Tho hydraulic resistance of such circuits is low. The hydros ta tic head in v ertical tubes, however, makes up a substantial portion of the total pressure gradient, especially at low loads when the contributio n of th e hydraulic res is tance is subs tantially lower. Thus, !::,.p = !::,.p hr

±

!::,.ph

{11.3)

The ratio between !::,.p ilr and !::,.ph in the total pressure gradient may have a vital effect on the hydraulic stability of the flow in straight-flow elements. The hydraulic s tability of flow is described by the hydraulic characteristic which gives -the total pressure gradient !::,.p as a function of the flow rate G of the working fluid: !::,.p = f (G). If all tubes in an element are of the same diameter , the hydraulic characteristic is a fu nc tion of the mass velocity: !::,.p = f (wp). The hydraulic characteris tic is u nambiguous if a particular to tal pressure gradient in the tube system has only one corresponding flow rate of the working flu id {curve 1 in Fig. 11 .2), and ambigt1ous if the same total pressure gradient can appear at two or more differen t flow rates {curve 2.) The hydraulic characteristic may b ecome ambiguous on a change i n the thermophysical properties of the working fluid, s ay, the unit volum e, which may be caus ed by a change from one flow ra te to anothe r (see S ec . 11.2) or u nder the effect of hydrostatic

11.2. Hydrod ynamic Stability of Flow In Horizontal Tubes

head {see Sec. 11. 3). The s ituation is complicated oven more by the fact t hat the effect of hydrostatic h ead in ascending and descending motion of the working fl uid is different. All this results in i ntri cate analytical relationships which are represented gra phically fo r each particular object: hyd raulic circuit , geom e Lrical pa rame ters of th e tube system , pressure, entha lpy of the working fluid at the inle t , etc. The hydra ulic operation of a ny circuit is de~cribed by its hydraulic ch aracteristic. H owever, becaus e of Lhe above-m entioned s pecifics of hydraulic s ystems with horizontal unc.l ver tical t ubes, t heir hydra ulic characteris tics will b e anal yser! separately. 1 1.2. llydrodynamic Stabilit y of Flow in Horizon tal Evaporating Tubes The decisive factor that determines the hydraulic characteristic of evaporating tubes is the temperature of the fluid at the tube inlet. It may be equal or close to the saturation temper a ture at the pressure at the tube inlet, t 1n ~ t', or be substantially lower than that temperature, t 1, < t'. If a tube is s upplied with subcooled water, st eam generation begins not immediately at fts inlet, but a certain dis tance downstream. The whole' length of the tube can then be divided into two portions: tho economizer portion and the evaporating portion {Fig. 11.3), their lengths being determined by the ratio between the heat rel"ase and flow rate of water.

c

UQ;i 0

t in
v';i';:r=O

x= :"

v;c;ix;:r"

t t t t t t h ley

lee

l

Fig. 11 .3. Vari ati ons of the working .fluid param eters in a heated tube a t tin < t'

For the same heat release rate, an increase in the water flow rate in·creas es the length of the economizer p ortion . while shortening that of the evaporating portion, which is associated with a lower quantity of s team produced by the heated tube. If a tube has an economizer portion {Fig. 11.3), we have:

{11.4) Tho unit volume of the working fluid in the economizer and evaporating portions is different. The lengths of these portions are different too. The unit volume of water varies only slightly along the economizer portion, and therefore, we have:

-

v0 + v'

2

{11.5)

-

In the evaporating portion, the unit volume of water changes sharply, and it is reasonable to take its mean value in calculations. For tubes that are heated uniformly along their le ngth, quite accurate results are obtained by assuming a linear law of variation of the steam content in the flow, and therefore:

-Veo = V '+

z(v· - rl) 2

{11.6)

With uniform heating, the heating load per m e tre of tube length, q 1 = = Qll, is constant. The length of the economizer portion will then be found as lee =

(i' - io)

QG l =

lH,ub (wp) f

(11. 7)

ql

and the length of the evapora ting portion: (11.8) Subs tituting Vee • z•• , Veo • and z•• into equation {11.4) and arranging the terms according to the powers of (wp), we obtain the cubic equation:

D. p = A (wp) 3

-

B {wp)2

+ C {wp) {11.9)

127

where

A=

~ (v" - v')

t.t!ubf

4dqt r

B = ifr[t.i~ub (v• - v')-v']

(11.10} (11.11)

and

C=

~ (v• -v') l2ql

4jdr

(11.12)

If there is no water subcooling at the inle t of an evaporating tube, !::,.i,ub = i' - i 0 = 0 and the coefficient A turns to zero and B changes. s ign. Equation (11.9) then takes the f o m: !::,.p = B {wp) 2 C {wp) {11.13)

+

i.e. it is a quadratic equation which expresses unambiguously tho hydraulic characteristic. Equa tion (11.9) is the equation of the hydraulic cha racteris tic of a n evaporating tube at !::,.i•ub > 0, i.e. in the presence of economizer portion. In this case, the h yd ra ulic charac teristic is described by a cubic equation whose solution can have either one· real and two imagin ary roots , or all three real roots. I n the former case, th e characteristic !::,.p = f (wp) has neither extrema nor common points with a h or izontal tangent {curve 1 in F ig. 11 .2) and is unambiguous, since any value of pressure gradient !::,.p has only one corresponding flow rate of the working fluid, wp. In the latter case, the characteris tic has an inflec-· tion point a nd two extrema; it is ambiguous, since any value of pressure gradient !::,.p has three corresponding different flow rates {curve 2). Ambiguity of the hydraulic characteristic will become clear from the following r easoning. For the same h eatrelease rate, as the flow rate of water s ubcooled ur1der the saturation point increas es, the volume v elocity of the mixture is not increased, but decreased . An increased flow rate of subcooled water leads to an increase in the le ngth of the economizer portion and in the quantity of heat spent for b eating the water to boiling . Accor-

128

11.2. H11d rodynamlc Stability of Flow in 1/orlzontal 1'ubcs

Ch. 11. Hydrodynamics of Open Hydraulic Systems

JJp

129

4p

IU')'

4

Fig. H .5. Region of the hydraulic characteristics of a tube producing steam-water mixture

lt!Cf

l

Fig. 11.4 . Effect of the flow rate of subGOo led water in a steam-generating tube on flow velocity at tube outlet 1-11- wnter now rates

I

0~------------~~

0

dingly, the re remains less h eat available for evaporation, and therefore, tho velocity at tho outlet decreases s ubs tantia lly {Fig. 11.4). This continues until the evaporating portion disappears in the tube. As soon as steam generation is interrupted, a further increase in the flow rate of water causes a proportional rise of the velocity along the whole length of the tube, and therefore, a change in the hydra ulic resis tance of the path. As the w a ter flow ra te increases, t he hydraulic resis tance of the economizer portion, !:.Pee• increases while that of the evaporating portion , t:.p •• , decreas es . Depending on t he ratio between !:.Pee and t:.p •• , t ho to tal hydraulic resistance of the path may either increase or d ecreas e with an inc rease of the load in a pa r t icular ra nge of flow rates , resulting in t he hydraulic characteris tic being either unambiguous (steady-s tate) or ambig uous (uns teady-state). L et us consider the hydraulic c haracteristic of a once- through element, which bas three real roots by equation (11.9), see Fig. 11.5. At v ery low flow rates of water, (wp) < (wp), and s uperheated s team will form at a g iven heat release rate , since the path

has neither an economizer nor evaporating portion, bu t is essentially a superheater along its whole length. At very high flow rates of water, (ri.lp) > (wp)w, the available heat is insufficient to heat the water to the s aturation point , there is no s team generation , and t he whole pa th is essentially an economize r which d elivers w a ter. For those extre me cases, the hydra ulic ch arac te ristic is described by the p arabolic equation:

b.p =

S (wp)2 2

l)

(11 14)

As follows fr om this equa tion, with the same mass v elocity o'f the flow , the resistance to the motion of ·steam is greater thari tha t to tho motion of water, Since V 1 is greater than V w • If a s team-water mixture forms in the path [the range of flow rates (wp). < (wp) < (wp)wl. the characteristic may become uns table and the flow rates ma y v ar y, so that a steam~ water mixture with widely different s team contents will be delivered periodicall y . S ome of t hese steam contents m ay turn out to be excessive for the give n heat release rate, meaning that the cooling of the tubes will become unreliab le. An unstable h ydr aulic charac teristic can result in differe nt flow rates through individual parallel tubes that are connected to a com mo n heade r. A m ore dangerous s i tua t i on , however, r~sults when the flow r a te through a tube c ha nges. For th e same heat re-

Fig. 11 .(). Effect o[ pressure on tll'e sta bility of hydra ulic charac teristic {P1 < P 2 < P 3 )

lease r a te, t his can cause t emperature varia tions in the tube wall and the a ppearance of critical fa tig ue s t resses and , eventuall y, d amage of the evapora ti ng tubes . Sin ce t he ambiguous hydraulic characte ris tic is mainly due to a l a rge d ifroronce between the u nit volumes of wntor a nd steam, a n incr·ease in pressuro can ma ke th e ch a rac teris ti c more s ta ble and tho motion of the wo rking fl uid more stead y-s late {F ig. 11. 6). As noted earlier, i nstability of t he hydL·a ulic characteris ti c is associated with the presence of an economizer po r·l ion in a tube. F or this reason , wi th a hig her degree of water s ubcooling a t t ile tube inle t , the ch aracteris tic is less s ta ble (Fig. 11.7). On th e o the r ha nd, if s t:ilicooling is decreased by rais ing the in le t te mpe rature c lose to . t', another proble m ma v result: as tho s team-water mix t.m·e enters the inle t header of a bank of evapora ting t ubes, i t will be separated into steam .tlp

~ig.

11 .7 . Elfcct of water 6ubcooling a t the •!llct of a steam-generating tu be on the slabilt ly or hydrauli c characteristic (p ~ consla nt) 9- 0 1524

Fig. 11.8. Tra nsfo rma t ion of an unsta ble hydraulic charac terist ic into stabl e by throttling subcoolcd water fl ow

a nd wa ter in t he header , so that some tubes will rece iv e more wa ter and less steam than othe rs, i.e. the flow r ates of water and s team in them will be sharply different and uncontrolla ble; this can lead to overheating, and even bur:n-throng h of tho ev apera ting tubes whic h a rc hea ted q uite intensively i n t he boiler furnaces. For the a bove reasons, once-through boilers are made with a non- boiling economizer section. N o-boiling conditio ns m ust be ensured no t onl y a t the r a ted load, but also in ope ration under nny load and with any kind of fu el for which the boiler has been designed. An unstable hyd raulic characteristic can be stabilized by introducing a n addition al res is ta nce in tho economizer portio n of the hydraulic s ystem, which will va ry by a parabolic law depend ing on the flow r ate of t he wo rking fl uid . As s hown in F ig. 11.8 , tho initial a nd the addi.tion al charact.e ri s ~i c a1·e added graphically so that t he resul Ling characteristic is stable. T he hydra\Jlic resistance of t he econom izer portio n is us ually increased by two me thods: by increas ing its resistance locally (moun t ing a n orifice plate) or at a n app reciable l en ~th (providing a s tepped tube) . "' Orifice pla tes. Orifi ce plates produce a pressure g radien t proport ional to the sq uaro of the flow r ate of t he s ingle-phase fl uid tha t passes t hroug h

Cll . 11. llydrodyllllmtcs of Open Hydraulic Systems

132 '-int

dp

1

u

'f.dphr

-tJ i

4i

\.

2

' In

obtain the hydrostatic head 'Ztlp 1, in a IT-sh aped section. The total hydraulic characteristic of a n-shaped section 'Ztlp = 'l:, tlphr + Li:!.ph {1'1.19}

ItJp "' rtJPnr ' '£Lip11

2 ioul

0 (a)

LJpf"

- 4p

u (6)

---- - ,v;o

l in

Fig. 11.1 3. Hydraulic cliamctcri s ti c of a ll-sbaped heated tube ~ i - J u• n t

:.hsorrltio n hy the tul>P

elemcn~

'I

I

in which enth alpy i 1n 1 var ies according to tho flow rate at a given heat release. With a smaller ~low. rate of the '~orking fluid, itnt 1s h•gher (curve I m F ig. 11.13b), so that the enthalpy of the working fluid sharply increases in th e descending element of the tube (curve II), especially in the outlet portion. With a decrease in the flow rate, this results in an essential decrease in the flow density and, accordingly, in the hydrostatic component tlp~"' . ·, ·; Adding together the h ydrostatic components fo r the ascending an d deseending element, tlp/!" and 6.1."' , we •

4p

t

llmhi!J.UOUS

lin lout

reg1011

o~~~=====~wp ~ r.tJp, - 4p

------~Lipf"

l:i g . 11.11t . H ydrau lic chamclerislic U-shaped heated tube

of

11

may be ambiguous in a wide range of flow ruLes of the working flu id {Fig. 11.13a). F ig ure 11.14. shows the total hydr aulic characteristic and its components for a U-shaped section, which has been constructed according to the method discussed above. Tho hydraulic resistances in tho ascending and descending branches arc positive aHd are thus added together ('l:,tlph,)· T he hydrostatic component in the descending branch, 6p1.''", is negative and tho enthalpy of flu id at the inlet to this hranch, i in, is assumed to be constant and independent of the flow rate . The beginning of th e ascendincr motion coincides with tho end of th~ descending motion. The hvdrostatic componen t of tho asccndin.g moti on is positive, but the en thalpy at the inlet to the ascending portion (in the point of transition from descending to ascending flow), i 1n, is dependent on the flow rate. A t low flow rates ' i ;n t is high and the density of th o fluid is low, which l1 as a retarding effect on the rise of the hydrostatic head with the increasing flow rate, in t h at zone. Only in the zone of large flow rates at which tho quantit y of heat por unit flow rate of the wot·king fluid is not as substantial, i. 1111 becomes lower and the hydrosta ti c head tlp;:• becomes much larger. Tho resulting curve of hydrost atic heads is denoted as L.tlph in the fi gure. As may be seen , the total hydraulic characteristic of a U-sh aped section is ambig uous in a w ide t·auge of flow r ates of the working fluid. Hydrau li c characteris tics of N-sh aped aod more complicated multi-pass systems can be obtained in a similar manner. As follows from tho above analys is, the hydrostatic pressure gradi ent improves the hydraulic characteristic of nn ascending flow and ad-

133

11.3. Hydrodynam ic Stability of Flow in Ve rtical Tubes .t.lp

tlp

3

6

J

- ..' ~ . '

4 I

0

1

wp (a}

4

0

{D)

Fig. 11.15. Ambiguous hydraul ic cha rac tcri~ti cs (a) o f·(l sing le lube: (b)

vorsoly affects that of a descending fl o.w . _In this respect, a U -sltapoc.l circu •L IS better than a fl-sbapcd, since t~o working . fluid in its outlet port•on, where •t has a high steam cont ent, and therefore, lower density moves upwards and the effect of th~ hy drostatic bond is positive. The hyd raulic characteristic is improved evcu more iu a u N -shaped system wi Lh an inlet header arr anged at the top and an outlot header at the bottom . In additi ou , this system has two ascending portions per one descending . I n general , tube sections w it h a low number or passes have either an ambiguous or insuHicien tl y stable hydraulic characteristic . With an increasing number of p asses in a h yd rau I ic system , tho effect of hydrostatic component on the total pressur e gradient d iminishes, but the role of hydraul ic resistance becomes more s_ignificant. When tho number of passes is greater than 8-10, the h ydraulic characteristic o[ a multipass s ystem approaches that of the horizon tal evaporating tubes. An unstable h ydraulic characteristic of a single tube (channel) with ascending or descending motion of the working fluid under the action of l1 ydrostatic head is realized in tho w hole region of ambiguity (curve 2-3-4-5 in Fig. 11.15a), s ince the flow rate is determined by external conditions, i.e. by pump capacity. R eal heating surfaces consist o[ a large number of pnm ll ol tubes. I n multi/.t.J,be systems, the total flow rate of the fluid is also determ ined by pump

or~

lube S)"Slem

capacity, but t he flow rates through individual tubes may tum out to be different. As has bo~n established by e~pcriment and confirmed by cxpen encc, hydraulic characteristics in such systems are r ealized only in ascend iug port ious (the leftmost curve 1-2-3 and tho righ trnost Clln·o tJ.-5-f} in Fig . 11.15b) . A decrease in t.he flow rate iu a tube uear a m iuimum of the h~draulic chnractoristic (at point 4) will cause a sudd en drop in the flow rate and a chango in the co nditions described by point 2 . A rise in !,he fl ow nlLe at a max imum o[ the eharacteristic (near po int 3) can load to a sudden increase in the fl ow rate in som o tubes of tho circuit and a ch ango in the conditions of point 5 . The descending brunch of the characteristic (portion 3-4) is no t realized, except for cases where the system docs not have many tubes (throe or four). rn th is region of flow rates. unambiguous motion cannot be ensured with the r esult that t he flow rotc either jumps into the region of lower values (left-hand branch) or inlo t hat of !1igher values (rigbt-hand branclt), while the to tal flow r ate of tbe working fluid remains unchanged. Operation on the left-hand branch of .the characted stic usual! y cannot malllLain tho required tompemturo cond it ions of inten sively heated eYaporating tubes. Tho sole real region of' operation of a once-throucrh circuit is tho right-hand branch of the clwracLeristic which inc] udes ambiguous and unombiguous por·tions. In the •u•ambiguous portion 5-{j of Lhal branch ..

134

Ch. 11. /Jydrodynamics of Open Hydraulic Systems

Ap

11.3. II ydrodynamic Stability of Flow in V ertical Tu bes

tlp 2

.

'

r

2'

0

(IU'J'Jwz

(a)

Fig. 11.16. EHccl of maldistribution on the workin~ fluid flow rate in pa rallel tubes with slnblc hydraulic characteristi cs 1- hydrnulic chnnH;I,I•ristic or un C h;Jlll'll t~ 2 -

hyrlra ullr. t' llru·aclf'rl~l ir of a m tlltuncLionlu g: lube

the v elocity of the working fluid is so high that iL often turns out to bo unfeas ible. Thus, one must opera te on the ambiguous portion 4-5 for which lhe bound:-.ries of reliability should be determined. In circuits co mposed of parallel tubes of the snme design and heated identicall y, st abil ity in the -region of an ambiguous hydraulic characterist ic can be ensw·ed for t he -entire external (ascending) branch of t he characteristic (portion 4-5 in Fig. ·1 1.15b). In a system of parallel tubes, heat -absorption is always different in view -of j the structural and hydraulic noni·dentity of the tubes (see Soc. 11.4). Therefore, somo tubes whose operating condi tions deviate from the average values for the syst em (malfunctioning tnbes) have a different hydraulic characteristic than is average for the system. As seen from F ig. 11.16, oven with an unambiguous ch aracteristic of' the system, differ on t (but s table) flow rates of the working fluid may appear in a reference tube and n malhmctioning tube. Vlith parallel operation of a system of tubes with different ambiguous characteristics, three cases arc possible: lhe pressure grad ient between the heAder of the system at the minimum of the hydraulic characteristic for a malfunctioning Lube can be e ithe r (a) sma ller than the pressure g radient in t.hc c.ircuil (Fig. 11 : 17a),

fw;p)wr

( 1/.J'f'Jmr

.dp

0

2 -~

~ ~

"" ""

on tho characteristic of the malfunctiGn ing tube). Points and 2" on the descending branches of the character istics cannot be realized (see Fig . 11.15b). Points 1' and 2' on the lefthand ascend irlg branch of the characteris tic are also not realizable as long as the condition t:..pw > llPmtn is satisfied. This means that unfavourabl e conditions in an element with a low flow rate of the worl~j.ng fluid can be prevented by providing a flow rate at wh ich the pressure gradient in the characteristic of the main number of tubes will be h igher than the pressure gr adient at the minimum of the hydraulic characteristic of a malfunctioning tube, (wp)101 > (wp)mf · With tlpw < LlPmtu• the1·e are no working points on the descending and right-band ascending branches of the characteristic of a malfunctioning tube. Under such conditions , tho only remaiuing points which correspond to the two different flow rates of the working fluid are 2' and r. At one of them (point 1"'), a high flow rate is establish ed, which ensures stable operation of the m ajority of the tubes in the circuit. At point 2', the flow rato is too low for proper cooling of the worst malfunctioning tube, with the result that the tube system will be unreliable. With tlp,0 = tlpm 111 , t he flow rate at the minimum of the characteristic of a malfunctioning tube is unstable. In this case , a pressure gradient in the tube system has two different corresponding flow rates. One of them, in the malfunctioning tube, has a low value , 2', which cannot ensure proper cooling conditions. The other flow r ate, 1"' on the characteristic of t he majority of the tubes, ensures proper cooling of these tubes. The flow rates in points 1' and 1" for an average element are unr eal for t he reasons given earlier. The condition tlpw = tlPmrn determines the boundary of stability of a malfunctioning tube, i.e. the extreme st ate up to which the operation of l,he wh ole circuit can be stable.

IUJ'

u (6)

(tu,DJ,,

Llp

(C)

Fig. 11.17. Hydraulic characteristi cs o f ITshaped circuits 1- element ; t-maltunclloning tube

(b) greater than that pressure gradient, (Fig. 11.17b), or (c) equal to tho pressure gradient in th e circuit. T hese cases are analysed with a gradually decreasing total fl.ow r ate of the working fluid, and therefore, a decreasing pressure gradient between the headers of tho hydraulic system. In th e first case, when the pressure gradient in the system exceeds the pressure gradient at the minimum of the characteristic of the worst malfunctioning tube, !J.p.., > llPmtn, i.e. of the malfunc tioning tube whose minimum of the characteristic is higher than the minimum of the characteristic of the main number of tubes , three operating poin ts are p ossible on each of tho curves (1', 1" and 1'" on the genera l charnct.cristic of the ma iu number of Lubes and 2', 2" and 2··

135

As regards the stability conditions, the lowest mass velocity allowed in a system is expressed by the inequality (wp)w ~ m (wp)m/ (11. 20) where (wp)m/ i.s the mass velocity in a system of tubes (channels) on an external branch of its hydraulic characteristic in the point corresponding to the minimum of the hydraulic characteristic of a malfunctioning Lube, (wp)w is the m ass velocity in the system, and m is Lhe reserve factor, whicl1 is assumed to be equal to 1.5. Stable opemtion in the region of ambiguous hydraulic characteristics can be ensured by throttling the flow bv means of orifice plates or stepped t ubes . Another method is to carefully select the mass velocity wp of the working fluid at the rated heating load of a particular water-wall section so as to ensure reliable cooling of the tubes at a given mini mal heating load in vi ew of their thet·mal and str uctural non-identity, Lhal is, to run the circuit on an external ascending branch of its hydraulic characteristic, d (t:..p)/d (wp) > 0. Tho latter method is preferabl e, since it does not require tube throttling. In most cases, orifice plates are employed when it is necessary to equalize the flo w rates of the fluid through parallel hydraulic elements, such as sections of furn ace water walls. In some cases, flow stability can be achieved more efficiently by changing over to another scheme of tube connection which bas no downtake tubes. The stabi liLy of the hydraulic characteristic call be affected not only by h eated (evaporating) tube sections, but also by unheated steam-circulating tubes. For instance, Fig. 11.18 shows an \lnlucky arrangement with external unheated steam-circulating tubes connected t o a lower header. The hydrostatic head in the tubes is negative and the circuit is essentially a IT-shaped system with an ambiguous hydraulic charact eristic . With hydraulic maldistribution and too low worki ng fluid veloci ties (due to a low

-

136

Ch. 11. Jlydrody11amlcs of Ope11 Hydraulic Systems

I f

I

2

Fig. 11.18. n-shaped circuit wit h un heated downtake Lubes (a) nnd its hydraulic characler isl ics (b)

dp

1-J-J'AI'IIIh•l heated wnlcr wn ll.• nnd lhciJ· n!spcct i vc h)'di'Clulic c h aract~rit-~Hcs: 4 unhcn lcd downtak•.. lubes; S- Uf•llt.'J' mixing

.i

h e aUt~r

31

\![ dp

(a)

gradient /::,.p1 ), ambiguity of tho characteristic may result in section 2 malfunctioning and havi ng a low flow rate on the lcfL-hand branch 2' of the characteristic. Thus, under completely favourable operating conditions in sections 1 and 3 (the corresponding fl ow r ates arc denoted as 1' and 3'), secti on 2 will operate unreliab ly. Reliable opemtion in the circui I, can be achieved by providing a pressu re gradient /::,.p 2 at which the working points will be on the extreme branches of the characteristics of all the sections and will have close velocit y values. It is also possible lo elim inate ambigu i ty of the hydraulic characteristics of the sections by providing a mixing header at the ir top. The operation of complex circu its with a number of once-through elements is analysed by using their r esulting hydraulic characteristics. Those are obtained graphically by adding the characteristics of individual elements together, taking account of the scheme of th eir connection. If some elements are connected in series, the flow r ate of the working fluid is the same in all of them. For this reason, the resulting characteristic is const-

ructed by ad ding together the chaL'acteristics o[ t he elements at the same flow rate of tho working fluid (Fig. 11.19a). Tho hydr aulic resistance of such a system is higher than the resistance of anyone of its clements. In complex once-tluough circuits composed of a number of elements corlnected in para llel, the hydraulic resistances of the elements are constant. The general char acteristic of the system is obtained by addi.ng together the characteristics of the elements taken at the same pressure gradient in the circuit (Fig. 11.19b) . In such oJ case, the hydraulic resis t ance of tho ci rcui t is smaller than that of anyone of its cleme nts. t 1.4. l\laldis tri bution of Heat

For reli able operation of a heating , surface (section), i t is extremel y important that all its parall el tubes oper ate und er roted (average) conditions. Jn practice, however, one has to b ear i n mi nd that the tubes of a section may have differen I. hydraulic characteris tics (d ue to Yllrialions in thei r diameter, length, surface roughness, effect of headers, or effect

dp. 2 I

0

WJ' (a)

1+2

0 (!>)

Fig. 11.19. S ummutiou or hydrauli c chamclcristics for series co nnec tion of once-through clements (a) and £o1· th eir parallel conn ection (b)

13T

11.4. M aldistrlbutioll of lleat

o( instability in opor·alion of evaporating tubes) and differe nt thermal cluwacteris tics (different heat absorp\.ion due to their dirforcnl. arrangement rei aU ve to the flow or combustion produels, uneven slagging. fouling, etc.). Differences in tho hyd r aulic and thermal charactel'istics of tuhes are especially evident in high-capacity plants with developed heating s urfaces where deviation of some ·..Plements from the rated conditions are· more probable. Undor s uch condi t ions, the working fluid m ay be distribu ted unevenly between the tubes, and thorofore its uni t enthalpy at the outlet or s~me tubes mny differ s ubstan tially from the average value. In some malfunction ing tubes , d angcrous \.em perat ure conditions m ay a ppear. T h e reliable operation of a oncethrough element can bo characterized by two groups of factors: (a) paramcl ors of an element tba t operates under average condi tions in the system: t~/' , !::,.i. 1,

0 . 11 q.z. z. 1, H e1

(b) parameters of a maHunctiouing tube (or channel) : t~~' /::,.iml• Gml• qml• Zmf• H m/ where G is the flow rate of the working fluid in an element, q is t~o quantity of beat transferred to 1t , /::,.i is the beat absorbed by an eleme nt , H is the area of a heating s urfa ce, z 1s the coeffi cient. of hydraulic resistance, and t"" is the temperature of the working fluid al. the outlet from parallel tubes; the s ubscrip ts 'el' and ·mf' s tand respccti vel y for an elemen t under average cond itions and a malfun ctioning tube. Let us i ntroduce some additiona l des ignations: p11 = Gm 11G. 1 (11.21) is the coeffi cient of hydraulic maldistri bution; p 9 = /::,.im / f>iel (11.22)

is the coefficient of maldistribution of h oa 1.; (11. 23)

is the coefficient of non-uniform heat absor ption ; (11 .24) is the coeffi cient of hydraulic nonun i rormity; (11.25)

is the coefficient of structuroJl nonidentity. . Taking the coefficient of maid is tribution of h eat. as a basis, we ca n establish the relations hip between the coc[ficients indicated: =

pq

tli,,J = t!.ier

qm!Hm/: qezffez

Gmf

(11 .26)

Gel

or, on substilu Ling Ll1e corresponding values from (11. 21), ~('11. 23) and (11 . 25): pq =

I] II a 1]< t r

p,

(11.27)

The coefficient of structura l nonident i ty 'll•tr is not associnlcd with tho processes wh ich occur in a I u be system ; it is usually estimated as l')•t r = 0.95-1. . The coefficients oi thermal and hydra ulic m aldistr ibut ion are in lorrelated as foll ows: (11 .28) Pq = 1l ha1Ph Maldistribution of heat m ay be· caused by different thermal characteristics of parallel tubes, while llydrau1ic rnaldistributioo is due to differences in their hydraulic characteristics. As follows from equation (11 . 28) , maldistribution of heat can be caused either by different beat absorption, by hydraulic maldistribution, or by both. Maldislribution of beat depends not only on the extent of thermal nonuniformity and hydraulic maldistribution, but al so on a particular combination of these t.wo fact ors. Tubes wbich are h ealed the most intensively and, on the other hand, with Lhe lowest flow rates tluough them , turn out to be operating under the most dangerous conditions. If the grealest d ifferences of various kinds (in h eat a bsorption, flow rnte, structura l d ifferences, etc.) arc found not in a s iugle·

•138

Ch. 11. Hydrodyn amics of Open Hydrau lic S y&te ms

tube, but in various t ubes, t hey a ll s hould be checked separately .for mal·dis tribul.ion of heat. Those with the worst mal d is tribution of heat should be checked additionally for l ongterm streng th and scaling con di t io ns . H ydraulic maldistribution. In a sys te m of parallel t ubes conn ected t o head ers at the inlet and outl et and with forced motion of the work i ng fluid, m easures s hould be take n to -distribute tho fluid evenly between the tubes. Unde r practical cond iUons, the distri bution of flow rates is alwa ys m ore o r l ess uneven, i.e. h ydraulic m aldist ri bution occurs [s ec formul a (11.21) 1. Hyd ra ulic mald ist ri bution m ay appear due t o h yd t·aulic non-i d enti ty of the parallel tub es, which may be caused by differences in their hydra ulic resis tance, due to th o h eader effect, o r through pressure variation a long the length of tho h eader (see Sec. 11 .5). T his ki nd of mal dis tribution appea rs mainl y i n the heating ·s urfaces of s uper heaLers a nd , to a lesser extent, in economizers. T o d eri ve t ho equation for hydraulic m aldis tribution, l et us write the ·equation for total pressure g t·adient in an element:

nec ted in pa ra llel in the system , it can be wl'itten that 6prn/ = 6 pel· Solving s imultaneous ly equations (11.29) and (11.30), we obtain the expression for hydraulic mald istribution:

_

(wP)mf (WPJel

Ph -

-=--+-2-(---:;·1--=-tt• )- ~

V. -

=

•ctVct

uin

Uf -

· mt'lrnt + 2(V_;m/ -

m/) V;n

•

(11.31)

X (1-

where ot..p,,cad = tlp'hfnd - tlp~1tad is the d ifference of pressure g rad i ents in the heade rs between t ho sections wi t h a m a lfunctioning tube a nd an ol em en t, Mp 11 = t..p;:•t - tlp•J. is the d i Herenee of h ydros ta tic h eads in the malfunc tio ning tube and el em ent, tlp 1" and tlp 0 0 nre ta ken for t ho c leme nt. H ydraulic maldistribution in s uperhealers a nd non-boiling economi ze rs. T he form ul a for the coeffici ent of hydra ulic maldistribu tion ac quires t he simplest for m for non- bo iling economizers in w hich the working fluid in a hori zonta l or ascending motion, has a n ampl e reserve of subcooli ng. In this cas e: 1 ~ ve 1 = cons tant; Vel ~ vm/; t he difference of h ydrostatic hea ds between a malfunctioning tube a nd cl em ent and the resistan ce due to fl ow acceleration a re neglig ible: Otlpll ~ 0, flPac ~ 0. With a uniform wa ter s uppl y to and removal from the head ers, the coeffici ent of hydra ulic mai d istributio n for these co nd itions is d etermined by tho ratio of the coefficients of hydrau l ic res istance:

ve

+ ll Pi/,ad + t..p~1

.I '

(H .29)

:and simil a rly for a malfun ctioning -tube (wp);nl -

A

-U PmJ = Zmj - -

2-

+ (wo) • m 1 (v"'' 2

1

Vm t

v~"') ' t.. prni 1n -r ltt ad

+

A

u

pmf II

(11.30) When thoro is a large n um ber of p a rall el tubes, the effect of a ch a nge in heal. absorption in a malfun ctioning tube on the total pressure g ra·dien t can be neg l ected , and t he re fore, tlpct = cons tan t . S ince the prcss ut·e gmdien t between the inl et sec tio n of the supply header and the outle t sect ion of t he d is charge head er is t he ·same for a ll t ho tubes whic h are co n-

p,, = 1./ v

Zet Zmf

= -_ ~ ( 'L1. 32)

,r,,,,

With t he workin g fluid s u pplied to and re mo ved fro m a head er t hro ugh its end faces, the head er effect is s ubstantial and the formula for h ydra ulic m aid is tri butio n t akes the form: Ph

=

V _1_ ( 1 l'JI,

139

11.4. Maldtstrlb ution of Heat

{jf:o.pr,.ad ) t!.f'ttr

( 11 .:!3)

ln non-boil i ng con vecti ve economize rs with hori zo nta l t ube coils , tho <:.oeffi c ion t of hyd rau I ic maid is tri bulion us ually does not exceed 0.9 . If d o wntake mo tio n t ak es place in a non-boiling econo m izor a nd the hydra ul ic characteristic m ay be ambiguous, t he hyd rostati c co mponent sh ould be t aken in t o acco un t; t his is a lso neccs~ a r y for all b oiling econom izers. Ilydraulic m a ldis tribution-jn superh eaters h eavily de pends on amorc nces in t h e total coeffi c ients of tube resis ta nce, and in the ir heat absorption , a ud o n pressure changes along t he 1tcade rs. Negl ecting I he pressure varial io n c aused by s tea m fl ow accel e ratio n , we got : ,.

,fl /, -=-

lI

-

VcJ 1 ·( l] 11. ''mI ·1

+

66Pil earl Ot!. p,, ) L\ Pil r

( 11.:-l4) The head er e ffect s hould be consid e red when ot..p,.eodi tlp ,, 0.05. The hyd rostatic co mponent of t h e total pressure g r ad ieut is s u bst an tia l o1d y in s ing l e- pass s uperheaters a l ?Jtlp,,! flphr 0.05. llydraulic maldis tribution in the zones o[ sudden c hanges in the uni t vol ume of the work ing fluid. In mode r n -once- thro ug h boi lers. stea m-ge ne rati ng ~l em en ts are usually arranged i n t he l ower rad i a tion section which is heal ed th e most iutcnsiv el y. The -combin a tion of a subs tantial hy draulic mald is tribution and_ inten·s ive hea ting can , uud er cert am cond il ions, resul t i n H s ha rp rise i n tempe rature and unit vol ume of t he working flu id in a malfuncti oning tu~e , reduct ion in the fl ow rate through 1L, and o verheati ug of t he tube m etal u ut i l bum-o ut occurs 1351. Hydra ulic mal d istri butio n in s uch tubes m uy nppear on nn a pprecia ble change iu ~h e uni t vo lume of tho hea ted wod(ln g fluid . It a ppears m a inly in tho h ea l,ing s urfaces of t he high-heat capac il y 1.oncs of su porc ri 1leal- pressure boi I ers , iu 1 he e \'nporat ing surfaces of s ub-cri t ic a l- press ure boil ers, and in boi J ing econo mize rs.

>-

>-

rn

some cases, t h e hydraulic c haracteristic is s t ab ilized by moun t ing orifice plates ut the entry to tubes. In steam genera tion zones , the press ure l osses 6Pa c a nd tlp,,cad are us ually negl ected . For s uch conditions, and no ting t he assumptions made earli er, formul a (11.31.) takes t he for m :

_ \

p,, --

, /

-

~ el

el

ZefVcl + 'oorVor . ( 1 _ Mpt. ) - .J. ~ '"'v"' ! t.p,, Zm jU ru j 1 'bor or

t11. 35) 1 ~ "' ' , v" l 1ere e-' ';:)on ~or or' v"' or cai·e tl1ecoeffi c icnts of resistance of t he o rifice plates a nd un it vol umes of the flu i d flow ing t h rough the orifice plates mounted in a n el ement and malfunct ion ing tube. The difference in hydros tatic heads be tween a mnl fun c tioning tube a nd an clement can be writte n in l,he foil owing for m: v..r

Mph= t..p;~ 1 - D.pf.' = -

:1 - z (l~;;r:o:., I

tl

-

- [ ~ (hp)~ . - Z (hp)fl-.1} g 1

- "' = {[l:" (hp) 01

-

"' "-'

(I~p )
- l ~ (hp)d:,~ - Z (hp)~}.. J} g

(11.36 )

W ater walls of st.carn bo ilers in high-capaci t y monobloc uni t;> are mad_e in t he form of ver tical sections , hon zontally coiled sections, U- and_ Nsh a ped sections , multi-p ass sect 1 o~s with horizonta l tubes , etc . H y dra ullc rnaldistribution in s ome of these types will b e discussed below. Horizontal circuit. In horizonta l tubes t here is no h y drosta tic head, an d therefore, the for mul a of hydraulic mal distribution with throttling t a kes the form:

(11.'37) If there is no t hrottling, t he fo r mula is s i mplified:

Ph -=

II

:dVe/ = =mJUmf

v

-

1 ll h

~cl Vmf

('11. 38)

140

Ch. JJ. ll ydrody na m ics o f Op e n /J ydrcwlic Syst ems

V cr t ical ci rcuil. I n ver tical c ircuits the hyd rostatic head h as a vital effect on hyd r a ulic m a ldist ribut ion. In some P.articu!ar cases, and t a ki ng into co.ns JdcratiOn th o cHect of va ri ous factors [in form u la (11.36)1, the eq uation for hydraulic maldislribulion (11.35) lakes tho form: s ing le-pass ascend ing section:

- ..VI r1 ;C/ _ _ . . :;. :_ lh "mf ., ZefVe / (u:p); l

..

I_..!_

v

lJ/1

2gh (Pel-

(1Ul0)

Umj

Pm f)

( 11.40)

'lh

;-

J If '

500 400

v

i

/

I

I I

I

I

/'It

-\ ! "'

)~

\

0.0

~

I I

r--......

T'0

I

0.8

I

\

T

l2

I

i\ •I

I

f.O

r

\

I

,3 0.5

~

t4

1\J 'It><

1.5

t.8

~~cl

ll.ae/• kJ/ kg : J-4 00; !!- GOO; J - 1

llm.f

'"'·' -

~· -

zo

Fig. 11.:1.0. Ma ldisl.ri bu tiou characteristics of a o ncc-tlu·oug h elemen t a t p = 24 J\1 Pa· . and iin = I 200 kJ / kg

-----~2g-,~.~ , (~-p7l,,,--= p'~'~)--~ (-~"~'/,-~fl p -~.",'.~'>-J

X \

/ 1-

_1_

~

-

0.5

t ,,·o- pass U-s h oped sect ion :

-11

500

lat

0. 7

_-;,r/

=et~e/ (wp); l

Pll

J

700

f.O

s ing le-pass descending sect i on:

Ph -

tu

300

P'' -

2gh (Pel - Pm/)

•c

~.•·•

ZeWe / (wp)~l

(11.41) As fol.l~ws from formu la (11.38), t he coefftcJen t o f hyd r a uli c maldistribu t ion .in h oriz onta l t ubes d e pends on th o r a t1 o be tween the the rm a l a nd t ho hydrau lic ch aracteristics i n a n cl emcnt and. a m a lfu ncti on ing t u be. I n vert1cal t ubes, h yd r aulic mal d isLdbuLion. also d e pends h eavily on the h yd rost.at1c com ponent and i ts con tr ibu I ion to t h e total pressure gradient betwee~ the dischnrgc and su pply headers, I.e. the r atio bot ween t.p and b Pirr· '' O n t.he o ther h and, the rel atio n be tween the increments of enth al pies in a malfunctioning lube and an aver age. e~ement is charactet·ized by the coefftc tent of mnl d is tr ib ution of hea t p , [see f ormuln (1 1.22)1. Assu m ing a nu mber of va lues of Pq = 6i m 1t U"i e /r , one ~a n d et~rm ~ ne the coefficient of hyd r au h c mal distnbution P~t=(wp)m tf(wp)c~

141

l1 .5. Distribution of Worlclng Fluid Be t ween T ubes

oou

and, f m m ?xpression p, = 1) 110 /p , th~ corres pondw g coe ff ici ents· of q nonuniform heat . a bsorp t io n 1l t.u· With the k ~1own values D.irnf and i 111 • we can futd the un it enthalpy of the ~vork ing fl ui d at the exit from a mal Iun ctioning tube, a nd the r efore, i ts tem per ature t~':J. Using these d a t a, we can construct maldistribution characteristics which show varialio ns in the coeffi c ien t of hydraulic mald is tri~ u ~ion a nd lem~ er a ture of the w orking n .Uld at t he e xt t from a m a lfu nctiolllng tube de pend ing on t h e coeffi cient of non- uniform h eat. absorption, i.e. Pt! = I (11110) and 1,:;1 = f (1lh 0 ), see F 1g. 11 .20. The mHI d istrib u ti o n ch a r acteristic:; o f mo re heavily heated Lubes (TJtra > 1) for Ph aro ~ailing a.nd for tex arc ris ing. In a part1cu l:u· porti011, curves of Ph fall off steeply t o a c l'i lical sLate wh en a slig ht i ncr ease o f 1'11," c an lead to a s h arp drop in the fl ow rate and an iucr e asl! in t ex· This is ass ociated with the fac t l h nt in the 1·egion of high

h ea t. capacit ies evcu a slight increase in the unit enthalpy. which can always occur in s ome t ubes, can cause a sh nr p i ncrease in the unit volu me of flui d, v, a nd t. h c1·e foro, in h y dr a ulic resis tnnce. T his i11 tu r n l eads to a l ower flow rate through the tube, i. e. mn l dislribution in the tube is aggrnvatccl furth e r , whil e the un i t h eal absorpti on 6i. inc reases al ong with the volu me of fl u id in il, IJ. The pro-cess c onti nues unti l a flow ···rate is -established which has a correspond i ng stat e of mal dislrihu tion of heal a t. a gi v on h e aling l oad. l n some cases, this m ay l ead t o emergen cy s ituations . T he a ll owabl e heat ing lond is established for the particular opornting cond itions of a heating s urface. For steam s upe rh eat ers in whic h the outlet port i ons o f tubes o per a te u n der almos t -ext re m e temperatu re condiUons, th e a ll owabl e h ealing load sh ould not -exceed '15% of the to t a l hcnl abso r pti on h y the s u perh eat er. F'or bett er rel i ability, tl1 e tube system of steam s u pcrh en ters is sect ion llli 7.od i n l.hc steam path. Eco u omizers arc mounted in r eg ions o[ m od er a t o h e at ing r ates, a nd wa t e r in them h as a r ei ali voly low t emperature. For econ omizers, thormal m aldistri bu t ion of 50% or more is allowabl e. F or this r eason, it is not n ecessary to sect io nalizc t h o econ om izers in tho wa t er path . I n ev!l porating t ubes, temper at ure is rel ativel y low (boiling point). there are no fl ow distu rbances, and th er efore, h eal is r emoved pro perly, the tem per ature of the l.u bc wa l Is usu a ll y exceeds t h at of t he worki ng

rr

flu id by only 20-30°C. An a pprocinb le maldis trihuLion o [ heat m ight be a llowed here, but adverso temperature condi tio ns are l ikel y to occu r in evaporating tubes, es pecially at hig h heating intensities. For this r eason, mu ld istribution of heat in evaporating 1uhes s hould be not more than 20-'tO%. In su percri lical -prcssure boi lers, t.he zo ne of high h eat. ca pacities is m ost sens i tiv e t o va ri a tions in h eat absor ption. In t his zone, tho ther m ophysic al proper ties of the wo rki ng flu id vary roost significantly w i th v ari a tions in enth alpy. T h er e for e, i t is es pecia ll y i rn por t a n l I o cons t r u c t t h e charuclcl'is lic t<;;;, = I (1l tra) for this zone.

11.5. E ffect of H eade rs o n t he Distribut ion ol' Wo rking fluid B elween T ubes There arc three main types of h eaders (Fig. '11.21): (a) sttpply, or distributin g, headers, l , whi ch d istrib utc 1he wo rking fJ u id between pa r allel tubes connected to thoro; (b) disc/wrge headers, 2, which rec eive fl uid from a number o £ paralle l t u bes; a n d (c) intermediate, or m ixing , hea.der.~, 3 , w hich are used t o equ a li ze n on - id entity i n tube oper ation. Mixing headers arc m ost effic ient when th o flow is single- phase (either steam or water). They nre ofte n em ployed to equa lize non- id entic al operation of tube coils in su perbeaters. Suppl y a nd d ischarge h eaders have d i£fercnt effects on the oper ation of h eat i ng s urfaces . 1 n s t eam s u perheaters, the ir effect d e pe nds h eav il y on I

I"'

J

3....._

l sl

section

'\-

Fig. '1 1.21. H eaders and the ir connect ion wit.h th e tube s ystem of h e.a Ling surfaces (u) lin~ar connection ; (b) Clow t mnst cr 1hrough mix in~ head~•..: (c) Cl ow transrcr :.hrou~ h t"rosso\'er t•a\}(·~

Zntl

X

1 I I I

'

.

section

2 (a)

2 (h)

(C)

•

142

11.6. Flow Pulsations

Ch. 11. Hydrodynamics oj Open Hydraulic Systems

~

.tfp[n

Steam inlef

I~-..,

IIJtn

% ...

·-"'~

":;

•

...

~

I ~ ~

0

~

I

11Pcotl Steam

outlet

0

we,

I

I

i!l l l l lll

I

I

i lli'l l ll

I

~· 0

, I

•

0

I

IITu I

0

0

'I (a)

Win

11111111111

111 11111111I

I

I

I

illiilnMrnl

(6}

••

Ill Uit n

~

I

111111111 11

1111111111 1 Ill

••

~

,,,

I

{d)

(C)

llp f,r

Fig. I 1.22. Hydraulic cbaructoristics of a Z-shnped superheater circuit (a) distribution or velocity nnd velocity head: (II) prORSUI'C dls trlbUtiOil

how sl,earn is su ppliod to, or t·emoved from, a header. ln some plants, headers were used in which steam was admiLLed and discharged as a concentrated jet through the end faces. For instance, in a Z-shapod superh eater circuit (Fig. 11. 22), as steam is distributed between the coils, its axial velocity in the supply header, w 1n, decreases, and accordingly, the pressure head wr.p/2 decr eases and changes to static pressure t.p;l' (Fig. 11. 22a). In con I rast to this, the static pressure at the outlet from the discharge header, t.p~~, decreases. As foll ows from Fig . 11.22/J, the leftmost coils of the superheater operate at a pressure grad ient 6P coii t.p!~' t.p~, which is higher than the pressure gradient 6P co il in the rightmost coils. The difference between the pressure heads in the coils is equal to the sum of static heads in the supply and discharge header, i.e. 6p!~1 t.p~~. The difference in pressure gradients appears not on ly between the extreme coils, but also between any two sections along tho l1oader length and is determined as t ho sum of the respective values of 6p $1 '1~ and t.pex for these , , sections. In other circui ts with concentrated supply and removal of steam through

+

+

+

+

~

Fig. H. 23. Pressure di s tribution in a nshapod hydraulic circuit containing a singlephose fluid

the header end faces, more favourable conditions can form, though the operation of parallel tubes in the sysLem may be substantia] ly non-identical. For instance, with concentrnted supply and removal of steam through the header end faces in a IT-shaped circuit (Fig. 11.23), the conditions of steam admission are similar to those of a Z-shaped circuit, so that the pattern of static pressure distribution a long the supply h eader remains essentially the same, i.e. pressure increases along the motion of fluid. ln the discharge header, static pressure decreases towards its outlet. As follows from Fig . 11 .23, the leftmost coils are at a pressure gradient 6p~~ 6P coil and the rightmost ones at 6Pcoil + t.p~~ . Tbe difference in the pressure gradients b etween them is determined by the difference in the static heads in the supply and discharge beadm·, t.p!? - t.p~"f. Since the unit volume of th e fluid in the coils is higher upon heating than before heating, the effect of the discharge headers on fluid distribution between parall el coils in any hydrau lic system is more substantial than the effect o£ supply headers. The header effect can be d ecreased either by increasing 6Pcoll or by decreasing t.p. 1 in the headers . Both measures are ineffi cient, however , sin-

+

Fig. 11.24. Static pressure di stributi on along the leng th of a header depending on the method', of supply and di scharge o£ a single-phase flow

ce the former requires a higher pres- multiple forced circulation boilers, sure in the boilet· and a higher auxi- tho resistance of evaporaLing tube liary flow rate, while the latter re- coils is high, so that the effect of presquires larger headers, and l,herefore, sure variations along the length of headers may be neglected. more metal. The effect of velocity head can be diminished if steam is introduced and 11.6. Flow Pulsations removed nol, through the end faces of Tho steady-state operating condia header (as in Fig . 11.24a and b), but from its middle (Fig. 11.24c). tions of a boiler may be disturbed by In this case the axial veloci ty of steam various factors, l eading to pulsations. decreases by 50% and the velocity in the working fluid flow. The disturhead by 75%. Still better r esults can bing factors are variations in the heabe obtained by distributed steam ad - ting intensity pressure, now rate and mission to a supply header and dis t- temperature of feed water. There aret•ibuted removal from a discharge hea- two main t ypes of flow pulsations: der. With two supply or discharge tu- boiler pulsations and intercoil pulsabes provided in a header (Fig. '11. 24d), tions . Boiler pulsations are oscillations in the a.xial velocity and velocity head can be diminished respectively to 1/4 tho £low rate of the working fluid in and 1/16 of th eir values in a header individual elements, circuits or the with a single tube. In modern boiler whole plant. They may be caused by plants, fresh steam sup01·hoaters are sharp variations in the indicated paraconnected to a number of supply and motet'S. In similar sections of parallel discharge tubes, so that headers have tubes, flow parameters vary synchonly a slight effect on steam d is tribu- ronously. Boiler pulsations are aLLetion. In reheaters, where the t·esi- nuating (Fig. 11.25a); they disappeat· stance o£ tube coils is relatively Jow as soon as the disturbance is elimiand the resistance of h eaders is high nated. Flow rate pulsations can attain a· (because of high steam velocities), the effect of headers can b e substantial. level characteristic for the gi von conIn economizers, where the unit volume ditions, after which they do not. disof water is naturally low and tho axial appear s pontaneously (Fig . 11.25b). velocity in headers inossen tiul, the This means that the flow rate of effect of velocity head presents no water (wp)w through some l.ubes firs!. problem. fn once-through bo il ers and increases to a maximum, then decrcaI

jft4

i

'

I

Ch. 11. llydrod!lnamics of Open Hydraulic Systems

upper surface of the tubes and forming fatigue cr acks. Intercoil pulsations can be characterized by the pulsation period -r and amplitude (see Fig. 11 .25). As has been established, intercoi I pulsations obey the follow ing regu(11) larities: (a) the~' can appear in some tubes (ruj')w.fu;pJs. r1 of an evapontliug section even under lt steady-state l1eating and hydraulic ~ conditions in a boiler; I :-_ I (b) pulsations of the flow rate of the fluid in pat·allel Lubes of a section are shifted in phase so that the total flow ra te and other· parameters of the fluid at the outlet from the heating sect ion remain cons tant; (c) the amplitude of flow rate pul'Fi g . 11.25. Pattern of attenuating oscilla- sations at the inlet of a tube is much tions (n) and sclf-s ustu ined osci llul.ions (b) higher than the amplitude at: its out·in 11 s ingle tube of a once-th rough element let, while tlto pu lsation period is the same; (d) the hig hest flow rate of water ses, passes llu-ough the average value. at the inlet corresponds to the lowest drops to a minimum (sometimes nega- steam fl ow rate at the outlet , i.e. tive), and then increases again. This the phases are d isplaced by 180°; (e) in intercoil pulsation, the presprocess may be repeated more than once. In this case, tho flow rate of sure in evapot·ating tubes oscillates water in other evaporating tubes also with a period equal to the period of pulsates, but the process is shifted in pulsation of the fluid flow rate. phase. Therefore, the flow rate of Since intercoil pulsations are caused wat or through some t.u bes i ncreascs primarily by variations in the phyperiodically while at tho same time s ical properLios of t;he working fluid docreasi ng in other tubes , wi Lh the in t.he zone where eva porat ion begins, total pressure gradient between the the probability of their appearance headers of these tubes remaining the decreases inversely with pressure. At same. This phenomenon, called in- supercrilical pressures, pulsations are tercoil pulsation, can occur ovoo at a less common and have a lower ampli·constant total flow rate through the tude, but the pattern of flow rate parallel tubes of a section. pulsations at su bcritical and supercriThe pulsation period in once-tluough tical pressure is essentially the same. boilers may sometim es be as long as Flow rate pu lsaLious diminish with fra<:l ious of a minute or a few minu- an increase in the muss velocity, owing t.es . With a high amplitude of flow to the resulting i ncroase in hydraulic rate pulsations, this may be harmful resistance in the econom izer porlion to evaporating tubes, since heat trans- of the tubes. The boundary mass velofer is impaired during tl1o periods of ci Ly in horizontal l.ubes at which pullow flow rate, resulting in tempera- sations can occur depends on pressure ture variations in t.he tube walls p, heating load q, length of the heated (curve t,) and fatigu e stresses in the portion l, and inner diameter of lumetal. In horizontal tubes, phase se- bes d: paration of t he flow may take pl ace . (11.42) -<·n·a t iug the risk of ovcd1oaLing the

12.1. Laws of Free Clrculatton

Ascending-Jl1Scending and slightly inclined tube elements in which fl.ph does not exceed 10% are calculated as are horizontal tubes, with (wph increased by 20% . With a higher contribut ion from tho hydrostatic component, the calculation is done as for vertical channels . Tho actual mass velocity should always be greater than tho boundary value: {wp)w > {wp)b. If this condition is not satisfied, the hydraulic resistance of the economizer porlion of tubes can be increased by mounling orifice plates. These are arranged at the inlet of tho tubes. For once-through boilers, throttling conditions to avoid pulsations should be calcul ated for the lowest value of (wp)g . mass velocity wp. The degree of throttC= I = f(l:!.tltlt p) ling required to prevent intercoil pul(wp)b sations also elim inates instability in where 6.iin i.s the degree of subcooling of the working fluid a t. tho inlet and p the hydraulic characteristic of the coils. is pressure.

where (wp} 0 is the boundary mass velocity at a pressure 10 MP.a which can be found on curves plotted for the given conditions of inlet throttling and subcooling, and kp is a correction factot· for the given pressure. In vertical tubes (coils), the hydrostatic componon t of the pressure gradient, 6.p 1., decreases the probabil ity of the appearance o[ pulsations, and therefore, decreases (wp) 1,: (wp)g = c (wp)~ · {11.43) where (wp)~ is the boundary mass velocity for a similar horizontal tube, Isee formula ('l1.'12) I and c is the coe£ficient for vertical tubes which can be found on curves:

HYDRODYNAMICS OF CLOSED HYDRAULIC SYSTEMS 12.1. Laws of F ree Circulation Free-circulation boilers usually have a circulation circuit of evaporating tubes arranged in the boiler furnace. Continuous motion of water and steamwater mixture {circulation) is organized in the circuit, which ensures continuous and ef£i.cient removal of heat from the heating surfaces. Tl1is makes it possible to maintain the temperature of the tube metal at a tolerable l evel and thus ensure the reliable and long operation of the circul ation circuit. Free circulation is produced by the driving circulating head S dr which 10 - 0 1524

appears from heating the vertical uptake lubes !see formula (1.1)1. Let us write Bernoulli's equation fot each section of the circulation circuit. I t is assumed (Fig. 12.1) that a steamwater mixture of density Pa moves in section 3-4, and, at the pressure in the drum, wal.er of density p' moves in the remaining path, i.e. in sections 4-1-2-3. The reference pl ane is taken at the header level (2-3). For section 1-2 (downtake tubes) we have: (H - h) p'g + p,

+ U~j• p' (12.1)

146

Ch. 12. Hydrodynamtcs of Closed Hydraulic Systems

. The ~ ifference b &tween the driving crrculatmg head and the resistance of ascending tubes is called the useful circulating head:

r''"~:

"

-7 f

Su, = Sdr - b.p 0

8

,.---'-----.,

Su 8 = 6-pdtu 2

J

Fig. 12.1 . To derivation of the principal equation of circulation

Similarly, for sections 2-3 (lower hendor), 3-4 (uptak e tubes), and 4-1 (water space of the drum) it can be written:

W:i•

,

Pl + TP =Hpag + p, '

•

A •TPo + uPJ-4

tr

1

IDe

(12.3)

w•

l.:.p g+ P4 +-f Pa = (H - h) pig + p1 (12.4)

Adding together the pressure gradients in the sections of the circulation circui t and noting formula (1.1), we obtain:

H (pi -

p~)

Sus

(12.7}

Comparing formul ae (12.6) and (12. 7), we fi pd the principal equation of circulation:

H

g = 'Lb.Phr = Sd, (12.5)

With steady-state motion, tho pressure difference between the water columns in the downtake t ubes and the columns of the steam-water m ixture in the uptake tubes is counterbalanced by the sum of hydraulic resistances to the motion of the working fluid in the circuit. Having related all resistances to the down take and up take tubes , we get : (12 .6)

147

12.1. Laws of Free Circulation

O L---------~-------~ ~0~

(12.8)

i.e. the useful circulating head is s pent to overcome the resistance of the descending sections . The driving circulating head, and therefore, the useful circulating head depend heavily on tho relative velocity of steam (w,) in the ascending tubes . With the same mass flow rate of the fluid, as the relative velocity of steam increases, the fraction of the tube cross section occupied by steam, ), increases; thus , the densit~ of the steam-water mi x ture in the ascend ing tu bes increases . In turn, the relative velocity of st eam, and therefore,
w;,

Pm = f (wo, w~, p, d) = f (w 0 , w~, p, d)

s...

These relationships aro rather complicated and hove not yet been solved anal ytically. Their graphical solution is also impossible in view of the large number of the parameters involved. For this reason , S,. , is usually represented graphically as a function of w 0 at various values of w~ and constant v~ues of the other parameters (p, d). Figure 12.2 sl'IOWS w 0 -Su. curves obtained at various values of resolved steam velocity w~. For the same calculation velocity w 0 , as the heating rate. is increased I (w~) 2 > (w;)1 l, the denstty o[ the steam-water mixture in ascending tubes decreases and Su, increases . Under identical condi t ions, the useful circul ati ng head depends on th e

Fig. 12.2. Effect of circula tion v
< (roOlt

pressur~

•

m the circuit (Fig. 12.3). At a h1gher pressure, the density of the steam-water mixture in ascending tubes is higher, and therefore, s~~. . is lower. As pressure approach es the critical value, s,.. decreases to a r elatively low value, so that free circul ation becomes ineffective. The ul timate pressure at which reliable free circul ation is still possible in boilers is equal to 18-19 MPa . On the other h and, even at supercritical pressures, but at low fri ction losses in the circuit of once-through boilers , free circulation turns out to be sufficient to fire the boiler at a water How rate that is reduced by 50% and wi th the recirculation pump switched off. The effect of relative steam v elocity depends on pressure in the circuit. At a low pressure, large formations of steam are possible in a tube (slug flow), and the relative s team velocity

f

is rather high. As the pressure and saturation tempernture increase sur~ace te~sion decreases and 'slugs' break mto fm ~r bubbles , thus decreasing th e rela ttve steam velocity. At piPer> > 0.7 (see Fig. 12.3), the effect of relative steam velocity can be neglect ed. Let us recall that the actual dens ity of \.he s team-water mixture p4 depends on \.he actual volume steam con te~t q> [see formula (9.40)1. Using equat ton (12.5), we obtain a convenient formula for th e driving ctrculaLing head: 1

Sdr = H (p = H

-

Pa )g p *) g

(12. 9)

Formula (12.9) h as been derived under the assumption that the ascending tubes of a circulation circuit con lain a steam-water mixture al ong their whole h eight. Actually, developed boiling b egins in the ascending tubes at a certain distance above the inlet and accordingly, the whole length of th~ tubes should be divided into an economizer portion H e c and evaporating portion H . 0 • Tho section in which developed boil.ing begin ~ . is called the boiling sectwn (or bo£lmg point). The height of the evaporating portion in ascending tubes can be found by the formula He o = H - flee (12.1 0) a nd subst!luted into formula (12.9) to determ1ne the driv ing circulating head. With a non-boiling economizer, tho water enthalpy at its outlet, i:C, is less than i and therefore, tho temperature of water in the boiler drum i~ b elow tl.1~ boiliog point. The quanttly of hodlfJg water that enters the rlrum is greater th an the quantity of feed water by the magnitude determil;led by tho cir culation ratio K, so that the degree of water subcooling in the drum can be found as 1

,

I

p

O.lJicr Per

Fig. 12.3. Effect of pressure on the useful circulating head 1-w, neglected; 1- w, considered

to•

(12.11)

148

Ch. 12. Hydrodynamics of Closed Hydraulic S ystems

149

12.2. Calculation of Circ ulation Circuits

Therefore, in the general case, water at tho inlet to the d escend ing tubes i s s ubcooled*. The degr ee of su bcooling increases as water moves in th o descend ing tubes, which i s du o to an increase in its hyd rostatic pressu re, and nllains a m aximum in the lower h eader of the circulation circuit:

+

lii' flp

P'g

(H _ flPde•) p'g d~·

(12. 12)

where Qc is the hea L absorption o f the c irc uit, kJ/h , and H e is lhe heated h eight or the c ircu i t, m. The height o[ the economizct· portion can be found [rom t h e IJal ance o£ heat fo r tha t portion, i.e. the quantity of heat that s hou ld be transferred t o w ater per unit time to preheat i t to boiliug in the economizer portion [formula (1.2.1 5)1 a ud the quautity of heat a bsorb ed for the same Lime by the ccouom izer portion from the furnace [formula (12.16)1. H ence we have:

flp

A •

Ll ~ce

H

=

•

61'

ecP g flp

(12.13)

Therefore, the degree of water subcooling per uni t flow rate (1 kg) up to t he boiling section is A .

Ll~c/

+

6i'

'

(H

P IJ1 des H , flt' ccP g t:.p

t:.p

t:.Prles \ p' g J

(12.14)

or for the total flow rate of c irculating water, G, kg/ h:

. + [ 6 ~d

l!.i' flp

,

Pg

H ec =

JG

(12.15)

Hc

• When all feed water is fed to a bubbleca p steam washer (see Sec. 15A), subcooli ng is climina led by Lhe partial condensa li on of bubbling steam.

6 i' ' flp p g

I

11 cG

(12:l 7)

12.2. Cnlculat ion of Cir cula tion Circu its Free-circulation circu i ts may be either s imple or compl ex. In a simple circuit (Fig. 12.4a a nd b), all ascendi ng tubes h ave the same geometrical characteristics