Boiling_Condensation- Heat and mass transfer)-2

Boiling_Condensation

C H A P T E R

Boiling and Condensation

10

Boiling_Condensation

654

Chapter 10

Boiling and Condensation

I

n this chapter we focus on convection processes associated with the change in phase of a uid. In particular, we consider processes that can occur at a solid–liquid or solid–vapor interface, namely, boiling and condensation. For these cases latent heat effects associated with the phase change are signi cant. The change from the liquid to the vapor state due to boiling is sustained by heat transfer from the solid surface; conversely, condensation of a vapor to the liquid state results in heat transfer to the solid surface. Since they involve uid motion, boiling and condensation are classi ed as forms of the convection mode of heat transfer. However, they are characterized by unique features. Because there is a phase change, heat transfer to or from the uid can occur without in uencing the uid temperature. In fact, through boiling or condensation, large heat transfer rates may be achieved with small temperature differences. In addition to the latent heat hfg, two other parameters are important in characterizing the processes, namely, the surface tension at the liquid–vapor interface and the density difference between the two phases. This difference induces a buoyancy force, which is proportional to g( l v). Because of combined latent heat and buoyancy-driven ow effects, boiling and condensation heat transfer coef cients and rates are generally much larger than those characteristic of convection heat transfer without phase change. Many engineering applications that are characterized by high heat uxes involve boiling and condensation. In a closed-loop power cycle, pressurized liquid is converted to vapor in a boiler. After expansion in a turbine, the vapor is restored to its liquid state in a condenser, whereupon it is pumped to the boiler to repeat the cycle. Evaporators, in which the boiling process occurs, and condensers are also essential components in vapor-compression refrigeration cycles. The high heat transfer coef cients associated with boiling make it attractive to consider for purposes of managing the thermal performance of advanced electronics equipment. The rational design of such components dictates that the associated phase change processes be well understood. In this chapter our objectives are to develop an appreciation for the physical conditions associated with boiling and condensation and to provide a basis for performing related heat transfer calculations.

10.1

Dimensionless Parameters in Boiling and CondensationIn our treatment of boundary layer phenomena (Chapter 6), we nondimensionalized the governing equations to identify relevant dimensionless groups. This approach enhanced our understanding of related physical mechanisms and suggested simpli ed procedures for generalizing and representing heat transfer results. Since it is dif cult to develop governing equations for boiling and

condensation processes, the appropriate dimensionless parameters can be obtained by using the Buckingham pi theorem[1]. For either process, the convection coef cient could depend on the difference between the surface and saturation temperatures, T Ts Tsat , the body force arising from the liquid–vapor density difference, g( l v), the latent heat hfg, the surface tension , a characteristic length L, and the thermophysical properties of the liquid or vapor: , cp, k, . That is, h h[ T, g( l v), hfg, , L, , cp, k, ] (10.1)

Boiling_Condensation

10.2

Boiling Modes

655

Since there are 10 variables in 5 dimensions (m, kg, s, J, K), there are (10 5) 5 pi-groups, which can be expressed in the following forms:3 2 hL f g( l v)L, cp T, cp, g( l v)L 2 k hfg k

(10.2a)

or, de ning the dimensionless groups, NuL f )L, Ja, Pr, Bo g ( l v 3 2

(10.2b)

The Nusselt and Prandtl numbers are familiar from our earlier single-phase convection analyses. The new dimensionless parameters are the Jakob number Ja, the Bond number Bo, and a nameless parameter that bears a strong resemblance to the Grashof number (see Equation 9.12 and recall that T / ). This unnamed parameter represents the effect of buoyancy-induced uid motion on heat transfer. The Jakob number is the ratio of the maximum sensible energy absorbed by the liquid (vapor) to the latent energy absorbed by the liquid (vapor) during condensation (boiling). In many applications, the sensible energy is much less than the latent energy and Ja has a small numerical value. The Bond number is the ratio of the buoyancy force to the surface tension force. In subsequent sections, we will delineate the role of these parameters in boiling and condensation.

10.2

Boiling ModesWhen evaporation occurs at a solid–liquid interface, it is termed boiling. The process occurs when the temperature of the surface Ts exceeds the saturation temperature Tsat corresponding to the liquid pressure. Heat is transferred from the solid surface to the liquid, and the appropriate form of Newton’s law of cooling is q s h(Ts Tsat ) h Te (10.3)

where Te Ts Tsat is termed the excess temperature. The process is characterized by the formation of vapor bubbles, which grow and subsequently detach from the surface. Vapor bubble growth and dynamics depend, in a complicated manner, on the excess temperature, the nature of the surface, and thermophysical properties of the uid, such as its surface tension. In turn, the dynamics of vapor bubble formation affect liquid motion near the surface and therefore strongly in uence the heat transfer coef cient. Boiling may occur under various conditions. For example, in pool boiling the liquid is quiescent and its motion near the surface is due to free convection and to mixing induced by bubble growth and detachment. In contrast, for fo

rced convection boiling, uid motion is induced by external means, as well as by free convection and bubble-induced mixing. Boiling may also be classi ed according to whether it is subcooled or saturated. In subcooled boiling, the temperature of most of the liquid is below the saturation temperature and bubbles formed at the surface may condense in the liquid. In contrast, the temperature of the liquid slightly exceeds the saturation temperature in saturated boiling. Bubbles formed at the surface are then propelled through the liquid by buoyancy forces, eventually escaping from a free surface.

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10.3

Pool BoilingSaturated pool boiling, as shown in Figure 10.1, has been studied extensively. Although there is a sharp increase in the liquid temperature close to the solid surface, the temperature through most of the liquid remains slightly above saturation. Bubbles generated at the liquid–solid interface rise to the liquid–vapor interface, where the vapor is ultimately transported across the interface. An appreciation for the underlying physical mechanisms may be obtained by examining the boiling curve.

10.3.1

The Boiling Curve

Nukiyama[2] was the rst to identify different regimes of pool boiling using the apparatus of Figure 10.2. The heat ux from a horizontal nichrome wire to saturated water was determined by measuring the current ow I and potential drop E. The wire temperature was determined from knowledge of the manner in which its electrical resistance varied with temperature. This arrangement is termed power-controlled heating, wherein the wire temperature Ts (hence the excess temperature Te) is the dependent variable and the power setting (hence the heat ux qs ) is the independent variable. Following the arrows of the heating curve of Figure 10.3, it is evident that as power is applied, the heat ux increases, at rst slowly and then very rapidly, with excess temperature. Nukiyama observed that boiling, as evidenced by the presence of bubbles, did not begin until Te 5 C. With further increase in power, the heat ux increased to very high levels until suddenly, for a value slightly larger than qm ax, the wire temperature jumped to the melting point and burnout occurred. However, repeating the experiment with a platinum wire having a higher melting point (2045 K vs. 1500 K), Nukiyama was able to maintain heat uxes above qm ax without burnout. When he subsequently reduced the power, the variation of Te with qs followed the cooling curve of Figure 10.3. When the heat ux reached the minimum point qm in, a further decrease in power caused the excess temperature to drop abruptly, and the process followed the original heating curve back to the saturation point. Nukiyama believed that the hysteresis effect of Figure 10.3 was a consequence of the power-controlled method of heating, where Te is a dependent variable. He also believed that b

y using a heating process permitting the independent control of Te, the missing (dashed) portion of the curve could be obtained. His conjecture was subsequently confirmed by Drew and Mueller[3]. By condensing steam inside a tube at

Vapor

Vapor bubbles

Liquid

y

T(y)

ySolid

Tsat T

Ts

FIGURE 10.1 Temperature distribution in saturated pool boiling with a liquid–vapor interface.

Boiling_Condensation

10.3

Pool Boiling

657

Vapor, 1 atm Water, Tsat

Wire, q" s, Te= Ts– Tsat

I E

FIGURE 10.2 Nukiyama’s powercontrolled heating apparatus for demonstrating the boiling curve.

different pressures, they were able to control the value of Te for boiling of a low boiling point organic fluid at the tube outer surface and thereby obtain the missing portion of the boiling curve.

10.3.2

Modes of Pool Boiling

An appreciation for the underlying physical mechanisms may be obtained by examining the different modes, or regimes, of pool boiling. These regimes are identi ed in the boiling curve of Figure 10.4. The speci c curve pertains to water at 1 atm, although similar trends characterize the behavior of other uids. From Equation 10.3 we note that qs depends on the convection coef cient h, as well as on the excess temperature Te. Different boiling regimes may be delineated according to the value of Te. Free convection boiling is said to exist if Te Te, A, where Te, A 5 C. The surface temperature must be somewhat above the saturation temperature in order to sustain bubble formation. As the excess temperature is increased, bubble inception will eventually occur, but below point A (referred to as the onset of nucleate boiling, ONB), uid motion is determined principally by free convection effects. According toFree Convection Boiling

q" max1062 q" s (W/m )

Heating curve with nichrome and platinum wires Absent in powercontrolled mode

q" max

Burnout of nichrome wire

q" minCooling curve with platinum wire 0

q" min

1

5

10

30

100

1000

Te (°C)

FIGURE 10.3

Nukiyama’s boiling curve for saturated water at atmospheric pressure.

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Boiling and Condensation

Boiling regimes Free convection Nucleate Transition Film

Isolated bubbles 107

Jets and columns

q" max 106

C

Critical heat flux, q" max Boiling crisis

E

P2 q" s (W/m )

105

B

q" min 104

DEquation 9.31 Leidenfrost point, q" min

AONB Te,A Te,B Te,C Te,D

103

1

5

10

30

120

1000

Te= Ts– Tsat (°C)

FIGURE 10.4 Typical boiling curve for water at 1 atm: surface heat ux q s as a function of excess temperature, Te Ts Tsat.1 1– whether the ow is laminar or turbulent, h varies as Te to the– 4 or 3 power, respectively, 5 4– in which case qs varies as Te to the– 4 or 3 power. For a large horizontal plate, the uid ow is turbulent and Equation 9.31 can be used to predict the free convection portion of the boiling curve, as shown in Figure 10.4.

Nucleate boiling

exists in the range Te, A Te Te,C, where Te,C 30 C. In this range, two different ow regimes may be distinguished. In region A–B, isolated bubbles form at nucleation sites and separate from the surface, as illustrated in Figure 10.2. This separation induces considerable uid mixing near the surface, substantially increasing h and qs . In this regime most of the heat exchange is through direct transfer from the surface to liquid in motion at the surface, and not through the vapor bubbles rising from the surface. As Te is increased beyond Te,B, more nucleation sites become active and increased bubble formation causes bubble interference and coalescence. In the region B–C, the vapor escapes as jets or columns, which subsequently merge into slugs of the vapor. This condition is illustrated in Figure 10.5a. Interference between the densely populated bubbles inhibits the motion of liquid near the surface. Point P of Figure 10.4 represents a change in the behavior of the boiling curve. Before point P, theNucleate Boiling

Boiling_Condensation

10.3

Pool Boiling

659

(a)

(b)

FIGURE 10.5 Boiling of methanol on a horizontal tube. (a) Nucleate boiling in the jets and columns regime. (b) Film boiling. (Photographs courtesy of Professor J. W. Westwater, University of Illinois at Champaign-Urbana.)

boiling curve can be approximated as a straight line on a log–log plot, meaning that n q s T e . Beyond this point, the heat ux increases more slowly as Te is increased. At some point between P and C, the decaying increase of the heat ux leads to a reduction of the heat transfer coef cient h q s/ Te . The maximum heat ux, q s,C q max, is usually termed the critical heat ux, and in water at atmospheric pressure it exceeds 1 MW/m2. At the point of this maximum, considerable vapor is being formed, making it dif cult for liquid to continuously wet the surface. Because high heat transfer rates and convection coef cients are associated with small values of the excess temperature, it is desirable to operate many engineering devices in the nucleate boiling regime. The approximate magnitude of the convection coef cient may be inferred by using Equation 10.3 with the boiling curve of Figure 10.4. Dividing qs by Te, it is evident that convection coef cients in excess of 104 W/m2 K are characteristic of this regime. These values are considerably larger than those normally corresponding to convection with no phase change. The region corresponding to Te,C Te Te, D, where Te, D 120 C, is termed transition boiling, unstable lm boiling, or partial lm boiling. BubbleTransition Boiling

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Chapter 10

Boiling and Condensation

formation is now so rapid that a vapor lm or blanket begins to form on the surface. At any point on the surface, conditions may oscillate between lm and nucleate boiling, but the fraction of the total surface covered by the lm i

ncreases with increasing Te. Because the thermal conductivity of the vapor is much less than that of the liquid, h (and qs ) must decrease with increasing Te. Film boiling exists for Te Te, D. At point D of the boiling curve, referred to as the Leidenfrost point, the heat ux is a minimum, qs , D qmin , and the surface is completely covered by a vapor blanket. Heat transfer from the surface to the liquid occurs by conduction and radiation through the vapor. It was Leidenfrost who in 1756 observed that water droplets supported by the vapor film slowly boil away as they move about a hot surface. As the surface temperature is increased, radiation through the vapor lm becomes more signi cant and the heat ux increases with increasing Te. Figure 10.5b illustrates the nature of the vapor formation and bubble dynamics associated with lm boiling. The photographs of Figure 10.5 were obtained for the boiling of methanol on a horizontal tube. Although the foregoing discussion of the boiling curve assumes that control may be maintained over Ts, it is important to remember the Nukiyama experiment and be mindful of the many applications that involve controlling qs (e.g., in a nuclear reactor or in an electric resistance heater) rather than Te. Consider starting at point P in Figure 10.4 and gradually increasing qs . The value of Te, and hence the value of Ts, will also increase, following the boiling curve to point C. However, any increase in qs beyond point C will induce a sharp increase from Te,C to Te, E Ts, E Tsat. Because Ts, E may exceed the melting point of the solid, system failure may occur. For this reason point C is often termed the burnout point or the boiling crisis, and accurate knowledge of the critical heat ux (CHF), qs ,C q max, is important. Although we may want to operate a heat transfer surface close to the CHF, we would rarely want to exceed it.Film Boiling

10.4

Pool Boiling CorrelationsFrom the shape of the boiling curve and the fact that various physical mechanisms characterize the different regimes, it is no surprise that a multiplicity of heat transfer correlations exist for the boiling process. For the region below Te, A of the boiling curve (Figure 10.4), appropriate free convection correlations from Chapter 9 can be used to estimate heat transfer coef cients and heat rates. In this section we review some of the more widely used correlations for nucleate and lm boiling.

10.4.1

Nucleate Pool Boiling

The analysis of nucleate boiling requires prediction of the number of surface nucleation sites and the rate at which bubbles originate from each site. While mechanisms associated with this boiling regime have been studied extensively, complete and reliable mathematical models have yet to be developed. Yamagata et al.[4] were the rst to show the in uence of nucleation sites on the heat rate and to demonstrate that qs is approximately propor

tional to 3 T e . It is desirable to develop correlations that re ect this relationship between the surface heat ux and the excess temperature.

Boiling_Condensation

10.4

Pool Boiling Correlations

661

In Section 10.3.2 we noted that within region A-B of Figure 10.4, most of the heat exchange is due to direct transfer from the heated surface to the liquid. Hence, the boiling phenomena in this region may be thought of as a type of liquid phase forced convection in which the uid motion is induced by the rising bubbles. We have seen that forced convection correlations are generally of the formmfc NuL Cfc ReL Pr nfc

(7.1)

and Equation 7.1 may provide insight into how pool boiling data can be correlated, provided that a length scale and a characteristic velocity can be identi ed for inclusion in the Nusselt and Reynolds numbers. The subscript fc is added to the constants that appear in Equation 7.1 to remind us that they apply to this forced convection expression. As we saw in Chapter 7, these constants are determined experimentally for complicated ows. Because it is postulated that the rising bubbles mix the liquid, an appropriate length scale for relatively large heater surfaces is the bubble diameter, Db. The diameter of the bubble upon its departure from the heated surface may be determined from a force balance in which the buoyancy force (which promotes 3 bubble departure and is proportional to Db ) is equal to the surface tension force (which adheres the bubble to the surface and is proportional to Db), resulting in the expression Db

g( )l v

(10.4a)

The constant of proportionality depends on the angle of contact between the liquid, its vapor, and the solid surface; the contact angle depends on the particular liquid and solid surface that is considered. The subscripts l and v denote the saturated liquid and vapor states, respectively, and (N/m) is the surface tension. A characteristic velocity for the agitation of the liquid may be found by dividing the distance the liquid travels to ll in behind a departing bubble (proportional to Db) by the time between bubble departures, tb. The time tb is equal to the energy it takes to form a vapor bubble (proportional to D3 b), divided by the rate at which heat is added over the solid–vapor contact area (proportional to D2 b). Thus, D Db q V tb s 3 b lhfg Db lhfg

q D s 2 b

(10.4b)

Substituting Equations 10.4a and 10.4b into Equation 7.1, absorbing the proportionalities into the constant Cfc, and substituting the resulting expression for h into Equation 10.3 provides the following expression, where the constants Cs, f and n are newly introduced and the exponent mfc in Equation 7.1 has an experimentally determined value of 2/3: q s l hfg ) g( Cl v 1/2

cp, l Ten s, f hfg Pr l

3

(10.5)

Equation 10.5 was developed by Rohsenow[5] and is the first and most widely used correlation for nucleate boiling. All p

roperties are for the liquid, except for v, and all should be evaluated at Tsat. The coefficient Cs,f and the exponent n depend on the solid–fluid combination, and representative experimentally determined values are presented in Table 10.1. Values for other surface–fluid combinations may be obtained

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TABLE 10.1 Values of Cs, for various surface– uid combinations[5–7]Surface–Fluid Combination Water–copper Scored Polished Water–stainless steel Chemically etched Mechanically polished Ground and polished Water–brass Water–nickel Water–platinum n-Pentane–copper Polished Lapped Benzene–chromium Ethyl alcohol–chromium Cs, 0.0068 0.0128 0.0133 0.0132 0.0080 0.0060 0.006 0.0130 0.0154 0.0049 0.0101 0.0027 n 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.7 1.7 1.7 1.7

from the literature[6–8]. Values of the surface tension and the latent heat of vaporization of water are presented in Table A.6 and for selected fluids in Table A.5. Values for additional fluids may be obtained from any recent edition of the Handbook of Chemistry and Physics. If Equation 10.5 is rewritten in terms of a Nusselt number based on an arbitrary length scale L, it will be in the form NuL Ja2 Pr 1 3n Bo1/2. Comparing with Equation 10.2b, we see that only the first dimensionless parameter does not appear. If the Nusselt number is based on the characteristic bubble diameter given in Equation 10.4a, the expression reduces to the simpler form NuDb Ja2 Pr 1 3n. The Rohsenow correlation applies only for clean surfaces. When it is used to estimate the heat ux, errors can amount to 100%. However, since Te (qs )1/3, this error is reduced by a factor of 3 when the expression is used to estimate Te from knowledge of qs . 2 Also, since qs h fg and hfg decreases with increasing saturation pressure (temperature), the nucleate boiling heat ux will increase as the liquid is pressurized.

10.4.2

Critical Heat Flux for Nucleate Pool Boiling

We recognize that the critical heat ux, q s,C q max, represents an important point on the boiling curve. We may wish to operate a boiling process close to this point, but we appreciate the danger of dissipating heat in excess of this amount. Kutateladze[9], through dimensional analysis, and Zuber[10], through a hydrodynamic stability analysis, obtained an expression which can be approximated as qm ax Chfg v

g( l v) 2 v

1/4

(10.6)

which is independent of surface material and is weakly dependent upon the heated surface geometry through the leading constant, C. For large horizontal cylinders, for spheres, and

Boiling_Condensation

10.4

Pool Boiling Correlations

663

for many large nite heated surfaces, use of a leading constant with the value C /24 0.131 (the Zuber constant) agrees with experimental data to within 16%[11]. For large horizontal plates, a value of C 0.149 gives better agreement with experi

mental data. The properties in Equation 10.6 are evaluated at the saturation temperature. Equation 10.6 applies when the characteristic length of the heater surface, L, is large relative to the bubble diameter, Db. However, when the heater is small, such that the Con nement number, Co /(g[ l v])/L Bo 1/2[12], is greater than approximately 0.2, a correction factor must be applied to account for the small size of the heater. Lienhard[11] reports correction factors for various geometries, including horizontal plates, cylinders, spheres, and vertically and horizontally oriented ribbons. It is important to note that the critical heat ux depends strongly on pressure, mainly through the pressure dependence of surface tension and the heat of vaporization. Cichelli and Bonilla[13] have experimentally demonstrated that the peak ux increases with pressure up to one-third of the critical pressure, after which it falls to zero at the critical pressure.

10.4.3

Minimum Heat Flux

The transition boiling regime is of little practical interest, as it may be obtained only by controlling the surface temperature. While no adequate theory has been developed for this regime, conditions can be characterized by periodic, unstable contact between the liquid and the heated surface. However, the upper limit of this regime is of interest because it corresponds to formation of a stable vapor blanket or lm and to a minimum heat ux condition. If the heat ux drops below this minimum, the lm will collapse, causing the surface to cool and nucleate boiling to be reestablished. Zuber[10] used stability theory to derive the following expression for the minimum , D qm in, from a large horizontal plate. heat ux, qs qmin C vhfg

g ( l v) ( l v)2

1/4

(10.7)

where the properties are evaluated at the saturation temperature. The constant, C 0.09, has been experimentally determined by Berenson[14]. This result is accurate to approximately 50% for most uids at moderate pressures but provides poorer estimates at higher pressures[15]. A similar result has been obtained for horizontal cylinders[16].

10.4.4

Film Pool Boiling

At excess temperatures beyond the Leidenfrost point, a continuous vapor lm blankets the surface and there is no contact between the liquid phase and the surface. Because conditions in the stable vapor lm bear a strong resemblance to those of laminar lm condensation (Section 10.7), it is customary to base lm boiling correlations on results obtained from condensation theory. One such result, which applies to lm boiling on a cylinder or sphere of diameter D, is of the form NuD

D3 g( l v)hfg hconv D C kv v k v(Ts Tsat )

1/4

(10.8)

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Boiling and Condensation

The correlation constant C is 0.62 for horizontal cylinders[17] and 0.67 for spheres[11]. The corrected latent heat h fg accounts for the sens

ible energy required to maintain temperatures within the vapor blanket above the saturation temperature. Although it may be approximated as h fg hfg 0.80cp,v (Ts Tsat ), it is known to depend weakly on the Prandtl number of the vapor[18]. Vapor properties are evaluated at the system pressure and the lm temperature, Tf (Ts Tsat )/2, whereas l and hfg are evaluated at the saturation temperature. At elevated surface temperatures (Ts 300 C), radiation heat transfer across the vapor film becomes significant. Since radiation acts to increase the film thickness, it is not reasonable to assume that the radiative and convective processes are simply additive. Bromley[17] investigated film boiling from the outer surface of horizontal tubes and suggested calculating the total heat transfer coefficient from a transcendental equation of the form4/3 h 4/3 hconv h rad h 1/3

(10.9)

If h rad h conv, a simpler form may be used: h hconv 3 4 hrad The effective radiation coef cient hrad is expressed as h rad 4 4 T sat ) (T s Ts Tsat

(10.10)

(10.11)

where is the emissivity of the solid (Table A.11) and is the Stefan–Boltzmann constant. Note that the analogy between lm boiling and lm condensation does not hold for small surfaces with high curvature because of the large disparity between vapor and liquid lm thicknesses for the two processes. The analogy is also questionable for a vertical surface, although satisfactory predictions have been obtained for limited conditions.

10.4.5

Parametric Effects on Pool Boiling

In this section we brie y consider other parameters that can affect pool boiling, con ning our attention to the gravitational eld, liquid subcooling, and solid surface conditions. The in uence of the gravitational e ld on boiling must be considered in applications involving space travel and rotating machinery. This in uence is evident from appearance of the gravitational acceleration g in the foregoing expressions. Siegel[19], in his review of low gravity effects, con rms that the g1/4 dependence in Equations 10.6, 10.7, and 10.8 (for the maximum and minimum heat uxes and for lm boiling) is correct for values of g as low as 0.10 m/s2. For nucleate boiling, however, evidence indicates that the heat ux is nearly independent of gravity, which is in contrast to the g1/2 dependence of Equation 10.5. Above-normal gravitational forces show similar effects, although near the ONB, gravity can in uence bubble-induced convection. If liquid in a pool boiling system is maintained at a temperature that is less than the saturation temperature, the liquid is said to be subcooled, where Tsub Tsat Tl . In the natural convection regime, the heat ux increases typically as (Ts Tl)n or ( Te Tsub)n, where 5/4 n 4/3 depending on the geometry of the heated surface. In contrast, for nucleate boiling, the in uence of subcooling is consider

ed to be negligible, although the

Boiling_Condensation

10.4

Pool Boiling Correlations

665

Vapor (a) (b)

Nucleation site (c)

FIGURE 10.6 Formation of nucleation sites. ( a) Wetted cavity with no trapped vapor. (b) Reentrant cavity with trapped vapor. (c) Enlarged pro le of a roughened surface.

maximum and minimum heat uxes, qmax and qmin , are known to increase linearly with Tsub. For lm boiling, the heat ux increases strongly with increasing Tsub. The in uence of surface roughness (by machining, grooving, scoring, or sandblasting) on the maximum and minimum heat uxes and on lm boiling is negligible. However, as demonstrated by Berenson[20], increased surface roughness can cause a large increase in heat ux for the nucleate boiling regime. As Figure 10.6 illustrates, a roughened surface has numerous cavities that serve to trap vapor, providing more and larger sites for bubble growth. It follows that the nucleation site density for a rough surface can be substantially larger than that for a smooth surface. However, under prolonged boiling, the effects of surface roughness generally diminish, indicating that the new, large sites created by roughening are not stable sources of vapor entrapment. Special surface arrangements that provide stable augmentation (enhancement) of nucleate boiling are available commercially and have been reviewed by Webb[21]. Enhancement surfaces are of two types: (1) coatings of very porous material formed by sintering, brazing, ame spraying, electrolytic deposition, or foaming, and (2) mechanically machined or formed double-reentrant cavities to ensure continuous vapor trapping (see Figure 10.7). Such surfaces provide for continuous renewal of vapor at the nucleation sites and heat transfer augmentation by more than an order of magnitude. Active augmentation techniques, such as surface wiping–rotation, surface vibration, uid vibration, and electrostatic elds, have also been reviewed by Bergles[22, 23]. However, because such techniques complicate the boiling system and, in many instances, impair reliability, they have found little practical application.Vapor Pore Liquid Vapor bubble

Liquid Sintered layer Tunnel Tube wall (a) (b)

FIGURE 10.7 Typical structured enhancement surfaces for augmentation of nucleate boiling. (a) Sintered metallic coating. (b) Mechanically formed double-reentrant cavity.

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EXAMPLE 10.1The bottom of a copper pan, 0.3 m in diameter, is maintained at 118 C by an electric heater. Estimate the power required to boil water in this pan. What is the evaporation rate? Estimate the critical heat ux.

SOLUTION Known: Water boiling in a copper pan of prescribed surface temperature. Find: 1. Power required by electric heater to cause boiling. 2. Rate of water evaporation due to boiling. 3. Critical heat ux corresponding to the burnout point. Schematic:Tsat= 100°C mb

Water-filled copper pan, D= 0.30 m Electri

cal heater

Ts= 118°C

q, electrical powerinput or heat transfer

Assumptions: 1. Steady-state conditions. 2. Water exposed to standard atmospheric pressure, 1.01 bar. 3. Water at uniform temperature Tsat 100 C. 4. Large pan bottom surface of polished copper. 5. Negligible losses from heater to surroundings. Properties: Table A.6, saturated water, liquid (100 C): l 1/vf 957.9 kg/m3, cp,l cp, f 4.217 kJ/kg K, l f 279 10 6 N s/m2, Prl Prf 1.76, hfg 2257 kJ/kg, 58.9 10 3 N/m. Table A.6, saturated water, vapor (100 C): v 1/ vg 0.5956 kg/m3. Analysis: 1. From knowledge of the saturation temperature Tsat of water boiling at 1 atm and the temperature of the heated copper surface Ts, the excess temperature Te is Te Ts Tsat 118 C 100 C 18 C According to the boiling curve of Figure 10.4, nucleate pool boiling will occur and the recommended correlation for estimating the heat transfer rate per unit area of plate surface is given by Equation 10.5. q s l hfg ) g( Cl v 1/2

cp,l Te

n s, f hfg Pr l

3

Boiling_Condensation

10.4

Pool Boiling Correlations

667

The values of Cs, f and n corresponding to the polished copper surface–water combination are determined from the experimental results of Table 10.1, where Cs, f 0.0128 and n 1.0. Substituting numerical values, the boiling heat ux is 6 q N s/m2 2257 103 J/kg s 279 10

9.8 m/s2 (957.9 0.5956) kg/m3 58.9 10 3 N/m

1/2

4.217 103 J/kg K 18 C 0.0128 2257 103 J/kg 1.76

836 kW/m3

2

Hence the boiling heat transfer rate is2 A qs D qs qs 4 (0.30 m)2 qs 8.36 105 W/m2 59.1 kW 4

2. Under steady-state conditions all heat addition to the pan will result in water evaporation from the pan. Hence˙ qs m bhfg˙ where m b is the rate at which water evaporates from the free surface to the room. It follows that˙ m b qs 5.91 104 W 0.0262 kg/s 94 kg/h hfg 2257 103 J/kg

3. The critical heat ux for nucleate pool boiling can be estimated from Equation 10.6: q max 0.149hfg v

g( ) l v 2 v

1/4

Substituting the appropriate numerical values,3 3 q max 0.149 2257 10 J/kg 0.5956 kg/m

58.9 10 3 N/m 9.8 m/s2 (957.9 0.5956) kg/m3 (0.5956 kg/m3)2

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2 q max 1.26 MW/m

Comments: 2 1. Note that the critical heat ux q max 1.26 MW/m represents the maximum heat ux for boiling water at normal atmospheric pressure. Operation of the heater at qs 0.836 MW/m2 is therefore below the critical condition. 2. Using Equation 10.7, the minimum heat ux at the Leidenfrost point is qm in 18.9 kW/m2. Note from Figure 10.4 that, for this condition, Te 120 C.

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EXAMPLE 10.2A metal-clad heating element of 6-mm diameter and emis

sivity 1 is horizontally immersed in a water bath. The surface temperature of the metal is 255 C under steady-state boiling conditions. Estimate the power dissipation per unit length of the heater.

SOLUTION Known: Boiling from outer surface of horizontal cylinder in water. Find: Power dissipation per unit length for the cylinder, qs . Schematic:

Ambient air p= 1 atm

Tsat= 100°C

Water

Electrical heater D= 6 mm

Ts= 255°C

Assumptions: 1. Steady-state conditions. 2. Water exposed to standard atmospheric pressure and at uniform temperature Tsat. Properties: Table A.6, saturated water, liquid (100 C): l 1/vf 957.9 kg/m3, hfg 2257 kJ/kg. Table A.4, water vapor at atmospheric pressure (Tf 450 K): v 0.4902 kg/m3, cp,v 1.980 kJ/kg K, kv 0.0299 W/m K, v 15.25 10 6 N s/m2. Analysis: The excess temperature is Te Ts Tsat 255 C 100 C 155 C According to the boiling curve of Figure 10.4, lm pool boiling conditions are achieved, in which case heat transfer is due to both convection and radiation. The heat transfer rate follows from Equation 10.3, written on a per unit length basis for a cylindrical surface of diameter D: qs qs D h D Te The heat transfer coef cient h is calculated from Equation 10.9,4/3 h 4/3 hconv h radh1/3

Boiling_Condensation

10.5

Forced Convection Boiling

669

where the convection and radiation heat transfer coef cients follow from Equations 10.8 and 10.11, respectively. For the convection coef cient: hconv 0.62 hconv 0.62

k3 v v( l v)g(hfg 0.8cp,v Te) v D Te

1/4

(0.0299 W/m K) 0.4902 kg/m 1(957.9 0.4902) kg/m 9.8 m/s3 3 3

2

(2257 103 J/kg 0.8 1.98 103 J/kg K 155 C) 15.25 10 6 N s/m2 6 10 3 m 155 C

1/4

hconv 238 W/m2 K For the radiation heat transfer coef cient: hrad hrad 4 4 T sat ) (T s Ts Tsat

5.67 10 8 W/m2 K4 (5284 3734)K4 21.3 W/m2 K (528 373) K

Solving Equation 10.9 by trial and error, h 4/3 238 4/3 21.3h 1/3 it follows that h 254.1 W/m2 K Hence the heat transfer rate per unit length of heater element is2 3 q m 155 C 742 W/m s 254.1 W/m K 6 10

Comments: Equation 10.10 is appropriate for estimating h; it provides a value of 254.0 W/m2 K.

10.5

Forced Convection BoilingIn pool boiling, uid ow is due primarily to the buoyancy-driven motion of bubbles originating from the heated surface. In contrast, for forced convection boiling, ow is due to a directed (bulk) motion of the uid, as well as to buoyancy effects. Conditions depend strongly on geometry, which may involve external ow over heated plates and cylinders or internal (duct) ow. Internal, forced convection boiling is commonly referred to as two-phase o w and is characterized by rapid changes from liquid to vapor in the ow direction

.

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10.5.1

External Forced Convection Boiling

For external ow over a heated plate, the heat ux can be estimated by standard forced convection correlations up to the inception of boiling. As the temperature of the heated plate is increased, nucleate boiling will occur, causing the heat ux to increase. If vapor generation is not extensive and the liquid is subcooled, Bergles and Rohsenow[24] suggest a method for estimating the total heat ux in terms of components associated with pure forced convection and pool boiling. Both forced convection and subcooling are known to increase the critical heat ux qm ax for nucleate boiling. Experimental values as high as 35 MW/m2 (compared with 1.3 MW/m2 for pool boiling of water at 1 atm) have been reported[25]. For a liquid of velocity V moving in cross ow over a cylinder of diameter D, Lienhard and Eichhorn[26] have developed the following expressions for low- and high-velocity ows, where properties are evaluated at the saturation temperature. Low Velocity: qmax 1 1 4 v hfgV WeD High Velocity: q ( / )3/4 ( l/ v)1/2 max l v v hfgV 169 19.2 We1/3 D (10.13)

1/3

(10.12)

The Weber number WeD is the ratio of inertia to surface tension forces and has the form WeD vV 2D (10.14)

The high- and low-velocity regions, respectively, are determined by whether the heat ux 1/2 1]. In most cases, parameter q max/ v hfgV is less than or greater than[(0.275/ ) ( l/ v) data within 20%. Equations 10.12 and 10.13 correlate q max

10.5.2

Two-Phase Flow

Internal forced convection boiling is associated with bubble formation at the inner surface of a heated tube through which a liquid is owing. Bubble growth and separation are strongly in uenced by the ow velocity, and hydrodynamic effects differ signi cantly from those corresponding to pool boiling. The process is accompanied by the existence of a variety of two-phase ow patterns. Consider ow development in a vertical tube that is subjected to a constant surface heat ux, through which uid is moving in the upward direction, as shown in Figure 10.8. Heat transfer to the subcooled liquid that enters the tube is initially by single-phase forced convection and may be predicted using the correlations of Chapter 8. Farther down the tube, the wall temperature exceeds the saturation temperature of the liquid, and vaporization is initiated in the subcooled o w boiling region . This region is characterized by large radial temperature gradients, with bubbles forming adjacent to the heated wall and subcooled liquid owing near the center of the tube. The thickness of the bubble region increases

Boiling_Condensation

10.5

Forced Convection Boiling

671

Vapor

Vapor forced convection

Liquid droplets

Mist

Liquid film Vapor core

Annular Saturated flow boiling

x

Vapor slug Core bubbles

Slug

Bubbly

Wall bubbles

S

ubcooled flow boiling

Liquid

Liquid forced convection

h

FIGURE 10.8 Flow regimes for forced convection boiling in a tube.

farther downstream, and eventually, the core of the liquid reaches the saturation temperature of the uid. Bubbles can then exist at any radial location, and the time-averaged mass fraction of vapor in the uid,1 X, exceeds zero at any radial location. This marks the beginning of the saturated o w boiling region . Within the saturated ow boiling region, the mean vapor mass fraction de ned as

X

Ac

u(r, x)XdAc˙ m

increases and, due to the large density difference between the vapor and liquid phases, the mean velocity of the uid, um, increases substantially. The rst stage of the saturated ow boiling region corresponds to the bubbly o w regime . As X increases further, individual bubbles coalesce to form slugs of vapor. This slug- ow regime is followed by an annular- ow regime in which the liquid forms a lm on the tube wall. This lm moves along the inner surface of the tube, while vapor moves at a larger velocity through the core of the tube. Dry spots eventually appear on the inner surface of the1

This term is often referred to as the quality of a two-phase uid.

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tube and grow in size within a transition regime. Eventually, the entire tube surface is completely dry, and all remaining liquid is in the form of droplets that travel at high velocity within the core of the tube in the mist regime. After the droplets are completely vaporized, the uid consists of superheated vapor in a second single-phase forced convection region. The increase in the vapor fraction along the tube length, along with the large difference in the densities of the liquid and vapor phases, increases the mean velocity of the uid by several orders of magnitude between the rst and the second single-phase forced convection regions. The local heat transfer coef cient varies signi cantly as X and um decrease and increase, respectively, along the length of the tube, x. In general, the heat transfer coef cient can increase by approximately an order of magnitude through the subcooled ow boiling region. Heat transfer coef cients are further increased in the early stages of the saturated ow boiling region. Conditions become more complex deeper in the saturated ow boiling region since the convection coef cient, de ned in Equation 10.3, either increases or decreases with increasing X, depending on the uid and tube wall material. Typically, the smallest convection coef cients exist in the second (vapor) forced convection region owing to the low thermal conductivity of the vapor relative to that of the liquid. The following correlation has been developed for the saturated ow boiling region in smooth circular tubes[27, 28]: h 0.6683 l v hsp or h 1.136 l v hsp

0.1

X 0.16 (1 X )0.64 f (Fr) 1058

qs ˙ m hfg q s˙ m hfg

0.7

(1 X )0.8 Gs, f

(10.15a)

0.45

X 0.72 (1 X )0.08 f (Fr) 667.2 0 X 0.8

0.7

(1 X )0.8 Gs, f

(10.15b)

˙˙ m/Ac is the mass ow rate per unit cross-sectional area. In utilizing Equation where m 10.15, the larger value of the heat transfer coef cient, h, should be used. In this expression,˙ / l)2/gD and the coef cient Gs, f depends on the the liquid phase Froude number is Fr (m surface– uid combination, with representative values given in Table 10.2. Equation 10.15 applies for horizontal as well as vertical tubes, where the strati cation parameter, f (Fr), accounts for strati cation of the liquid and vapor phases that may occur for horizontal tubes. Its value is unity for vertical tubes and for horizontal tubes with Fr 0.04. For horizontal tubes with Fr 0.04, f (Fr) 2.63 Fr0.3. All properties are evaluated at the saturation temperature, Tsat. The single-phase convection coef cient, hsp, is associated with the liquid forced convection region of Figure 10.8 and is obtained from Equation 8.62 with properties evaluated at Tsat. Because Equation 8.62 is for turbulent ow, it is recommended that Equation 10.15 not be applied to situations where the liquid single-phase convection is laminar. Equation 10.15 is applicable when the channel dimension is large relative to the bubble diameter, that is, for Con nement numbers, Co /(g[ l v])/Dh 1/2[3]. In order to use Equation 10.15, the mean vapor mass fraction, X, must be known. For negligible changes in the uid’s kinetic and potential energy as well as negligible work, Equation 1.12d may be rearranged to yield X(x) qs Dx˙ m hfg (10.16)

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10.6

Condensation: Physical Mechanisms

673

TABLE 10.2 Values of Gs, f for various surface– uid combinations[27, 28]Fluid in Commercial Copper Tubing Kerosene Refrigerant R-134a Refrigerant R-152a Water For stainless steel tubing, use Gs,f 1. Gs, 0.488 1.63 1.10 1.00

where the origin of the x-coordinate, x 0, corresponds to the axial location where X begins to exceed zero, and the change in enthalpy, ut pv, is equal to the change in X multiplied by the enthalpy of vaporization, hfg. Correlations for the subcooled ow boiling region and annular as well as mist regimes are available in the literature[28]. For constant heat ux conditions, critical heat uxes may occur in the subcooled ow boiling region, in the saturated ow boiling region where X is large, or in the vapor forced convection region. Critical heat ux conditions may lead to melting of the tube material in extreme conditions[29]. Additional discussions of ow boiling are available in the literature[7, 30–33]. Extensive databases consisting of thousands of experimentally measured values of the critical heat ux for wide ranges of operating conditions are also available[34, 35].

10.5.3

Two-Phase Flow in Micro

channels

Two-phase microchannels feature forced convection boiling of a liquid through circular or noncircular tubes having hydraulic diameters ranging from 10 to 1000 m, resulting in extremely high heat transfer rates[36, 37]. In these situations, the characteristic bubble size can occupy a signi cant fraction of the tube diameter and the Con nement number can become very large (Co 1/2). Hence, different types of ow regimes exist, including regimes where the bubbles occupy nearly the full diameter of the heated tube[38]. This can lead to a dramatic increase in the convection coef cient, h, corresponding to the peak in Figure 10.8. Thereafter, h decreases with increasing x as it does in Figure 10.8. Equation 10.15 cannot be used to predict correct values of the heat transfer coef cient and does not even predict correct trends for microchannel ow boiling cases. Recourse must be made to more sophisticated modeling[36, 39].

10.6

Condensation: Physical MechanismsCondensation occurs when the temperature of a vapor is reduced below its saturation temperature. In industrial equipment, the process commonly results from contact between the vapor and a cool surface (Figures 10.9a, b). The latent energy of the vapor is released, heat is transferred to the surface, and the condensate is formed. Other common modes are homogeneous condensation (Figure 10.9c), where vapor condenses out as droplets suspended in a gas phase to form a fog, and direct contact condensation (Figure 10.9d ), which occurs when vapor is brought into contact with a cold liquid. In this chapter we will consider only surface condensation.

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