Condensation Heat Transfer Fundamentals

CONDENSATION HEAT TRANSFER FUNDAMENTALS

J. W. ROSE

Department of Engineering, Queen Mary and West® eld College, University of London, London, UK

The paper gives an outline and discussion of those aspects of condensation heat transfertheory which are relatively well understood. For free convection condensation, theNusselt approximations (neglect of inertia, convection and surface shear stress) are

discussed separately. The effects of interphase matter transfer and variable wall temperature,not considered by Nusselt, are also discussed. Complications arising in forced convectioncondensation, where the condensate ® lm is signi® cantly affected by surface shear stress, areoutlined. The present status of dropwise condensation theory and measurements is also brie¯ yreviewed.

Keywords: condensation; ® lmwise; dropwise; interface resistance

INTRODUCTION

Many factors in¯ uence heat-transfer coef® cients duringcondensation. The condenser surfaces may be wetted by thecondensate, when ® lm condensation (the normal mode)occurs, or non-wetted when dropwise condensation occurs.The vapour may be quiescent or moving across thecondensing surface at signi® cant velocity. The condensateand vapour ¯ ows may be laminar or turbulent. The vapourmay comprise more than one molecular species and not allspecies may condense. Condensate from higher or upstreamsurfaces will generally impinge on lower or downstreamsurfaces (inundation) and thereby affect the heat transfer.The problems encountered in practical condensers are toocomplex to permit detailed and accurate modelling, andcondenser design generally incorporates a substantialamount of idealization and empiricism. Even for condensa-tion of a pure (single constituent) vapour, there existdif® cult or intractable problems such as those associatedwith geometry (3-D ¯ ow with free-stream vapour velocityaligned neither with gravity nor the condensing surface),turbulence, calculation of vapour-condensate interface shearstress, irregularity (rippling) of vapour-condensate interfaceand inundation. Before these problems can be properlyaddressed, it is necessary to have a good appreciation ofthose aspects which are better understood, namely gravity-controlled laminar ® lm condensation with simple geometryand, to a lesser extent, forced convection condensation anddropwise condensation.

The Nusselt1 models for laminar free convection ® lmcondensation on a vertical plate and horizontal tube areoutlined and the Nusselt approximations (neglect of inertia,convection and surface shear stress) are discussed indivi-dually. The effects of interface resistance, arising frominterphase matter transfer, and variable wall temperature,not considered by Nusselt, are also discussed. As is wellknown, the Nusselt model is remarkably accurate in thepractical ranges of the relevant parameters. The forcedconvection condensation problem is less straightforward inthat the surface shear stress evidently cannot be neglectedand in typical practical circumstances neither of the extreme

(high and low condensation rate) approximations for thesurface shear stress is generally valid. Moreover, typicallythe condensate ¯ ow is neither gravity- nor shear stress-dominated and, in the case of the tube, vapour boundarylayer separation adds signi® cant complication. Approachesto the laminar forced-convection condensation problem arealso discussed. Finally, in view of recent reports ofpromising new techniques for promoting dropwise con-densation of steam, dropwise condensation theory andmeasurements are also brie¯ y reviewed.

THE NUSSELT THEORY

The problem considered is shown in Figure 1. Nusseltanalysed the case of condensation of a pure, quiescent,saturated vapour on an isothermal plate and on anisothermal horizontal cylinder. He neglected shear stressfrom the vapour at the surface of the condensate ® lm as wellas inertia/acceleration and convection in the ® lm. Equili-brium was assumed at the condensate-vapour interface sothat the temperature at the outer surface of the condensate® lm was taken as the vapour temperature. Condensateproperties were assumed independent of temperature and,for the case of the horizontal tube, the ® lm thickness wasassumed small in comparison with the tube radius.

For the vertical ¯ at plate and with these assumptions,equating the net downward force on an element ofthe condensate ® lm at distance x down the surface andhaving width dy and height dx to zero (rather than mass ´acceleration) gives:

m¶2u

¶y2 + gD q

q = 0 (1)

Here the vertical pressure gradient term (equal to pressuregradient in the remote vapour) omitted by Nusselt, has beenincluded and leads to the buoyancy term with D q rather thanthe condensate density q ; this only affects the result forcondensation near the critical point. Integration of equation(1) twice with boundary conditions of zero velocity at thewall and zero velocity gradient at the outer surface of the® lm gives the velocity pro® le across the ® lm. The product of

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Trans IChemE, Vol 76, Part A, February 1998

velocity with condensate density and plate width may beintegrated to give the mass ¯ ow rate in the ® lm. When theincrement in mass ¯ ow rate over dx is equated to thecondensation rate on the surface of the element (the latterwritten in terms of the heat ¯ ux obtained by assuming pureconduction across the ® lm equation) a simple differentialequation for ® lm thickness d in terms of distance down theplate is obtained. This may be integrated to give the localand mean heat ¯ ux. The result for the mean Nusselt numberis

NuL = Åa L

k =ÅqL

k D T = 2 2

3

GrL

J

1 / 4

(2)

where

J = k D T / g hfg (3)

and

GrL = q D q gL3 / g 2 (4)

When the heat ¯ ux rather than D T is regarded as uniform(independent of height), the result for the mean Nusseltnumber is identical to equation (2) except that the constant qreplaces the mean value Åq (in NuL) and the mean D Treplaces the constant D T.

In the case of the horizontal tube, the same approach, withthe additional assumption that the condensate ® lm thicknessis small compared with the tube radius, leads to thefollowing differential equation for the ® lm thickness

sin hdz

d h +4

3z cos h - 2 = 0 (5)

where the dimensionless ® lm thickness

z =q D q ghfg

g k d D Td 4 (6)

The solution of equation (5) subject to the condition that z is® nite at h =0, or by symmetry dz/d h = 0, is

z = 2

sin4 / 3 h

h

0

sin1 / 3 h d h (7)

as may be readily veri® ed by differentiation.Using equations (6) and (7), the local heat ¯ ux q = k D T / d

may be written

q =q D q ghfg k

3 D T3

g d

1 / 4

2-1 / 4 sin1 / 3 h

h

0

sin1 / 3 h d h

-1 / 4

(8)

The mean heat ¯ ux up to angle h is then given by

Åq h = 1

h

h

0

qd h =q D q ghfg k

3 D T 3

g d

1 / 4

w (h ) (9)

where

w (h ) = 1

21 / 4

1

h

h

0

sin1 / 3 hh

0

sin1 / 3 h d h

1 / 4 = d h

=4

21 / 43

1

h

h

0

sin1 / 3 h d h

3 / 4

(10)

so that

Åq h =q D q ghfg k

3 D T3

g d

1/ 44

21/ 43

1

h

h

0

sin1 / 3 h d h

3/ 4

(11)When h = p we obtain the mean ¯ ux for the tube

Åq = 4

21 / 43 p

q D q ghfg k3 D T3

g d

1 / 4p

0

sin1 / 3 h dh

3 / 4

(12)

144 ROSE


Figure 1. Condensation on a vertical plate and horizontal tube.

andp

0

sin1/ 3 h d h = 27 / 3 p 2 / C (1 / 3)[ ]3 (13)

where C is the gamma function.Thus

Åq = Kq D q ghfg k

3 D T3

g d

1/ 4

(14)

where

K = (8 / 3)(2 p )1 / 2 C (1/ 3)[ ]-9 / 4= 0.728 018¼ (15)

and

Åa =Åq

D T = Kq D q ghfg k

3

g d D T

1 / 4

(16)

Nu =Åa d

k = Kq D q ghfgd

3

g k D T

1 / 4

(17)

It is interesting to note that Nusselt used planimetry toevaluate the integral in equation (7) to obtain values of z as afunction of h . These were used to obtain the local heat ¯ ux

q =k D T

d =q D q ghfg k

3 D T3

g d

1 / 4

z-1/ 4 (18)

and hence, again using planimetry, the mean heat ¯ ux forthe tube

Åq = 1

p

p

0

qd h (19)

It is remarkable that Nusselt’ s results, obtained by multipleplanimetry, give a value of 0.725 for the constant K, an errorless than 0.5%.

Before leaving the Nusselt solution, the situation near thebottom of the tube warrants comment. Here the ® lmthickness predicted by the theory increases to in® nity as happroaches p and the assumption that d is small comparedwith the tube radius is invalid. In the case of the Nusseltproblem with an isothermal surface, when D T is uniform,this is not too serious since the erroneous heat ¯ ux valuesnear the bottom of the tube where the ® lm is thick are smalland make relatively small contribution to the total heattransfer and hence to the mean heat ¯ ux for the tube. Forthe case of uniform heat ¯ ux considered by Fujii et al.2,the mean value of D T is evaluated to obtain the meanheat-transfer coef® cient, otherwise using the Nusseltassumptions. Here

D T = q

p k

p

0

d d h (20)

when, as may be seen from Figure 2, the contribution to theintegral in equation (20) from the erroneous values as happroaches p , is signi® cant. Fujii et al.2 obtained

Nu = 0.695q D q ghfgd

3

g k D T

1 / 4

(21)

In practice neither uniform D T nor uniform q is found and,in order to determine total or mean heat-transfer rate for a

condenser tube, it is strictly necessary to calculate the localheat-transfer rate from vapour to coolant using a value forthe coolant-side heat-transfer coef® cient and taking accountof two dimensional heat transfer in the tube wall. Separatesolutions are needed for all condensing ¯ uids, vapour andcoolant temperatures, coolant-side heat-transfer coef® cientsand tube diameters and thermal conductivities. Theprocedure is described by Honda and Fujii3. Memory andRose4 noted that measured wall temperature pro® les areclosely approximated by cosine curves (see Figure 3). Thevapour-to-surface temperature difference in this case maybe written

D T = D T(1 - A cos h ) (22)

where A is a constant which may take values between 0(when the temperature difference is uniform around thetube, i.e. the Nusselt case) and unity (the extreme casewhere the temperature difference is zero at the top of thetube). Treating the problem otherwise, as in the Nusseltsolution, the following differential equation is obtained forthe ® lm thickness

dz

d h +4

3z cot h -

2(1 - A cos h )sin h = 0 (23)

The solution of equation (23) with the condition that z

145CONDENSATION HEAT TRANSFER FUNDAMENTALS


Figure 2. Condensation on a horizontal tube. Dependence of ® lm thicknesson angle.

Figure 3. Surface temperature variation around a horizontal tube.Condensation of ethylene glycol (see Memory and Rose4). The brokenlines are cosine ® ts.

remains ® nite at the top of the tube is

z =2

h

0

sin1 / 3 h d h -3A

2sin4 / 3 h

sin4 / 3 h(24)

De® ning the dimensionless mean heat ¯ ux

q* = qg d

q D q ghfg k3 D T

3

1 / 4

(25)

we have, from the de® nition of z and for radial conductionacross the ® lm

q* = (1 - A cos h )z-1 / 4 (26)

Figure 4 shows the variation of q* around the tube forvarious values of A. It is seen that for values of A greaterthan zero the heat ¯ ux at ® rst increases, where the effect ofthe increasing value of D T outweighs that of increasing ® lmthickness, before passing through a maximum and decreas-ing to zero at the bottom of the tube where the ® lm thicknessbecomes in® nite. The area under the curves, which isproportional to the mean heat ¯ ux, is found to be constant tothe fourth signi® cant ® gure for all values of A so that, in allcases, the mean Nusselt number is given by equation (17)with mean values of both q and D T, and the leading constantis 0.7280. On this basis it is suggested that, unless aconjugate vapour-to-coolant solution is carried out, theNusselt equation should yield an accurate mean heat-transfer coef® cient when the wall temperature is non-uniform and, in particular, should be better than equation(21) for the uniform heat ¯ ux case.

Nusselt’ s neglect of the temperature dependence of theviscosity, density and thermal conductivity of the con-densate is also not too serious, and the variation ofproperties across the condensate ® lm is usually small withthe exception, in some cases, of viscosity. Even in the caseof viscosity, it is seen from equations (2) and (17) that theheat-transfer coef® cient depends only on the quarter powerof viscosity. By using the approximation that reciprocalviscosity is linear in temperature, and otherwise adoptingthe Nusselt approach, it is readily shown (see Mayhew

et al.5) that the appropriate mean temperature at which theviscosity should be evaluated is

T * = (3 / 4)Tw + (1 / 4)T y (27)

The density and thermal conductivity may be taken at thearithmetic mean of the wall and vapour temperature.

INTERFACE TEMPERATURE DROP

As noted above, Nusselt assumed that the temperature atthe outer surface of the condensate ® lm was equal to that ofthe vapour. Only at equilibrium, when there is no netcondensation, are the condensate surface and vapourtemperatures equal. Although the interphase matter transferproblem cannot yet be said to be fully understood,theoretical expressions for the temperature drop betweenthe vapour and condensate surface at the interface areavailable. Apart from other uncertainties, theoretical modelsincorporate a condensation coef® cient de® ned as thefraction of vapour molecules striking the condensate surfacewhich remain in the liquid phase. It was earlier thought thatthe condensation coef® cient might take values less than0.01. In this case the predicted interface temperature dropcould be signi® cant in comparison with the temperaturedrop across the condensate ® lm. However, it is generallynow thought that the condensation coef® cient is close tounity, in which case the interface temperature drop isgenerally negligible. A notable exception is the case ofliquid metals. Figure 5 shows experimental data forcondensation of mercury. The temperature drop across thecondensate given by the Nusselt theory in this case is in therange 0.1±0.5 K at the lowest vapour temperature and 2.5±5.5 K at the highest vapour temperature. It is evident that forliquid metals the Nusselt theory cannot be used withoutallowance for interphase mass-transfer resistance; in fact, inmany cases, the temperature drop across the condensate ® lmis negligible in comparison with the interface temperaturedrop.

146 ROSE


Figure 4. Condensation on a horizontal tube. Dependence of dimensionlessheat ¯ ux on angle for cosine surface temperature distributions (see Memoryand Rose4).

Figure 5. Condensation of mercury on a vertical plate at various vapourtemperatures (Niknejad and Rose23).

CORRECTIONS TO NUSSELT MODEL

Effect of Vapour Shear Stress

As the condensate ® lm falls, it is retarded by the `stationary’vapour and consequently the ® lm will be thicker than whenvapour shear stress is neglected, as in the Nusselt theory. Toincorporate vapour shear stress in the Nusselt model, theconditionof zero velocitygradient at the outer edgeof the ® lmis replaced by continuity of velocity and shear stress at theinterface and the condensate problem must be solvedsimultaneously with the momentum equation for the vapour:

U ¶U

¶x + V ¶U

¶y = m y

¶2U

¶y2(28)

subject to the conditions in the remote vapour

U ! 0 and¶U

¶y ! 0 for y ! ¥ (29)

and at the interface

u = U = ui (30)

g¶u

¶y = gy

¶U

¶y(31)

m = - qyV (32)

It may be noted that strictly it is the velocities and shearstresses tangential to the condensate surface which shouldbe equated and the condensation mass ¯ ux is moreaccurately given by

m = - q y V + q y Udd

dx(33)

Since the ® lm thickness increases slowly with distancedown the plate (1/4 power dependence) these approxima-tions should not lead to appreciable error.

Boundary layer theory indicates that, for in® nitecondensation rate, the surface shear stress is given by

s = -mui = -(q/ hfg)ui (34)

With this approximation it is no longer necessary toconsider the vapour momentum equation and the Nusselttheory with the surface shear stress given by equation (34)yields

Nux

Nux,Nu= J + 4

4(J + 1)

1 / 4

(35)

Equation (35) is shown in Figure 6 where it is seen that the

Nusselt number decreases to an asymptotic value 2- 1 /2 ofthe Nusselt value with increasing J.

Comparison of equation (35) with a more accuratesolution where the interface velocity and shear stress aredetermined by simultaneous solution of the condensate andvapour momentum equations (see Maekawa and Rose6) isshown in Figure 7. It is seen that the more accurate solutioninvolves the additional parameter R(= q g / q v g v)

1 / 2. Notingthat in practice R is generally larger than 50, it is evident thatthe approximate solution is very accurate. Then, referring toFigure 6, and noting that in practice the upper limit of J isaround 0.01, it is clear that the effect of vapour shear stressin free convection condensation is very small.

Effects of Inertia and Convection in the Condensate Film

These effects may be considered by treating thecondensate ® lm using the boundary-layer equations for thecondensate ® lm rather than the Nusselt approximations.Thus, for the ¯ at plate case, we have

¶u

¶x+ ¶m

¶y = 0 (36)

u¶u

¶x + m¶u

¶y = gD q

q+ m

¶2u

¶y2(37)

u ¶T

¶x+ m

¶T

¶y = j ¶2T

¶y2(38)

expressing the conservationof mass, momentum and energyin the ® lm. It is seen that when the inertia or accelerationterms on the LHS of equation (37) and the convection termson the LHS of equation (38) are omitted, the equations arethose of Nusselt. Numerical solutions of these equationshave been obtained by Sparrow and Gregg7, Koh et al.8,Chen9 and Fujii10. These yielded virtually identical resultsexcept at low condensate Prandtl number where the Nusseltnumbers obtained by Sparrow and Gregg, who did notconsider surface shear stress, are higher.

As discussed by Maekawa and Rose6, when only theinertia effects are included the normalized Nusselt numberdepends only on the parameter J and the solution for thiscase is shown in Figure 8. Since, as noted above, the upperlimit of J is around 0.01, it is clear that inertia effects areunimportant. Higher (erroneous) values of J may be foundin the literature for condensation of liquid metals. These aredue to temperature drops in the vapour, resulting from the



Figure 6. Effect of vapour shear stress. Figure 7. Effect of vapour shear stress.

presence of non-condensing gas, or signi® cant interfacetemperature drops being included in D T, the temperaturedrop across the condensate ® lm.

When only convection effects are included, i.e. theboundary-layer energy equation is used with the Nusseltmomentum equation, the normalized Nusselt numberdepends only on the parameter H(= cP D T / hfg) as indicatedin Figure 9. Since in practice H does not exceed a value ofaround 0.1, it is clear that convection is also unimportant.

For the general case where both inertia and convectionare included and the boundary-layer equations are solvedsubject to boundary conditions of zero velocity at the wall,continuity of velocity and shear stress at the interface,uniform wall and vapour temperatures, results of the form

Nu / NuNu = w(J, H) (39)

or, since

Pr = H / J (40)

Nu / NuNu = w (J, Pr) (41)

or

Nu / NuNu = n (H, Pr) (42)

are obtained.As indicated by Maekawa and Rose6, the result obtained

when only inertia effects are included corresponds to thegeneral case when the condensate Prandtl number is zeroand the result when only convection effects are includedcorresponds to the general case in the limit of in® nitecondensate Prandtl number.

Chen9 has given the approximate equations

Nu

NuNu= 1 + 0.68H + 0.02HJ

1 + 0.85J - 0.15HJ

1 / 4

(43)

Nu

NuNu= 1 + 0.68PrJ + 0.02PrJ2

1 + 0.85J - 0.15PrJ2

1/ 4

(44)

Nu

NuNu= 1 + 0.68H + 0.02H2 / Pr

1 + 0.85H / Pr - 0.15H2 / Pr

1 / 4

(45)

which summarize the numerically-obtained results to within1% to well beyond the practical ranges of J and H.

As noted above, when the convective terms on the LHS ofequation (38) are omitted, we have the Nusselt equationwhich leads to a linear temperature distribution across thecondensate ® lm and consequently no account is taken of theaddition to the wall heat ¯ ux due to condensate subcooling;omission of the convection terms automatically precludessubcooling. Approximate corrections of the Nusselt theoryto include the effect of condensate subcooling have beengiven. A simple approach, retaining the Nusselt lineartemperature pro® le, indicates that the speci® c latent heatin the Nusselt expression should be replaced byhfg + (3 / 8)cP D T . When the boundary-layer energy equationis used, as in Chen9, the heat ¯ ux, and hence the heat-transfer coef® cient and Nusselt number, are evaluated fromthe temperature gradient in the condensate at the wall so thatcondensate subcooling is included. The temperature gra-dient and heat ¯ ux (equal to the condensation mass ¯ uxmultiplied by the latent heat) at the outer surface of thecondensate ® lm are smaller. It is interesting to note that amore accurate subcooling correction by Rohsenow11 giveshf g + 0.68cP D T to replace hf g . This is the same as equation(43) with J =0, i.e. Chen’ s solution when convection butnot inertia effects are taken into account. It should be notedthat when using equations (43)±(45), no adjustment to hf g

should be made.The case of the horizontal tube has been treated by

Sparrow and Gregg12 and Chen9 and is less straightforwardbecause true similarity only exists near the top of the tube.However, based on the ¯ at plate results, inertia andconvection effects are expected to be small also for thetube. This is shown by Chen9 who shows that the pro® les arequite similar for angles up to 1508 and, since there isrelatively little heat transfer on the lower part of the tubewhere the ® lm becomes thick, concludes that equations(43)±(45) may also be used for the mean Nusselt numbersfor the tube. Chen’ s equations give a correction factor to theNusselt theory which include both the negative effects ofvapour shear stress and condensate inertia and the positiveeffects of convection and subcooling.

FORCED CONVECTION CONDENSATION

The simplest case, namely condensation with vapour ¯ owparallel to a horizontal, isothermal plate has been studied byCess13 and Koh14; gravity is not involved and the vapourshear stress, which is responsible for the condensate motion,evidently cannot be neglected. When the Nusselt approachis used for the condensate ® lm, i.e. inertia and convectionare ignored, and with continuity of velocity and shear stressat the condensate surface (assumed parallel to the plate),

148 ROSE


Figure 8. Effect of inertia.

Figure 9. Effect of convection.

together with the assumption that the vapour free-streamvelocity is much greater than the condensate surfacevelocity, the following results are obtained:

For condensation rate approaching zero:

NuxÄRe-1 / 2

x = 0.436G-1 / 3 (J ! 0) (46)

where

G = RJ (47)

and for condensation rate approaching in® nity

NuxÄRe-1 / 2

x = 0.5 (J ! ¥) (48)

Equation (46) is obtained when the surface shear stress isgiven by

s / q vU2

¥ Re1 / 2v,x = 0.332 (49)

and equation (48) when

s = mU¥ (50a)

which leads to

s / q U2

¥ÄRe1 / 2

x = J / 2 (50b)

or

s / q vU2

¥)Re1 / 2v,x = G/ 2 (50c)

(note that q v and g v cancel in equation (50c)).In the general case

NuxÄRe-1 / 2

x = w(G) (51)

For the practical range of G (0.01±10) an approximateexpression for equation (51) has been given by Fujii andUehara15:

NuxÄRe-1 / 2

x = 0.45 1.2 + G-1 1 / 3(52)

which is shown in Figure 10 and is in good agreement withthe numerical solution.

It may be seen from the above that the ® lm thickness(= x/Nux ) varies as x1 /2 so the approximation that thecondensate surface is parallel to the plate is less accuratethan for the free convection case for the vertical plate.

When the condensate ® lm is treated on the basis of theboundary-layer equations so that the inertia and convectionterms are included, it is found that inertia effects terms arenot important and convection is only important at highvalues of G and R, which rarely occur in practice. In thesecases the results given by equation (52) are conservative. Toallow for variable viscosity in forced convection, the factors3/4 and 1/4 in the reference temperature equation (27)

should be replaced by 2/3 and 1/3 respectively (see Mayhewet al.5 and Fujii10).

Figure 11 illustrates the dependence of surface shearstress on G. It may be noted that, unlike the case of freeconvection condensation, the simple asymptotic expressionfor the surface shear stress given by equation (50) is notgenerally valid in the practical range of G (0.01±10).However, owing to its simplicity, this has been widely used,particularly in treating the practically more important caseof the horizontal tube. It may be seen from Figure 11 thatthis will underestimate the shear stress and therefore giveincreasingly conservative results at low values of G as maybe seen from equation (48) in Figure 10.

For condensation on a vertical plate or horizontal tube,not only is neither asymptotic expression for the surfaceshear stress valid, but gravity also has to be taken intoaccount. This adds no dif® culty in principle (except in thecase of vapour up¯ ow when instability of the condensate® lm may arise from the opposing forces of gravity andvapour shear stress). When equation (50) is used for thevapour shear stress it is found that

Nu ÄRe-1 / 2 = w(F) (53)

where

F = J-1(Gr / ÄRe2) (54)

F is sometimes written Pr/Fr H. This notation is somewhatmisleading since cP , and hence Pr, are not involved in theproblem and the fact that F measures the relative importanceof gravity (through Gr) and vapour velocity (through ÄRe2) isless apparent. Roughly speaking, gravity effects begin todominate when F >about 1 and vapour shear stress becomesmore important than gravity when F <about 1. Unfortu-nately, F =1 lies roughly in the centre of the practical range.For condensation on a horizontal tube, the problem is furthercomplicated by other factors (variation of surface shearstress and pressure around the tube and vapour boundarylayer separation). Results of various investigators arediscussed by Rose16.

Figure 12 shows representative results for condensationof steam on a horizontal tube obtained by somewhatdifferent approaches. Sugawara et al.17 evaluated the shearstress at the condensate surface as in the case of single-phase¯ ow, i.e. they neglected the effect of condensation(equivalent to equation (49) for the ¯ at plate case); theslope discontinuity in Figure 12 shows the point of vapourboundary-layer separation beyond which the surface shear



Figure 10. Solution for forced convection condensation on a horizontalplate (after Cess13).

Figure 11. Shear stress on condensate surface for condensation on ahorizontal plate (after Cess13)Ð the broken lines are the high and lowcondensation rate asymptotes.

stress was set to zero. Shekriladze and Gomelauri18 used theopposite extreme (in® nite condensation rate) to approximatethe surface shear stress (equivalent to equation (50) for the¯ at plate case). As noted in the ¯ at plate case, both of theextreme shear stress approximations are conservative i.e.they underestimate the shear stress and hence also the heat-transfer coef® cient. Fujii et al.15 attempted to evaluate thesurface shear stress correctly using the interface conditionsequations (30±32) (where y is measured radially outwards).Fujii et al. used an approximate quadratic pro® le for thevelocity distribution across the vapour boundary layer; withthis approximation the radial velocity gradient in the vapouris always positive so that boundary layer separation isprecluded in this approach and the curve in Figure 12 has noslope discontinuity. For most of the tube surface it is seenthat the approach of Fujii et al. yields the highest heat-transfer coef® cient, as expected.

DROPWISE CONDENSATION

Dropwise condensation, which may occur when thecondensing surface is not wetted by the condensate, was ® rstnoted by Schmidt et al.19 but only became better understoodin the 1960s. The fact that the heat-transfer coef® cientduring dropwise condensation of steam is much higher thanfor ® lm condensation has continued to stimulate interest inthis topic. Early experimental data were widely scatteredand not until 1964 was there a successful theory of dropwisecondensation heat transfer.

Here the focus is on what is known. The broad facts arethat dropwise condensation has only been obtained with afew relatively high surface tension ¯ uidsÐ notably steam, afew organic ¯ uids and mercury. Except in the case ofmercury, metal surfaces, in the absence of impurities, arewetted by the condensate and ® lm condensation ensues;non-wetting agents (`promoters’ ) are needed to promotedropwise condensation without themselves offering sig-ni® cant thermal resistance and thereby offsetting the

advantage of dropwise condensation. For steam, goodpromoters, such as dioctadecyl disulphide, which formmonomolecular layers on the condenser surface withnegligible resistance to heat transfer, have been found.These have given lifetimes of thousands of hours on copperor copper-containing surfaces under clean laboratoryconditions, thereby permitting experimental determinationof heat-transfer coef® cients for dropwise condensation.Mercury condenses in the dropwise mode on some metallicsurfaces such as stainless steel but owing to its high liquidthermal conductivity, the advantage of dropwise over ® lmcondensation is much less marked than is the case for steam.With other ¯ uids such as ethylene glycol, dropwisecondensation has been obtained in the laboratory butgenerally with more dif® culty and with shorter lifetimesthan for steam. Despite the early promise of the monolayerpromoters, no satisfactory method for promoting dropwisecondensation industrially have yet been found. The factthat for steam the heat-transfer coef® cient for dropwisecondensation is known to be around ten times that for ® lmcondensation at power station condenser pressures (andaround 20 at atmospheric pressure) continues to stimulatethe search for a practical means of promoting this mode ofcondensation.

This brief review is prompted by the fact that in somerecent studies heat-transfer coef® cients are reported whichare evidently inaccurate (as were the early measurements).Figure 13 shows reliable data from several investigations atatmospheric pressure and Figure 14 shows data for lowerpressures. It is evident that the heat-transfer coef® cientdepends on pressure and D T or q (as shown by the curvatureof the q- D T graphs). In contrast to ® lm condensation, theheat-transfer coef® cient increases with vapour-to-surfacetemperature difference or heat ¯ ux reaching an almostconstant value. The data for steam are summarized by theempirical equation

q

kW / m2 = t0.8 5D T

K+ 0.3

D T 2

K2 (55)

where t is Celsius temperature, or

a

kW / m2K = t0.8 5 + 0.3D T

K(56)

150 ROSE


Figure 12. Forced convection condensation of steam on a horizontal tubewith vertical vapour down¯ owÐ predictions of various models (after Fujiiet al.15).

Figure 13. Dropwise condensation of steam at atmospheric pressure (fromRose22 ).

It is suggested that equation (55) be used to validateequipment and techniques. In order to assess the effective-ness of new promoters, the apparatus should ® rst be usedwith a copper condensing surface and a recognizedmonolayer promoter to check that results conforming (towithin say 1 K) with equation (55) can be obtained. Lowerheat-transfer coef® cients which decrease with increasing D Tare caused by the presence in the steam of non-condensinggasesÐ even minute gas concentrations can give largeincreases in D T unless steps are taken to avoid this.

There is still some debate regarding the importance orotherwise of `constriction resistance’ caused by non-uniformity of surface heat ¯ ux and consequent dependenceof heat-transfer coef® cient on conductivity of the surfacematerial. It seems clear, however, that this effect is smallexcept at low heat ¯ ux (see Rose20) and would be of littleimportance in a power station condenser.

Outline of Theory

The general approach used in the model of Le Fevre andRose21 and outlined by Rose20,22 is to combine an expressionfor the heat transfer through a single drop with an expressionfor the distribution of drop sizes to give the mean heat ¯ uxfor the condenser surface as a function of vapour-to-surfacetemperature difference. The range of drop sizes (typicalradius of the largest drops, around 1 mm, is 106 times that ofthe smallest drops) is such that for the smallest, the

dependence of saturation pressure on surface curvaturedominates (the vapour must be cooled below its normalsaturation temperature before condensation can occur on theconvex liquid surface). For the drops a little larger than thesmallest, the interphase matter transfer resistance becomesimportant (this is responsible for the dependence of heat-transfer coef® cient on pressure), while for the larger drops,the conduction resistance dominates. A general theory,therefore, must include all three effects. Le Fevre and Roseobtained, for the heat ¯ ux through the base of ahemispherical drop (a good approximation for water)having radius r,

q =D T - 2 r T

r q hfg

C1

r

k+ C2

T

h2fg q m

c + 1

c - 1

RgT

2 p

1 / 2 (57)

where C1 and C2 are shape factors (for steam C1 =2/3 andC2 = 1/2).

The terms in the denominator are the conduction andinterface resistances, while the numerator is the `drivingforce’ given by the vapour-to-surface temperature differ-ence less the amount by which the vapour must be cooledbefore condensation can occur and the convex surface.

For the distribution of drop sizes Le Fevre and Rose used

f = 1 - (r / Ãr)n (58)

where f is the fraction of surface area covered by dropshaving base radius greater than r. It is seen that equation(58) indicates that no area is covered by drops larger thanthe largest and that if the smallest drop had zero radius, all ofthe area would be covered. n is a constant (for steam n= 1/3)and Ãr is the base radius of the largest drop. It follows fromequation (58) that the fraction of the surface area covered bydrops in the size range r, r + dr is

A(r)dr = nr

Ãr

n-1dr

Ãr(59)

and, the mean heat ¯ ux for the surface is given by

q = n

Ãr n

Ãr

Ïr

D T

T -2 r

r q hfg

C1r

k T +C2

q m h2fg

c + 1

c - 1

RgT

2 p

1 / 2rn-1dr

(60)

The lower and upper limits of integration are

Ïr = 2 r / q hfg D T (61)

and

Ãr = C3(r / q g)1 / 2 (62)

respectively; the former is given by the radius of thesmallest viable drop (owing to surface curvature smallerdrops would evaporate) and the latter was obtained fromdimensional analysis, the constant C3 (= 0.4) being foundempirically.

The integral in equation (60) can be evaluated and theresult obtained given in closed form. As may be seen fromRose20,22 the results are in very good agreement withexperimental measurements, predicting the correct depen-dence on D T and pressure. It seems probable, however, thatif dropwise condensation becomes a practical proposition



Figure 14. Dropwise condensation of steam at sub-pressures (from Rose22).

it is only likely to be used for steam. For pressures notexceeding atmospheric the simple empirical equations (55)and (56) are adequate. The theory may be used to determineheat-transfer coef® cients for steam at higher pressures or forother ¯ uids.

NOMENCLATURE

A constant in equation (22)A(r)dr fraction of surface area covered by drops in the size range r, r + drC1 constant in equation (57)C2 constant in equation (57)C3 constant in equation (57)cP isobaric speci® c heat capacity of condensated diameter of tubeF de® ned in equation (54)Fr U2

¥/ gx or U2¥/ gd

f fraction of surface area covered by drops having base radiusgreater than r

Gk D T

g hf 8

q g

qyg

y

1 / 2

Gr q D q gd3/g 2 or q D q gxc3 / g 2

g speci® c force of gravityH cPD T/hfg

hfg speci® c latent heat of evaporationJ k D T/g hfg

K constant, see equation (15)L height of platem condensation mass ¯ uxNu Nusselt numberNuNu Nusselt number given by simple Nusselt theoryNux local Nusselt numberNux,Nu local Nusselt number given by simple Nusselt theoryNuL mean Nusselt number for plate of height Ln constant, see equations (58) and (59)Pr Prandtl numberq local heat ¯ uxÅq mean heat ¯ uxq* dimensionless heat ¯ ux, de® ned in equation (25)q h local heat ¯ ux for tubeR ( q g / q

yg

y)1/2

ÄRe U¥q d / g or U¥ q x / gÄRex U¥q x/ g

Rev,x U¥qyx / g y

Rg speci® c gas constantr radius of tube, radius of dropÃr base radius of largest dropÏr radius of smallest dropT temperatureTv vapour temperatureTw wall temperatureT* reference temperaturet Celsius temperatureU x-wise vapour velocityU¥ vapour free-stream velocityu x-wise condensate velocityui x-wise interface velocityV y-wise vapour velocityy y-wise condensate velocityx distance from top of plate or tube (measured around surface)y distance normal (outward) from plate or tubez dimensionless ® lm thickness (see equation (6))

Greek lettersa heat transfer coef® cientÅa mean heat-transfer coef® cientc ratio of principle speci® c heat capacities of vapour

D T Tv - Tw

D T Tv - Tw

D q q v - qd thickness of condensate ® lmg viscosity of condensateg

yviscosity of vapour

h angle with vertical measured from top of tube

j k / q cP

k thermal conductivity of condensatem kinematic viscosity of condensatem

ykinematic viscosity of vapour

q density of condensateq

ydensity of vapour

r surface tensions shear stress at condensate surface

REFERENCES

1. Nusselt, W., 1916, Die Ober¯ aÈ chenkondensation des Wasserdampfes,Z Vereines Deutsch Ing, 60: 541±546, 569±575.

2. Fujii, T., Uehara, H. and Oda, K., 1972, Film condensation on a surfacewith uniform heat ¯ ux and body force convection, Heat Transfer JpnRes, 4: 76±83.

3. Honda, H. and Fujii, T., 1984, Condensation of a ¯ owing vapour on ahorizontal tubeÐ numerical analysis as a conjugate heat transferproblem, J Heat Transfer, 106: 841±848.

4. Memory, S. B. and Rose, J. W., 1991, Free convection laminar ® lmcondensation on a horizontal tube with variable wall temperature, Int JHeat Mass Transfer, 34: 2775±2778.

5. Mayhew, Y. R., Grif® ths, D. J. and Philips, J. W., 1965, Effect ofvapour drag on laminar ® lm condensation on a vertical surface, Proc IMech E, 180: 280±287.

6. Maekawa, T. and Rose, J. W., 1997, Laminar natural convection ® lmcondensation. (In preparation).

7. Sparrow, E. M. and Gregg, J. L., 1959, A boundary layer treatment oflaminar-® lm condensation, J Heat Transfer, 81: 13±18.

8. Koh, J. C. Y., Sparrow, E. M. and Hartnett, J. P., 1961, The two phaseboundary layer in laminar ® lm condensation, Int J Heat Mass Transfer,2: 69±82.

9. Chen, M. M., 1961, An analytical study of laminar ® lm condensation:part 1Ð Flat plates and part 2Ð Multiple horizontal tubes, J HeatTransfer, 83: 48±54, 55±60.

10. Fujii, T., 1991, Theory of Laminar Film Condensation, (Springer-Verlag).

11. Rohsenow, W. M., 1956, Heat transfer and temperature distribution inlaminar ® lm condensation, Trans ASME, 78: 1645±1648.

12. Sparrow, E. M. and Gregg, J. L., 1959, Laminar condensation heattransfer on a horizontal cylinder, J Heat Transfer, 81: 291±295.

13. Cess, R. D., 1960, Laminar-® lm condensation on a ¯ at plate in theabsence of a body force, Z Angew Math Phys, 11: 426±433.

14. Koh, J. C. Y., 1962, Film condensation in forced convaction boundary-layer ¯ ow, Int J Heat Mass Transfer, 5: 941±954.

15. Fujii, T., Uehara, H. and Kurata, C., 1972, Laminar ® lmwisecondensation of ¯ owing vapour on a horizontal cylinder, Int J HeatMass Transfer, 15: 235±246.

16. Rose, J. W., 1988, Fundamentals of condensation heat transfer:Laminar ® lm condensation, JSME Int J Series 2, 31: 357±375.

17. Sugawara, S, Michiyoshi, I. and Minamiyama, T., 1956, Thecondensation of vapour ¯ owing normal to a horizontal pipe, ProcSixth Jap Nat Cong App Mech paper III-4, 385±388.

18. Shekriladze, I. G. and Gomelauri, V. I., 1996, The study of laminar ® lmcondensation of ¯ owing vapour, Int J Heat Mass Transfer, 9: 581±591.

19. Schmidt, E, Schurig, W. and Sellschop, W., 1930, Versuche uÈ ber dieKondensation von Wasserdampf in Film- und Tropfenform,Tech Mechu Thermodynamik (Forschung Ing Wes), 1: 52±63.

20. Rose, J. W., 1994, Dropwise condensation, Heat Exchanger DesignHandbook Update, 2.6.5.

21. Le Fevre, E. J. and Rose, J. W., 1966, A theory of heat transfer bydropwise condensation, Proc Third Int Heat Transfer Conf, Chicago,Vol. 2: 362±375.

22. Rose, J. W., 1988, Some aspects of condensation heat transfer theory,Int Communications in Heat and Mass Transfer, 15: 449±473.

23. Niknejad, J. and Rose, J. W., 1981, Interphase matter transferÐ anexperimental study of condensation of mercury, Proc R Soc Lond,A378: 305±327.

ADDRESS

Correspondence concerning this paper should be addressed to ProfessorJ. W. Rose, Department of Engineering, Queen Mary and West® eldCollege, University of London, Mile End Road, London E1 4NS.

This paper was presented at the 5th UK National Heat TransferConference, held at Imperial College London 17± 18 September 1997.

152 ROSE


Condensation Heat Transfer Fundamentals

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