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C HAPTER 11 Condensation 11.1 I NTRODUCTION Condensation occurs when a vapor is exposed to a surface that,is at a temperature below the saturation temperature of the vapor. The latent heat of the vapor is transferred' to the surface and condensate, i.e., a liquid, is formed on the surface. Condensation thus involves a change in phase from vapor to liquid at a surface. The process is shown schematically in Fig. 11.1, see [1] to [12]. The prediction of the heat transfer rates associated with condensation is an essential stage in the design of a wide range of industrial devices and processes. For , example, in a modem steam-powered electricity generating plant, steam that has expanded through the turbine is condensed before being pumped back to the boiler via the feedwater heating system. Similarly, in a refrigeration plant, the refrigerant is condensed,: in this way providing the cooling. The design of the condenser is critical to the proper operation and to the achievement of a high'thermal efficiency in,both of these situations. Condensation is also involved in many chemical process plants, e.g., i n an oil refinery. I t has been observed that when a vapor condenses on a surface the condensate can fonn in two possible ways or modes, these two modes of ,condensation being tenned Film (or Filmwise) Condensation and Dropwise Condensation. In film condensation~ the condensate forms as a continuous layer (or film) on the surface, this layer draining continuously off the surface under the influence of gravity. I n dropwise condensation, the vapor condenses i n the fonn of individual droplets. These droplets grow and sometimes coalesce with other nearby droplets to form larger droplets. Sporadically, as individual droplets become large enough, they drain off the surface under the influence of gravity. The two modes o f condensation are shown i n Fig. 11.2. The way i n which the condensate fonns on the surface, i.e., in the fonn o f a continuous film or in the form of discrete droplets, has a strong influence on the heat transfer rate. I n film condensation, the latent heat released by the vapor must be 555 \ 556 Introduction to Convective Heat Transfe.r Analysis FIGURE 11.1 Condensation of a vapor on a surface. Drops o f Con(fensed Liquid 1 ·1 1 : Film Continuous Film o f Condensed Liquid Dropwise Condensation ' Drains Continuously Off Surface Film Condensation F IGURE 11.2 Film and dropwise condensatiol on a vertical surface; transferred to the surface through the liquid film. The film f onns a thermal "barrier" between the warm. vapor a nd the cool surface, i.e., i t offers substantial resistance to heat transfer. I n contrast, during dropwise condensation, parts o f the surface are directly exposed to the vapor. For example, as a large dropl~t runs down the surface under the influence o f gravity, it coalesces with other droplets i n its path, leaving a dry surface i n i ts wake. Many new droplets form and grow rapidly on the newly exposed surface, producing a high heat transfer rate. Also, t he droplets tend to form a t distinct sites and parts o f t he surface may as a result always b e directly exposed to the vapor. As a result o f t hese differences, heat transfer rates w ith dropwise condensation can be more than t en times higher than those with film condensation under the same conditions [13]. Typical values for film and dropwise condensation are shown in Fig. 11.3. The high heat transfer rates associated with dropwise condensation make it very attractive for engineering applications. High heat transfer rates lead to high condensation rates allowing the use o f smaller condensers. T he controlling factor for CHAPI'ER 300r-------~------~--~--~ ·11: Condensation 557 I' 200 100 Condensation Condensation of Steam at Atmospheric Pressure o "___ _____-'--______ o 10 20 . (Ts-Tw)-OC - ..J-_ _ _ _ _ _- - - - ' Heat transfer rates with film. 30 ,and dropwi$e condensation of steam on a vertical surface.; F IGURE 11.3 promoting dropwise condensation is the wettability of the surface as determined by the contact angle of a droplet. As shown in Fig. 1104, a liquid/surface combination is considerednonwetting i f the contact angle (8) is greater than 90°. I f (J > 90°, the condensate will tend to form.as discrete droplets, i.e., dropwise condensation will tend to occur~ while i f (J < 90°, the droplets will tend to merge into a continuous filril, i.e., filin condensation will tend to occur, e.g. See [14] to [29]. Considerable research on creating long-lasting nonwettable surface-liquid combinations that can be used to produce dropwise condensation in industrial devices has been undertaken. Surface coatings (such as grease, Teflon®, and nonoxidizing "noble" metals) ang.additives to the working fluid have been used to promote drOpwise condensation. A detailed review of much of this work can be found in reference 13. Although some Success in snstaining dropwise condensation has been obtained using thin Teflon®and gold coatings, in most practical sitUations the process eventually reverts to filril condensation as unavoidable contaminants build up on the surface. For this reason, dropwise condensation will not be considered further here and this chapter will, therefore, focus on the analysis of filril condensation. Condensers are most often designed based on the assumption of filril condensation, this design practice being adopted because of the difficulty.of sustaining dropwise condensation and because of the lower heat transfer rates associated with filril, condensation. These lower rates yield a more conservative, safer design. 8 =Contact Angle " . " Surface Non-Wetting, 8 > 90° Surface F IGURE 11.4 Wetting, 8 < 90° / Wettability of a surface. 558 Introduction to Convective Heat Transfer Analysis 11.2 LAMINAR F ILM CONDENSATION O N A V ERTICAL PLATE T he situation shown in Fig. 11.5 is considered here, i.e., consideration is given to film condensation on a cold isothermal vertical plate, held a t temperature Tw, w hich i s e xposed to a reservoir o f saturated vapor a t saturation temperature, Ts. I t s hould b e noted that the x-coordinate i s m easured vertically downward along t he p late surface and the y-coordinate i s m easured pe~ndicular to the plate surface. F or c ondensation to occur, the wall temperature, Tw, m ust b e lower than the saturation temperature, Ts , corresponding to t he v apor reservoir pressure. Vapor condens~s o n t he p late forming a thin film o f l iquid that flows down the plate under the influence o f gravity. T he thickness o f the film, 8, a nd t he l ocal m ass flow rate increase with distance down the plate as condensate f oims continuously along the entire film/vapor interface.. S teady film condensation on an isothermal surface c an b e analyzed using an a pproximate analysis tirst proposed b y N usselt [30], s ee also [31] to [40]. To s implify t he problem,_ t he following assumptions are made: 1. T he flow i n t he liquid film is laminar and t he condensed liquid has constant properties. 2. T he v apor reservoir is at rest and i~·ev~rywhere a t t he saturation temperature, Ts. 3. M omentum changes (Le., inertia forces) within the film are negligible, i.e., i n w riting the momentum equation for t he film, the viscous, gravitational, a nd p ressure forces acting on the film are assumed to b e f ar g reater than the rate o f c hange o f l iquid momentum within the film. . 4. T he temperature o f t he film/vapor interface is the saturation temperature, Ts ,. corresponding to the vapor reservoir pressure. T he t emperature within the liquid film layer varies linearly from the wall tem. perature, Tw, a t y = 0 to the saturation temperature, Ts , a t y = 8. 6. T he viscous shear force o f the vapor o n t he film a t y = 8 is negligible, i.e., s. auloyl y =8 =0. 7. P ressure changes i n the y-direction across the film are negligible. x r ....---1..- Y . J , , . Cold Wall at Unifonn Temperature T w«Ts) Liquid Film u \ I Vapor at Saturation Temperature Ts Condensate F ilm I . \ .I ) J.J F IGURE 11.5 Film condensation on a vertical . isothermal surface. Q, Lk t &£; CHAPI'ER 11: Condensation 559 While this is a very simplified model of the process, i t does allow much insight to be gained into factors that govern film condensation and does, in fact, give results that are, in many situations, in good agreement with experiinent. More sophisticated models that relax some of the assumptions on which the Nusselt· analysis is based will be discussed later in this chapter.. . The analysis is, of course, based on the application o f the conservation of mass, momentum, and energy principles. These conservation principles:are applied to a control volume spanning all or part of the liquid film such as that shown in Fig. 11.6 and, because two-dimensional flow is being assumed in the following analysis, a control volume with unit width will be considered. Conservation of momentum is first applied. Consider the forces on the control volume. Because thechanges.ofmomentum within the film are being neglected (assumption 3 above), conservation of momentum requires that the forces·on the control volume must balance. Thus, considering the control volume shown in Fig. n .7, the s umo! the net pressure, and shear and gravitational forces i n the x-direction on the element mustbe 0, i.e.: ~ g dm - ILt a u d x ~ d X)(8-- Y ) = 0 . aydx .. (t!.E (11.1) where d m is the mass o f the Jiquid in the control volume which -is given by: dm = P tdV = pt(8 - -y)dx (11.2) x Two. Dimensional Film Flow Control Volume With Unit Width Wide .Surface . dx III 1YPe of control volume across condensed film used. in . Nusselt analysis. FIGURE 11.6 OllU . FIGURE 11.7 9>ntrol value used in applying conservation of momentum. / S60 Introduction to Convective Heat Transfer Analysis w heredV is the volume of the liquid in the control volume and where It has been noted that a unit width of the control volume is being considered. The subscript, e, in the above equations refers to the properties of the liquid. The subscript, v, will be used below to, denote the properties of the vapor in the reservoir. Because the vapor in the reservoir is assumed to be at rest, i.e., to be stagnant, the pressure distribution in the vaporteservoir will be given,by the hydrostatic pressure equation. The vertical pressure gradient in the vapor is therefore given by: dp d x = Pvg (11.3) In writing this equation it has been noted that the coordinate, x, is being measured in the vertically downwards direction, i.e., in the direction of increasing hydrostatic pressure. This pressure gradient is imposed on the outer edge of the liquid film. However, . the liquid film on the surface is assumed to remain thin and the velocity component normal to the surface in the film is therefore assumed to be very small. As a result, as mentioned above, "pressure changes across the liquid film are neglected: This is why a total derivative was used for the pressure in Eq. (11.1). Eq. (1l.3) therefore gives the pressure gradient everywhere in the condensed film. Substituting Eqs. (11.2) and (1l.3) into Eq. (11.1) gives: g pl(8 - y) d x . - ILl a y d x - Pvg dx(8 - y) = 0 au' (11.4) This first term in this equation is the result of the gravitational force on the liquid in the control volume, the second term is due to the viscous shear force, while the third represents the pressure exerted on the control volume due to the gravitational force on vapor. . DividingEq. (11.4) by d x and rearranging then gives: ILe au a .= y g(8 - y)(pe - Pv) . (11.5) This equation indicates that the shear force is equal t o the net gravitational force, i.e., the gravitational force on the·liquid less the gravitational force that would have acted on the control volume had it contained vapor and not liquid. Integrating Eq. (11.5) with respect to y, i.e., across the liquid film and applying the no-slip condition at the surface, i.e, setting u = 0 at y = 0, gives the velocity distribution in the film as: u = (pe - Pv)g ( 8y _ y 2) ILe \ 2 (11.6) The mass flow rate per unit width o f the plate, denoted here by r, at any x location can be then be calculated by integrating this velocity distribution. This gives: r= I: p ,udy = I: p, [(P' : ;)g (IlY ~ ; )] d y (11.7) CHAPTER 11: Condensation 561 from which it follows that: f =. P t(pt - Pv)g8 3 . 3f.Lt (11.8) It will be noted that both Eqs. (11.6) and (11.8) contain the film thickness 8 which a t this stage o f the analysis is not known. Conservation o f mass and conservation o f energy are next considered. Consider the control volume shown in Fig. 11.8. IJetween x a nd x + d x, i.e., over the length o f the control volume, the mass· flow rate in the film ( f) increases because· vapor condenses at the rate dlhv on the film and is absorbed into it.·Applying conservation o f mass to the control volume gives: Rate mass is condensed Rate mass leaves on the outer surface control volume o f control volume Rate mass enters control volume Le.: i.e.: dnL = d f d x . . "V d x· (11.9) A s t he vapor condenses onto the film, its latent heat (hf g p er unit mass) must be transferred to the liquid film. The latent heat plus the sensible heat associated with the cooling o fthe liquid film below the saturation temperature is then transferred to the wall. I n m ost situations, the amount o f sensible heat associated with the subcooling, i.e., thec()oling o f the liquid below the saturation temperature o f the liquid i n the condensed film, is very small compared to the released latent heat. Therefore, to simplify the analysis, the effects o f this subcooling on the energy balance will be neglected. This will be discussed i n m ore detail later i n this section. This assumption that subcooling effects are negligible implies that at any value o f x, the rate at which heat is transferred into. the outer surface o f the liquid layer is equal to the r ate heat is transferred to the wall. The assumption o f a linear temperature profile in the film is consistent with this assumption. F IGURE 11.8 / II U11 Control volume useq in applying conservation of mass .. 562 Introduction t o Convective Heat Transfer Analysis Because the temperature profile is assumed to be linear (assumption 5 above), the heat transfer rate, which is given by Fourier's law, is the same at all points across the filin.1t follows therefore that the heat tranSfer rate to the wall from the liquid film is given by: _ qx - kla T - ay y=o I - k (Ts - T w) - . t 8 . (11.10) . Because sensible heat effects are being neglected, the rate oflatent heat release by the condensing vapOf can be equated to heat transfer rate through the film to the wall, i.e., considering the control volume of length dx: h Ig d' mv k (Ts - Tw) d l 8 x (11.11) Substituting Eq. (11.9) into Eq. (11.11) then gives: hI d r d - k (Ts - Tw) d x g_ X . .. l dx 8 (11.12) Dividing this equation through by d x and substituting the expression for given by Eq. (11.8) then gives: h Ig d x r ~ [Pl(Pl - pv)g8 3 ] = ' k /Ts - Tw) 3P-l 8 (11.13) Because the liquid properties in the film are being assumed constant, thisequa. tion can be written as: . (11.14) i.e.: (11.15) Integrating this equation and noting that the film thickness must beO at the top of the. plate, i.e.,that 8 = 0 at x = 0, gives: "4 i.e.: 84 [ = P-lkl(Ts - Tw) ] hlgPl(Pl - Pv)g x 8 = [4P-lkt(Ts - Tw)x]li4 ghlgPl(Pl - Pv) .(11.16) Using this expression for the film thickness, 8 , E q. (11.t(» gives the local heat transfer rate at the wall as: . (11.17) CHAPI'BR 11: Condensation 563 This can be written i n terms o f a local heat transfer coefficient which, since q x is t he local heat rate from the film to the wall, is conventionally defined by: . . - qx = h (Ts - Tw) U sing Eq. (11.17), this gives: (11.18) This can be written i n terms o f t he local Nusselt number as: . 1/4 N ux = h x = [ gh/gPl(Pl - Pv)x3] kl 4 p,lkl(Ts ,..... T w) T he m ean h eat transfer rate from a wall o f height L i s given by: (11.19) if = ! (L qx dx LJo (11.20) which c an b e rewritten in terms o f t he mean Nusselt number as: Nu = h L = 0.943 [ g h/gPl(Pl:- Pv)L3]114 kl.· P ,lkl(Ts - T w) . (11.21) W hen applying Eq. ( IJ.19) o r Eq. (11.21) the effects o f temperature dependent liqUid properties can normally be adequately accounted for b y evaluating these· liquid properties a t the mean temperature in the film, i.e., a t (Ts + T w)l2. T he l atent heat, h/ g , i s evaluated at the saturation temperature. I n s ome cases, i n order t o achieve greater acc~y, i t m ay b e necessary to account for the effect o f variable liquid properties b y using a more complex procedure to find the temperature at which the properties are evaluated. T he appropriate temperature w ill t hen d epend on the specific liquid involved. This is 4 iscussed i n references [41] to [43]. E XAMPLE 1 1.1. Stagnant saturated steam at 100°C qondenses on a O.S m high vertical plate with a surface temperature of 95°C. Assumfug steady laminar flow, calculate the heat transfer rate and condensation rate per m width of the plate. Also find the maximum film thickness. ' Solutio". The water properties in the film are evaluated at the mean film temperature, i.e., at: Tj = (Ts~ Tw) = 97.50C / At this temperature for water: . Pl = 960 kglm3, P,l = 2.89 X 10-4 kg/m s: ke = Ol680 W/m K 564 Introduction to Convective Heat Transfer Analysis The vapor density and latent heat are evaluated a t the saturation temperature,· Ts which is 1oo°Cand are: Pv = 0.598 kg/m3, hfg = 2257 kJ/kg In the situation being considered it will be seen that the vapor density is very small compared to the liquid density so: Pe(pe - Pv) = p l is a good approximation. This is usually true for the condensation of steam. The average Nusselt number is calcullited by applying Eq. (11.21), i.e.: Nu . = 0.943 [ghfgpl(pe .. pv)L3]1I4 JLeke(Ts - Tw) ... 114 i.e.: . N u = 0.943 [9.81 X 2,257,000 X 96()2 . 0.000289 X 0.68 X (100 - 95) X 0.5 3 ] = 6730 The average he~t transfer coefficient is then given by: L 0.5 Considering both· sides of the plate, the total heat transfer rate per m plate width is . then given b y : ' Q h = Nuke = 6730 x 0.68 = 9152 W/m2 0C = hA(Ts - Tw) = 9152 X (0.5 X 2) X (100 - 95) = 45,760 W = 45.76 kW The condensation rate will be equal to the total heat transfer rate divided by the latent heat, i.e.: m= hQg = 2,~~;:o = 0.0203 kg/s = 73.0 kg/h f , The maximum film thickness is found using Eq. (11.16), i t being noted that the maximum film thickness will occur at the bottom of the plate, i.e., a t ~ = L. Hence, the . niaximum film thickness is given by: /) = [4 JLe ke(Ts = Tw)L]1I4 ghfgP"e(Pe - Pv) = [4 JLe ke(Ts - Tw)L]1/4 ghfgP~ I) [ 4 X 0.000289 X 0.68 X (100 - 95) X 0.5]1/4 . 9.81 X 2,257,000 X 9602 .. Q :: = 0.0000991 m 0.1 mm , This illus.trates the fact that condensate films are usually very thin. I t was assumed in this example that the flow in the liquid film remained laminar. This may not be true, however, as will be discussed later. Although not necessary, some insight can be gained by writing Eq. (11.21) in terms of commonly used dimensionless products, i.e., to write Eq. (11.21) as: (11.22) I CHAPI'ER ·11: C ondensationS65, where GrL is the Grashof number based on the plate length and Pre is the fluid Prandtl number. The dimensionless group, J a, is termed the Jakob number and is defined as: J a = cpe(Ts - Tw) = sensible heat hf g latent heat (11.23) The Jakob number is basically a measure o f the importance of subcooling ex:.. pressing, as it does, the change in the sensible heat.per unit mass of condensed liquid in the film relative to the enthalpy associated with the phase change. The Jakob number is small· for many problems, i.e the sensible heat change across the liquid fihn is small compared to the latent heat release. For example, for cases involving the condensation of steam, J a, is typically of the order o f 0.01. The definition of the Grashof number, G rt, in Eq. (11.22) is: (11.24) i.e., the density change i n this Grasho(number is associated with the density change that results from ·the change in phase whereas in the conventional Grashof number used in the analysis of natural convection, the density change is associated purely with the temperature changes in the fluid. The density changes associated with a liquid-vapor phase change are usually much greater than those associated purely with a temperature change in a fluid. It should be noted that the Eqs. (11.19) and (11.21) are also valid for condensation on the outside surface of vertical tubes [44], provided the radius of the tube is large compared to the film thickness as indicated in Fig. 11.9. Since condensate films are typically very thin in practical situations, this condition is usually met. Vapor F IGURE 11.9 Condensation on the outer surface o f a. vertical cylinder. 566 Introduction to Convective Heat Transfer Analysis The above analysis of condensation on a vertical. plate can be. easily extended to condensation on an inclined plate. Consider a plate inclined at an angle, 6, ,with respect to the gravity vector as shown in Fig. 11.10. The x-coordinate is again measured in the flow direction along the plate and the y-coordinate is measured normal to the plate surface as indicated in Fig. 11.10. When the plate is inclined, there are components of the gravitational force both parallel to and nOl;mal to the direction of film flow. The. norm~, i.e., y-:-direction, component of the gravitational force will, in g~neral, produce a pressure ~hange across the ~lm. Hovvever, because the film is by a~sumption thin, this pressure·cbange across the film js here.assumed to be negligible. lienee, as shown previously, the net boqy force on a fluid element.will ~ g(pe ""7 Pv) d V. However, only the' ~ component of th~ net body force acts in the direction o f the flow, so, in tbis case, a force balance i n ,the x-direction (see Fig. 11.11) gives: . IJ-e a y = g(u - y)(pe - Pv)cos8 a u., ~ . (11.25) This is the same'as Eq. (11.5) except that gcos8occurs i n place of g. The remainder of, the analysis is identical to that givell,above for a vertical plate and the same result therefore applies to an inclined plate except that g i s replaced b y g cos 8. Vapor at Saturation Temperature T, Temperature ~g FIGURE 11.10 . Condensation on an inclined plate. Surface-. ; FIGURE 11.11 Control volume used in setting up force balance for a condensed film on an inclined plate. CHAPI'ER 11: Condensation 567 For example, for an inclined plate, Eq. (11.21) becomes: N u = 0.943 gcos Ohjgpt{pt - Pv)L3]1~ ILtkt{Ts - Tw) [ (11.26) The effect of subcooling in the film will next be considered. As indicated previously, the sensible heat transfer associated with the subcooling of the liquid film is often small compared to the latent heat release and has been neglected in the above analysis. However, for working fluids with low values of latent heat, such as most refrigerants, a correction to the above analysis to account for subcooling may sometimes be needed. . To illustrate how the effect of subcooling is accounted for, consider an energy bal~ce for an elemental control volume such as that shown in Fig. 11.8. When subcooling of the liquid film is allowed for, the total enthalpy flowing into and out of the control volume must be inCluded in the energy balance. In this case, conservation of energy requires for the control volume: Rate of mass . Heat transfer = mfl ux at the outer X Latent.heat . rate to wall . edg eof the film per urnt mass _ (Rate enthalpy leaves _ Rate enthalpy enters) \the control volume control volume the enthalpy being that of the liquid. This equation gives: q. dx Hence: ~ dm.h,g - U: Ptuht dy + : x a: Ptuh, dY)dX '" f: PiU'" d Y] ~~ L .ax ~ dm"h,g - :x a: PtU'" d Y) d x (11.27) The enthalpy of the subcooled liquid is measured relative to that existing at the saturation temperature, i.e., assuming that the liquid specific heat is constant, the enthalpy of the subcooled liquid film h t can be expressed as: h t = cpt{T - Ts) {I 1.28) Now, it will be recalled that it was shown [see Eq. (11.9)] that the mass flux of vapor into the control volume, dmv, is given by: . , . dmv dd f = -x x = - x d d d'(i 8 0 Ptu. d y ) d x (11.29) Substituting Eqs. (11.28) and (11.29) into Eq. (11.27) gives: a TI df k t a y y=o = h jg d x d + cpt d x (J(8 ptu(Ts - T)dy )'. o (11.30) 568 Introduction to Convective Heat Transfer Analysis As discussed before, the temperature distribution in the film is being assumed to be linear and can therefore be expressed as: (Ts - T) = (Ts - Tw)(I-~) Hence, substituting this expression for the temp~rature profile and the expression for the velocity profile given in Eq. (11.6) into the integral into Eq. (11.30) gives: cptPt : x a: u(T, - T)dY)dX = cptPt(p, - ::(T,,.. Tw)g : X {8' L' [(i) (11.31) -H~)}- (~)Hi)} dx Carrying out the integration then gives: d C plPl- (1 dx _ 1 d8 3 s - T )d y ) d' x - CplPl(Pl - Pv)(Ts - Tw)g - - d x OJLl . 8 dx U 8 (T But Eq. (11.8) gives: . f = P l(Pl ~ Pv)g8 3 . 3JLl from which it follows that: Hence: CptP, : x [Io" u(T, 8 TJdY] d x = cpt(Ts - Tw)~~ dx (11.32) Substituting this back into Eq. (11.30) then gives: k l(Ts - Tw) = [ h Ig + 8 cpl ~ (T _ T )] d f s w dx (11.33) This result is the same as Eq. (11.12) which was obtained by ignoring subcooling except that h ig has been replaced by: h lg + g Cpl(Ts - 3 Tw) Therefore, to include subcooling effects, h ig is simply replaced by a modified latent heat, g ' which is given by: hi hi. = hf' + ~Cp«Ts - Tw) = hf,(l + ~Ja) / (11.34) CHAPI'ER 11: Condensation 569 ·Rohsenow [32] has improved on the above analysis by performing an.integral analy~is that drops the assumption of a linear temperature profile. Based on this analysis, the following equation for the modified latent heat is recommended: .... h jg = hfg (1 + 0.68Ja) '(11.35) Now the heat transfer rate depends on I f it is assumed, therefore, that subcooling effects are negligible i f the term (1 + 0.68Ja) 114 is 1.01 or less, it will ,be seen that subcooling effects are negligible i f J a is approximately less than 0.06, i.e., i f: cpe(Ts - Tw) s~turated hy;. < 0.06hfg E XAMPLE 1 1.2. Consider the situation described in Example 11.1 i n which stagnant p~rature steam at 100°C condenses on a O.5-m high vertical plate with a surface temo f 95°C. Detel1lli.q.e whether subcooling effects have a significant influence on the condensation rate. Solution. Using the water and vapor properties given in Example 11.1, the. following value for the Jakob number is obtained: J a = c pi{Ts - Tw) = 4217 X (100 - 95) . h fg 2257000 = 0 00934' . Eq. (11.35) therefore indicates that the effect of film 'subcooling will be small. The .' modified latent heat is given by: h jg = h fg{l + 0.68Ja) = 2257 X (1 + 0.68 X 0.00934) = 2271 kJ/kg N u = 0.943 [ g h* gPi ( P i - P v )L f J.Like(Ts - Tw) The average Nusselt number is then given by: 3]114 . i.e.: n2 N = 0 943 [ 9.81 x 2,271,000 x 96v- u . X 0.5 . 0.000289 X 0.68 X (100 - 95) 3]114 = 6740 The average heat transfer coefficient is then given by: h = N uki = L 6740 X 0.68 = 9166 W /m 2 0C 0.5 . T he total heat transfer rate p er meter width, o f the plate is then given by: Q = hA{Ts - Tw) = 9166 X (0.5 X2) X{lOO - 95)= 4 5,800W = 45.8 k W T he condensation rate'per meter plate width is. therefore given by: rh = h~g = 2,~~:':O = 0.0202 kgls = 72.7 kglhr When undercooting effects were ignored in Example 11.1, the conden~ation rate was found to b e 7 3.0 k glhr which differs by approximately 0.5% from the above value. I n the situation here being considered, therefore, the effects o f subcooling are very small. ·570 Introduction to Convective Heat Transfer Analysis The effect of superheat of the vapor will next be .consideted. I n the previous analysis i t was assumed that the vapor reservoir was a t the saturation temperature. I f the vapor reservoir is superheated to temperature, Too, the enthalpy of the vapor, hv, condensing onto the film will be: (11.36) where Cpv is the constant pressure specific heat of the vapor. Using theabove·equation, the energy balance for the control volume in the Nusselt analysis when both superheat and subcooling effects are included is: i t ~~ 1,.0 dx = :x {fPtU[h'B + cpv{Too - T,) + cpt{T, - T)j dY} d x (11.37) Comparing Eqs. (11.30) and'(11.37) shows that the effect of superhe~t can be ac.counted for by replacing h fg with h fg + cpv(Too - Ts)~ In most applications, this . correction will b e very snutl\. 11.3 . WAVY AND TURBULENT F ILM C ONDENSATION O N A V ERTICAL SURFACE Transition from laminar to turbulent flow within. the condensed film can occur when the vapor is condensed on a tall surface or on a tall vertical bank of horizontal tubes [45] to [47]. It has been found that the film Reynolds nuinber, based on the mean velocity i n the film, Um, and the hydraulic diameter, Dh, can be used to characterize the conditions under which transition from laminar flow occurs. The mean velocity in the film is given by definition as: . Um = - r . P€8 (11.38) The hydraulic diameter is defined by: 4A Dh =p - (11.39) where A is the flow area and P is the wetted perimeter, i.e., the length of the line of contact between the fluid and the solid surfaces over which it is flo~ing. Consider a portion of a plane film as shown in Fig. 11.12. The only wetted surface is that in contact with the wall. Hence, for a unit width of film, the wetted periIneter is unity and the hydraulic diameter is given by: Dh = 4A = 4 x ( 8X 1) = 48 P 1 (11.40) Therefore, the film Reynolds number is:. R~ = UmDhP€ = (rtp€8)(48p€) /Le /L'e == 4 f /Le (1'1.41) CHAPI'ER' 11: Condensation 571 T woDimensional Film Flow , Section Considered A A =Wetted Surface Section o f F ilm Considered FIGURE 11.12 Quantities used i n defining the hydraulic diameter of a condensed film. Consider the condensed film on a vertical,plate as shown in Fig. 11.13. There are three possible distinct flow regimes, as shown in Fig. 11.13. Near the top of the plate the mass flow rate is low and the flow is huninar. In this regime, corresponding fu approximately Re < 30,' the surface of the film is smooth. Further down' the: plate, as the mass flow rate of condensate increases, a series of regular ripples appear on the surface of the film. These ripples 'promote mixing in the film, enhancing the local heat transfer and condensation rates. The wavy region has been found to correspond approximately to 30 < R e < 1800. Eventually, i f the wall is sUfficiently high 'such that R e > 1800, the flow in the liquid film becomes fully turbulent. ' The, analysiso{the wavy and turbulent regions is very difflculfbecause o f the random naiure o f the flow. For engineering design purposes, it is usually necessary to resort to empirical correlations. Many correlation equations can be found in the literature. Mter an extensive review, Chen, Gerner, and Tien [4] recommend the ~ ~ " .. Laminar F il mFlow " " Vertical Vapor ( Isothermal Swface ( :) ,) W avy p- lmFlow i .. , ~~ ..:~ "~I .•.:~ , >' '~}1 ~J ~:.~ , ~. Turbulent F ilm Flow FIGURE 11.13 Flow regimes in a condensed film on a vertical fiat plate. 572 Introduction to Convective lieat Transfer Analysis following correlation for the wavy and turbulent regions: ! (~)113 = (Re -fi44 + 5.82 X 0-6 eO.' Pr}/3)lfl. Re > 30 1R (11.42) The quantity ( vilg)l/3 in this equation has the dimensions of length. Therefore, the left-hand side of Eq. (11.42) has the form of a modified average Nusselt number with characteristic length ( viIg )113. The heat transfer rate for the laminar region can also be cast in terms of this modified average Nusselt number, the equation having the form: . .! (~r = 1.468Re-1I3.Re < 30 (11.43) Figure 11.14 shows the variation of modified av~rage Nusselt number with Reynolds number for the laminar, wavy, and turbulent regimes. I n the laniinar region, the average Nusseit number·decreases with Reynolds number beca~e of the thermal resistance of the thickening film. I n the turbulent regime, however, the Nusselt number increases. as Reynolds number increases despite the thickening of .the film. I n this regime, turbulent "mixing" reduces the thermal resistance o f the film and enhances the heat transfer and condensation rates. Eqs. ( 1t:42) and (11.43) are very convenient for design calculations when ~e mass flow rate of condensate is specified and the required temperature difference is to be determined. However, when the condensation rate is not specified, the solution ofEq. (11.42) requires an iterative procedure since the Reynolds number cannot be calculated a priori. A simple iterative approach is described in Example 1 1.3.In the laminar regime, if the condensation rate is not known and the temperature difference is specified, iteration can be avoided by using Eq. (11.21) instead of13q. (11.43). :i 1 r-----~----~----~----_n , •, ke E(1)113 g 0.1 ' --___"""---...._ _- 1.-_ _.....1-_ _- - ' l e+ 1 l e+2 l e+3 Re(= 4rtfJe) l e+4··· l e+5 FIGURE 11.14 Variation of modified average Nusselt number with Reynolds number for the laminar, wavy, and turbulent regimes. CHAPTER E XAMPLE 1 1.3. 11: Condensation 573 Stagnant saturated steam at 100°C condenses on a vertical plate with surface temperature of 95°C. The plate is 0.5 m high and 2.0 m wide. Find the condensation rate on the plate. Solution. The water properties i n the film are evaluated at the mean film temperature T f = (Ts + Tw)12 = 97.5°C. At this temperature for water: Pl = 960 kg/m3, I Ll = 2 .8910 X 1 0-4 kg/ms k l = 0.680 W /mK, Cp,l = 4217 J/kg The vapor density and latent heat are evaluated at the saturation temperature, Ts = 100°C and are: Pv = 0.598 kg/m3, h lg = 2257.0 kJ/kg III the situation being considered, the vapor density is very small compared to the liquid density so P l(Pl - Pv) = ~ is a good approximation. This is usually true for the condensation of steam. The Jakob number is given by: J a = .cp,l{Ts - Tw) h fg = (4217)(100 - 95) 2257 x 103 1.0063hfg = 0 00934 . Hence: h ig = hfg{1 + 0.68Ja) = = 2271 kJ/kg 6740 The average Nusselt number is then calculated using Eq. (11.21), Nu = hL kl - 2 \~ = 0 943 [h*f gPl L3 ]114 = [9.81)(2271 x hIr)(960)2(0.5)3]114 == g (. . I Llkl(Ts - T w) (2.89 x 1O-4 ){0.680)(100 - 95) The average heat transfer coefficient is then given by: h = N ukl = L 6740 x 0.680 0 .5· = 9166 W/m2 0C From which it follows that the total heat transfer rate is given by: Q = hA(Ts - T w) = (9166)(0.5 x 2.0)(100 - 95) = 45.8 k W The cond~nsation rate is again found using the total heat transfer rate divided by the latent heat, i.e.: . Q 45.8 X 103 m = hf. = 2271 X 103 = 2.02 g x 10 -2 ~gls = 72.7 kglh . 1 0-2 Therefore, the condensation rate per unit width is r = (2.02 x 1 0- 2 )/2 = 1.01 X kg/s and the Reynolds number o f the film at the bottom of the plate is: Re = 4 r = 4{1.01 I Ll 1 0-2 ) = 140 2.89 X 1 0-4 X This Reynolds number is greater than 3 0 so the film will not be entirely laminar and i t is therefore necessary to use Eq. (11.42), Le.: ! (7 r ~ (Re- M4 + 5:82 X 1O-6Re" pr r)'" / 574 Introduction to Convective Heat Transfer Analysis While the actual Reynolds number willbe greater than 140 the value Re = 140 is a ,goO'dinitialvalue to' use in the abO've equation. Using the liquid properties given above, the liquid kinematic visCO'sity and Prandtl number are: ILt 2.89 X 1 0-4 = ' 301 X 10-7 Vt ~ P t = '960' ,. ms . 2/' P rt = J.LtCpt = 2.89 X 1 0- (4217) = 1 79 kt 4' , 0.680 Hence, Eq. (H.42), gives: r. - k,(~ (Re-· r 44 + 5.82 x 1O-6Re··pr,'''r -= 0.68 [(',' 9.810-0.7p ]113 [(140)-0.44 + "5 82 X : 0-6 )(140)°.8(1.79)113]112 , ( ,1 , 3.01 X 1 , ,, = 1 0,950 W/m2K Now the ReyriO'lds number can be recalculated using the fO'llO'wing relatiO'nship: hL(Ts - Tw) = f hig i.e. = J.L~Re hig = 167 R e = 4hL(Ts - T~) J.Lthig = 4(10,950)(0.5)(100 - 95) (2.89.x 1 0-4)(2271 X 1()3) This improved estimate O'f the ReynO'lds number can nO'w be used in Eq. (11.42) to' recalculate the average heat transfer cO'efficient, h = 0;68 [ (3.01 9X 1 0-7)2 ]113 [(167)-0.44 + ( 582 X 1 0-6)(167)°.8(1 .79)113]112 .81 . . = 10,520 W/m2K" The ReynO'lds number can then be recalculated using this value of the heat transfer coefficient which then allO'ws' a new value O'f the cO'ndensatiO'n rate and therefore a new value of the Reynolds number to' be fO'und a.nd sO' O'n. Repeating this iterative procedure gives cO'nverged values fO'r the ReynO'lds number and heat transfer coefficient O'f 162 and 10,590 W/m2K respectively. Using thes~ values, the tO'tal heat transfer rate is, Q = hA(Ts - Tw) = (10,590)(0.5 X 2.0)(100,- 95) _ = 52.6 kW and the condensation rate is, . ~ Q :;,.. 5.26 X 103 m - h* - 2271 103 Ig X - 2 3' 0"':2 kg! - 83 4' kglh • 2 XIs . a4 I. , F ILM CONDENSATION ON HORIZONTAL TUBES There are a number of very important practical situations i n which film condensation occurs on the outside of horizontal tubes. An obvious example is a shell. and-tilbe , I / CHAPl'ER 11: Condensation 575 condenser used i n steam power plants. The analysis o f film condensation on a horizontal tube proceeds along the same lines as.that for a vertical plate. Figure· 11.15 shows the coordinate system used and the control volume considered in the analysis. In this case, then, the x-coordinate is measured circumferentially around the tube and the y-coordinate is measured normal to the tube surface at all points. The tube has radius, R, a nd is assumed to b e maintained at a uniform surface temperature Tw ' The gravitational force on the liquid in the control volume has components in b oth the x - and y-directions. The y-direction component will, i n general, produce a pressure change across the film. However, i t will again be assumed that the film is thin a nd this pressure change across the film will therefore again b e neglected. As shown previously, the net body force on the liquid in the control volume is g(pe Pv) d V. T he x-direction (or tangential) component o f this force i sg(Pl- Pv) d V sin O. Therefore, because d V = (8 - y)RdO, a tangential force balance for the control volume gives: g(8,."": y)RdO(pe ~ pv)sinO = ILe ~~RdO (11.44) Integrating this equation with respect to y across the liquid film and applying the no-slip condition on velocity at the surface, i.e., setting u = 0 a t y = 0, gives the velocity distribution in the film as: u = (pe - Pv) sinOg (8; ILe \ _ y2) 2 Control Volume (11.45) ' .-' Vapor -- .... Horizontal Cylinder a tUnifonn Surface Tempera~ FIGURE 11.15 Condensation on a horizontal tube. / 576 Introduction to Convective Heat Transfer Analysis T he mass flow rate in the film c an then b e found by again using: r= f p ,udy = r p ,[ ( p, - ~~Sin8g (BY- ~)] d y (11.46) Carrying out the integration gives: 3 f 112 = P€(P€ - Pv)g sin 8 8 . 31l-€ (11.47) Because only one side o f the tube is being considered, the local mass flow rate is denoted b y f 1l2. Next, noting that x = R8, i t will b e seen that an energy balance for the control volume gives: (11.48) Solving Eq. (11.47) for 8 and substituting the result into Eq. (11.48) gives: fHi dfl12 R = [gP€(P€ - Pv) ]113 k€(Ts - Tw) sin l13 8 d8' 31l-€ h fg (11.49) The analysis here differs from that given previously for condensation on a flat plate. Eq. (11.49) is expressed in terins o f the mass flow, f1/2' rather than the film thickness, 8, because the film thickness is not known at 8 = O. However, fl12 is known at x = 0, i.e., by symmetry f1l2 = 0 a t x = O. Therefore, Eq. (11.49) can be integrated from 8 = 0 to 8 = '1T to give: 4 ~f4/3 112,total = [gP€(P€ - pv)R3 ]1I3 k€(Ts - Tw) (1T sin 1l3 8 d 8 31l-€ h fg Jo (11.50) Hence, 'noting that: Jo sin l13 (J d8 = 2 J osin1l3 8 d8 and noting that: (1T (1T12 (11.51) (1T12 Jo sin l13 8 d 8 = 1.2936 (11.52) i t follows from Eq. (11.50) that the total condensation rate for one side o f the tube is given by: f l l2,total = 1.923 [ gP€(P€ - Pv)R3 I d(T - T )3 ]114 € s h3 w Il-€ f g (11.53) Now, the average heat transfer coefficient for the entire tube can be evaluated b y noting that: (11.54) CHAPI'ER 11: Condensation 577 Hence, using Eq. (11.53) 'and Eq. (11.54), the average Nusselt number based on the tube diameter ( D) becomes: Nu = hD = 0.729 [ghfgpe(pe - PV)D3]1I4 ke /Leke(Ts - Tw) (11.55) For problems in which the temperature difference is unknown, i t is useful to cast Eq. (11.55) i n terms o f the condensation Reynolds number. I t will be recalled that the Reynolds number is defined as Re = 4 f1 /Le, where f is the total condensation rate from the entire tube. Using this definition and noting that f = 2 f1/2,totah Eq. (11.54) can be written as: (T - T ) = /Lehfg~e s w 87TRh 113 = 1.52Re-l13 (11.56) S olvingEq. (11.53) for (Ts -:- Tw) a nd substituting the result into Eq. (11.56) gives: (v2) ke g h (11.57) I n this equation it ha$ been assumed that Pe(Pe - Pv) is effectively equal to pe2 • Eqs. (11.55) and (11.57) are valid for laminar flow. I f flat plate flow is used as a rough guide, i t will· be seen that these equations 'can b e expected to yield accurate res~ts for about R e < 60. This Reynolds number is twice the value for a flat plate because condensation occurs on both sides o f the tube. In many practical applications, condensation occurs on a column o f vertically aligned horizontal tubes. In such a case, as illustrated in Fig. 11.16, the condensate cascades f rom t ube to tube down the colwrin o f tubes. I f the flow remains laminar, the heat t ransfer r ate from lower tubes will decrease because o f t he thickening condensate film. Referring to Fig. 11.16, Eq. ( ll.49) CalJ. b e applied to the n th t ube in the vertical array. Now the mass flow rate at (J = 0 for the n th tube is the total mass flow rate Tube! Tube 2 Column of Horizontal . Tube 3 Tubes Tuben F IGURE 11.16 Condensation on a column o f horizontal tubes. 578 'Introduction to Convective Heat Transfer Analysis from the (n - l)th,tube, denoted f 1n,n-l. So, Eq. (11.53) gives: f 4/3 f 4/3 112,n 112,n-l- B (11.58) where: B = 2.394 keR(Ts - Tw) [gPi(Pi - pv)]I13 h jg ILe :, . For tube 1, f1n,n-1 is Q, so: '413 ' f l12,1 = B (11.59) The following is then obtained for tube 2: fln,~4/3 = f t/i,l + B = 2B n th (11.60) tube the (11.61) Continuing down the column of cylinders, it can be seen that for the following will be obtained: f1n,n , '4/3 = nB The average heat transfer coefficient for the tube bank can then be obtained by noting that: '' 2 fln,nhjg =n2'1TRh(Ts - Tw) ,. (11.62) Usmg Eq. (11.58) and Eq. (11.62) the average Nusselt n~ber for the entire column of tubes is therefore given by: ', , , N u = h(nD) = 0.729 [ghjgPi(Pi - pv)(nD)3]1I4 ki ' ILiki(Ts - Tw) , " (11.63) It will be noted that the above equation is identical to the equation for condensation on a single horizontal cylinder, i.e., Eq. (11.55), except that Dois replaced with nD. Experiments have shown that Eq. (11.63) tends to underestimate the heat transfer rate from a column of tubes, and is thus conservative for design purposes. The actual heat transfer r~tes are enhanced by several factors not accounted for in the theoretical model discussed above. These factors include such effects as splashing of the film when it impinges on a lower tube, additional condensation on the subcooled film as it falls between tubes, and uneven run-off because of bowed or slightly inclined tubes. A compact condenser has 100 horizontal tubes arranged in a square array. Saturated steam at 30°C condenses onto the tubes. Each tube has, an outside diameter of 1.5 cm and has a wall temperature of 15°C. Assuming laminar flow, calcuJ.ate the condensation rate per unit length of the tubes. E XAMPLE 1 1.4. Solution. The fluid properties are evaluated at the film tempera~ T I 22. SoC and the latent heat is evaluated at the saturation temperature T, Pl =' (Ts +T w)12 = ::::i 30°C. Hence: / = 997lcglm3 cpl I Ll = 9.82 X',10-4 kg/~s K, his, k = 0.602 W /mK = 4181 J/kg = 2430.0 kI/kg CHAPTER 11: Condensation 579 For this problem the vapor density is very small, so it is assumed that Pl(Pl - py} is equal to pl. The effect of film subcooling is small, but is included here for illustration purposes. h jg = hfg + 0.68cpt(Ts - Tw) = (2430 X 103) + 0.68(4181)(30 - 15) = 2473 kJlkg . There are 10 tubes in each column. So, using n = 10 in Eq. (11.63), the following is obtained: Nu = hnD = 0.729 [ kl g h*gPl nD)3 ]114 f _2( P -lkl(Ts - Tw) = 0729[(9.81)(2473 X . (9.82 X 103)(997)2(10 X 0.015)3]114 1 0-4 )(0.602)(30 - 15) = 1270 The average heat transfer coefficient is then: h = N~kl ,. nD = (1270)(0.6024) = 5 096 W/m2 0C 10(0.015) So, the total heat transfer rate (per unit length) is: Q = hA(Ts - T w) = (5096)(100}7T(0.015)(30 360 X 103 2473 X 103 ' 15) = 36.0 kW and the total condensation rate (per unit length) is therefore: m = Q h jg = = 0.145 kg/s = 524 kglh The Reynolds number must now be checked to ensure that the flow is laminar as assumed: Re = 4 r = 4(0.'145110) = 59 ' P-l . 9.82 X 1 0-4 Because this value is, less than 60, the assumption that the flow is laminar is reasonable. 11.5 . E FFECT O F SURFACE S HEAR S TRESS O N F ILM CONDENSATION ON' A V ERTICAL PLATE In the analysis o f film. condensation given in the previous sections it was assumed that the shear stress at the outer edge of the film negligible. In some situations, however, particularly when the vapor velocity is high, this assumption may not be was justified, i.e., the shear stress exerted on the outer surface of the condensed liquid film may have a significant influence on the heat transfer rate. The action of the shear stress on the surface of the liquid film is illustrated in Fig. 11.17. I f 'Tv is the shear stress acting on the outer surface of the liquid film at any distance down the surface, then the momentum balance for the .control volume shown in Fig. 11.18 (it is the same as the control volume shown before i n Fig. 11.7) gives for the situation here being considered: g dm - 1', / ~; d X7 {~~dX)(8 - Y) -I; 'T, d x = 0 (11.64) 580 IntrodQction to Convective Heat Transfer Analysis i l l Vapor m Flow Flow in Liquid Film FIGURE 11.17 , Shear stress on outer surface of liquid film on a vertical plate. FIGURE 11.18 Control volume used in analysis. Here, Tv h as been taken as p ositivefu the d ownward direction, i.e., i n the direction in which gravity acts. Using Eqs. (11.2) and (11.3) and dividing b y d x, E q. (11.64) gives after rearrangement: /Xt ~y uu = g(8 - Y)(Pi . - Pv) + Tv (11.65) Integrating this equation across the film gives~ s ince u " ',' = 0 at y = 0: u ' " { p, __ p ,)g (Sy _ y2)+ Tv y 2 (U.t;6) I Lt· .' ILt 3 2 ."..', T he mass flow rate in t he film is .then g iven by: r = r ptudy =. .Pt(Pt 3-ILt . + ITLt ~2Pt" 'pv)g8 v . Jo [)' .: (11..67) B ut Eq. (11.12) gives: CHAPI'ER 11: Condensation 581 Hence, using Eq. (11.67), the following is obtained: h fg [ Pt(Pt - Pv)g8 2 ILt Tv ~ _ (Ts - Tw) + P t- u ] -d 8 - kt-'--~ILt dx 8 (11.68) Integrating this equation and again noting that the film thickness must be 0 a t the top o f t he plate, i.e., that 8 = 0 a t x = 0, gives: h fK[Pt(Pt - Pv)g8 4 4ILt i.e.: + Pt~ 8 3 ] ILt 3 = k t(Ts - Tw)x . . 84 Pt(Pt - Pv)g h 4 ILt f~ [1 + (Pt -TvPv)g ~8] 3 = k t(Ts - Tw)x . (11.69) I f t he shear at the surface o f the liquid film can be ignored, this equation gives: 8 Pt(Pt - Pv)g h fg = k t(Ts - Tw)x 4 ILt Hence, Eq. (11.69) can be ·written as: 6 (11.70) 84 [ 1 + (Pt ~vPv)g 3~] = «1 86 (11.71) T he effect o f t he shear stress on the surface o f the liquid layer will be small i f: Tv ~ (Pt - Pv)g 38 (11.72) This will b e true i n many real situations and i n this case Eq. (11.71) can be approximated by: i.e.: -= 8 80 [ 1+--~-- ]-114 4Tv 3 (pt - Pv)g80 (11.73) Now the heat transfer rate is given by: qx = (Ts - Tw) kt-'----:-~ 8 . (11.74) Hence, i f qxO i s t he heat transfer rate that would exist a t the same position i f t he effects o f t he shear stress were negligible i t follows that: (11.75) R 1M 582 Introduction to Convective Heat Transfer Analysis Using Eq. (11.73) then gives: qx qxO [1. 3(pe -4TvPv)g8 ]114 =+ 0 (11.76) It will be seen from Eq. (11.76) that i f Tv acts in the same direction as the gravitational force the heat transfer is increased while i f Tv acts i n the opposite direction to gravity the heat transfer rate is decreased. I t might be thought that the value o f the shear stress acting on the surface of the film could be obtained by assuming there is effectively a boundary layer flow in the vapor over the liquid film and that Tv could then be obtained using standard boundary layer type relations for the wall shear stress. However, when condensation is occurring, vapor is being removed from the vapor flow over the liquid film. I f the rate o f momentum change in the flow direction within the vapor boundary layer on the outer surface o f the liquid layer is small, the shear stress on the surface o f the liquid film will be equal to the rate o f decrease o f vapor momentum. In this case, i f t he vapor has a downward Velocity o f Uv parallel to t he surface and i f the velocity o f the outer surface o f the liquid film is Ue, there is momentum flux from the vapor to t he outer surface of the liquid film that is equal to mv(Uv - ue) where mv is the rate o f vapor condensation per unit surface area at the point being considered. Momentum considerations indicate that there must be a shear stress acting on the vapor at the liquid/vapor interface that is equal to this momentum flux, i.e., that: But: and, i n m any situations: Uv » Ue qxUv h fg I n this case: Tv = I f the effects o f t he shear stress are small, this can b e approximated by: qxOUv Tv = - hfg Substituting this into Eq. (11.76) then gives: qx qxO [1 + 3(Pe -4qxo Uv = Pv)g8 ]114 oh fg I n general the vapor could be flowing upwards or downwards and this equation should be written as: CHAPI'ER 11: Condensation 583 . where t he sign o f the shear stress t enn depends on whether the vapor flow is vertically downwards or vertically upwards. Saturated steam at 100°C condenses on a 0.3-m high vertical plate with a surface temperature of 95°C. Assuming steady laminar flow, find the heat transfer rate at the bottom of the plate i f the steam has a velocity of 10 rn/s parallel to the surface of the plate: E XAMPLE 1 1.5. Solution. The water properties in the film are evaluated at the mean film temperature, i.e., at: At this temperature for water: Pl = 960 kglm3, JLl = 2.89 X 10-4 kglm s, k l = 0.680 W/m-K The vapor density and latent heat are evaluated at the saturation temperature, Ta, which is 100°C and are: Pv = 0.598 k glm3 ,hjg = 2257 kJ/kg I n the situation being considered it will b e seen that the vapor density is very small compared to the liquid density so: P l(Pl - Pv) = P~ is a good approximation. I f the steam is stagnant, Eq. (11.17) gives at the bottom of the plate, i.e., at x = 0.3m: . i.e.: qxO = 0.68 [ 4 x 0.000289 x 0.68 x (100 _ 95) x 0.3 9.81 x 2257000 x 9602 ]114 x (100 - 95) = 38997 W /m 2 When the vapor shear stress on the liquid film is accounted for, the heat transfer rate is given by: i.e.,. in the situation here being considered: !!:... = [1 + 4qxa Uv ] qxa - 3Plg80 hjg i.e.: 114 !!:... qxO = [1+ 1/4 4 x qxO x 10 ] - 3 x 960 x 9.81 x 80 x 2257000 = [I + ]114 qxa - 159420000080 ( i) S84 Introduction to Convective Heat Transfer Analysis But Eq. (11.16) gives at the bottom of the plate: 80 = [4J.l-e ke(Ts - Tw)x]1I4 = [4J.l-e k e(Ts - Tw)X]1/4 ghfgpe(pe - Pv) ghfgP~ = [ 4 X 0.000289 X 0.68 X (100 - 95) X 0.3 ]114 = 0.000087 m 9.81 X 2,257,000 X 9 602 Substituting this into Eq. (i) then gives: qx qxO = [ 38,997 ]114 + 114 1 ± 1,594,200,000 X 0.000087 = [1 - 0.281] Hence, i f the vapor flow is downwards: !k. = [1 + 0 .281t4 qxo = 1.064 In this case then: qx = 1.064 X 38997 = 41,490 W/m2 Similarly, i f the vapor flow is upwards: !k. qxO = [1 - 0.281t4 = 0 .921 In this case then: qx = 0.921 X 38,997 = 38210 W/m2 Therefore, the heat transfer rates with the vapor flow in the downward and upward directions are 41,490 W/m2, and 38,210 W/m2, respectively. E XAMPLE 1 1.6. In the discussion of the effects of vapor shear stress on the heat transfer rate presented above it was assumed that the shear stress effects on the flow in the liquid film were small compared to the effects of gravity. Derive an expression for. the heat transfer rate for the opposite case where the effects. of gravity are negligible compared to the effects of the vapor shear stress. Solution. I f gravitational effects are negligible, Eq. (11.68) gives: Te ~ .(Ts-Tw) hIg [P e- U ]d8 _ ke J.l-e d x 8. Integrating this equation and again noting that the film thickness must be zero at the top of the plate, i.e., that 8 = 0 at x = 0, gives: T, 8 3 V h f gPe- - 3 = ke(Ts - T w)x J.l-e i.e.: But: C HAPfER 11: Condensation 5 85 Hence: h fgpe'Tv ]113 qx = ke(Ts - Tw) [ 3/Le ke(Ts - Tw)X This c an b e written as: The right-hand side o f this equation can be written as: where: Re* . = VP; /Le f !iPt X i s the Reynolds number based on the "shear velocity" nUIilber a nd Pre is the liquid Prandtl num~r. Therefore, the heat transfer rate is given by: J'TvIPe and where J a is the Jakob 11.6 E FFECT OF NONCONDENSmLE GASES O N F ILM CONDENSATION In practice, the vapor that is to be condensed sometimes contajns noncondensible gases such as air. The presence of these noncondensible gases can significantly lower the heat transfer rate from that which would exist under the same circumstances with . a pure vapor. A common example is the build-up of a ir in power plant condensers. These condensers usually operate at a substantial vacuum and some air entrainment is unavoidable. The continuous removal of air by specially designed ejector systems is essential to maintain the condenser vacuum and to maintain acceptable conden- . sation rates. In some chemical plants, the. separation of constituents is sometimes produced by condensing one gas from a mixture of gaSes and in such cases the presence o f a noncondensible gas is unavoidable. ,Even relatively low concentrations of noncondensible gases can substantially reduce the condensation rate. The main reason for this is that·as vapor condenses, noncondensible gases get carried with the vapor toward the surface of the film. Since the film is impermeable, the concentration and partial pressure of noncondensible gases build up near the film to levels much higher than those far from the film. Since the total pressure is constant, the noncondensible gases suppress the partial pressure of the vapor at the edge of the film. This reduces the saturation temperature locally at the film/vapor interface as illustrated in Fig. 11.19. I n turn, the reduction in the driving temperature difference leads to a reduction in the heat transfer and condensation 586 Introduction to Convective Heat Transfer Analysis Condensate Ftlm A A Total Pressure on AA Vapor Pressure Across AA Inert Gas Pressure Across AA ' 0. F IGURE 11.19 Changes near liquid film when a noncondensible vapor is present i n vapor. rates. A more detailed discussion and analysis of the effects of noncondensible gases is given by Rohsenow and Griffith [13]. 11.7 I MPROVED ANALYSES O F L AMINAR F ILM CONDENSATION The basic Nusselt analysis ignores inertial effects ~ the condensed film and subcooling effects. Approximate methods of accounting for subcooling were discussed above. A method of accounting for both effects is discussed in this section. To illustrate the method, condensation on an isothermal vertical plate is again considered [52] to [54]. Interfacial shear stress effects will be neglected. Assuming that the film is thin and that the pressure changes across the film are, . therefore, negligible and that the velocity gradients in the cross-film' direction (ydirection) are much greater than in the flow direction (x-direction), the flow in the film will be governed by the following set of equations: AU + ov ax oy = 0 (11.77) (11.78) (11.79) . The coordinate system and velocity components are shown i n Fig. 11.20. The above set of equations are, of course, very similar to the boundary layer equations governing free convective flow over a plate. .. The boUndary conditions on the solution are: y = 0: u = 0, v = 0, T = Tw y = 8: oy = 0, T = Tsat au (11.80) / where Tw is the uniform plate temperature. CHAPI'BR 11: ,Condensation 587 y x Wide Vertical Plate • v +g Condensate F llmTwO-DimensionaJ Flow u HJ FIGURE 11.20 Coordinate sys~m used. The zero velocity gradient at the outer edge of the film requirement comes from the assumption that the shear stress at the outer edge of the film is zero. The basic assumpti.,on i n the solution procedure is that the velocity profiles in the film at all values o f x are similar, i.e., that: ~ U8 = function (~) 8 (11.81) where U8 is the velocity at the outer edge of the film. Now, all terms i n Eq. (11.78) must have the same order of magnitude, i.e.: and The first of these gives: and the second then gives: Hence, Eq. (11.81) can be written as: [(Pi ~ Pv )gXr U = fUflCtiOn{y[(P€ - ~XPt . fJv)g]1I4} 114 = funCtiOn{..L [ (P€ - Pv)g ] . x1l4v~P€ } (11.82) 588 Introduction to Convective Heat Transfer Analysis Le.: (11.83) where the similarity variable, 11, is here given by: 11 = L x1l4 114 [ (pe - Pv)g ] v~pe (11.84) I f the velocity profiles are similar, it seems logical to assume that the temperature profiles are also similar, i.e., that: Tw - Tsat T - T sat = 8(11) (11.85) the function 8 thus depending only on 11. Rather than working-with u and v it is, in the present situation, more convenient to express these velocity components in terms of a stream function, I/J, which is defined as before such that: (11.86) I f the u velocity profiles are similar, the stream function profiles in the film must be similar, Le.: I.e.: I/J = [(pe - pv)gx] 112 [ 2 VexPe Pe i.e.: '" = (pe - Pv)g ]114 F(l1) Wi;' p, )gv~xlr F('1) (11.87) the function F, as indicated, depending only on 11. Of course, by introducing the stream function, the continuity equation is always satisfied. The momentum equation, i.e., Eq. (11.78), can be written in terms of the stream function as: 2 al/J a21/J _ al/J a 1/J = (pe - Pv) + Ve a3 1/J a yayax a x ay2 Pe g a y3 I t is convenient to define: A= (11.88) [(pe - Pv)g ] Pe . 114 (11.89) C HAPTER 11: Condensation 589 Eqs. (11.84) and (11.87) can then b e w ritten as: A 1'/ = y 11112 X 1l4 (11.90) e and (11.91) i.e.: Fill - ~FF" - (F,)2 + 1 = 0 4 2 (11.92) In terms o f the stream function, the boundary conditions on velocity are: y = 0: ay = 0, ax = 0, l.e., ' " = 0 (11.93) ' " b eing arbitrarily taken as 0 o n the surface. In terms o f the variables introduced above, these boundary conditions become: 1'/ = 0: F ' a", a", . == F = 0 ( q.94) 1'/ = 1'/8: F " = 0 1'/8 b eing the value o f 1'/ a t the outer e dge o f t he film. Next, consider the energy equation, i.e., Eq. (11.79). In t erms o f t he s tream function a nd t he temperature function, (J, t his equation becomes, because T w a nd T s at are constants: a", a(J a", a(J ke a2(J a yax - ax ay = PeCpe ay2 i.e., using Eqs. (11.89), (11.90), and (11.91): (11.95) A ]( A [A 11 112 X 3/4F' 11112 x 1 lJ' [ - 411112 xY] 4 5/4 - [ 3FA1I 4x1l4 1I2 + A 11 112 3/4 X F ,( - Ay )~' A 4 JiI12 x S/4 ~ (J 11112 x 1l4 590 Introduction to Convective Heat Transfer Analysis i.e. - 3/4FO' i.e.: = a eO" Ve 0 " + 3 Pr FO' 4 =0 (11.96) I n terms o f 0, the boundary conditions on this equation are: TJ = 0: 0 = 1 TJ = TJ13: 0 = 0 (11.97) Eqs. (11.92) and (11.96), along with the boundary conditions, constitute a pair o f simultaneous ordinary differential equations i n F and O. However, the value o f TJ13 m ust be found in order to derive the solution. To do this, it is noted that i f the flow up to any value o f x from the top o f the plate is considered, the overall energy balance reqUIres: X L t th t Rate o f h eat transfer to plate = Mass flow rate . · "d " cons! . at section " dered . a e n ea up to p omt cons! ered Enthalpy flux relative to that + at the saturation temperature for the film at the section considered i.e.: JoJo = ke aaT I (X qw dx = h jgpe (13 u dy + Pecpe (13 u (Tsat Jo - T)dy (11.98) where qw is the local heat transfer rate to the plate at any position on the plate. It is given by: . qw y y=o = keO~ aaTJ (Tw y x Tsat ) = ke ve x 11: 1I4 00 (Tw - Tsat ) (11.99) B ecause 00is n ot dependent on x, this gives: Io o qw d x = ke 4Ax3/4 3ve 1/2 O~(Tw - T sat ) (11.100) Hence, using this and the other variables previously introduced and noting that: 13 113 al/l 1 u dy = . - dy = I/Iaty = 0 ay ~ CHAPI'ER 11:' Condensation 5 9i' Le.: -1 A +PlCpl ( V 112 x 114 ) ( T. - Tw) sat 18 F Av , 0 1/2 3/4 X A vx 1/2 1I4 fJd 'T/ i.e.: . ~kl80(Tw - Tsat ) = h jg PlVlF(718) + Plcpe(Tsat - Tw)Vl Jo"S F'8d'T/ (11.101) Integrating by parts gives: "S F 1o'8 d'T/ = F81"s - 1"s 8,'F d71 ' 0 0 But F (11.102) = 0 at 'T/ = 0 and 8 = 0 at 71 = 'T/8 so: F81:s = 0 Hence: Jllll F ' 8d 71 O But rearranging Eq. (11.96) gives: 8 'F = = Jo"l1 8'Fd'T/ (11.103) -~8" 3 Pr Hence: - L"11 8 'F d71 '4 kl(Tw 3 But: = 3 !r o 8 " d71 = - 3!r [8'(718) - 8'(0)] llS J (11.104) Using these results, Eq. (11.101) gives: , 4 Plcpe(Tw - Tsat)Vl [ , ,] Tsat)8 0 = hfgPlVlF('T/8) - '3 Pr 8 ('T/8) - 80 PlCp{ _ P k _ ke - - - -ecpex -e- - - Pr Ve Pecpe Ve so the above equation gives: o= i.e.: hjgpeVlF(718) - ~kl(Tw - T sat )8'('T/8) I F('T/8) _ 4 ke cpe(Tsat - Tw) _ 4 cpt(Tsat - Tw) 8'('T/8) - '3 pevecpe . h fg - 3Pr' h fg 592 Introduction to Convective Heat Transfer Analysis Hence: (11.105) This shows that, in a given situation, F(TJ8)/(J'(TIB) is a constant. To review, the governing equations and boundary conditions are: Fill' + 4' ~FF" 4 (F')2 2 +1=0 (11.106) (11.107) (11.108) (11.109) (11.110) (J" + 3 Pr F(J' =0 - -'-- = - - F(TJ8) (J'(TJ8) 4 Ja 3 Pr TJ = 0: F' = F = 0, (J = 1 TJ = TJ8: F " = 0, (J = 0 I f the inertial terms i n the momentum equation are ignored as in the Nusselt analysis, i.e., i fEq. (11.78) is approximated by: P€ - P.,,) jJ2u ( - ---'-- g + v €- = 0 , P€ a y2, Fill + 1 = 0 Integrating this o nce gives: (11.111) a consideration o f the derivation o f Eq. (11.92) shows that it becomes i n this case: (11.112). ' But F " = 0 w hen TJ = TJ8, so: Cl = TJ8 Hence: F" = TJ8 - TJ Integrating again then gives: But F ' = 0 w hen TJ = 0, so C2 = 0, i.e.: F" = Integrating again then gives: 2 T TJ8TJ - - J 2 CHAPTER 11: Condensation 593 B ut F = 0 w hen 11 = 0, so C3 = 0, Le.: F = 'YJ8T - "6 l = !L 'YJ8 'YJ2 'YJ3 (11.113) Defining the following for convenience: (11.114) Eq. (11.113) c an b e written as: l2 F::;: 11~ ( - - 26· Substituting Eq. (11.115) into Eq. (11.107) then gives: (3) (11.115) .8 " 8' Integrating this gives: = _ ~Pr'YJ~ ( l2 4 2 _ ( 3) 6 (11.116) i.e.: (11.117) Integrating again gives, because 8 8- 1 B ut 8 = 80 ')6 I: {exp HpT'J6 e xp - = 1 at 'YJ 4 = 0: (~,3 - ~ ')4)]} d, (11.118) = 0 a t 11 = 'YJ8 so: I - 1 = 80 'YJ8 Jo e{ [3'4 4(1'6 Pr'YJ8 e xp P r')6 4 y l 3 - 24 'YJ ~ d~ 1 4)1} D ividing the above two equations then gives: 8 = 1- H, d f.' {eXpHpT'J84(~,3 - 2~')4)]} d, ~,3 - 2~ ')4 . 3 ':"'::""'~---==----~-----'-..:;;...:-.-- I: { H (11.119) N ow b y setting 'YJ = 'YJ8, Eq. (11.113) gives: F(118) = 'YJ8 3 (11.120) 594 Introduction to Convective Heat Transfer Analysis Substituting this into Eq. (11.108) then gives: 7J~=~Ja(}'(1J8)=~Jad(}1 3 3 Pr 3 Prd( ,=17J8 ~ i.e.: Pr1/: = 4Ja~n~, (11.121) For any chosen value of Pr7J~, Eq. (11.119) can easily be integrated to give the variation of (} with ( . Eq. (11.121) then allows the value of J a corresponding to the " chosen value of Pr7J~ to be found. Eq. (11.99) gives: Le.: Nux = - [(Pi - Pv)g:i3]114 (J' PiP~ 0 Hence, using Eq. (11.118) the following is obtained: (11.122) Nux [ ( p, Le.: ~l::lgx' . r 1+.f: { = l /4 exp[_~P",.4(~ -~)]} dC] Nu [ x ( Pi - Pv)gx3 P r PiV~· ]V4 = lJ[(1/: prl fJ expH(p",:l(~ - ~)]} d'] [ p eke· ]114 x U (Pi - Pv)gx3cpe (11.123) The left-hand side of this equation can be written as: N (11.124) Therefore, for any chosen value of Pr7J~, the value of J a and then the co~­ spondfug value of the quantity given in Eq. (11.124) can b e found. In this way, the· variation of : N [ pike ]114 x U (Pi ~ Pv)gx3C i p / With J a can be determined. The variation so found is given in Fig. 11.21. . CHAPTER 11: Condensation 5 95 0.4 ' - - - - " ' - - - _ . . ! . . -_ _. ...L-_ _. .J 0.0 0.5 1.0 Ja 1.5 2.0 F IGURE 11.21 Variation o f Nuxlveke/(pe .:... Pv) g.x3cpt]1I4 with J a when inertial effects are neglected. . Now, the result o f the Nusselt analysis which ignoressubcooling effects can be written as: N ux [ veke (Pe - Pv)gx cpe 3 ]114 =[ h fg 4 cpe(Tsat - Tw) ]114 = ( 4Ja)-1/4 (11.125) T he results given by this equation agree with the results derived using the procedure described in this section for values o f J a less than approximately 0.1. The approximate solution o f Rohsenow [32] that was discussed above, see Eq. (11.35), gives: N ux [ veke (Pe - pv)gx3 cpe ]114 = (4Ja)-1I4(1 + 0.68Ja)1I4 (11.126) The results given by this equation agree with the results derived using the procedure described in this section for essentially the entire range o f values o f J a considered. The inertial terms i n the momentum equation were neglected in the above analysis, i.e., the terms: ~FF" _ (F')2 4 2 were neglected compared to t he terms: F ill (11.127) +1 (11.128) i n Eq. (11.106). But the solution with these inertial terms ignored, i.e., Eq. (11.113), gives: F= 1]2 1]3 1 ]8- - - 2 6 (11.113) 596 Introduction to Convective Heat Transfer Analysis Hence, the terms that were ignored, i.e., the terms given in Eq. (11.127), are to first order of accuracy, given by: i.e., by: (11.129) The terms that were ignored in the momentum equation thus have the order o f magnitude 7]8 4 • But Eq~ (11.121) gives: .4 ' r. 4 Ja dO 718 = P r d , '=1 I (11.130) This indicates that the inertial terms will only have a significant effect on the heat transfer rate i f J a is large and P r is small. This is illustrated by the results given il), Fig. 11.22 which shows th!! variation of: with J a for various values of P r taking account of the inertial terms together with the variation obtained by ignoring these inertial terms, this latter variation being independent of Prandtl number. I t will be seen that inertial effects only have a sigJiificant influence ott the/heat transfer rate i f P r < 10. P r= 100 and . I o. 81l nng Inertia Terms P r= 10 P r= 1 0.4 " --_ _I ..-_---'_ _- --L_ _- -J 0.0 0.5 1.0 Ja 1.5 2.0 FIGURE 11.22 Variation of NU x [ veke/(pe - Pv)gx3 cpe ]1I4 with Ja for various Pro CHAPTER 11: Condensation 597 11.8 NONGRAVITATIONAL·CONDENSATION The rate of condensation on a vertical surface is controlled by the force of gravity acting on the condensed liquid film. A consideration of Eq. (11.20) shows for example that for a vertical plate the mean heat transfer rate from the plate with laminar flow in the film is proportional to g1l4. Attempts have therefore been made to increase condensation rates by using centrifugal forces instead· of the .gravitational force to drain the condensed liquid film from the cold surface [55]. The simplest example of this would be condensation on the upper surface of a cooled circular plate rotating in a horizontal plane. This situation is shown in Fig. 11.23. A Nusselt-type analysis of this situation will be considered in the present section. Consider the control volume shown i n Fig. 11.24. The same assumptions as used in the basic Nusselt analysis are used here. With these assumptions, the forces acting on the control volume shown. in Fig. 11.24 must balance, i.e.: J.te ay27Trdr au = pe(8 - y)27Trdrw r 2 i.e.: JLe - au = Pe(8 ay Control Volume y)w r 2 (11.131) Rotating Cooled P late Condensate Film F IGURE 11.23 Laminar film condensation on a rotating horizontal circular plate. Vapor Control Condensate Film r F IGURE 11.24 Control volume used in the analysis of laminar film condensation on a rotating horizontal· circular plate. 598 Introduction to Convective Heat Transfer Analysis Integrating Eq. (11.131) with respect to y, i.e., across the liquid film and applying the no-slip condition at the surface, i.e., setting u = 0 at y = 0, ·gives the veloCity distribution in the film as: (11.132) The mass flow rate i n the liquid film, r, is thengiyen by integrating ,this. velocity distribution. This gives: r = 21T'r · i I) . ~w2r83· o PtU d y = 21T'r t ... 3JLt . (11.133) I t is next noted that an energy balance requires that: hf ! d r d r - k·(Ts - TW )2· T'r d r dr -t 8 1 (11.134) Dividing this equation through by d r and substituting the expression for r given by Eq. (11.133) then gives: (11.135) i.e.: / [3r83d8 dr + 284] = 3lLt kt(Ts - Tw) . hfgPlW 2 This equation indicates that 8 is not dependent on r and therefore that: 84 = [3ILtkt(Ts - Tw)] 2hfg Plw 2 Because the local heat transfer rate per unit surface area is given by: . kt(Ts - Tw) q = -:.......;...."..-;..;.. 8 (11.136) (11.137) it follows that q is the same everywhere on the plate and that the Nusselt number based on the radius o f the plate R is therefore given by: (11.138) This equation can be written in the following form: Nu . =[~ . hfg - 3 c pt(Ts ....2 PtW R R JLtCpt Tw) IL~ kt 222 ]114 (11.139) CHAPTER I I: Condensation 599 i.e.: Nu = [ ~Re2Pre ] 3 Ja 1/4 (11.140) Here Re is defined by: Re = pe(~R)R ILe I t) (11.141) R being, of course, the rotational velocity of the outer edge of the plate. The condensation rate for the rotating plate is given by: m" = q'11"R2 = '11"R2ke(Ts - Tw) [ h lg h lg . ]114 3ILeke(Ts - Tw) 2hlgP~lt)2 (11.142) From this equation it will be seen that mv is proportional to I t) In. The condensation rate can therefore be increased by increasing the rotational speed of the plate. ~ , . E XAMPLE 1 1.7. Derive an expression for the ratio of the heat transfer rate to a rotating horizontal circular plate on which a vapor is condensing to the heat transfer rate that would exist i f the vapor were condensing on a stationary vertical square plate. Both pla:tes have the same surface temperature and the same surface area. Consider only one side of the square plate. Solution. The rotating plate has a radius R and the square plate has a side length of H . Because the two plates have the same area it follows. that: I f it is assumed that: Pe(pe - Pv) = p~ then Eq. (11.20) gives for the vertical square plate: Dividing the above two equations then gives: ~: = ~ ~H~ [ .But: 2]114 Hence: Qrot Qsq = 1.106 [ (() 2R ]114 g . 600 Introduction to Convective Heat Transfer Analysis 11.9 CONCLUDING REMARKS Vapor can condense on a cooled surface i n two ways. Attention has mainly been given in this chapter to one o f these modes o f condensation, i.e., to film condensation. The classical Nusselt-type analysis for film condensation with laminar film flow has been presented and the extension o f this analysis to account for effects such as subcooling i n the film and vapor shear stress at the outer edge o f the film has been discussed. The conditions under which the flow in the film becomes turbulent have also been discussed. More advanced analysis o f laminar film condensation based on the use o f the boundary layer-type equations have been reviewed. PROBLEMS· 11.1. Consider condensate forming in an insulated container with the lower surface held at constant temperature, Tw , as shown in Fig. PI1.1. Since the condensate cannot drain.. away, the rate of condensation will decrease as the layer thickness increases. Assuming that the heat transfer is by quasi-steady conduction derive an expression for film thickness as a function of time and for the heat transfer rate as a function of time: Vapor at Ts Insulated Sides a Condensate Bottom at Tw F IGURE P11.1 11.2. Consider condensate forming on a cold tube of radius, R, in a micro-gravity environment. In such an environment, the condensate will not run off the tube. Assuming that the heat transfer is by quasi-steady conduction show that the film radius as a function of time, r, is given by: ! + (!..)2 (~In!.. _!) = 2ke(Ts - Tw)t 2 2 R R pehfgR2 11.3. Do film subcooling and vapor superheat cause the overall heat transfer and condensation rates to increase or decrease? Explain the answers you give. 11.4. There are several substances available commercially that can be applied to glass surfaces to prevent ''fogging''. The promotional literature for one of these substances for use on eye glasses states that "this long-lasting anti-Jog treatment prevents surface con- . densation". Discuss the validity of this claim. How does the ~ti-fog treatment work? / CHAPTER 11:· Condensation 601 11.5. Stagnant saturated steam at 100°C condenses on the outside o f a horizontal copper tube with outside diameter 5.0 cm. The copper tube has a wall thickness o f 1.5 m m and a thermal conductivity o f 3 90 WIm- K. At the axial location being considered, the bulk: temperature o f the water flowing inside the copper tube is 80°C and the inside. heat transfer coefficient is 3500 W1m2 • Assuming film condensation, calculate the heat transfer and condensation rate per unit length o f pipe. 11.6. Stagnant saturated steam at 100°C condenses onto a vertical plate with a surface temperature o f 70°C. Approximately how far down the plate does the film become wavy? A t approximately what distance does the film become fully turbulent? 11.7. Stagnant saturated steam at 100°C is to be condensed on a 15-cm long vertical tube with an outside diameter o f 25 mm. What tube surface temperature is needed to maintain a condensation rate o f 0 .6 kg per hour? 11.S. A 30-cm high by 150-cm wide plate is maintained at 5°C and is inclined at 45° from the vertical. Calculate the rate o f condensation, the heat transfer rate, and the maximum . film thickness-when the plate is exposed to stagnant saturated water vapor at 20°C. Do the calculation both with and without the effect o f film subcooling. What is the value o f the Jakob number for this problem? ·11.9. An 8-m high by I.5-m wide vertical plate is maintained at 5°C.. Calculate the rate o f condensation and the heat transfer rate, when the plate is exposed to stagnant saturated water vapor at 20°C. Include the effect o f film subcooling. 11.10. Steam is to be condensed on 25 horizontal tubes in a square array. The two configurations shown in Figure P II.IO are being considered. Explain, using only physical arguments, which configuration will give the highest condensation rate. Assume that the flow is laminar. 11.11. Assuming identical conditions,calculate the percent difference in the condensation rates for the two horizontal tube bundle configurations shown in Figure P H.H. 11.12. Steam at a saturation temperature o f 50°C condenses on a wide flat vertical wall which has a height o f 30 m m and which is kept at a surface temperature o f 46°C. Find the heat transfer rate to the plate per unit width o f the plate. 11.13. I f the plate described in Problem 11.12 was 3 m high instead o f 3 0 rom high, would the flow in the condensed film on the surface become turbulent? I f i t does become turbulent, find the distance from the top o f the plate at which transition occurs. 0 0000 0 0000 0 0000 00000 0 0000 Configuration 1 0 000 o00a 000 a a 000 0 000 o0 o o o Configuration 2 F IGURE P11.11 . 602 Introduction to Convective Heat Transfer Analysis 11.14. Steam at a saturation temperature of 60°C condenses on a wide flat vertical wall which has a height of 0.15 m and which is kept at a surface temperature of 55°C. Plot the variatio:n. o f the heat transfer rate per unit wall area with distance along the plate. 11.15. Stagnant saturated steam at 80°C condenses on a O.4-m high vertical plate with a surface temperature of 75°C. Calculate the heat transfer rate and.condensation rate per meter width of the plate assuming that the flow i n the liquid film is steady and laminar. Also find the maximum film thickness. 11.16. Consider the situation described i n Problem 11.15 in which stagnant saturated steam at 80°C condenses on a O.4-m high vertical plate with a surface temperature of 75°C. Calculate the heat transfer rate per meter width of plate with and without taking subcooling effects into account. Do subcooling effects have, a significant influence on the condensation rate in this situation? 11.17. Stagnant saturated steam at 95°C condenses onto a vertical plate with a surface temperature of 90°C. The plate is 0.6 mhigh and 1.5 m wide. Determine whether the flow in the condensed liquici film remains laminar and then find the rate of condensation on the plate. 11.18. In a steam condenser there are 81 tubes arranged in a square array, i.e;, there are 9 columns of tubes with 9 tubes in each column. Saturated steam. at 40°C condenses on the tubes. Each tube has an outside diameter of 1 cm and has a wall temperature of 35°C. Assuming laminar flow, calculate the condensation rate per meter length of the tubes. 11.19. Consider laminar ·film condensation on a vertical plate when the vapor is flowing parallel to the surface in a downward direction at velocity, U. Assume that a turbulent boundary layer is formed in the vapor along the outer surface of the laminar liquid film. Determine a criterion that will indicate when the effect of the shear stress at the outer edge of the condensed liquid film on the heat transfer rate is less than 5%. Assume that Pv « Pt. 11.20. Steam at a saturation temperature o f 50°C condenses on the upper surface of a rotating circular horizontal plate which has a diameter o f 30 cm and which is kept at a surface temperature of 46°C. Plot the variation condensation rate with rotational speed for values between 50 and 1000 revolutions per minute. REFERENCES .1. Bell, K.J. and Panchal, C.B., "Condensation," 6th. Int. Heat Transfer Conf., Toronto, Vol. 6,pp.361-375,1978. 2. Burmeister, L.C., Convective HeatTransfer, 2nd ed., Wiley-Interscience, New York, 1993. 3. Carey, v.P., Liquid- Vapor Phase-Change Phenomena, Hemisphere PubliShing, Washington, D.C., 1992. 4. Chen, S.L., Gerner, F.M., and Tien, C.L., "General Film Condensation Correlations," Experimental Heat Transfer, Vol. 1, pp. 93-107, 1987. 5. Collier, J.G., Convective Boiling and Condensation, McGraw-Hill, New York, 1972. CHAPI'ER 11: Condensation 603 6. Collier, J.G., "Heat Transfer in Condensation,"in Two-phase Flow a nd Heat Transfer in the Power a nd Process Industries, Bergles, A.E., Collier, J.O., Delhaye, 1.M., Hewitt, G.F., and Mayinger, F., Eds., pp. 330-65. Hemisphere Publishing, Washington, D.C., 1981. 7. Butterworth, D., "Condensers: Basic Heat Transfer and Fluid Flow," in H eat Exchanger Sourcebook, Palen, J.W., Ed., pp. 389-413, Hemisphere Publishing, Washington, D.C., 1986. 8. Eckert, E.R.G., "Pioneering Contributions to our Knowledge o f Convective Heat Transfer," A SME J. H eat Transfer, Vol. 103, pp. 409-414, 1981. 9. Griffith, P., "Condensation," i n Handbook o f Multiphase Systems, Hestroni, G .,Ed., Hemisphere Publishing, Washington, D.C., 1982. 10. Merte, H., Jr., "Condensation Heat Transfer," Adv. Heat Transfer, Vol. 9, pp. 181-272, 1973. 11. Silver, R.S., " An Approach to a General Theory o f S urface Condensers," Proc. Inst. Mech. Eng., Vol. 178, Part 1, No. 14, pp. 339-376, 1964. 12. Whalley,P.B., Boiling, Condensation, and Gas-Liquid Flow, Oxford University Press, New York, 1987. 13. Rohsenow, W.Nt. and Griffith, P., "Condensation/' H andbook o f H eat Transfer Fundamentals, Rohsenow, W.M., Hartnett, J.P., Gani, E.N., Eds., McGraw-Hill, New York, pp. 11.1-11.50, 1985. . 14. Citakoglu, E. and Rose, J.W.,· "Dropwise Condensation: Some Factors Influencing the Validity o f Heat Transfer Measurements," Int. J. H eat Mass Transfer, Vol. 11, p. 523, 1968. 15. Drew, T.B., Nagle, W.M., and Smith, W.Q., ''The Conditions for Dropwise Condensation o f Steam," Trans. AIChE, Vol. 31, pp. 605-621, 1935. 16. Rose,I.W., "DropwiseCondensation Theory," Int. J. H eatMass Transfer, Vol. 2 4,p.191, 1981. 17. Rose, J.W., " On the Mechanism o f Dropwise Condensation," Int. J. H eat Mass Transfer, Vol. 1 0,pp. 755-762,1967. 18. Tanner, D.W., Pope, D., Potter, C.J., and West, D., " Heat T ranfer in Dropwise Condensation o f Low Pressures in the Absence and Presence o f Non-Condensable Gas," Int. J. H eat Mass Transfer, Vol. 11, p. 181, 1968. 19. Tanasawa, I.T., "Dropwise Condensation - The Way to P ractical Applications," Proc. 6th. lnt. Heat Transfer Conf., Toronto, Vol. 6, pp. 393-405, 1978. . 20. Takeyama, T. and Shimizu, S ., "On the Transition o f D ropwise-Film Condensation," Prot:. 5th. Int. Heat Trans. Conf., Tokyo, Vol. III, 274, 1974. 21. Griffith, P., "Dropwise Condensation," in Handbook o f H eat Transfer, Rohsenow, W.M. e t al., Eds., 2nd. ed., McGraw-Hill, New York, 1985. 22. Song, Y., Xu, D., and Lin, J., " A Study o f the Mechanism o f D ropwise Condensation," Int. 1. H eat Mass Transfer, Vol. 34, pp. 2827-2831, 1991. 23. Graham, C. and Griffith, P., "Drop Size Distributions and H eat Transfer in Dropwise Condensation," Int. J. H eatMass Transfer, Vol. 16, p. 337, 1973. 24. Merte, H., Jr. and Son, S., "Further Consideration o f Two-Dimensional Condensation Drop Profiles and Departure Sizes," Warmeund StofJubertragung, Vol. 21, pp. 163-168, 1987. 25. Merte, H., Jr., Yamali, c., a nd Son, S., "A Simple Model for Dropwise Condensation Heat Transfer Neglecting S weeping," Proc. Int. Heat Trans. Conf., Vol. 4, pp. 1659-1664, San Francisco Hemisphere Press, New York, 1986. 26. Griffith, P. and L ee, M.S., " The Effect o f Surface Thermal Properties and Finish on Dropwise Condensation," Int. J. H eat Mass Transfer, Vol. 10, pp. 697-707,1967. 604 Introduction to Convective Heat Transfer Analysis 27. Hannemann, R. a nd Mikic, B., " An Experimental Investigation into the Effect o f Surface Thermal Conductivity on the Rate o f H eat Transfer in Dropwise Condensation," Int. 1. Heat Mass Transfer, Vol. 19, p. 1309, 1976. 28. Haragucbi, T., Shimada, R., Humagai, S., and Takeyama, T., " The E ffect o f Polyvinylidene Chloride Coating Thickness on Promotion o f Dropwise Steam Condensation," Int. 1. Heat Mass Transfer, Vol. 34, pp. 3047-3054,1991. 29. de Gennes, P.G., Hua, X., and Levinson, P., "Dynamics o f Wetting: Local Contact Angles," 1. Fluid Mech., Vol. 212, pp. 55-63, 1990. 30. Nusselt, W., " Die Oberflachenkondensation des Wasserdampfes," Zeitschrift des Vereines Deutscher Ingenieure, Vol. 60, pp. 541-569, 1916. 31. Rose, J.W., "Fundamentals o f Condensation Heat Transfer: Laminar Flow Condensation," l SME Int. 1., Series n, Vol. 31, pp. 357-375, 1988. 32. Rohsenow, W.M., " Heat Transfer and Temperature Distribution i n L aminar Film Condensation," Trans. ASME, Vol. 78, pp. 1645-1648, 1956. 33. Butterworth, D., "Filmwise Condensation," i n Two-Phase Flow and Heat Transfer, Butterworth, D. and Hewitt, G.F., Eds., Oxford University Press, London, pp. 426·-'462,1977. 34. Butterworth, D., " Film Condensation o f Pure Vapor," i n Heat ExchangerDesign Handbook, Schlunder, E.U., E"d.-in-Chief, Vol. 2, Chapter 2.6.2, Hemisphere Publishing,New York,1983. 35. Chen, M.M., " An Analytical Study o f Laminar Film Condensation, P art 1. F lat Plates; Part 2. S ingle and Multiple Horizontal Tubes," 1. Heat Transfer, Vol. 83, pp. 48-60, 1961. 36. Fujii, T., Theory o f Laminar Film Condensation, Springer-Verlag, New York, 1991. 37. Kutateladze, S.S., "Semi-Empirical Theory o f F ilm Condensation o f P ure Vapours," Int. 1. Heat Mass Transfer, 25, pp. 653-660, 1982. 38. Labuntsov, D.A., " Heat Transfer in Film Condensation of Pure Steam on Vertical Surfaces and Horizontal Tubes," Teploenergetika, Vol. 4, p. 72,1957. 39. Rohsenow, W.M., " Film Condensation," in Handbook o fHeat Transfer, Rohsenow; W.M. and Hartnett, J.P., Eds., Chapter 12A, McGraw-Hill, New York, 1973. 40. Shekriladze, I.G. and Gomelauri, V.I., "Theoretical Study o f L aminar Film Condensation o f Steam," Int. 1. HeatMass Transfer, Vol. 9, pp. 581-591, 1966. 41. Denny, V.E. and Mills, A.F., "Nonsimilar Solutions for Laminar Film Condensation on a Vertical Surface," Int. J. Heat Mass Transfer, Vol. 12, pp. 965-979, 1969. 42. Minkowycz, W.J. and Sparrow, E.M., "Condensation H eat Transfer i n t he Presence o f Non-condensables, Interface Resistance, Superheating, Variable Properties and Diffusion," Int. 1. Heat Mass Transfer, Vol. 9, p. 1125, 1966. 43. Poots, G. a nd Miles, R.G., "Effect o f Variable Properties on Laminar Film Condensation o f Steam," Int. 1. Heat Mass Transfer, Vol. 10, p. 1677, 1967. 44. Kirkbridge, C.G., " Heat Transfer b y Condensing Vapors on Vertical Tubes," Trans. AIChE, Vol. 30, p. 170, 1934. 45. Colburn, A.P., "The Calculation of Condensation Where a Portion o f t he Condensate Layer Is in Turbulent Flow," Trans. AIChE, Vol. 30, p. 187,1933. 46. Hirshburg, R.I. and Florschuetz, L.W., " Laminar Wavy-Film Flow: P art I, Hydrodynamics Analysis, P art n, Condensation and Evaporation," J. Heat Transfer, Vol. 104, pp. 452-464, 1982. 47. Nakorykov, V.E., Pokusaev, B.G., and Alekseenkon, S.V., "Stationary Rolling Waves on a Vertical F ilm o f Liquid," J. Eng. Phys., Vol. 30, pp. 517-521,1976. 48. Mills, A.F., Heat Transfer, Richard D .lrwin, Homewood, n, 1992. 49. Sparrow, E.M., Minkowycz, W.J., and Saddy, M., "Forced Convection Condensation in the Presence o f Non-Condensibles and Interfacial Resistance," Int. 1. Heat Mass Trans.fer, Vol. 10, pp. 1829-1845, 1967. CHAPTER 11: Condensation 60S 50. Rohsenow, W.M., Weber, I.M., and Ling, A.T., "Effect of Vapor Velocity on Laminar and Thrbulent Film Condensation," Trans. ASME, Vol. 78, pp. 1637-1644~ 1956. 51. Carpenter, E.F. and Colburn, A.P.• "The Effect of Vapor Velocity on Condensation Inside Thbes," in Proceedings, General Discussion on Heat Transfer, Inst. Mech. Eng.-ASME, New York, pp. 20-26,1951. 52. Sparrow, E.M. and J.L. Gregg, "A Boundary-layer Treatment of Laininar Film Condensation," J. o f l Ieat Transfer, Vol. 81,pp.13-18, 1959. 53. Stuhltrager, E., Narldomi, Y., Miyara, A., and Uehara, H., "Flow Dynamics and Heat Transfer of a Condensate Film on a Vertical Wall. I. Numerical Analysis and Flow Dynamics," Int. J. Heat Mass Transfer, Vol. 36, pp. 1677-1686, 1993. 54. Koh, J.C. Y., Sparrow, E.M., and Hartnett, J.P', "The Two-Phase Boundary Layer in Laminar Film Condensation," Int. J. Heat Mass Transfer, Vol. 2, p. 69, 1961. 55. Dhir, Y.K. and Lienhard, J.H., "Laininar Film Condensation on Plane and Axisymmetric Bodies in Non-:-uniform Gravity," J. Heat Transfer, Vol. 93, p. 97, 1971.