Heat Mass Transfer (29) 45:355 361 DOI 1.17/s231-8-436-y ORIGINAL An analytical investigation into filmwise conensation on a horizontal tube in a porous meium with suction at the tube surface Tong Bou Chang Æ Fu Jen Wang Receive: 28 nuary 28 / Accepte: 27 August 28 / Publishe online: 13 September 28 Ó Springer-Verlag 28 Abstract An analytical investigation is performe into the problem of steay filmwise conensation flow over the outsie surface of a horizontal tube embee in a porous meium with suction at the tube surface. As in classical film conensation problems, an assumption is mae that the conensate an vapor layers meet at a common bounary rather than being separate by an intermeiary two-phase zone. Furthermore, it is assume that the conensate film has constant properties an conforms to Darcy s law within the porous meium. By introucing an effective suction function to represent the effect of the wall suction on the thickness of the liqui film, both the local conensate film thickness an the local Nusselt number are erive using a simple numerical shooting metho. The analytical results inicate that the mean Nusselt number epens on the Darcy number, the kob number, the Rayleigh number an the suction parameter. Furthermore, it is foun that the local Nusselt number has a maximum value at the upper surface of the horizontal tube an reuces towar zero at the lower surface as a result of the finite thickness of the conensate layer. T. B. Chang Department of Mechanical Engineering, Southern Taiwan University, Tainan, Taiwan F. J. Wang Department of Refrigeration, Air Conitioning an Energy Engineering, National Chin-Yi University of Technology, Taichung, Taiwan T. B. Chang (&) 1, Nan-Tai Street, YungKang, Tainan, Taiwan e-mail: tbchang@mail.stut.eu.tw List of symbols Cp specific heat at constant pressure Da Darcy number efine in Eq. 1 f effective suction function efine in Eq. 12a g acceleration of gravity h heat transfer coefficient h fg heat of vaporization kob number efine in Eq. 1a k thermal conuctivity K permeability of porous meium Nu Nusselt number efine in Eq. 19a Pr e effective Prantl number efine in Eq. 1c R raius of circular tube Ra Rayleigh number efine in Eq. 1b Re w Reynols number efine in Eq. 1e Sw suction parameter efine in Eq. 1f T temperature DT saturation temperature minus wall temperature u velocity component in x-irection v velocity component in y-irection Greek symbols conensate film thickness l liqui viscosity q liqui ensity a thermal iffusivity h angle measure from top of tube Superscripts average quantity * imensionless variable Subscripts sat saturation property w quantity at wall e effective property
356 Heat Mass Transfer (29) 45:355 361 1 Introuction The problem of laminar film conensation on a horizontal tube in a porous meium has receive extensive attention ue to its wie range of practical applications, incluing heat exchange systems, thermally enhance oil recovery processes, waste isposal, chemical engineering processes, heat pipe esign, geothermal energy utilization, an so forth. Laminar film conensation on a vertical surface was first analyze by Nusselt [1] in 1916. In performing his analysis, Nusselt mae three basic assumptions, namely that the conensate film was very thin, the convective an inertial effects were negligible an coul be ignore, an the temperature within the conensate layer varie linearly with the film thickness. Following the publication of Nusselt s finings, many researchers attempte to refine the original analysis by implementing more realistic assumptions. An excellent review of the relate stuies is presente by Merte in [2]. In practical engineering applications, the problem of laminar conensation is not restricte to vertical surfaces only, an thus researchers have explore a variety of conensation systems, incluing horizontal plates [3, 4], horizontal cyliners [5 8], an spheres [9], for example. The problem of conensate flow on surfaces embee in porous meia has attracte particular attention. For example, Cheng et al. [1, 11] use the Darcy moel to analyze the conensate flow along incline surfaces in a porous meium. Chiou et al. [12] employe a novel transformation metho to investigate the problem of film conensation flowing over the external surface of a horizontal elliptical tube in a porous meium. The problem of transient film conensation on a vertical plate embee in porous meium has been stuie by Al-Nimr et al. [13, 14]. Char et al. [15] consiere the case of a mixe convection conensate flow along a vertical wall embee in a porous meium. Wang et al. [16, 17] investigate the problem of film conensation on flat an wavy horizontal plates in porous meia an showe that the wavy surface yiele an effective improvement in the heat transfer performance. More recently, Chang [18, 19] analyze the effects of capillary forces an wall suction on the heat transfer characteristics of a conensate layer on a horizontal flat surface embee in a porous meium an emonstrate that the wall suction effects significantly enhance the conensation heat transfer performance. Accoringly, the present stuy examines the heat transfer performance of a conensate layer flowing over the external surface of a horizontal tube embee in a porous meium with a suction force acting at the tube surface. By introucing an effective suction function to moel the effect of the wall suction on the conensate film thickness an using the separation of variables metho, analytical expressions are obtaine for both the Nusselt number an the conensate film thickness as a function of the Darcy number, the kob number, the Rayleigh number an the suction parameter. 2 Analysis The current analysis consiers a horizontal, permeable tube with a wall temperature of T w embee in a porous meium fille with a pure ry vapor of uniform temperature T sat. Figure 1 presents a schematic illustration of the physical moel an coorinate system, in which the curvilinear coorinates (x, y) are aligne along the tube wall surface an the surface normal, respectively. As shown, when the wall temperature, T w, is lower than the saturation temperature, T sat, the liqui wets the tube surface ieally, an a conensate film is forme on the tube surface. Uner steay-state conitions, the thickness of the liqui film bounary layer,, has a minimum value at the top of the tube an increases graually as the liqui flows ownwar over the tube surface. In analyzing the heat transfer characteristics of the conensate film, the following assumptions are mae: 1. The flow is steay an laminar, an thus the effects of inertia an convection are negligible an can be ignore. 2. The wall temperature, vapor temperature an properties of the porous meium, ry vapor an conensate, respectively, remain constant. Porous meium δ R y Wall temperature T w Vapor temperature T sat x g Liqui-vapor interface Fig. 1 Schematic iagram of current physical moel an coorinate system
Heat Mass Transfer (29) 45:355 361 357 3. The conensate film has negligible kinetic energy. 4. The effective pore raii of the porous meium are not very small, an thus the effects of capillary suction in the porous zone can be neglecte. 5. The flow of the liqui film in the porous meium is governe by Darcy s law. Note that the neglecting of capillary suction effects will inuce an error when the effective pore raii are very small. However, in practical problems involving conensation in porous meia, the effective pore raii are not very small. Consequently, capillary suction effects on the heat transfer performance were neglecte in most of the previous papers [1 17]. Uner these assumptions, the governing equations for the liqui film subject to bounary layer simplifications are given as follows: Continuity equation ou ox þ ov oy ¼ ð1þ Momentum equation in x-irection ¼ ðq q v Þg sin h l e K u ð2þ Energy equation o 2 T ¼ a e oy 2 ð3þ where K is the intrinsic permeability of the porous meium, u an v are the Darcian velocity components in the x- an y-irections, respectively, a e is the effective thermal iffusivity of the porous meium when saturate with conensate, an l e is the effective ynamic viscosity of the liqui-saturate porous meium. The bounary conitions are as follows: At the tube surface, i.e., y = v ¼ v w an T ¼ T w ð4þ where v w is the wall suction velocity. At the liqui vapor interface, i.e., y = T ¼ T sat at y ¼ ð5þ Since the thickness of the liqui film is very small relative to the iameter of the tube, the temperature profile within the conensate layer varies in accorance with the following linear function: T ¼ T w þ DT y ð6þ where DT = T sat T w. The term (q q v ) in Eq. 2 can be approximate as q because the vapor ensity is relatively small compare with the liqui ensity. Note that the approximation of neglecting the vapor ensity relative to the liqui ensity is acceptable only at low an moerate pressures but not at the high pressures. Rearranging Eq. 2, the following velocity istribution equation can be erive: u ¼ qgk sin h ð7þ l e In accorance with Nusselt s classical analysis metho, the overall energy balance in the liqui film can be written as 8 9 Z < = qu h fg þ CpðT sat TÞ y x : ; x ot þ q h fg þ CpDT vw x ¼ k e oy x ð8þ where k e is the effective thermal conuctivity of the liquisaturate porous meium. The right han sie of Eq. 8 represents the energy transferre from the liqui film to the tube surface. Meanwhile, the first term on the left han sie of the equation expresses the net energy flux across the liqui film (from x to x? x) while the secon term expresses the net energy sucke out of the conensate layer by the permeable tube. Substituting Eqs. 6 an 7 into Eq. 8 yiels q 2 gk h fg þ 1 2 CpDT l e x f sin h DT ¼ k e gþ q h fg þ CpDT vw ð9þ For analytical convenience, the following imensionless parameters are efine: ¼ CpDT h fg þ 1 2 CpDT Ra ¼ q2 gpr e R 3 l 2 e Pr e ¼ l ecp k e Da ¼ K R 2 Re w ¼ qv wr l e ð1aþ ð1bþ ð1cþ ð1þ ð1eþ Sw ¼ 1 þ 1 2 Pr e Re w ð1fþ where R is the tube raius; is the kob number; Pr e is the effective Prantl number; Ra is the effective Rayleigh number; Da is the Darcy number; Re w is the Reynols number, an Sw is the suction parameter. Substituting the imensionless equations give in Eqs. 1a 1f into Eq. 9 an applying the relationship x = Rh, with
358 Heat Mass Transfer (29) 45:355 361 introucing a imensionless liqui film thickness parameter * = /R, Eq.9 can be rewritten in the form ð h sin hþþsw ¼ ð11þ It can be seen that the imensionless liqui film thickness, *(h), varies as a function of the kob number, the Rayleigh number, the Darcy number, an the suction parameter. However, it is ifficult to solve * irectly, an thus an effective suction function, f, is introuce to represent the effect of the wall suction on the thickness of the conensate layer, i.e., j Sw¼ ¼ 1 þ f ð12aþ or ¼ ð1 þ f Þ j Sw¼ ð12bþ where j Sw¼ is the imensionless liqui film thickness in the absence of a wall suction effect, i.e., Sw =. From Eq. 12b, it can be seen that f = when Sw =. Now, the present stuy becomes to solve j Sw¼ an f. When the wall suction effect is ignore, Eq. 11 can be rewritten as j Sw¼ h j Sw¼ sin h ¼ ð13þ By using the separation of variables metho, the analytical solution of the imensionless film thickness for the case of Sw =, can be erive in the form 1 2¼ 2 j Sw¼ sin h 1 cos h ð Þ ð14þ or 2 1 j Sw¼ ¼ ð15þ 1 þ cos h Substituting Eq. 15 into Eq. 12b, the imensionless liqui film thickness, *, can be rewritten as ¼ ð1 þ f ¼ ð1 þ f Þ Þ j Sw¼ 2 1 1 þ cos h Substituting Eq. 16 into Eq. 11 yiels ð16þ 2 sin h 1 þ cos h ð1 þ f Þf h þ 2 þ Sw 2! f þ 1 þ cos h f 2 2 þ Sw ¼ ð17þ 1 þ cos h Equation (17) is a first-orer ifferential equation. The initial bounary conition for this ifferential equation can be obtaine simply by setting h = in Eq. 17, i.e., f ðþ ¼ 2 þ Sw qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2þSw þ 2 2 ð18þ In this stuy, the numerical solution for the effective suction function, f(h), is obtaine using the shooting metho. The solution proceure commences by substituting the initial bounary value at the top of the tube (i.e., f()) given in Eq. 18, into Eq. 17. The variation of f in the h-irection can then be calculate. The local Nusselt number is efine as Nu h ¼ h hd ð19aþ k e where h h ¼ k e ð19bþ Substituting Eqs. 16 an 19b into Eq. 19a, the local Nusselt number can be rewritten in the form Nu h ¼ 2 ¼ 2 1 þ cos h ð2þ 1 þ f Meanwhile, the mean Nusselt number is efine as Nu ¼ 1 Z p Nu h h ¼ 1 Z 2 p 1 þ cos h h p p 1 þ f ð21þ For the particular case of Sw =, the local Nusselt number can be erive by substituting f = into Eq. 2, i.e., 2 Nu h j Sw¼ ¼ ð1 þ cos hþ ð22þ Similarly, the mean Nusselt number for the case of Sw = can be erive by substituting f = into Eq. 21, i.e., Nu Sw¼ ¼ 1 Z 2 p ð1 þ cos hþ h ð23þ p 3 Results an iscussions Ignoring the wall suction effect, Chiou et al. [12] use a novel transformation metho to investigate the problem of film conensation on a horizontal elliptical tube embee
Heat Mass Transfer (29) 45:355 361 359 in a porous meium. Accoring to the erivations presente in their stuy, the mean Nusselt number for a circular tube can be estimate as: Nu ¼ :9 ð24þ In the present stuy, an explicit formulation for the mean Nusselt number can be obtaine for the case of Sw = by using a simple numerical integration metho to eal with the integration term R p ð1 þ cos hþ h in Eq. 23, yieling Nu Sw¼ ¼ :8996 ð25þ It is evient that a goo agreement exists between Eq. 25 an that presente by Chiou et al. in Eq. 24. In simulating practical engineering problems, the physical parameters use in Eq. 11 (i.e.,, Ra, Da, an R) must be assigne reasonable values. The present analyses assume the working liqui to be water-vapor an use the imensional an imensionless parameter values summarize in Table 1 (reprouce from [2, 21]). Figure 2 illustrates the variation of the effective suction function, f(h, Sw), with h as a function of the wall suction parameter, Sw. Note that the remaining parameters are assigne the typical values shown in Table 1, i.e., =.5, Ra =291 11 an Da = 6.4 9 1-1. As shown, the effective suction function has a negative value, which inicates that the wall suction effect reuces the thickness of the liqui film. Figure 3 presents the corresponing profiles of the imensionless film thickness, *. The results confirm that the imensionless film thickness ecreases with an increasing suction effect at the wall. It can also be seen that the imensionless film thickness has a minimum value at the top of the tube (h = ) an increases with increasing h. This result is to be expecte since the problem consiere in the current analysis is one of falling film conensation, an thus the effects of gravity cause the film to have a minimum thickness on the upper surface of the horizontal tube, but to increase in thickness as the liqui flows ownwar over the tube surface. In Table 1 Physical parameters use in present analyses Symbol Interpretation Typical value K Permeability 6.4 9 1-12 m 2 R Raius of tube.1 m Pr e Da Ra CpDT h fgþ 1 2 CpDT.5 l e Cp k e 1.76 K R 2 6.4 9 1-1 q 2 g Pr e R 3 l 2 e 2 9 1 11 f (,Sw) -.2 -.4 -.6 : Sw=. : Sw=.1 : Sw=.5 : Sw=.1 =.5, Ra=2 * 1 11, Da=6.4 * 1-1 2 6 1 14 18 4 8 12 16 Fig. 2 Variation of f(h, Sw) in h-irection as function of Sw for =.5, Ra =29 1 11 an Da = 6.4 9 1-1 δ/r 1.5.2.1.5.2.1 =.5, Ra=2 * 1 11, Da=6.4 * 1-1 : Sw=. : Sw=.1 : Sw=.5 : Sw=.1 2 4 6 8 1 12 14 16 18 Fig. 3 Variation of imensionless film thickness in h-irection as function of Sw for =.5, Ra =29 1 11 an Da = 6.4 9 1-1 contrast to the problem of falling film conensation on a vertical tube [22], it can be seen that in the current case, the liqui film thickness tens towar an infinite value as h?18 (i.e., towar the lower surface of the tube). Equation (2) shows that the value of the local Nusselt : 2 number varies as a function of 1þcos h 1þf Figure 4 plots the variation of the local Nusselt number over the tube surface as a function of the wall suction parameter,
36 Heat Mass Transfer (29) 45:355 361 Nu 5 2 1 an ecreases to zero at the lower surface as a result of the finite thickness of the conensate layer. Finally, for the particular case where a wall suction effect is not present at the tube surface, it has been shown that the mean Nusselt number is given by Nu ¼ :9 : Acknowlegments This stuy was supporte by the National Science Council of Taiwan uner Contract No. NSC 96-2221-E-218-34. 1 5 2 Sw, for constant values of =.5, Ra =29 1 11 an Da = 6.4 9 1-1. The results emonstrate that the local Nusselt number has a maximum value at the top of the tube (h = ) an ecreases with increasing h. It is observe that the local Nusselt number has a value of zero at the lower surface of the tube (h = 18 ) since the liqui film has an infinite thickness at this position. Moreover, the results show that the local Nusselt number increases with an increasing suction effect at the wall, i.e., because a greater suction effect reuces the thickness of the conensate layer. 4 Conclusion : Sw=.1 : Sw=.5 : Sw=.1 : Sw=. =.5, Ra=2*1 11,Da=6.4*1-1 2 6 1 14 18 4 8 12 16 Fig. 4 Variation of local Nusselt number in h-irection as function of Sw for =.5, Ra =29 1 11 an Da = 6.4 9 1-1 This stuy has performe an analytical investigation into the problem of laminar film conensation on a horizontal tube embee in a porous meium with suction at the tube surface. The motion of the liqui film within the porous meium has been moele using Darcy s law. Moreover, an effective suction function, f, has been introuce to moel the effect of wall suction on the conensate film thickness, thereby allowing both the local conensate film thickness an the local Nusselt number to be erive using a simple numerical shooting metho. The results have shown that the local Nusselt number varies as a function of the kob number,, the Rayleigh number, Ra, the Darcy number, Da, an the effective suction function, f, in accorance with Nu h ¼ 2 1þcos h 1þf : Moreover, it has been shown that the local Nusselt number has a maximum value at the upper surface of the horizontal tube References 1. Nusselt W (1916) Die oberflachen Konensation es Wasserampes. Z Verein Dtsch Ingenieure 6:541 546 2. Merte H Jr (1973) Conensation heat transfer. Av Heat Transfer 9:181 3. Nimmo B, Leppert G (197) Laminar film conensation on a finite horizontal surface. In: 4th International heat transfer conference, pp 42 43 4. Chiou JS, Chang TB (1994) Laminar film conensation on a horizontal isk. Warme Stoffubertragung 29:141 144 5. Sparrow EM, Gregg JL (1959) Laminar conensation heat transfer on a horizontal cyliner. Trans ASME J Heat Transfer 81:291 296 6. Chen MM (1961) An analytical stuy of laminar film conensation. Part 1 Flat plates. J Heat Transfer 83:48 54 7. Ali AFM, McDonal TW (1977) Laminar film conensation on horizontal elliptical cyliners: a first approximation for conensation on incline tubes. ASHRAE Trans 83:242 249 8. Yang SA, Chen CK (1993) Roll of surface tension an ellipticity in laminar film conensation on a horizontal elliptical tube. Int J Heat Mass Transfer 36:3135 3141 9. Yang JW (1973) Laminar film conensation on a sphere. Trans ASME J Heat Transfer 95:174 178 1. Cheng P (1981) Film conensation along an incline surface in a porous meium. Int J Heat Mass Transfer 24:983 99 11. Cheng P, Chui DK (1984) Transient film conensation on a vertical surface in a porous meium. Int J Heat Mass Transfer 27:795 798 12. Chiou JS, Yang SA, Chen CK (1994) Filmwise conensation on a horizontal elliptical tube embee in porous meium. Chem Eng Comm 127:125 135 13. Al-Nimr MA, Alkam MK (1997) Film conensation on a vertical plate imbee in a porous meium. Appl Energy 56:47 57 14. Masou S, Al-Nimr MA, Alkam M (2) Transient film conensation on a vertical plate imbee in porous meium. Transp Porous Me 4:345 354 15. Char MI, Lin JD, Chen HT (21) Conjugate mixe convection laminar non-darcy film conensation along a vertical plate in a homogeneous porous meium. Int J Eng Sci 39:897 912 16. Wang SC, Chen CK, Yang YT (25) Film conensation on a finite-size horizontal plate boune by a homogenous porous layer. Appl Therm Eng 25:577 59 17. Wang SC, Chen CK, Yang YT (26) Steay filmwise conensation with suction on a finite-size horizontal flat plate embee in a porous meium base on Brinkman an Darcy moels. Int J Thermal Sci 45:367 377 18. Chang TB (28) Effects of surface tension on laminar filmwise conensation on a horizontal plate in a porous meium with suction at the wall. Chem Eng Commun 195:721 737 19. Chang TB (28) Laminar filmwise conensation on horizontal isk embee in porous meium with suction at wall. ASME J Heat Transfer 13:7152
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