An analytical investigation into the Nusselt number and entropy generation rate of film condensation on a horizontal plate

Transcription

1 Journal of Mechanical Science and Technoloy (8) 4~4 Journal of Mechanical Science and Technoloy.sprinerlink.com/content/78-494x DOI.7/s An analytical investiation into the Nusselt number and entropy eneration rate of film condensation on a horizontal plate Ton Bou Chan, and Fu Jen Wan Department of Mechanical Enineerin, Southern Taian University, Tainan, Taian. Department of Refrieration, Air Conditionin and Enery Enineerin, National Chin-Yi University of Technoloy, Taichun, Taian. (Manuscript Received November 8, 7; Revised April, 8; Accepted Auust, 8) Abstract This study investiates the heat transfer characteristics and entropy eneration rate of a condensate film formed on a horizontal plate ith suction at the all. Applyin the minimum mechanical enery principle, the dimensionless liquid film thickness alon the plate is found to vary as a function of the Rayleih number, the Jakob number, the Prandtl number and the suction parameter. The overnin differential equation of the condensate thickness is solved numerically by usin a finite-difference shootin method. Closed-form analytical expressions are derived for the Nusselt number and the dimensionless overall entropy eneration number. When there is no suction at the all, the results obtained from the analytical expression for the Nusselt number are found to be in ood areement ith those presented in the literature. Keyords: Film condensation; Minimum entropy eneration; Wall suction Introduction Laminar film condensation is of practical importance in a variety of enineerin and industrial applications ranin from heat exchaners to dryin and coolin processes, packed-bed heat exchaners, distillation processes, chemical vapor deposition processes, and so on. The problem of laminar film condensation on a vertical plate as first addressed by Nusselt [] in 96 and has been the subject of intense study ever since. Folloin the publication of Nusselt s oriinal study, many researchers attempted to improve the accuracy of the analytical results by incorporatin more realistic assumptions. The earliest study of film condensation heat transfer on a horizontal surface as performed by Popov []. In a later study, Nimmo and This paper as recommended for publication in revised form by Associate Editor Dae Hee Lee Correspondin author. Tel.: , Fax.: address: tbchan@mail.stut.edu.t KSME & Spriner 8 Leppert [, 4] conducted experimental and theoretical investiations into film condensation on a finite-size horizontal plate. In the theoretical study of Nimmo and Leppert [4], the boundary condition at the ede of the plate as simply assumed by a particular boundary condition. By contrast, in investiatin the same problem, Shiechi et al. [5] improved the accuracy of the numerical results by adjustin the inclination anle of the vapor-liquid interface at the plate ede. Hoever, the precise nature of the boundary condition at the plate ede as not fully clarified. Adoptin a special transformation method ith certain approximations, Yan and Chen [6] conducted an analytical investiation into the problem of condensation on a horizontal plate ith suction effects actin at the all. The results indicated that the presence of a suction force yielded a sinificant improvement in the condensation heat transfer performance. Hoever, the closed-form analytical expression for the Nusselt number proposed in their study did not consider the

2 T. B. Chan and F. J. Wan / Journal of Mechanical Science and Technoloy (8) 4~4 5 all suction parameter. In practical thermal systems, irreversibilities due to friction and heat transfer inevitably exist. These irreversibilities induce an entropy eneration effect and destroy the enery available in the system to perform ork. Bejan [7] proposed a method for minimizin the rate of entropy eneration in sinle-phase convection heat transfer systems. Adeyinka and Naterer [8] analyzed the problem of entropy eneration in a vertical condensation film subject to phase chane. Recently, Dun and Yan [9] applied the entropy eneration minimization method proposed by Bejan to optimize the heat transfer in a condensate film on a horizontal tube. Althouh many researchers have studied the problem of film condensation heat transfer on a horizontal plate, a comprehensive thermodynamic analysis of the detailed characteristics of this particular heat transfer problem has yet to be presented. Accordinly, the current study performs thermodynamic analyses of laminar film condensation on a horizontal plate ith suction effects actin at the all. A closed-form analytical expression for the Nusselt number is derived as a function of the Jakob number Ja, the Rayleih number Ra and the suction parameter S. In addition, an approach is proposed for minimizin the overall entropy eneration induced by the condensation heat transfer and condensate film flo friction irreversibilities.. Analysis The current analysis considers a pure quiescent vapor in a saturated state ith a uniform temperature T sat and pressure P sat condensin on a horizontal, clean, permeable flat plate ith a idth L and a constant temperature T. Fi. presents a schematic diaram of the physical model and the correspondin coordinate system. Under steady-state conditions, a filmise condensate layer forms on the plate and has a maximum thickness at the plate center. The flo rate of this condensate layer is overned by the variation in the hydrostatic pressure actin upon it. The momentum boundary layer is subject to a uniform suction force hich removes the condensate at a constant velocity. In investiatin the heat transfer characteristics of the condensate film, the folloin assumptions are made: () the inertia and kinetic terms ithin the liquid film are neliible, () the physical properties of the dry vapor and condensate layer are constant, Fi.. Condensate film flo on finite-size horizontal plate ith suction at all. and () the all and the vapor both have a uniform temperature. Given these assumptions, the overnin equations for the liquid film ith boundary layer simplifications have the form: continuity equation: u v + =, () x y x-direction momentum equation: P = +, () x y u µ y-direction momentum equation: sat ( ) P = P + ρ y, () enery equation: u x y ρcp y y T T k T + v = x Liquid film V x=l, (4) here u and v are velocity components in the x- and y-directions, respectively, P is the liquid pressure, is the thickness of the liquid film, and ρ, Cp, µ and k are the density, specific heat, dynamic viscosity and thermal conductivity of the saturated liquid, respectively. The boundary conditions imposed at the plate surface (y=)are iven by u= (5) v=v, (6) T= T, (7) hile those at the liquid-vapor interface (y= )have the form u =, (8) y

3 6 T. B. Chan and F. J. Wan / Journal of Mechanical Science and Technoloy (8) 4~4 T= T sat. (9) Substitutin Eq. () into Eq. () and applyin the boundary conditions iven in Eqs. (5) and (8), the velocity profile can be expressed as ρd y u = y. () µ Since the thickness of the liquid film is very small relative to the idth of the plate, the temperature profile can be expressed in the folloin linear form: y = +, () T T T here T = Tsat T. Accordin to Bejan [7], the local entropy eneration rate for to-dimensional convection heat transfer can be ritten as k T T µ u S = + +. () T x y T y The overall entropy eneration rate in the current problem is derived by the folloin three-step procedure. Eqs. () and () are substituted into Eq. () to yield the local entropy eneration rate: k T µ ρ d S = + ( y ) T T µ. () Eq. () is then interated ith respect to y from zero to to obtain the entropy eneration rate across the film thickness: ρ S = = + T T. (4) k T d S dy µ Finally, Eq. (4) is interated from x= -L to x=l to obtain the overall entropy eneration rate across the idth of the plate. L L = = L k T L ρ L d = T + µ T S S S. (5) In Eq. (5), the thickness of the liquid film,, is the only unknon. From the first la of thermodynamics and the Fourier conduction la, the thermal enery balance beteen the condensate layer and the plate surface can be iven as d { ρ ( ( )) f + sat } u h Cp T T dy. (6) k T + ρv( hf + Cp T) = The riht hand side of Eq. (6) represents the enery transferred from the liquid film to the solid plate. Meanhile, the first term on the left hand side expresses the net enery flux across the liquid film (from x to x+)hile the second term describes the net enery sucked out of the condensate layer by the permeable plate. The assumption is made that the suction velocity, v, is constant. Substitutin Eqs. () and () into Eq. (6) and rearranin yields d d k Tµ = dr ρ hf + Cp T 8. (7) hf + Cp T vµ + h f + Cp T ρ 8 For analytical convenience, let the folloin parameters be defined: Cp T Ja =, (8) hf + Cp T 8 ρ Pr L Ra =, (9) µ µcp Pr = k () Pr ρvl 5 S = + Ja Ja µ 8, () here Ja is the Jakob number, Ra is the Rayleih number, Pr is the Prandtl number and S is the suction parameter. With these parameters, the thermal enery balance iven in Eq. (7) can be reritten as

4 T. B. Chan and F. J. Wan / Journal of Mechanical Science and Technoloy (8) 4~4 7 d d Ja = ( SL L ). () Ra The liquid film has its maximum thickness at the center of the place (i.e., x=) and the film thickness radually reduces in the x-direction toard a minimum value at the plate ede. The boundary conditions of the film thickness can therefore be ritten as d, at x = () = min. at x = L (4) Hoever, it is still impossible to solve Eq. () since min is unknon. In practice, the condensate film cannot have exactly zero thickness at the plate ede. Chan [, ] suested that the critical condensate thickness could be derived by usin the minimum mechanical enery principle presented by Bakhmeteff []: d 5 Pr = 6 ( ) x = Ra x = / at x =. () The overnin differential equation of the condensate thickness (Eq. (9)) is solved numerically subject to the correspondin boundary conditions (Eqs. () and ()) by usin a finite-difference shootin method. The overall solution procedure is illustrated schematically in Fi.. When the final value of has been derived at each rid point alon the x- direction, the mean Nusselt number is computed in accordance ith Nu here hl k = =, () u P + y + ρudy =, (5) ρ mc here m c is the critical value of the mass flo over the plate ede. Substitutin Eqs. () and () into Eq. (5) yields the ne boundary condition as d 5 Pr L = x= L 6Ra ( x= L) /. (6) For convenience, the folloin normalized variables are introduced: x = xl /, (7) = /L. (8) Eq. () and the correspondin boundary condition equations (Eqs. () and (6)) can then be reritten as d d Ja = ( S ), (9) Ra d at x =, () Fi.. Numerical solution procedure.

5 8 T. B. Chan and F. J. Wan / Journal of Mechanical Science and Technoloy (8) 4~4 L L k k ( ) -L. () h= h x = = L L L Usin the normalized variables and dimensionless parameters defined in Eqs. (8)-(), (7) and (8), the overall entropy eneration rate iven in Eq. (5) can be reritten as ( ) ρ d. (4) k T S = + T µ T Let the folloin dimensionless parameters be defined: T ψ =, (5) T Lµ Br =, kt (6) ρ L Gr =, µ (7) here ψ is the dimensionless temperature difference, Br is the modified Brinkman number, and Gr is the modified Grashof number. Eq. (4) can then be reritten in dimensionless form as d. (8) S = kψ + kbrgr The characteristic entropy eneration rate can be defined as friction irreversibility, respectively. It is observed that the term describin the heat transfer irreversibility is identical to Eq. (), i.e., the analytical expression for the mean Nusselt number.. Results and discussion Fi. illustrates the variation of the dimensionless liquid film thickness alon the x-direction for four different combinations of Ja/Pr and S. As expected, the thickness of the liquid film reduces as the value of the suction parameter, S, increases. Additionally, it can be seen that the film thickness near the plate ede increases as the ratio Ja/Pr decreases. This result is reasonable since a larer hydrostatic pressure radient is required to drive the flo over the plate ede hen the fluid has a hiher Prandtl number (i.e., a smaller Ja/Pr ratio). Hoever, in the non-ede reion of the plate, it is apparent that the film thickness is insensitive to Ja/Pr. As shon in Eq. (), the Nusselt number varies inversely ith the film thickness since the thermal resistance of the condensate layer increases as its thickness increases. Furthermore, Eqs. (9) and () sho that the value of the Nusselt number also varies as a function of Ja, Ra, Pr and S. Fi. 4 shos that the mean heat transfer coefficient, Nu( Ja ) /5, Ra increases sinificantly as the suction parameter S is increased. Hoever, Nu( Ja ) /5 increases only Ra slihtly ith increasin Ja/Pr (as observed also in Fi. ). From inspection it is found that if the effects of the S k T = = kψ. T (9) The dimensionless overall entropy eneration S number, N =, can then be ritten as S d ψ S N = = + BrGr S. (4) In Eq. (4), the first and second terms on the riht hand side correspond to the entropy induced by the heat transfer irreversibility and the liquid film flo Fi.. Variation of dimensionless film thickness in x-direction.

6 T. B. Chan and F. J. Wan / Journal of Mechanical Science and Technoloy (8) 4~4 9 Prandtl number, Pr, are nelected, the correspondin error in the computed value of the mean heat transfer coefficient is less than %. Accordinly, the remainin analyses nelect the effects of Pr and simply assin a constant value of Pr=.76 (calculated from ater at ). Fi. 5 plots the variation of the mean heat transfer coefficient as a function of the all suction parameter. Applyin a curve-fittin technique, it can be shon that Nu and S are related as follos: Nu =.8 (.78) Ja S Ra 5. (4) From inspection, the maximum error beteen the results predicted by Eq. (4) and those determined numerically is found to be less than 4%. In their study of the heat transfer characteristics of a condensate film on a finite-size horizontal plate, Yan and Chen [6] shoed that in the absence of all suction effects, the Nusselt number is iven by Nu.8 Ra = Ja 5. (4) d Substitutin a value of S= into Eq. (4), it is found that the expression developed in the current study for the mean heat transfer coefficient of a condensate layer on a finite-size horizontal disk is consistent ith that presented by Yan and Chen [6]. Fi. 6 shos that the liquid film flo friction irreversibility (as iven by the second term on the riht hand side of Eq. (4)) reduces as the all suction parameter increases. Applyin a curve-fittin technique, it is found that and S are related approximately as follos: Fi. 4. Variation of heat transfer coefficient ith Ja/Pr as function of S. d 4. Ja = (.) S Ra. (4) x - 9x -4 8x -4 7x -4 6x -4 5x -4 4x -4 : Numerical results : Correlation, Eq.(4) Nu(Ja/Ra) / : Correlation, Eq.(4) : Numerical results )(Ra/Ja) ( (d / ) x -4 x -4 x -4 9x -5 8x -5 7x -5 6x -5 5x -5 4x -5 x -5 x -5.5 Pr=.76 x -5 Pr= S S Fi. 5. Comparison of Nusselt number correlation results and numerical results. Fi. 6. Comparison of liquid film flo friction irreversibility correlation results and numerical results.

7 4 T. B. Chan and F. J. Wan / Journal of Mechanical Science and Technoloy (8) 4~4 Substitutin Eqs. (4) and (4) into Eq. (4) yields the folloin expression for the dimensionless overall entropy eneration number: S Ra N =.8 (.78) Ja. (44) 5 S Ja + 7 BrGrψ (.) Ra In eneral, for a specified value of the parameter roup BrGrψ, the dimensionless overall entropy eneration number, N, can be minimized by differentiatin N ith respect to Ra/Ja and settin it equal to dn zero (i.e. = ). The optimal value of Ra/Ja d ( Ra Ja) can be obtained as Ra Ja opt 5 5 ( BrGrψ ) ( ) S = (45) Fi. 7 shos the variation of the dimensionless total entropy eneration number, N, ith the ratio Ra/Ja 8 as a function of S for constant BrGrψ =. Ra From Eq. (45), it is calculated that the =77, Ja opt 646 and 95 for all suction parameter values of S=, 5 and, respectively. Then correspondin minimum values of N are also derived as 5.7, 5.6 N x x x BrGr - = 8 x x x x 4 x 5 x 6 x 7 Ra/Ja : S=. : S=5. : S=. : minimum point Fi. 7. Variation of entropy eneration parameter ith Ra/Ja as function of S. and 5.5 for all suction parameter values of S=, 5 and, respectively. 4. Conclusion This study has investiated the heat transfer characteristics and the overall entropy eneration rate in a condensate layer on a finite-size horizontal plate ith suction effects actin at the all. The minimum mechanical enery principle has been applied to determine the boundary condition at the plate ede. The results sho that the mean Nusselt number varies as a function of the ratio Ra/Ja and S. Moreover, the overall entropy eneration rate induced by the heat transfer irreversibility effect is equivalent to the Nusselt number. Applyin a curve-fittin method, closedform analytical expressions have been developed to predict the mean Nusselt number and the entropy eneration rate. In both cases, a ood areement has been observed beteen the predicted results and the numerical solutions. Finally, the value of Ra/Ja hich minimizes the dimensionless overall entropy eneration number has been derived as a function of the parameter roup BrGrψ. Acknoledments This study as supported by the National Science Council of Taian under Contract No. NSC 96-- E-8-4. Nomenclature Br : Modified Brinkman number (defined in Eq. (6)) Cp : Specific heat at constant pressure : Acceleration of ravity Gr : Modified Grashof number (defined in Eq. (7)) h : Heat transfer coefficient h f : Vaporization heat Ja : Jakob number (defined in Eq. (8)) k : Thermal conductivity L : Half-idth of plate m : Condensate mass flux N : Dimensionless overall entropy eneration number (defined in Eq. (4)) Nu : Nusselt number (defined in Eq. ()) P : Pressure Pr : Prandtl number (defined in Eq. ())

8 T. B. Chan and F. J. Wan / Journal of Mechanical Science and Technoloy (8) 4~4 4 Ra : Rayleih number (defined in Eq. (9)) S : Local entropy eneration rate S : Overall entropy eneration rate S : Characteristic entropy eneration rate S : Suction parameter (defined in Eq. ()) T : Temperature T : Saturation temperature minus all temperature x, y : Horizontal and vertical coordinates u, v : Horizontal and vertical velocity components Greek symbols : Condensate film thickness µ : Liquid viscosity ρ : Liquid density ψ : Dimensionless temperature difference (defined in Eq. (5)) Superscripts : Averae quantity : Dimensionless variable Subscripts min : Minimum quantity sat : Saturation property : Quantity at all Reference [] W. Nusselt, Die oberflachen kondensation des asserdampes, Zeitschrift des Vereines Deutscher Inenieure, 6 () (96) [] V. D. Popov, Heat transfer durin vapor condensation on a horizontal surfaces, Trudy Kiev. Teknol. Inst. Pishch, Prom. (95) [] G. Leppert and B. Nimmo, Laminar film condensation on surface normal to body or inertial forces, J. Heat Transfer, 8 (968) [4] B. Nimmo and G. Leppert, Laminar film condensation on a finite horizontal surface, 4th Int. Heat Transfer Conference, (97) 4-4. [5] T. Shiechi, N. Kaae, Y. Tokita and T. Yamada T, Film condensation heat transfer on a finite-size horizontal plate facin upard, JSME series B, 56 (99) [6] S. A. Yan and C. K. Chen, Laminar film condensation on a finite-size horizontal plate ith suction at the all, Appl. Math. Modellin, 6 (99) 5-9. [7] A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, FL, (996) chapter 4. [8] O. B. Adeyinka and G. F. Naterer, Optimization correlation for entropy production and enery availability in film condensation, Int. Commun. Heat Mass Transfer, (4) [9] S. C. Dun and S. A. Yan, Second la based optimization of free convection film-ise condensation on a horizontal tube, Int. Commun. Heat Mass Trasnsfer, (6) [] T. B. Chan, Effects of capillary force on laminar filmise condensation on a horizontal disk in a porous medium, Applied Thermal Enineerin, 6 (6) 8-5. [] T. B. Chan, Laminar film condensation on a horizontal avy plate embedded in a porous medium, International Journal of Thermal Sciences, 47 (8) 5-4. [] B. K. Bakhmeteff, Hydraulics of Open Channel, McGra-Hill, Ne York (966) 9-4 Ton-Bou Chan received the Ph.D. deree in Mechanical En-ineerin from National Chen Kun University, Tainan, Taian, in 997. From 997 to, he as a researcher at Yuloon-Motor Group (Taian), hose job function includes desin and characterization of the thermal and fluid flo systems for vehicle. Since, he has been as a Professor at the Department of Mechanical Enineerin, Southern Taian University. His current research interests include heat transfer ith phase chane, enery-system optimization, heat and mass transfer in porous medium, enhancement heat transfer and hih performance heat exchaners.