MODELING AND DESIGN OF MICRO-GROOVED FLAT PLATE EVAPORATOR

Transcription

1 Proceedings of ICMM005 3rd Internationa Conference on Microchannes and Minichannes June 13-15, 005, Toronto, Ontario, Canada ICMM MODELING AND DESIGN OF MICRO-GROOVED FLAT PLATE EVAPORATOR Naoki Shikazono 1, Yasushi Suehisa 1, Nobuhide Kasagi 1 and Hiroshi Iwata 1 The University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan Nichirei Industries Co. Ltd., 1570 Mayumi, Ohhira-machi, Tochigi-ken, Japan ABSTRACT A micro-grooved fat pate evaporator is modeed and its heat transfer characteristics are investigated numericay and experimentay. A test mode is deveoped for the vapor compression cyce evaporator, where pressure gradient drives the vapor and the iquid fow. In this study, the effect of pressure gradient is impicity introduced through the Smith s equation for predicting void fraction from given quaity. The fim thickness profie in the micro near the contact ine is obtained by soving the 4th order differentia equation. Then the oca heat fux is obtained by assuming that the heat conduction through the iquid is one dimensiona in the wa norma direction. The shape of iquid-vapor interface is assumed to be a circuar arc in the macro, whose radius is directy inked to the void fraction. This curvature radius is used as the boundary condition for the micro mode at the micro-macro interface. Finay, the heat transfer coefficient on a micro-grooved fat pate evaporator is measured in a HFC134a experimenta oop and compared with the numerica prediction. The present mode assumptions are vaidated and assessed. Keywords: Evaporator, Micro-Groove, Thin Fim, Surface Tension, Disjoining Pressure. INTRODUCTION Since thin iquid fims can promote very rapid and intensive heat and mass transfer, micro-grooved surfaces have been attracting arge attention in the area of refrigeration and air conditioning industries. In fact, micro-fin tubes are recognized as one of the most successfu heat transfer enhancement technoogies. On the other hand, for the temperature contro of eectronic or biotechnoogy appications, it is common that the heat has to be removed from one side of the device. Thus, fat pate evaporators are considered to be an important technoogy for such appications. In case of micro-grooved evaporators, it is known that intensive evaporation can be achieved in a near the contact ine where very thin iquid fims are formed adjacent to the adsorbed fim (see, e.g. Wayner [1,]). However, this strong evaporation occurs at a very imited area ony in the vicinity of the contact ine. Thus, the main chaenge in deveoping an efficient evaporator is to spotight such microscopic strong evaporation on a macroscopic heat transfer surface. It is expected that numerica simuation can be a powerfu design too for this purpose. In the present study, a micro-grooved fat pate evaporator is simuated anayticay extending the method proposed by previous researchers, e.g. Wayner, et a. [3,4], Kamotani [5] and Stephan and Busse[6] etc. In the foowing sections, the mode equations for the micro and macro s are presented. Finay, the mode is compared with the experimenta resuts and the mode assumptions are vaidated and assessed. NOMENCLATURE A dispersion constant, J or area, m f. accommodation coefficient, dimensioness G. macro mass fow rate, kg/s g macro evaporation rate per unit ength, kg/(m s) h channe height, m h fg specific heat of evaporation, J/kg K. curvature of the meniscus, 1/m M. micro mass fow rate, kg/s m micro evaporation rate per unit ength, kg/(m s) p. pressure, Pa q heat fux, W/m Q heat transfer, W R curvature radius, m 1 Copyright 005 by ASME

2 R g gas constant, J/(kg K) S(x) wet area width at position x, m T temperature, K w groove width, m x coordinate aong the groove, m z transformed variabe for soving Eq. (10) α heat transfer coefficient, W/(m K) β void fraction, dimensioness δ fim thickness, m γ groove ange, rad η viscosity, Pa s λ therma conductivity, W/(m K) ν kinematic viscosity, m /s ρ density, kg/m 3 σ surface tension, N/m ξ fow direction in the micro, m ζ distance from the wa, m Intensive evaporation w Vapor Liquid γ Wa Micro Fig. Meniscus of iquid in a trianguar groove ξ SUBSCRIPT c capiary end end boundary iv vapor side of the iquid-vapor interface in inet boundary iquid m mean s soid sat saturation v vapor ζ T sat δ T iv. m. M T w Intrinsic meniscus Adsorbed iquid fim Interine MODELING OF A FLAT PLATE EVAPORATOR Figure 1 shows the cross section of the micro-grooved fat pate evaporator. Heat is removed from one side of the wa. The fow inside the groove is divided into two s, i.e., micro and macro s. These two s are modeed individuay as is commony done for soving thin fim evaporation. Foowing the method proposed by previous researchers, e.g. Wayner, et a. [3,4], Kamotani [5] and Stephan and Busse[6] etc., a differentia equation of the iquid fim thickness is soved for the micro. On the other hand, very simpe heat conduction equations have been appied for cacuations of the macro. This approximation is thought to be acceptabe because oca heat fux in the macro is an order of magnitude smaer than that in the micro. The iquid fim thickness, fim incination ange and Liquid Fow Insuated Wa Vapor Fow Micro-Grooves Heated Wa Fig. 1 Cross section of the micro-grooved fat pate evaporator w h Fig. 3 Enarged view of the micro curvature are connected smoothy at the boundary. Ajaev and Homsy [7] combined these two s using asymptotic matching. In the present study, the micro and macro s are simpy combined at the position at the imit where the tota heat transfer rate is not affected by its ocation. In the foowing sections, the modes for micro and macro s are described. Micro Region Mode Foowing previous study of a faing fim evaporator (Hasebe et. a. [8]), the anaytica mode described by Stephan and Busse [6] is adopted for the micro. Since the iquid fim is very thin and the surface tension becomes dominant in this scae, gravitationa effects are ignored in the micro mode. Figure shows the meniscus of the iquid captured in a trianguar groove, and Fig. 3 shows the enarged view of the micro. In this study, trianguar shape is adopted for the cross section of the groove because of the simpicity of anaytica treatment. Intensive evaporation takes pace at the so-caed interine where the iquid fim gets very thin. The heat fux q& in the iquid fim is assumed to be one dimensiona in the wa norma direction and can be written as Copyright 005 by ASME

3 p c TW T sat 1 + h fg q ρ & =, (1) δ Tsat (πrgtsat ) ( f ) + λ h ρ f fg v where the first and the second terms in the denominator correspond to the heat conduction resistance and interfacia heat resistance, respectivey. Constant wa temperature T w is assumed in this study. The effect of wa conductivity is reported to be ess than 15 % in case of auminum groove with heat fux of 30 kw/m [6]. Thus, constant wa temperature assumption seems reasonabe for cupper wa with moderate heat fux. The accommodation coefficient f expresses the fraction of moecues actuay vaporized at the surface (the remaining 1-f is due to the refection of vapor moecues). In the present study, f = 1 is used uness otherwise noted. The pressure difference between vapor and iquid phases p c is expressed as p c = p v p = Kσ + A δ 3. () The first and the second terms describe the surface tension and the disjoining pressure effects, respectivey. The dispersion 1 constant was set as A =.0 10 J [6]. The curvature K can be expressed using the fim thickness δ as d δ dξ K = 1+ dδ. (3) 3 dξ This pressure gradient due to the fim thickness profie pus the iquid to the strong evaporating interine and baances with the viscous drag of the fow. The oca and mean veocity profies u and u in a aminar boundary ayer are given as u = 1 dp µ dξ δζ, (4) ζ 1 δ δ dp u = udζ δ =. (5) 0 3ν ρ dξ Then the mass fow rate M & ( = u δρ ) can be obtained using Eq. (5); 3 δ dp M& =. (6) 3ν dξ Thus, the evaporation rate m& is obtained as foows: dm& 1 d 3 dp m& = = δ. (7) dξ 3ν dξ dξ Assuming the vapor pressure p v to be constant, the iquid phase pressure gradient is reated to the capiary pressure gradient as dp dξ = dp c dξ. (8) Thus, one obtains the expression for the heat fux as hfg d 3 dp q& c = δ. (9) 3ν dξ dξ Combining Eqs. (1), (), (3) and (9), the fourth order differentia equation for the iquid thickness δ is obtained; d δ σ d 3µ dξ δ 3 d dξ dξ 1+ dδ 3 + A σδ 3 dξ d δ T sat T w + T satσ dξ h fg ρ 1+ dδ + A 3 σδ 3 dξ. (10) = δ ρ h fg + T sat πr g T sat f λ h fg ρ v f In order to sove Eq. (10), foowing four variabes are introduced: z 1 = δ, (11) z = dδ dξ, (1) d δ dξ z 3 = 1+ dδ + A 3 σδ, (13) 3 dξ d δ z 4 = σ δ 3 d dξ 3µ dξ 1+ dδ + A 3 σδ 3, (14) dξ where variabes z 1, z, z 3 and z 4 correspond to fim thickness, fim incination, capiary pressure and fow rate, respectivey. Then Eq. (10) can be rewritten as a system of four first order differentia equations as foows: 3 Copyright 005 by ASME

4 dz 1 dξ = z, (15) dz dξ = z 3 A 3 ( 1+ z ) 3, (16) σz 1 dz 3 dξ = 3µ z 4 σ z, (17) 3 1 T sat T w + T satσ z 3 dz 4 dξ = h fg ρ δ ρ h fg + T. (18) sat πr g T sat f λ h fg ρ v f This set of equations is integrated by fourth order Runge- Kutta method. The boundary conditions are shown in Fig. 4. At the macro and micro boundary, the fim thickness z 1in, and z 3in (=p c /σ) are set as h w Liquid Fow R macro (x) x=0 S(x) x z z 1 in = δ in, (19) 3 in K = 1 Rin = 1 Rmacro( x). (0) The boundary fim thickness δ in, which determines the ocation of the micro-macro boundary, can be chosen arbitrariy. It is confirmed that the choice of δ in does not affect the tota heat transfer rate. The iquid fim incination ange z in and fow rate z 4in at the macro-micro boundary are chosen by iteration so that both fim sope z end and fow rate z 4end become zero at the micro and adsorbed fim boundary, i.e. x x=x end Fig. 5 Schematic view of the fow inside the groove θ z end = 0, (1) z 4 end = 0. () Micro R macro (x) Micro Macro Region Mode Then, the micro mode is combined with the macro mode. The schematic view of the fow aong the groove is shown in Fig. 5. The shape of iquid-vapor interface in the macro is assumed to be a circuar arc. This Fig. 6 Macro R out (i) Macro modeing z in ζ R in = R macro (x) = 1/z 3in R macro =1/ z 3in tan -1 (z in ) R macro sin(θ) θ = π/ γ/ tan -1 (z in ) z 1in =δ in Macro Region z 4in Micro Region z 1in =δ in z 3in =K = 1/R in = 1/R macro (x) z end z 4end z 1end ξ Adsorbed Fim Region z end =0: Sope is zero z 4end =0: Fow rate is zero A macro A micro z 1in =δ in γ tan -1 (z in ) R macro sin(θ)/sin(γ/) Fig. 4 Boundary conditions for the micro Fig. 7 Cross section of the trianguar groove 4 Copyright 005 by ASME

5 curvature radius R macro is used as the boundary condition for the micro cacuation at each cross section of the groove. In case of evaporators used in vapor compression cyces, iquid fim on the wa is pushed with the vapor fow by the pressure gradient aong the channe. Assuming arge Froude number, gravity is aso ignored in the macro mode. In principe, the fim thickness is determined by the baance of streamwise pressure gradient and the shear stress in both fuids. However, it requires time consuming numerica procedures to capture the iquid-vapor interface. In the present study, the position of iquid-vapor interface is directy obtained from the Smith s[9] equation which predicts void fraction β from given quaity χ; 1 ρv χ ρ 1 v χ ρ χ β = (3) ρ χ χ χ 1 This simpe equation is originay derived for smooth pipe fows. In addition, Eq. (3) is independent on fow rate. However, Eq. (3) is known to give fairy good void fraction predictions for micro-fin tubes. And most thankfuy, the fim thickness can be obtained expicity without soving a system of equations. This is a very attractive advantage from an engineering point of view. Thus, this simpe approach is adopted in this study as a first order approximation. The macro is divided into concentric strips as shown in Fig. 6. The amount of heat transferred from each strip is cacuated assuming one dimensiona radia heat conduction, i.e. λ θ Q = ( ) ( ) ( T ) sat Tw. (4) n Rout ( i) Rmacro ( x) The tota macro is divided into 00 strips in this study. The simuation starts at x=0 where the tip of the micro touches the crest of the groove, and ends where micro s from both sides meet each other at the center of the groove as shown in Fig. 5. Once quaity χ is known at any streamwise position x, the micro mode is cacuated with an assumed curvature R macro (x). The vaue of curvature R macro (x) is modified iterativey unti the cacuated iquid area (A macro +A micro ) shown in Fig. 7 satisfies the void fraction given from Eq.(3). With. converged R macro (x), the tota amount of iquid evaporated g (x) at position x is known from the sum of micro and macro resuts. Thus, we can know the iquid. fow rate G (x+ x) (aso quaity) at the next position x+ x from Eq. (5). This procedure is continued unti the iquid fow is dried out. ( x + x) = G& ( x) g& ( x) x G&. (5) Fig. 8 Predicted oca heat transfer coefficient and fim thickness in the micro Preiminary Evauation of the Mode Figure 8 shows the predicted fim thickness and heat fux profie in the micro for HFC134a with T w =53.15K and T sat =48.15K. The groove width is w=0.9mm and the channe height is h=1.6mm. As the fim thickness δ decreases, the χ=0.5 Fig. 9 Effect of fim thickness δ in at macro-micro boundary Fig. 10 Effect of accommodation coefficient f 5 Copyright 005 by ASME

6 Evaporator P Compressor Condenser Pre-heater 1 Pre-heater Separator Bypass expansion vave Main expansion vave T P Fow meter Fig. 1 Experimenta setup Fig. 11 Effect of dispersion constant A therma resistance of iquid fim is reduced and thus the heat fux increases. However, as the iquid fim gets much thinner, the capiary pressure p c increases dramaticay due to the disjoining pressure effect. This reduces the equiibrium vapor pressure at the interface which is the driving force of evaporation. Extremey high heat fux ( q& peak > 0 MW/m ) is obtained at a very narrow area of the interine, even for HFC134a which has reativey ow therma conductivity compared to water or ammonia. The fim thickness is ess than 100nm at the interine. The iquid fim thickness at the ocation of macro-micro boundary, δ in, can be chosen arbitrary as noted earier. Figure 9 shows the predicted heat transfer coefficient for different boundary fim thickness δ in. Projected area of the groove is used for the definition of the heat fux, and the groove shape is equiatera triange. The micro, macro and tota heat transfer coefficients are defined as foows: Qmicro Qmacro α tota = α micro + α macro = +. (6) w x T w x T Athough the ratio of micro to macro contributions varies with δ in, the tota heat transfer coefficient remains neary constant. It is confirmed that the tota heat transfer prediction is not affected by the ocation of micro macro boundary. Thus, the fim thickness boundary vaue was fixed as δ in = 1µm in this study. Figures 10 and 11 show the effects of accommodation coefficient f and dispersion constant A on the predicted tota heat transfer coefficient. As it can be seen from the figures, the tota heat transfer coefficient is quite robust against wide variations of f and A. It is obvious that the accommodation coefficient f has arge infuence on the prediction near the contact ine in the micro. However, sma f makes the iquid fim incination (dδ/dξ) much more moderate in the micro. And this makes the effective thin fim onger in the macro. It shoud be emphasized that this Fig. 13 Fig. 14 Wet area width S(x) and oca heat transfer coefficient. Heat transfer coefficient against quaity. 6 Copyright 005 by ASME

7 R=0.67mm R=0.94mm z in = 9.7 z in = kw/m 38.3 kw/m Fig. 15 Liquid Soid Average heat transfer coefficient against heat fux Fig. 16 Predicted iquid-vapor interface profie robustness of the predicted tota heat transfer coefficient against these microscopic parameters is a very desirabe feature as an 1 engineering design too. Thus, f = 1 and A =.0 10 J are used as reference vaues in this study. EXPERIMENTAL VERIFICATION The schematic of the HFC134a experimenta oop is shown in Fig. 1. Main and bypass expansion vaves are controed to get required fow rate at the evaporator test section. The size of the fat pate evaporator is 5mm (width) 40mm (ength). The width of the micro-groove is 0.9mm, and the shape is equiatera triange (γ=60deg). Averaged wa temperature from 9 K-type thermocoupes and the saturated temperature from the pressure gauge were used for the heat transfer coefficient cacuations. Figure 13 shows the predicted wet area width S(x) (see Fig. 5 for its definition) and oca heat transfer coefficient against fow direction. The macro heat transfer coefficient and wet area width S(x) decrease aong the fow direction. On the other hand, the micro heat transfer coefficient remains neary constant. Thus, the decrease of the tota heat transfer coefficient is due to the degraded heat transfer in the macro. Figure 14 shows the heat transfer coefficient against quaity. Predicted heat transfer coefficient decreases as quaity increases. Heat transfer coefficient is overpredicted neary 70% when heat fux is sma. For arger heat fux, prediction curve gets much coser to the experimenta data even the absoute heat fux vaue is different. The average heat transfer coefficient is potted against heat fux in Fig. 15. Cacuated heat transfer coefficients for both micro and macro s decrease as heat fux increases, whie experimenta resut shows neary no dependence on heat fux. No boiing bubbes were observed from the sight gass visuaization. Thus, experimenta data impy that the mechanism of micro groove evaporation has ony weak dependence on the heat fux variation. Figure 16 shows the predicted iquid-vapor interface profies in the macro. The micro is hardy recognizabe when potted in this size since its range is so sma. Both incination ange at the micro-macro boundary and the radius of curvature are arger for higher heat fux case (38.3kW/m ). This change in the iquid-vapor interface profie deteriorates heat transfer for arger heat fux cacuation. The agreement of cacuation and experiment at high heat fux impies that the experimenta iquid-vapor interface profie shoud be quite simiar to that for high heat fux prediction curve, i.e. the broken ine, shown in Fig. 16. Furthermore, this experimenta iquid-vapor interface profie shoud remain neary unchanged regardess of heat fux. It is considered that the difference between experiment and prediction can be attributed to the error in predicting the iquid-vapor interface profie. More eaborate modeing of iquid-vapor interface profie wi be necessary for vapor compression cyce microgrooved evaporators. CONCLUSIONS A micro-grooved fat pate evaporator is modeed and investigated by numerica anaysis and experiment. The foowing concusions are derived: 1. Variation in accommodation coefficient f and dispersion constant A has reativey sma effect on tota heat transfer coefficient prediction. This feature is reay desirabe for engineering appications.. Predicted heat transfer coefficient decreases as quaity increases. 3. The predicted heat transfer coefficient shows strong dependence on heat fux, which is not the case with experimenta data. It is considered that this difference can be attributed to the error in predicting the iquidvapor interface profie. 7 Copyright 005 by ASME

8 ACKNOWLEDGMENTS The authors thank Mr. Y. Mukasa for his hep during constructing the experimenta setup. REFERENCES [1] Wayner, P. C., Jr., 1999, Intermoecuar forces in phasechange heat transfer: 1998 Kern Award Review, AIChE J., 45, pp [] Wayner, P. C., Jr., 1997, Interfacia Forces and Phase Change in Thin Liquid Fims, in Microscae Heat Transfer, Edited by C.L. Tien, F.W. Gerner, and A. Majumdar, Tayor & Francis, New York, Chapter 6, pp [3] Wayner, P. C., Jr., Kao, Y. K. and LaCroix, L. V., 1976, The interine heat-transfer coefficient of an evaporating wetting fim, Int. J. Heat Mass Transfer, 19, pp [4] Wayner, P. C., Jr., 198, Adsorption and capiary condensation at the contact ine in change of phase heat transfer, Int. J. Heat Mass Transfer, 5, pp [5] Kamotani, Y., 1978, Evaporator fim coefficients of grooved heat pipes, Proc. 3rd Int. Heat Pipe Conf., Pao Ato, CA. [6] Stephan, P. and Busse, C. A., 199, Anaysis of the heat transfer coefficient of grooved heat pipe evaporator was, Int. J. Heat Mass Transfer, 35, pp [7] Ajaev, V. S. and Homsy, G. M., 001, Steady Vapor Bubbes in Rectanguar Microchannes, J. Cooid Interface Sci., 40, pp [8] Hasebe, S., Shikazono, N. and Kasagi, N., 004, Modeing and Design of Micro Groove Faing Fim Evaporators, Proc. nd Int. Conf. Microchannes & Minichannes, ICMM , RIT, Rochester, NY. [9] Smith, S. L., 1970, Void fraction in two-phase fow: A correation based upon an equa veocity head mode, Heat and Fuid Fow, 1, 1, p.. 8 Copyright 005 by ASME