Analysis of loop heat pipe performance under varying wick load

Transcription

1 - Journal of Mechanical Engineering 51(57-8, UDK - UD Original scientific paper (1.1 Analysis of loop heat pipe performance uner varying wick loa A. Heitor Reis 1,, António F. Miguel 1, an Murat Ayin,3 1 hysics Department, University of Évora, R. Romão Ramalho, 59, Évora, ortugal Évora Geophysics enter, R. Romão Ramalho, 59, Évora, ortugal 3 Dep. of Mech. Engineering, Istanbul echnical University, Gumussuyu, Istanbul, urkey ahr@uevora.pt Abstract Loop heat pipes (LH are heat transfer evices use to enhance cooling of small spaces an basically consist of seale tubes connecting a heat source, the evaporator, whose major part is a porous wick, with a conenser that operates as heat sink. In this paper we analyse the effect of curvature of the liqui vapor interface upon the vapor pressure within wick pores. e show how this effect affects start-up by requiring a ifference between wick an conenser temperatures as higher as wick pore with becomes smaller. e analyse also transient operation an foun that iealy LH are self-ajusting systems that ten to stable operation. e present a formula to escribe the transient regime. he analysis provies also optimization of wick pore with for maximum heat transfer. Optimal pore with is shown to vary with temperature ifference between wick an conenser. It is envisage how this feature may help in LH esign. Introuction Loop heat pipes (LH are heat transfer evices in which a working flui transfers continuously heat from the evaporator to the conenser. LH operate uner a pressure ifference generate at a porous wick (the evaporator an that rives the working flui within the loop (see Fig.1. he pressure ifference is ue to the capillary forces rives the liqui from the conenser into the evaporator while the ifference between the saturation pressures corresponing to the evaporator an conenser temperatures rive the vapor back into the conenser. LH have been broaly use in electronics cooling, spacecraft an other areas. Numerous papers ealing with various aspects of LH performance characteristics have been publishe so far [1-1]. Our purpose is not to review this abunant literature extensively but just to focus on a feature whose complete comprehension remains rather elusive: LH require a pressure ifference across the wick in orer to start properly. his is usually overcome through the use of active evices to assist in the start-up. Although some structural aspects of LH may contribute to start-up problems, we believe that thermoynamics play here the major role. hen analyzing LH cycles it is commonly assume that wick pressure follows lausius-lapeyron equation that relates pressure to temperature at an equilibrium planar liqui-vapor interface. However liqui-vapor interfaces in the wick pores are far from being planar. In fact such interfaces are in a shape of meniscus which implies that the equilibrium vapor pressure follows Kelvin s equation [13,14]: s 4 cos exp LR g where s is the saturation pressure corresponing to a planar interface, is surface tension, is soli/liqui contact angle, is pore with, L is ensity of liqui, R g is the specific gas constant an is temperature. Eq. (1 inicates that liqui can boil in the wick pores even if pressure in the overlying vapor is well bellow the saturation pressure corresponing to a planar interface that is given by lausius-lapeyron equation: (1 h LV s exp 1 1 ( R g where h LV stans for the enthalpy of evaporation an the subscript for reference values. By efining the imensionless temperature = (R g /h LV an imensionless pore iameter 451

2 Nomenclature D uct iameter (m ensity (kg m -3 pore with (m soli/liqui contact angle (ra h LV enthalpy of evaporation (J kg -1 L uct lenght (m Superscript m mass flow rate (kg s -1 non-imensional n number of wick pores pressure (a Subscripts s saturation pressure (a conenser Q heat current ( s -1 eq equilibrium R g specific gas constant (J kg -1 K -1 L liqui temperature (K max maximum value t time (s op optimal wick loa (liqui V vapor surface tension (N m -1 wick kinematic viscosity (m s -1 reference value = ( L h LV /cos, an combining Eqs. (1 an ( we obtain: exp 1 (3 For ifference in temperature such that / <<1, the vapor pressure ifference between two successive menisci an (see Fig. an [13] for etails is: (4 where <> represents the average pressure of the vapor between the menisci. ompensation chamber rimary wick Seconary wick Evaporator Bayonet hen (planar interface Eq. (3 reuces to lausius-lapeyron equation. However, when is finite the vapor pressure is lower than saturation pressure at the same temperature. his effect is as much important as the pore with becomes smaller. As a consequence when both the evaporator an conenser are at the same temperature, vapor pressure in the wick is lower than that in the conenser (see Eq. (4 an [13] for etails: (5 D where D >> are the curvature raiuses of the liquivapor interfaces in the conenser an the wick, respectively. o start LHs one has to surpass this negative pressure ifference by increasing the wick temperature. By using Eq. (4 again, we see that LH start-up can only occur for values of the ratio of wick temperature to conenser temperature higher than: 1 4 (6 his is one of the reasons why LHs require a significant grooves Liqui line line liqui vapor liqui onenser Fig. 1. Schematic representation of a LH (major parts. Fig.. Liqui/vapor interfaces of ifferent curvature raiuses. ressure an temperature of the vapor in equilibrium with the liqui epens upon the raius of curvature of the interface. 45 Reis A.H. - Miguel A.F. - Ayin M. A. H. Reis et al.

3 ifference of wick to conenser temperature in orer to start properly. Otherwise LHs can even start with vapor conensation in the wick, which is recognize as the ability to occasionally start in the reverse irection [1]. Authors of recent LH literature [1-1] o not consier the effect of the curvature of the interface upon the vapor pressure an usually restrict the explanations of start-up ifficulties to features of LH operation. In this paper we aress the problem LH operation uner varying heat loa by taking into account the effect of interface curvature upon pressure of the vapor in the wick. q L Analysis of transient LH operation he sum of the pressure heas an rops along a close path within a LH must be zero. herefore for a LH in a horizontal plane if, V, an L enote the pressure ifferences across the liqui-vapor interface at the wick, between vapor at the conenser an the wick, across the liqui-vapor interface at the conenser an between liqui in the wick an the conenser, respectively (see Fig. 3, this conition reas: + V + + L = (7 he pressure ifference across the liqui-vapor interface in the wick is given by the Young-Laplace equation as [13]: 4cos (8 he vapor pressure ifference between conenser an wick is the sum of the pressure hea given by Eq. (4 with the pressure rop ue to vapor flow (see Fig. 3: V 1 D V D L L (1 LV 3m V V (9 4 4 n DV L V V L L L Fig. 3. Schematic representation of the liqui an vapor phases within the LH as well as pressure ifferences an iameter an length of the liqui an vapor lines. Fig.4 Schematic representation of the liqui vapor interface in the wick. ick loa is represente by that is the wick s pore filling level. In Eq. (9 both imensional an imensionless variables coexist as an intermeiary step towars a final imensionless equation to be presente further on. Also in Eq. (9 m V is vapor flow rate, an are pore with an imensionless pore with, n is the number of wick s pores an the subscripts, V, represent conenser, vapor an wick. e assume planar liqui-vapor interface in the conenser an therefore = (1 By assuming Hagen-oiseuille flow along the liqui line, the pressure rop is given by: L LL L 3m L L ( n DL where m L is liqui flow rate an the subscript L stans for liqui. In the wick, heat ( Q is absorbe almost totally by the liqui phase (see Fig. 4 therefore we assume that Q Q (1 where Q represents the heat current absorbe by the wick at maximum loa (=1. e efine wick s varying loa as: m m Liqui L V (13 n L L By combining Eqs. (7-(13 an aing to the alreay efine an the following imensionless variables: - /( L hlv - t t(q /(n L LhLV Q Q 3 /( L h - L L LV A. H. Reis et al. 3 Analysis of loop heat pipe performance uner varying wick loa 453

4 we arrive at the following equation: L nl L L Q 4 D L L nl n L Q L V (1 L V V L D L L D V (14 In Eq. (14 the first an secon terms represent liqui an vapor pumping heas, respectively while the remaining terms represent pressure rops ue to flui flow. Since D V ;D V an V L Eq. (14 can be simplifie to: H (15 where H 1 (16 Q L is the imensionless total pumping hea. he general solution of Eq. (15 is H A exp( t (17 where A is a constant to be etermine from initial conitions. At t =, H A an when t, the imensionless pumping hea equals equilibrium wick loa, i.e. H eq (18 which implies that for reaching stable (equilibrium operation the total pumping hea H cannot be negative. his equilibrium conition also follows from Eq. (17 with that correspons to liqui flow rate being equal to vapor flow rate. herefore rearranging Eq. (17 one obtains: eq exp( t eq (19 which by taking into account Eq. (1 an the efinitions of Q an t may be also expresse in terms of the heat current absorbe at the wick as Q eq Q Q eq Q Q exp nlh LV t ( Fig. 5. Variation of wick loa with time. Lines correspon to ifferent ratios of start-up loas to equilibrium wick loa. Stable operation may be reache by starting either from above or from bellow equilibrium loa ( that matches stable operation. he variation of wick loa, with time is shown in Fig. 5. e see that stable operation (equilibrium may be (ieally reache either from above or from bellow the equilibrium wick loa. In practice, often the wick is not in equilibrium itself what may lea to aitional troubles in LH operation [1]. However as Fig. 5 shows, LHs have a self-ajusting capacity that enable them to reach stable operation. As H eq >, (see Eq. 18 the total pumping hea has to be always positive in orer LH to start an keep on functioning. From Eq. (16 this implies that start-up must comply with the following inequality: eq.8.4 eq = t tttt 1 /4 (1 / /4 For a planar surface, as the start-up conition woul be > simply. By consiering the efinitions of an we conclue that 3 / is of orer V / L 1 what enables us to consier that the general conition expresse by Eq. (1 is not impeitive of LH functioning. Although Eq. (1 is a necessary conition for LH start-up it oes not represent a sufficient conition. In fact, by Eq. (16 H is the sum of the liqui an the vapor pumping hea. Since the first, 4/, is always positive the secon one may be negative. Nevertheless it is sufficient for LH start-up that wick an conenser temperatures aitionally comply also with Eq. (6. From Eq. (6 we conclue that curvature effects of the wick s liqui-vapor interface become important for = ( L h LV /cos smaller than 1 3 (see Eq. 6. Since ( L h LV /cos 1 1 m -1 significant ifference between wick an conenser temperatures is 454 Reis A.H. - Miguel A.F. - Ayin M. A. H. Reis et al. 4

5 require to start LHs with wick pore withs smaller than.1m. Optimization of wick s pore with for LH esign he analysis of the preceing section may become useful in LH esign as it allows for the optimization of wick pore with. By combining Eqs. (1 an (18 one obtains: 1/ Q eq Q 1 L 4 ( By analyzing Eq. ( we can see that heat absorbe at stable operation Q eq, can be maximize with respect to wick s pore iameter. he optimal pore iameter is a function of wick an conenser temperatures of the form: (3 op which in imensional form reas: op LR g cos Lh LV (4 Eq. (4 shows that wick pore iameter that allows for maximum heat transfer is as smaller as the ifference between wick an conenser temperatures gets higher. On the other en we see that op (planar surface as. In Eq. (4 the term within brackets is of orer 3 V / L 1 while (cos/ L h LV 1-1 m -1. herefore for 1 K an 3K, op is of orer 3m. herefore Eq. (4 may help LH manufacturers in choosing the wick pore iameter that matches optimal performance at prescribe wick an conenser temperatures. onclusions he analysis of loop heat pipe (LH carrie out in this paper shows that curve liqui vapor interfaces may change significantly equilibrium vapor pressure in wick pores of withs of orer 1 m. an smaller. his affects LH start-up by requiring significant ifference between wick an conenser temperatures for start-up to occur. his ifference in temperature gets higher as the wick pore with becomes smaller. Solution of a ifferential equation escribing transient functioning shows that LH are self-ajusting systems that ieally ten to stable operation either if starte from above or from below stable operation points. However LH start-up from below stable operation points are unlikely to occur in LH whose wick pores are of orer 1 m an smaller ue to the above mentione ifference between wick an conenser temperatures require to start-up. Optimization of performance of LH with respect to wick pore with shows that optimal pore with can be relate to wick an conenser temperatures an vary linearly with the arnot coefficient of performance of heat pumps. his aspect may help in choosing the appropriate wick structure for prescribe LH operating conitions. References 1. Ku, J. Operating characteristics of loop heat pipes, roc. 9 th Int. onf. Of Environ. Systems, paper No. 9811, (1999 Denver, USA.. Kamotani, Y,hermocapillary flow uner microgravity Experimental results, Av. Space Res. 4, 1 ( Kaya an. Hoang, Mathematical moeling of loop heat pipes an experimental valiation, J. of hermophysics an Heat ransfer 13, 3, ( I. Muraoka, F. M. Ramos, V. V. Vlassov, Analysis of the operational characteristics an limits of a loop heat pipe with porous element in the conenser, Int. J. of Heat an Mass ransfer 5, 8 ( R. hanratilleke., H. Hatakeyama, an H. Nakagome, Development of cryogenic loop heat pipes, ryogenics 38, ( N. Zhang, Innovative heat pipe systems using a new working flui. Int. omm. Heat Mass ransfer, 8 ( H. F. Smirnov an B. V. Kosoy, Refrigerating heat pipes, Appl. herm. Eng. 1 ( Q. Liao. an. S. Zhao, Evaporative heat transfer in a capillary structure heate by a groove block, J. of hermophysics an Heat ransfer 13, No. 1 ( J. S. Allen, K..Hallinan an J. Lekan, A stuy of the funamental operations of a capillary riven heat transfer evice in both normal an low gravity, AI onference roceeings 4, ( , E. American Institute of hysics, oobury, New York. 1.. Figus, Y. Le Bray, S. Bories, an M. rat Heat an mass transfer with phase change in a porous structure partially heate: continuum moel an pore network simulations, Int. J. of Heat an Mass ransfer 4 ( J. Ku, Operating characteristics of loop heat pipes, Int. onf. on Env. Syst., paper J. Baumann, B. ullimore; J. Ambrose, E. Buchan an B. Yenler, A methoology for enveloping reliable start-up pf LHs. aper nº -85, ( Am. Inst. of Aeronautics an Astronautics, Inc. 13. A. H. Reis, hermoynamics of fluis in mesoporous meia, in D. B. Ingham (E. roc. of NAO Av. St. Inst. on orous Meia, 9- June 3, Oviius Un. ress ( A. H. Reis an R. Rosa, Sorption isotherms as a funamental tool for the analysis of couple heat an mass fluxes in porous meia, J. of orous Meia (to appear in 4. A. H. Reis et al. 5 Analysis of loop heat pipe performance uner varying wick loa 455