International Journal of Heat and Mass Transfer 42 (1999) 3405±341 Analysis of cylindrical heat pipes incorporating the e ects of liquid±apor coupling and non-darcian transportða closed form solution N. Zhu a, K. Vafai b, * a Lucent Technologies, Mount Olie, NJ 072, USA b Department of Mechanical Engineering, The Ohio State Uniersity, Columbus, OH 43210, USA Receied 10 June 199; receied in reised form 4 January 1999 Abstract This paper presents a two-dimensional analytical model for low-temperature cylindrical heat pipes. A closed-form solution which incorporates liquid±apor interfacial hydrodynamic coupling and non-darcian transport through the porous wick for the rst time, is obtained for predicting the apor and liquid elocity and pressure distributions. In addition, the steady-state apor and wall temperatures for a gien input heat load in the eaporator region and a conectie boundary condition in the condenser region, are obtained. The e ects of liquid±apor interfacial hydrodynamic coupling and non-darcian transport through the porous wick on the apor and liquid elocity and pressure distributions as well as the heat pipe capillary limit are discussed and assessed. The analytical solutions of the axial apor and wall temperature distributions, the apor and liquid pressure distributions, and the centerline apor elocities compare ery well with both experimental and numerical results. This work constitutes for the rst time a comprehensie analytical solution which proides closed form solutions for the apor and liquid ow as well as the operating temperature and the maximum heat remoal capability of the heat pipe. # 1999 Elseier Science Ltd. All rights resered. 1. Introduction Heat pipes are currently used in a wide ariety of heat transfer related applications. Analyses of heat pipe operations, both analytical and numerical, hae been performed extensiely by many inestigators. Almost all of the analytical studies hae been concentrated on the dynamics of apor ow. Liquid ow and the liquid±apor coupling were mostly neglected in * Corresponding author. Tel.: +1-614-292-6560; fax: +1-614-292-3163. E-mail address: afai.1@osu.edu (K. Vafai) analytical studies because of their complexity. A comprehensie analytical model for the oerall simulation of steady-state heat pipe operation is not aailable in open literature. Cao and Faghri [1] inestigated the e ects of heat pipe wall and the porous wick on the heat pipe operation. They concluded that it is important to include the porous wick and the wall in heat pipe analysis and to treat the entire heat pipe as a single system rather than to analyze the apor ow alone. Rosenfeld [2] also reported the importance of heat transfer within the wall and the porous wick in the case of an asymmetric heat input. Due to the di culty of obtaining an analytical solution for oerall heat pipe operation, more and more 0017-9310/99/$ - see front matter # 1999 Elseier Science Ltd. All rights resered. PII: S0017-9310(99)00017-4
3406 N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 Nomenclature A 1 constant de ned in Eq. (44) A 2 constant de ned in Eq. (45) B constant de ned in Eq. (22) C constant de ned in Eq. (26) D constant de ned in Eq. (30) G 1 constant de ned in Eq. (2) G 2 constant de ned in Eq. (34) h conectie heat transfer coe cient, [W/m 2 K] h fg latent heat of working uid [kj/kg] K permeability of the wick [m 2 ] k e e ectie thermal conductiity of the liquid-saturated wick [W/m K] k wall thermal conductiity of the heat pipe wall [W/m K] L length of heat pipe [m] L a length of the adiabatic section [m] L c length of the condenser section [m] L e length of the eaporator section [m] M 1 constant de ned in Eq. (29) M 2 constant de ned in Eq. (35) p pressure [Pa] p c capillary pressure [Pa] Q input heat [W] r radial coordinate [m] r c e ectie pore radius of the wick [m] R o heat pipe wall outer radius [m] R apor core radius [m] R w heat pipe wall inner radius [m] Re injection Reynolds number (r 1 R /m ) T temperature [K] T b bulk temperature of the coolant in cooling jack [K] u axial elocity [m/s] u i axial interfacial elocity [m/s] U mean axial apor elocity [m/s] U l maximum axial liquid elocity [m/s] radial elocity [m/s] 1 apor injection elocity [m/s] 2 apor suction elocity [m/s] x axial coordinate [m]. Greek symbols E porosity of the wick p g porous wick shape parameter ( E=K ) m dynamic iscosity [N s/m 2 ] r density [kg/m 3 ] s l surface tension of the working liquid [N/m]. Subscripts i liquid±apor interface l liquid phase apor phase. Superscript + dimensionless quantity.
N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 3407 numerical models hae been deeloped. Some comprehensie numerical models [3±5] coer both the apor ow and the liquid ow. The boundary and inertial e ects were included in these numerical models by applying the generalized momentum equation in porous medium to describe the liquid ow in heat pipes. The coupling of the liquid and apor momentum equations was also incorporated in these models by applying either the Laplace±Young equation [3,4] or the momentum jump condition [5] at the liquid±apor interface. In these models, the matching conditions of elocity and shear stress at the liquid±apor interface were neglected by assuming a non-slip condition and neglecting the interfacial drag. The e ects of these matching conditions as well as the boundary and inertial e ects on heat pipe operation hae not been inestigated in any of the preious studies. Vafai et al. [6±12] hae deeloped comprehensie pseudo-three-dimensional analytical models for asymmetrical at-shaped, including both disk-shaped and at-plate, heat pipes. They incorporated liquid ow, secondary apor ow and the e ects of liquid±apor hydrodynamic coupling and non-darcian transport in their models [±11]. Their results show that, for the at-shaped heat pipes, while the e ects of liquid±apor interfacial hydrodynamic coupling are negligible, neglecting the boundary and inertial e ects can lead to signi cant error in predicting liquid ow and the maximum heat transfer capability of the heat pipe. In the present work, a two-dimensional analytical model is deeloped for the oerall simulation of the steady-state cylindrical heat pipe operation. This analytical model employs matched asymptotic expansions for the liquid ow to incorporate liquid±apor interfacial hydrodynamic coupling and the boundary and inertial e ects. A closed-form solution is then obtained based on an in-depth integral method. The e ects of liquid±apor hydrodynamic coupling and the boundary and inertial e ects on cylindrical heat pipe operation and operating limit are also inestigated. The analytical prediction of apor and liquid elocity and pressure distribution as well as apor and wall temperature distributions compare ery well with both the experimental data reported by Huang et al. [13] and the numerical results reported by Tournier and El-Genk [5]. 2. Mathematical modeling The schematic of the cylindrical heat pipe and the coordinate system used in the present analysis is shown in Fig. 1. Heat applied at the eaporator section causes aporization and subsequent pressurization of the working liquid. The apor ows to the condenser section and releases latent heat as it condenses. The heat is then remoed from the condenser wall surface by conection. In the present analysis, apor and Fig. 1. Schematic of the heat pipe and the coordinate system used in the analysis.
340 N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 liquid ows are assumed to be steady, laminar and incompressible. The wick is assumed isotropic and saturated with the working liquid. The liquid and apor phases are coupled at the liquid±apor interface. The apor injection and suction at the liquid±apor interface are assumed to be uniform. 2.1. Goerning equations for apor ow Based on the aboe assumptions, the continuity and the momentum equations goerning the apor ow are gien by @u @x @ @r r ˆ 0 1 2.3. Vapor±liquid hydrodynamic coupling The apor and liquid phases are coupled at the apor±liquid interface. The continuity of mass uxes in the r-direction at the liquid±apor interface yields r x, r ˆ R ˆr l l x, r ˆ R < r 1, 0RxRL e ˆ 0, L : e RxRL e L a r 2, L e L a RxRL The continuity of axial elocity and shear stresses at the liquid±apor interface yields u x, r ˆ R ˆu l x, r ˆ R ˆu i x 6 7 @u r u @x @u ˆ @p @ 2 @r @x m u @r 2 1 r @u @r 2 and m @u @r @u l m l rˆrˆ @r rˆr @p @r ˆ 0 The x-direction shear stress in the momentum equations is neglected based on the work of Busse and Prenger [14] which shows the alidity of the boundary layer approximation for the apor ow in long cylindrical heat pipes. 2.2. Goerning equations for liquid ow The liquid ow within the porous wick is modeled using the generalized momentum equation [15] which accounts for the boundary and inertial e ects. The conectie term in the generalized momentum equation is dropped in accordance with the analysis gien in Vafai and Tien [15]. The goerning equations for the liquid ow are @u l @x 1 r m l @ 2 u l E @r 2 1 r @ @r ru l ˆ0 @u l m l @r K u l r lfe K 1=2 j u l j u l @p l @x ˆ 0 In the aboe goerning equations, r is the apor density and r l the liquid density, m the apor dynamic iscosity, m l the liquid dynamic iscosity, K and E are the permeability and porosity of the wick and F is a geometric function based on the porous wick structure and is calculated using the expression outlined in Vafai [16], i.e., F=1.75/Z150E 3/2. 3 4 5 where u i (x ) is the interfacial elocity at the liquid± apor interface. The apor injection elocity, 1 is related to the input power Q by the following relation: Q 1 ˆ 2r pr L e h fg where h fg is the latent heat of the working uid. The apor suction elocity 2 is determined by the mass balance which requires that the uid entering the apor region in the eaporator section to ow out through the condenser section. 2.4. Boundary conditions The boundary conditions are as follows: x ˆ 0: u ˆ ˆ u l ˆ l ˆ 0 10 x ˆ L: u ˆ ˆ u l ˆ l ˆ 0, p ˆ p l 11 r ˆ 0: ˆ 0, @u @r ˆ @p @r ˆ 0 9 12 r ˆ R w : u l ˆ l ˆ 0 13 The aboe goerning equations, coupling equations and boundary conditions are used to obtain a closedform analytical solution for the apor and liquid elocity and pressure distributions which accounts for the e ects of liquid±apor coupling and non-darcian transport through the porous wick.
N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 3409 3. Analytical solution An in-depth integral analysis along with the method of matched asymptotic expansions is employed to obtain the closed-form analytical solution for the apor and liquid elocity and pressure distributions. The temperature alues are then obtained through the use of a simple conduction model. 3.1. Vapor elocity pro le The following elocity pro le is utilized for the apor ow within the heat pipe: u x, r ˆU x a 0 a 1 r a 2 r 2 where U x ˆ 1 pr 2 R 0 u x, r 2pr dr 14 15 three parts: an inner solution for the interface zone between the liquid-wick and the apor phase, an outer solution for the main wick region and an inner solution for the interface zone between the liquid-wick and the heat pipe wall. This results in the following elocity pro le: u l x, r ˆ c 1 c 2 >< >: c 3 c 4 exp r R p U l x r Rw exp p 0R r R p R1 outer solution 1R r R w p R0 17 where U l (x ) is the maximum liquid elocity. Applying the boundary conditions gien by Eqs. (7) and (13) and matching the inner and outer solutions yields U l x u i x U l x Š exp r R p >< u l x,r ˆ U l x r Rw >: U l x exp p 0R r R p R1 outer solution 1R r R w p R0 Eq. p (1) can be written in a more compact form, by utilizing the fact that the thickness of the interface regions,, is much smaller than (Rw R )/2. This results in >< U l x u i x U l x Š exp r R p u l x,r ˆ r Rw >: U l x exp p R RrR R w R 2 R w R RrRR w 2 1 19 is the mean apor elocity. Applying the boundary conditions gien by Eqs. (7) and (12) to Eq. (14) yields the following apor elocity pro le: " u x, r ˆu i x 2 U x u i x Š 1 # r 2 R 16 3.3. The mean apor elocity The mean apor elocity U (x ) is determined by integrating the apor continuity equation. Utilizing the apor elocity pro le and the boundary conditions gien by Eqs. (6)±(), (10) and (11), the integration of the apor continuity Eq. (1) with respect to r from 0 to R yields 3.2. Liquid elocity pro le Vafai and Thiyagaraja [17] hae shown that the momentum boundary layer thickness at the interface between a porous medium and a uid or an impermeable medium is of the order of (K/E ) 1/2. Utilizing the method of matched asymptotic expansions the liquid elocity pro le within the wick region is deried in >< U x ˆ >: 2 1 R x, 2 1 R L e, 2 2 R L x, 0RxRL e L e RxRL e L a L e L a RxRL 20
3410 N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 3.4. The maximum liquid elocity The maximum liquid elocity U l (x ) is determined by integrating the liquid continuity Eq. (4) with respect to r from R to R w. Applying the liquid elocity pro le gien by Eq. (19), the mean apor elocity gien by Eq. (20), and the boundary conditions gien by Eqs. (6)±(), (10) and (13) within the integrated liquid continuity equation yields < B 1 x, U l x ˆ B : 1 L e, B 2 L x, 0RxRL e L e RxRL e L a L e L a RxRL 21 where! 2R B ˆ r R 2 w R2 1 r m g 2 R 2 22 p where g= E=K is the shape parameter of the porous wick, and r + =r l /r, m + =m l /m. If the hydrodynamic coupling is neglected at the liquid±apor interface, the following expression is obtained for B based on a noslip condition at the liquid±apor interface: B ˆ 2R r R 2 w R2 3.5. Interfacial elocity 23 The interfacial elocity is determined by utilizing the < G 1 2 1 M 1 1 x 2, p x ˆp 0 G : 1 2 1 M 1 1 L 2 e 2M 1 1 L e x, G 1 2 2 M 1 2 x L 2 M 1 2 L L a L c, where G 1 ˆ r 3R 2 D 2 7D 16 apor and liquid elocity pro les in Eq. (). This yields u i x ˆU l x 4 m U x 24 gr p The fact that 1/g= W(Rw R )/2 is used in the deriation of Eq. (24). Substituting Eqs. (20) and (21) into Eq. (24) results in the following expression for u i (x): < C 1 x, u i x ˆ C : 1 L e, C 2 L x, where 0RxRL e L e RxRL e L a L e L a RxRL 25 C ˆ m gr 2 B 26 Neglecting the hydrodynamic coupling at the liquid± apor interface leads to C=0. 3.6. Vapor pressure distribution The pressure distribution in the apor phase is obtained by integrating the apor momentum equations with respect to r from 0 to R. Introducing apor continuity equation, apor elocity pro le, apor-liquid coupling conditions and the related boundary conditions into the integrated momentum equations results in the following deried expression for apor pressure distribution: 0RxRL e L e RxRL e L a L e L a RxRL 27 2 M 1 ˆ 4m R 3 D 2 29 D ˆ CR If the hydrodynamic coupling at the liquid±apor interface is neglected, Eq. (27) reduces to >< p x ˆp 0 >: 16r 3R 2 16r 3R 2 16r 3R 2 2 1 m R 3 1 2 2 m R 3 2 2 1 m R 3 1 x 2, 0RxRL e L 2 e 16m 1 L e x, R 3 x L 2 m 2 L L a L c, R 3 L e RxRL e L a L e L a RxRL 30 31
Eq. (31) is similar to that obtained by Faghri [1], which in turn was in agreement with similar relations deeloped by Busse [19] earlier. It should be noted that in Ref. [1], in addition to not accounting for the liquid±apor coupling, the non-darcian e ects and for that matter the liquid ow transport was not taken into account. In fact, Ref. [1] considered only the apor ow. 3.7. Liquid pressure distribution The liquid pressure distribution is determined by integrating the generalized momentum Eq. (5) with respect to r from R to R w. Integrating Eq. (5) and utilizing all necessary boundary conditions and the cited relationships results in the deriation of the following expression for the liquid pressure distribution: N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 3411 3.. Vapor and wall temperatures < G 2 1 x 2 M 2 2 1 x 3, p l x ˆp l 0 G : 2 1 L e 2x L e M 2 2 1 L2 e 3x 2L e, G 2 2 x L 2 L L a L c Š M 2 2 2 x L 3 L 2L a L 2 c Š, For a fully-thawed low-temperature heat pipe, the apor phase can be assumed to be saturated and uniform. Therefore, there is no need to sole the energy equation for the apor phase. The alidity of this approximation was examined by Huang and El-Genk [20] experimentally. They measured the axial apor and wall temperature distributions of a copper heat pipe with water as the working uid. Their results show that the apor temperature is uniform along the heat pipe. They also found that the wall temperature in the eaporator and the condenser sections is almost uniform, except near the interfaces with the adiabatic section where axial conduction in the wall is most pronounced. In the present study, the heat pipe is heated 0RxRL e L e RxRL e L a L e L a RxRL 32 where p l 0 ˆp L G 2 2 L L a L c M 2 2 2 L and 2L a L 2 c 33 G 2 ˆ mlb 1 1 2 2K gr 1 R w =R M 2 ˆ rlfe " 3K 1=2 B 2 1 # C C 2B gr 1 R w =R 2 34 35 If the boundary and inertial e ects are neglected, G 2 and M 2 are reduced to the following relationships: G 2 ˆ ml 2K B 1 gr M 2 ˆ 0! 2C 1 R w =R 2 36 37 uniformly oer the eaporator section and conectiely cooled in the condenser section. The apor temperature is assumed uniform along the heat pipe, and a one-dimensional heat conduction model is used for the wall and liquid-wick regions. For steady-state operation, the conectie boundary condition at the wall outer surface (r=r o ) is: Q ˆ 2pR o L c h T wall,c T b 3 where Q is the heat input, h is the conectie heat transfer coe cient, T wall,c is the wall temperature in the condenser section, and T b is the bulk temperature of the coolant. The apor and wall temperatures are obtained as follows: T ˆ T b Q ln Ro =R w ln R w=r 2pL c k wall k eff 1 hr o and 39 T b Q 2pL c >< T wall x ˆ T b Q 2pL c >: " ln Ro =R w k wall ln R w=r k eff 1 L c 1 L e hr o ln Ro =R w k wall ln R w=r k eff 1 hr o Q T b 2phR o L c # 0RxRL e L e RxRL e L a L e L a RxRL 40
3412 N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 where k wall is the thermal conductiity of the heat pipe wall, and k e is the e ectie thermal conductiity of the liquid-saturated wick. 3.9. The maximum heat transport capillary limit For a heat pipe under steady-state operation, stable circulation of the working uid in the heat pipe is achieed through the capillary pressure established by the wick structure. In conentional heat pipes, there exists a maximum capillary pressure that can be deeloped for a liquid±apor pair. The maximum heat transport capillary limit for a heat pipe is achieed when the sum of the pressure losses along the circulation path of the working uid reaches the maximum capillary pressure; that is Dp x max x min Dp l x min x max p c x min ˆ 2s l r c 41 where p c (x min )=p (x min ) p l (x min ) is the minimum capillary pressure, s l is the surface tension of the liquid, r c is the e ectie pore radius of the wick, and x max /x min denotes the location where the capillary pressure is maximum/minimum. The notation Dp(x max x min ) refers to ealuation of Dp oer the distance (x max x min ). In the present study, as stated in Eq. (11) the condenser end (x=l ) is assumed to be the wet point ( p (L )=p l (L )) during steady-state operation. This reduces Eq. (41) to p 0 p l 0 ˆ2s l 42 r c Substituting Eqs. (9), (31) and (33) into Eq. (42) and soling for the maximum heat transport capillary limit yields Q max ˆ 1 q A 2 2 2A A 1s l =r c A 2 43 1 where Q max is the maximum heat transport capillary limit, and A 1 ˆ M2 L 2L a 12p 2 R 2 r2 h2 fg A 2 ˆ G 2 2M 1 L L a 4pR r h fg 44 45 When boundary and inertial e ects are neglected, Eq. (43) reduces to Q max ˆ 2s l A 2 r c 46 4. Results and discussion To erify the model predictions, results are compared with the experimental data reported by Huang et al. [13] and the numerical results reported by Tournier and El-Genk [5], for a copper heat pipe with water as the working uid. The physical dimensions of the heat pipe are chosen as: R o =9.55 mm, R w =9.4 mm, R =.65, L e =60 cm, L a =9 cm, and L c =20 cm which are based on the physical dimensions used in Refs. [5] and [13]. The wick e ectie pore radius, porosity and permeability are chosen as 54 mm, 0.9 and 1.5 10 9 m 2, respectiely. The e ectie thermal conductiity of the liquid-wick is calculated to be 1.965 W/m K, using the equation gien by Chi [21]. The thermophysical properties of the working uid are obtained at the calculated apor temperature and are assumed to be constant along the heat pipe. The input heat is taken as 455 W. The cooling water enters the condenser cooling jacket at 294.5 K and 11.33 g/s. The temperature of the cooling water at the exit of the cooling jacket is calculated using the equation Q=mÇc p (T out T in ). The bulk temperature of the cooling water is taken as the arithmetic mean of T in and T out. The conectie heat transfer coe cient in the cooling jacket is taken as 100 W/m 2 K [5]. It should be noted that a great adantage of the presented closed form analytical solution is the ease and speed in which a comprehensie solution can be obtained subject to changes in arious parameter and properties. In here some representatie alues were chosen for the simulations. We were able to show that the main conclusions remain unchanged for wide range of ariation of these parameters. Fig. 2 compares the model predictions with the numerical and experimental results of the apor and wall temperature distributions along the heat pipe. The model predictions of both the apor and the wall temperatures agree well with the numerical and the experimental alues. As can be seen in Fig. 1, the assumption of uniform apor temperature along the heat pipe is good for low-temperature heat pipes. Although there is a discrepancy between the calculated wall temperature and the experimental data at the beginning and the end of the condenser section, the model predictions agree ery well with the numerical results reported by Tournier and El-Genk [5]. Since same parameter alues for the conectie cooling boundary condition are used in the present analytical model as those used in the numerical model, the good agreement between the analytical and the numerical results of the wall temperatures demonstrates that the one-dimensional approximation for the heat transfer within the wall and liquid-wick regions is adequate for low-temperature heat pipes. This is in agreement with the obseration of Tournier et al. [22].
N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 3413 Fig. 2. Comparison of the calculated apor and wall temperature distributions with the numerical results gien by Tournier and El- Genk [5] and the experimental results gien by Huang et al. [13]. Fig. 3 compares the analytical apor and liquid pressure distributions along the heat pipe with the numerical results of Tournier and El-Genk [5]. The absolute alue of the apor pressure at the eaporator end, p (0), is related to the calculated apor temperature based on the assumption that the apor phase is saturated for a low-temperature fully-thawed heat pipe. The analytical results agree ery well with the numerical results. As the results show, the apor pressure ariation along the heat pipe is small, which means that a uniform apor temperature pro le is expected. The ariations of the mean apor elocity, the centerline apor elocity, the maximum liquid elocity and the interfacial elocity along the heat pipe are shown in Figs. 4 and 5. The positie elocity alue denotes that the ow is along the x direction and the negatie elocity alue denotes that the ow is along the negatie x direction. The analytical centerline apor elocity at the exit of the eaporator is compared to the numerical result by Tournier and El-Genk [5]. Again, as seen in Fig. 4, the analytical predictions agree ery well with the numerical results. This further demonstrates that the analytical model proides accurate predictions of the low-temperature heat pipe operations. The e ects of liquid±apor interfacial hydrodynamic coupling and the boundary and inertial e ects are analyzed next. Results were obtained for the radial Reynolds number (Re=r 1 R /m ) alues of 2.0, 5.0 and 10.0 and are displayed in Figs. 6±. To accommodate apor and liquid pressures at di erent Re alues in the same gure and to make the comparison more meaningful, the apor pressure at the eaporator end, p (0), is taken as zero in Figs. 7 and for all the Re alues. The e ects of liquid±apor interfacial hydrodynamic coupling on the maximum liquid elocity along the heat pipe is shown in Fig. 6. It should be mentioned that liquid±apor interfacial hydrodynamic coupling has no e ect on the mean apor elocity, as indicated in Eq. (20). This is because the apor ow rate at steady-state operation is purely determined by the apor injection/suction elocities, i.e., the eaporation/condensation rates. Howeer, liquid±apor interfacial hydrodynamic coupling do a ect the apor elocity pro le, as illustrated in Eq. (16). For example, the interfacial e ects result in a larger centerline apor elocity. Since the interfacial elocity is negligible compared to the mean apor elocity, as can be seen in Figs. 4 and 5, the interfacial e ects on the apor elocity pro le is negligible. The analytical results in Fig.
3414 N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 Fig. 3. Comparison of the calculated apor and liquid pressure distributions with the numerical results gien by Tournier and El- Genk [5]. Fig. 4. Variations of the mean apor elocity and the centerline apor elocity along the heat pipe.
N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 3415 Fig. 5. Variations of maximum liquid elocity and interfacial elocity along the heat pipe. Fig. 6. The e ects of liquid±apor interfacial hydrodynamic coupling on the maximum liquid elocities.
3416 N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 Fig. 7. The e ects of liquid±apor interfacial hydrodynamic coupling on apor and liquid pressures. Fig.. Boundary and inertial e ects on the apor and liquid pressure distributions.
N. Zhu, K. Vafai / Int. J. Heat Mass Transfer 42 (1999) 3405±341 3417 6 show that the interfacial e ects lead to a larger maximum liquid elocity. The larger the injection Reynolds number, the larger the di erence caused by the interfacial e ects. Fig. 7 shows the e ects of liquid±apor interfacial hydrodynamic coupling on the apor and liquid pressures. The interfacial e ect on the apor pressure is negligible. Howeer, neglecting liquid± apor interfacial hydrodynamic coupling will lead to a smaller liquid pressure drop along the heat pipe. The boundary and inertial e ects on apor and liquid pressure pro les are shown in Fig.. As indicated in Eq. (27), the pressure distribution in the apor phase is not a ected by the boundary and inertial e ects. Howeer, neglecting the boundary and inertial e ects leads to an underestimation of the liquid pressure. The larger the Reynolds number, the larger the error inoled in the calculation of liquid pressures by using a Darcian model. This is in agreement with the results by Vafai and Tien [15]. Fig. 9 shows the interfacial and the boundary and inertial e ects on the heat transport capillary limit of the heat pipe. It can be seen in Fig. 9 that either neglecting the interfacial e ects or neglecting the boundary and inertial e ects can lead to an oerestimation of the capillary limit. This is because a smaller oerall liquid pressure drop along the heat pipe is expected when the interfacial e ects or the boundary and inertial e ects are neglected. While the interfacial e ects on the maximum heat remoal capability is relatiely small, the error inoled in neglecting the boundary and inertial e ects is substantial. 5. Conclusions The steady-state operation of a low-temperature cylindrical heat pipe has been studied analytically. An analytical model, which incorporates liquid±apor interfacial hydrodynamic coupling and non-darcian transport through the porous wick, was deeloped for predicting the apor and liquid ow and the maximum heat transfer capability of the heat pipe. A closed-form solution was obtained for the apor and wall temperatures as well as the apor and liquid elocity and pressure distributions for a conectie cooling condition in the condenser region. A closed-form solution of the heat pipe capillary limit during steady state operation was also obtained. These closed-form analytical solutions, proide a quick, accurate prediction method for low-temperature heat pipe operation and were found to be in ery good agreement with both experimental and numerical results. The e ects of liquid± Fig. 9. The e ects of liquid±apor interfacial hydrodynamic coupling and the boundary and inertial e ects on the heat pipe capillary limit.
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