Analysis of overloaded micro heat pipes: Effects of solid thermal conductivity

International Journal of Heat and Mass Transfer 81 (2015) 737–749

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Analysis of overloaded micro heat pipes: Effects of solid thermal conductivity Kek-Kiong Tio a, Yew Mun Hung a,b,⇑ a b

Faculty of Engineering and Technology, Multimedia University, 75450 Malacca, Malaysia School of Engineering, Monash University, 46150 Bandar Sunway, Malaysia

a r t i c l e

i n f o

Article history: Received 14 July 2014 Received in revised form 12 September 2014 Accepted 24 October 2014 Available online 20 November 2014 Keywords: Overloaded micro heat pipe

a b s t r a c t Starting from the principles of mass, momentum, and energy conservations as well as the Young–Laplace capillary equation, a mathematical model of triangular MHPs has been developed, primarily to investigate the effects of the thermal conductivity of the solid wall on their performance under overloaded conditions. For this purpose, two solids of significantly different thermal conductivities, copper and nickel, have been selected for the wall material. Using the model, a map encompassing the various possible operation zones of an MHP was constructed, to provide insight into the modes of operation under a given operating temperature but with varying heat loads and charge levels of the working fluid. The model predicts that in the overloaded zones, the dryout and flooded lengths increase with the applied heat load, resulting in a decrease in the effective length of the MHP. Moreover, it is also observed that the existence of dryout is accompanied by a large temperature rise over the dry region and, thus, a large total axial temperature drop, which increases rapidly with the applied heat load. Finally, comparison between copper and nickel MHPs on how the key performance indicators, such as the dryout and flooded lengths, the effective length, and the total axial temperature drop, response to the applied heat load shows that an MHP of solid wall of higher thermal conductivity out-performs one with lower solid thermal conductivity. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Owing to their ability to deliver high heat fluxes, micro heat pipes (MHPs) are efficient heat transfer devices, especially in applications where space constraint is a major consideration. However, for an MHP to operate effectively, it must not be overloaded. Otherwise, adverse operating conditions such as dryout, flooding, or the combination of both may set in [1]. In the existing literature, most analytical studies do not include overloaded MHPs. While the relatively small number of studies which deal with overloaded MHPs have provided useful insights, many questions remain unanswered. It is, therefore, the purpose of this paper to address some of those issues. Specifically, we shall investigate the effects of the thermal conductivity of the solid wall of overloaded MHPs. One of the most important problems associated with an overloaded MHP is the possible existence of a dryout region at its evaporator section, resulting in a high solid wall temperature there. Peterson and co-workers [2,3] and Suman and co-workers [4,5] ⇑ Corresponding author at: School of Engineering, Monash University, 46150 Bandar Sunway, Malaysia. E-mail address: [email protected] (Y.M. Hung). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.10.060 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

have investigated overloaded MHPs with a focus on dryout. However, the occurrence of dryout at the evaporator is not the only possible manifestation of overload, because flooding may also exist in an overloaded MHP. In fact, flooding at the condenser section may take place in the absence of or simultaneously with the occurrence of dryout at the evaporator. Kim and co-workers [6,7] and Launay et al. [8], respectively, have conducted analytical studies pertaining to these two cases. In this paper, we shall cover all the three cases and elaborate on the conditions for the prevalence of each of them. Since MHP was first conceptualized in the 1980s, it has been a common practice by MHP investigators, in earlier studies [9,10] as well as the more recent ones [11,12], to exclude from their analyses axial heat conduction in the solid wall. This practice is justified on the ground that heat transport by axial solid conduction is negligible compared to that by phase change of the working fluid. The absence of axial solid conduction then implies that the temperature of the solid wall does not vary in the axial direction; in other words, an MHP may be regarded as an isothermal device. However, a few recent analytical studies [7,13–15] show that the temperature of the solid wall varies continuously along the axis of an MHP. Moreover, while the heat transport by phase change usually dominates that by axial solid conduction, the exclusion of

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Nomenclature A C Ca DH F f G h hfg K k L L⁄ L0 M b M b opt M _ m b m N Nu P p Q_ Q_ c Q_ cap Q_ p q_ r Re s T T T0

cross-sectional area; cross-sectional area of micro heat pipe, m2 constant; geometrical parameter, Eqs. (8)–(12), (45), (46) capillary number, defined in Eq. (28) hydraulic diameter, Eqs. (47), (48), m function defined in Eq. (30) friction factor function defined in Eq. (31) heat transfer coefficient, W m2 K1 1 latent heat of evaporation, J kg constant, Eqs. (25), (26) thermal conductivity, W m1 K1 length; length of micro heat pipe, m effective length; effective length of micro heat pipe, Eq. (33), m axial distance from evaporator end where solid and liquid temperatures are equal, m mass of working fluid, kg charge level optimal charge level mass flow rate, kg s1 dimensionless mass flow rate, defined in Eq. (21) number of corners/vertices of a polygon Nusselt number, defined in Eq. (2) length of interface, m pressure, N m2 rate of heat transport; applied heat load, W rate of heat transport by axial conduction in solid wall, W heat transport capacity, W rate of heat transport by phase change of working fluid, W rate of heat transfer per unit axial length, W m1 meniscus radius of curvature, m Reynolds number, defined in Eq. (19) volume fraction occupied by liquid phase temperature, °C average temperature, °C solid temperature at evaporator end, °C

the latter may incur significant errors in the calculation of the heat transport capacity of the MHP [13]. To properly model an MHP, it is therefore necessary to include axial solid conduction and the associated axial temperature variation of the solid wall. This is especially true for an overloaded MHP, since large axial temperature drops may be involved. In fact, Lin et al. [16] have experimentally observed axial temperature drops as large as 50 °C for MHPs operating with, presumably, dryout. For the liquid and vapor phases of an MHP which is not overloaded, uniformity in temperature can be assumed, since their total axial temperature drops are much smaller than that of the solid wall [15]. This isothermal assumption may still be valid for overloaded MHPs, since the liquid and vapor phases which participate in phase change and circulation are both saturated. Fig. 1 is a schematic illustration of the heat and fluid flows together with the axial volume distribution of the liquid phase inside an overloaded MHP. In this paper, the incoming heat flux at the evaporator and the outgoing heat flux at the condenser are both assumed to be uniform, but their respective values may be different from each other. However, the model developed based on this assumption can be very easily modified for cases of nonuniform fluxes. In Fig. 1, the evaporator length Le , condenser length Lc , and the length of the adiabatic section, La , are all prescribed

T1 T op DT u w We x b x

solid temperature at condenser end, °C operating temperature, °C total axial temperature drop, °C velocity, m s1 side width of cross section of flow channel, m Weber number, defined in Eq. (29) axial distance from evaporator end, m dimensionless x, in units of L

Greek symbols c vapor-to-liquid kinematic viscosity ratio e vapor-to-liquid density ratio g coefficient defined in Eq. (23), K1 H quantity defined in Eq. (5), K h contact angle, rad k coefficient defined in Eq. (4) l dynamic viscosity, kg s1 m1 nc ;np fractions of applied heat load, defined in Eq. (36) e x angular parameter, defined in Eq. (41) q density, kg m3 r coefficient of surface tension, N m1 s shear stress, N m2 / half corner angle at vertices of polygon, rad Subscripts a of c of cl of d of e of f of fl of l of lv of s of sl of sv of v of

adiabatic section condenser section capillary limit dried out segment of solid wall evaporator section flooded segment of solid wall onset of flooding liquid phase of working fluid liquid–vapor interface solid wall solid–liquid interface solid–vapor interface vapor phase of working fluid

geometric parameters. On the other hand, the dryout length Ld and the length of the flooded region, Lf , are not pre-set constants; rather, they are obtained by solving the governing equations subject to the heat load and, therefore, are functions of the heat input to the MHP. The three cross sections depicted in Fig. 1 illustrate the changes in the respective areas occupied by the liquid and vapor phases as we proceed from the evaporator toward the condenser, the third being an illustration of the onset of flooding. This cross section corresponds to the furthest point of the vapor flow, since the flow channel beyond is completely flooded by stagnant liquid [6–9]. The scenario depicted in Fig. 1 is the most general case, in the sense that both dryout and flooding occur. However, as we shall see later, either dryout or flooding can take place without the other. The MHP model developed in this paper is a steady-state, onedimensional construction. All the pertinent variables, such as the temperature of the solid wall or the liquid and vapor velocities and pressures, are assumed to have been properly averaged over the cross-section and, therefore, are functions of the axial position only. As remarked previously, we shall assume a uniform temperature for both the liquid and vapor phases, with the understanding that the stagnant liquid occupying the flooded region, if any, may have a slightly lower temperature than the circulating fluid. This

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Fig. 1. A schematic diagram of an overloaded micro heat pipe with dryout and flooded regions. Also shown are three cross-sectional profiles of the spatial distribution of the liquid and vapor phases, the last depicting the onset of flooding.

uniform temperature will be regarded as the operating temperature of the MHP, at which all the relevant thermo-physical properties of the solid wall and the liquid and vapor phases of the working fluid will be evaluated. In conjunction with the assumption of uniform temperature for the working fluid, axial conduction and convection in the liquid and vapor phases are ignored. At any axial position where evaporation takes place, the heat transfer from the solid to the liquid is all taken up as the latent heat of evaporation; on the other hand, all the heat released by condensing vapor is transferred to the solid through the liquid. In view of the fact that the thermal conductivity of vapors is generally much smaller than that of solids, we assume that no heat transfer occurs between dry solid wall and the adjacent vapor. Finally, we assume that the contact angle with which the liquid–vapor interface touches the solid wall is uniform throughout the MHP. This assumption is, admittedly, an idealization of the actual situation, but is needed to keep our model as simple as possible. As in previous studies [13–15], we shall set the contact angle at 15°. The MHP selected for our study has a cross section in the shape of an equilateral triangle of sides w equal to 1.04 mm. Its total length L is 50.0 mm, consisting of an evaporator section of length Le = 12.7 mm, an adiabatic section of length La = 24.6 mm, and a condenser section of length Lc = 12.7 mm. The solid wall has a thickness of 0.14 mm. In order to elucidate the thermal effects of the solid wall, two types of solid materials of significantly different thermal conductivities are considered in this paper: copper and nickel. For the working fluid, water has been selected exclusively. All the thermo-physical properties of copper, nickel, and water which are needed in our calculations will be retrieved from the monograph of Dunn and Reay [17]. 2. Governing equations and solution method The steady-state, one-dimensional mathematical model for the study of an overloaded MHP in this paper takes into account the

MHP’s solid wall as well as the liquid and vapor phases of its working fluid. The model is developed by applying energy conservations to the solid wall and the liquid phase, momentum and mass conservations to both liquid and vapor phases, and the Young–Laplace capillary equation to the liquid–vapor interface. It is similar to the model developed previously by Hung and Tio [13], but is more general since it allows dryout and flooding. Applying the principle of energy conservation to the solid wall, we obtain the governing equation for axial conduction as 2

ks As

d Ts 2

dx

 hP sl ðT s  T l Þ þ q_ ¼ 0;

ð1Þ

where the first term on the left is associated with the axial conductive flux, the second term accounts for the heat transfer between _ denotes the the solid and the working fluid, and the third term, q, rate of heat transfer per unit axial length between the MHP and its surroundings. In Eq. (1), T s and T l are the solid and liquid temperatures, ks is the thermal conductivity of the solid, As is the cross-sectional area of the solid wall, and Psl is the length of the solid–liquid interface of the cross section at the axial position x, which is measured from the evaporator end of the MHP. As noted earlier, the heat transfer between the solid wall and the vapor phase of the working fluid over the dryout region, 0 < x < Ld , is assumed to be negligible; therefore, the heat transfer coefficient h between the solid wall and the vapor phase is equal to zero. On the other hand, the flooded region at the condenser section is completely occupied by liquid, thus blocking the flow of vapor and acting as an impediment to condensation heat transfer. Since the liquid-tosolid heat transfer is solely provided by the condensation of vapor, no heat transfer takes place between the liquid and solid in the flooded region. This implies that their temperatures are equal and, under the assumption of uniform liquid temperature, that the solid temperature is also uniform over the flooded region. It follows that no axial solid conduction takes place in the flooded region and, con-

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sequently, there is no heat transfer between the solid wall and the surroundings either. Effectively, the condenser length of the MHP has been reduced to Lc ¼ Lc  Lf by the presence of flooding [9]. For the non-flooded wetted region, we follow Hung and Tio [13] and prescribe a value of 2.68 for the Nusselt number, which is defined as

hDH;l ; kl

Nu 

Ce ¼

ð2Þ Cd ¼

where kl is the thermal conductivity of the liquid phase, and DH;l is the cross-sectional hydraulic diameter of the liquid phase, taking into account the solid–liquid interface only. Considering the consequences of dryout and flooding discussed above, we rewrite Eq. (1) as 2

d ðT s  T l Þ  k2 ðT s  T l Þ þ H ¼ 0; d^x2

ð3Þ

where ^ x ¼ x=L is the dimensionless distance from the evaporator end,

8 < 0;

0 6 ^x 6 LLd ;

k ¼ C 2  k  2  : sl l L Nu; 4 ks As 2

Cc ¼

Ld L

ð4Þ

6 ^x 6 1  LLf ;

and

8 _ 2 H ¼ QL ; 0 6 ^x 6 LLe ; > > < e ks As Le Le 6 ^x 6 1  LLc ; H ¼ 0; L > > _ 2 : QL Hc ¼  ks As ðLc L Þ ; 1  LLc 6 ^x 6 1  LLf :

ð5Þ

f

The constant C sl in Eq. (4) is a geometrical parameter which incorporates the effect of contact angle, and is given in Appendix A. Associated with the heat load Q_ , the heat fluxes entering and leaving the MHP are assumed to be uniformly distributed over the entire length of the evaporator and the effective length of the condenser, respectively. Eq. (3), valid over the interval 0 6 ^x 6 1  Lf =L, is a second order differential equation and thus requires two boundary conditions; they are:

  dT s  dT s  ¼ ¼ 0: d^x ^x¼0 d^x ^x¼1Lf =L

ð6Þ

These boundary conditions ensure that the rates of heat energy entering and leaving the MHP are the same and equal to the heat load Q_ . Integrating Eq. (3) piecewise, imposing the requirement of continuity of temperature and heat flux at ^ x ¼ Ld =L, ^ x ¼ Le =L, and ^ x ¼ 1  Lc =L, and then applying the boundary conditions of Eq. (6), we obtain

8 1 > ^2 0 6 ^x 6 LLd ; > >  2 He x þ C d ; >  > kLd =L > H L þC kLe > ekðLd =L^xÞ þ Hk2e ; LLd 6 ^x 6 LLe ; > C e ek^x þ e d kLe > < Le T s  T l ¼ C a1 ek^x þ C a2 ek^x ; 6 ^x 6 1  LLc ; L > >  k^x  Hc > ^ x Þ kð22L =L > f > 1  LLc 6 x^ 6 1  LLf ; þ k2 ; Cc e þ e > > > > : 2C ekð1Lf =LÞ þ Hc ; 1  LLf 6 ^x 6 1; c k2

  k2 C a1 ekð1Lc =LÞ þ C a2 ekð1Lc =LÞ  Hc k2 ekð1Lf =LÞ ðekðLc =LLf =LÞ þ ekðLc =LLf =LÞ Þ

    k2 C a1 ekLe =L þ C a2 ekLe =L  He 1 þ ðkLd =LÞekðLe =LLd =LÞ k2 ðekðLe =L2Ld =LÞ þ ekLe =L Þ h i 4C e k2 ekLd =L þ He 2 þ 2kLd =L þ ðkLd =LÞ2

T s ¼ T l ;

He 2kðLd =LÞekð1Ld =LLf =LÞ þ ekð1Le =LLf =LÞ  ekð12Ld =LþLe =LLf =LÞ 2

2k

þ

C a1 ¼

ðekð1Lf =LÞ



 ekð12Ld =LLf =LÞ Þ

Hc ðekð1Lc =LLf =LÞ  ekð1Lc =LLf =LÞ Þ 2k2 ðekð1Lf =LÞ  ekð12Ld =LLf =LÞ Þ

  2C a2 k2 e2kLd =L þ He 2kðLd =LÞekLd =L þ ekLe =L  ekðLe =L2Ld =LÞ 2k2

ð8Þ

;

;

ð9Þ

ð11Þ

ð12Þ

T s 

1 1  Ld =L  Lf =L

Z

1Lf =L

T s ð^xÞd^x:

ð13Þ

Ld =L

Eq. (13) can be verified by actually calculating the average temperature T s , or by integrating Eq. (3) subject to the boundary conditions given by Eq. (6). From a practical standpoint, Eq. (13) allows us to determine the temperature of the circulating fluid of an MHP and, thus, its operating temperature by simply taking the average temperature of the non-flooded wetted segment of the solid wall. For an MHP of known geometrical parameters and solid wall material, filled with a given type of working fluid, and operating under a heat load Q_ at a given temperature, the constants C a2 , C a1 , C c , C e , and C d can be successively calculated using Eqs. (8)– (12), thus yielding the axial temperature distribution of the solid wall, provided the lengths of the dryout and flooded regions, Ld and Lf , are known. However, these quantities are not known in advance but constitute an integral part of the solution to the governing system of equations. In this paper, we use a trial-and-error method, starting with both Ld and Lf , to obtain the solution. Irrespective of the method used, the solution must satisfy the constraint

b ¼ e þ ð1  eÞ M

"Z

1Lf =L

sð^xÞd^x þ

Ld =L

# Lf ; L

b  M ; M ALql

ð14Þ

since the mass M of the working fluid inside the MHP, or its dimenb for a given operating temperature, is fixed. sionless counterpart M In Eq. (14), ql is the density of the liquid phase, e is the vapor-toliquid density ratio, and s ¼ Al =A is the volume fraction occupied by the liquid phase, A and Al being the cross-sectional area of the flow channel and the area of the portion occupied by the liquid, respectively. At any given axial position ^ x, the liquid volume fraction s can be related to the radius of curvature r of the liquid–vapor interface and, through the Young–Laplace capillary equation, to the liquid and vapor pressures, pl and pv , as well:

pv  pl ¼ 

C a2 ¼

:

2k2

;

It should be noted that the solid wall temperature in the flooded region, 1  Lf =L 6 ^x 6 1, is a constant independent of the axial distance. This constant is determined from Eq. (7) by setting ^x ¼ 1  Lf =L, the end point of the non-flooded region of the condenser section. Since no overall heat transfer takes place between the liquid phase and the non-flooded wetted region of the solid wall, the average temperature of this segment of the solid wall, T s , is equal to that of the liquid:

ð7Þ where

ð10Þ

;

r r

¼

~ rx A1=2 s1=2

;

ð15Þ

~ is the angular where r is the coefficient of surface tension and x parameter relating the volume fraction s and the radius of curvature r (see Appendix A). In Eq. (15), we have considered only the curvature of the liquid–vapor interface in the cross-sectional plane. This is justified by the fact that the curvature in the longitudinal plane is sub-dominant compared to that in the cross-sectional plane, as the cross-sectional length scale of the MHP selected for our study is much smaller than the axial length scale. The liquid and vapor pres-

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sures can be determined by applying momentum conservation to the liquid and vapor phases, respectively. The resulting equations are given by

ql

d  2 dp Al ul ¼ Al l þ ssl Psl þ slv P lv ; dx dx

ð16Þ

qv

 d  dp Av u2v ¼ Av v  ssv Psv  slv Plv; dx dx

ð17Þ

where qi and ui denote the density and axial velocity of phase i, respectively, Ai the area of the cross-sectional region occupied by phase i, and P ij the length of the phase i-phase j interface of the cross section. The interfacial stresses sij are given by (see Hung and Tio [13])

sij

1 ¼ qj u2j f ; 2

14:7 f ¼ ; Rej

ð i ¼ s; l; j ¼ l; v; i–j Þ;

ð18Þ

where f is the friction factor. The Reynolds number is defined as

qj uj DH;j Rej  ; lj

_ m Q L=hfg Le

;

ð21Þ

ð22Þ

where

k k s L e As : Q_ L2

ð23Þ

Substituting Eq. (7) into Eq. (22) and integrating the resulting equab d =LÞ ¼ 0, we obtain tion subject to the boundary condition mðL 8    He   g k^x kð^x2Ld =L > Þ þ kL L^x  Ld ekð^xLd =L Þ LLd 6 ^x 6 LLe ; > < k C e e  e  Le b ¼ gk C a2 ek^x  C a1 ek^x þ K 1 m 6 ^x 6 1  LLc ; L > > : g C ek^x  ekð^x2þ2Lf =L Þ þ Hc ^x þ K  1  LLc 6 ^x 6 1  LLf ; c 2 k k

ð24Þ where

  K 1 ¼ C e ekLe =L  ekðLe =L2Ld =LÞ þ C a1 ekLe =L  C a2 ekLe =L  He  þ Le  Ld ekðLe =LLd =LÞ ; kL     K 2 ¼ C e ekLe =L  ekðLe =L2Ld =LÞ þ C c ekð1þLc =L2Lf =LÞ  ekð1Lc =LÞ  kLe =L     ekð1Le =LÞ þ C a2 ekð1Le =LÞ  ekLe =L þ C a1 e   He Le  Ld ekðLe =LLd =LÞ  Hc ðL  Lc Þ : þ kL

Ca 

"

L A1=2

We 

# ðQ_ L=ql hfg Le AÞ ; r=ll

ð28Þ

ql ðQ_ L=ql hfg Le AÞ A1=2 : r

ð25Þ

ð26Þ

ð29Þ

The two functions FðsÞ and GðsÞ are given by

FðsÞ ¼

C 2sl ; 8s2 h

GðsÞ ¼

ð30Þ

NwA1=2 þ ðC lv  C sl Þs1=2

i

NwA1=2 þ C lv s1=2  C sl s1=2

8ð1  sÞ3

 ; ð31Þ

where N is the number of corners of the cross section (N ¼ 3 for a triangle) and w is the length of the cross section’s sides. To integrate Eq. (27) numerically, the fourth order Runge–Kutta method with a step size of 0.001, which was tested to be adequate, has been selected. If dryout is involved, the boundary condition of sðLd =LÞ ¼ scl , which corresponds to the onset of dryout at the evaporator section, must be satisfied. Following Hung and Tio [13], we have selected the value of 0.0001 for scl . In cases where flooding is involved, the boundary condition of sð1  Lf =LÞ ¼ sfl must be satisfied. This liquid volume fraction, which corresponds to the onset of flooding, is given by

sfl ¼

2

g¼

where c is the vapor-to-liquid ratio of kinematic viscosity. The capillary number Ca and Weber number We are defined, respectively, as

ð20Þ

hfg being the latent heat of evaporation of the working fluid, changes along the axis of the MHP. Under the assumption of negligible axial conduction and convection in the liquid phase, the heat transfer between the solid wall and the liquid is equal to the latent heat taken up by evaporation or released by condensation. Thus,

^ dm ¼ ðT s  T l Þg; d^x

ð27Þ

2

_ denotes the mass flow rate of either phase. Eq. (20) allows us where m to express the axial velocities ul and uv in terms of the mass flow rate _ effectively reducing the number of unknown variables by one. m, _ Owing to evaporation and condensation, the mass flow rate m, or its dimensionless counterpart

b  m



 2  2 ^ ^ ds 2s3=2 1 d 1 d m m We  ¼ ~ ^ ^ ^ dx s dx s x eð1  sÞ dx 1  s

 þ Ca½f Rel FðsÞ þ cf Rev GðsÞ ;

ð19Þ

where DH and l denote hydraulic diameter and dynamic viscosity, respectively. At any given axial position ^x, the mass flow rate of the liquid must be equal to that of the vapor in the opposite direction, since, under steady-state condition, there can be no mass accumulation anywhere inside the MHP; that is,

_ ql ul Al ¼ qv uv Av ¼ m;

Since the flooded region is completely occupied by stagnant liquid, _ must be zero at ^x ¼ 1  Lf =L. This condition is the mass flow rate m ^ d =LÞ ¼ 0, automatically satisfied with the boundary condition of mðL as is clearly seen from Eqs. (13) and (22). _ and, through Eq. (20), the axial velocities ul Having obtained m and uv as well, we can now combine Eqs. (15)–(17) to derive the equation for the liquid volume fraction s,

~ 2 tan / x 4cos2 ð/ þ hÞ

;

ð32Þ

where h is the contact angle of the liquid–vapor interface with the solid wall, and / is the half corner angle at the vertices of the cross section (/ equals 30o for an equilateral triangle). With the liquid volume fraction thus calculated, we now return to Eq. (14) to complete the iteration cycle and determine whether this constraint is satisfied. For simplicity and accuracy, Simpson’s Rule has been selected for the evaluation of the integral in Eq. (14). 3. Results and discussion Before we examine the performance of an overloaded MHP, the validity of the model employed in the investigation must be confirmed. Since the heat transport capacity of an MHP is one of its most important attributes, we shall, following Hung and Tio [13], compare first the heat transport capacity predicted by the MHP of this study with readily available experimental results. Thus, we plot in Fig. 2 the predicted heat transport capacity as a function of the operating temperature of the MHP, the contact angle h being a parameter, and compare it with the experimental data of Babin et al. [18], who used a trapezoidal copper MHP of the same

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Fig. 2. Heat transport capacity Q_ cap of a copper–water MHP as a function of operating temperature T op for different values of contact angle. Experimental data of Babin et al. [18] are included for comparison.

Fig. 3. Heat transport capacity Q_ cap of a copper–water MHP as a function of contact angle h, the operating temperature T op being a parameter.

cross-sectional area as the MHP of the present study and filled with 3.2 mg of water. We observe from Fig. 2 that the predicted heat transport capacity compares favorably, in trend and order of magnitude, with the experimental results. However, as pointed out previously [13], it would be meaningless to go beyond the orderof-magnitude comparison, since the heat transport capacity of and MHP is very sensitive to its cross-sectional shape [12]. Another important attribute of an MHP is how it responses in terms of its total axial temperature drop when it is operating under overloaded conditions. A comparison between the predicted temperature drops and experimental results will therefore be presented later, when we discuss the temperature drops of overloaded MHPs, to validate the model of the present study. From Fig. 2, we observe that the contact angle between the liquid and the solid wall has significant effects on the heat transport capacity of an MHP. Moreover, the close agreement between experimental results and those obtained theoretically for h ¼ 45 may lead one to conclude that the contact angle should be set at 45° in all subsequent calculations. However, we must refrain from doing this, for two main reasons. The first is that, as noted above, the heat transport capacity of an MHP is very sensitive to its cross-sectional shape, and that we are comparing MHPs of different cross-sectional geometries. Secondly, the wetting of a solid surface by a liquid is a complex phenomenon and the associated contact angle may be affected by numerous factors such as evaporation of the liquid and the solid’s surface properties, as noted previously by Hung and Tio [15]. It is thus very likely that the contact angle changes along the axis of an MHP, but how it varies remains an open question. In this paper, the contact angle is therefore assumed uniform throughout the flow channel of the MHP. To see how the heat transport capacity depends on this uniform contact angle, we plot it as a function of the latter in Fig. 3. For a given operating temperature, we observe that as the contact angle is increased from zero degree, so does the heat transport capacity, until a threshold value of the contact angle is reached; beyond this threshold value, the heat transport capacity decreases as the contact angle is increased. As pointed out by Tio et al. [1], this threshold contact angle results from the balance between two opposite trends. On the one hand, an increasing contact angle has a positive effect on the heat transport capacity through the optimal charge level, which increases in tandem with the contact angle and thus provides capillarity with more working fluid to function effectively. On the other hand, as the contact angle increases, the ability of capillarity to circulate the working fluid is reduced. Apparently, the positive effects of the former dominate the adverse effects of the latter for small contact angles. However, for contact angles larger than the threshold value, which is about 15°, the ability of cap-

illarity to circulate the working fluid is so much reduced that an increase in the charge level cannot compensate for the reduction in the strength of capillarity, thus culminating in a decreasing trend of the heat transport capacity. In the previous discussion on contact angle, we have excluded the possible thin film extending from the intrinsic meniscus region of the liquid phase. Typically, an extended meniscus consists of the intrinsic meniscus, an evaporating thin film and a non-evaporating adsorbed layer (see, for example, references [19–21]). As the evaporating film is very thin, the heat flux at the solid surface in contact with it can be large and may, therefore, have a significant contribution to the total solid–liquid heat transfer coefficient. However, this contribution cannot be so large that it qualitatively alters the results presented in this paper. The reason is that the solid–liquid interface at the evaporating thin film is only a small fraction of the total area of the interface. Since the main objective of this paper is to elucidate the effects of solid conductivity on the overall performance of an MHP, we shall not include the evaporating thin film in our analysis, to avoid the additional complexity resulting from its inclusion. Moreover, inclusion of the evaporating thin film may still leave the question on the actual value of the contact angle unresolved. As stated Section 1, the contact angle will be fixed at 15°. For an MHP to operate effectively at a given operating temperature, neither dryout nor flooding is allowed to occur anywhere inside the MHP. As pointed out by Tio et al. [1], this requires a comb Q_ Þ, which is confined bination of charge level and heat load, ð M; within Zone I of the operation-zone map depicted in Fig. 4, where, for the purpose of illustration, the operating temperature has been set to 60 °C. This zone is bounded by the curves cl and fl, which correspond to the first occurrence of dryout at the evaporator end (i.e., capillary limit) and the onset of flooding at the condenser end, respectively. The intersection of these curves, which corresponds to the simultaneous onsets of dryout and flooding, then yields the heat transport capacity Q_ cap and the associated optimal charge b opt . For a given operating temperature of an MHP, its heat level M transport capacity is the maximum allowable heat load that can be applied if it is to operate without dryout and flooding. Furthermore, the amount of the working fluid required by the MHP to achieve its heat transport capacity is unique and is given by the optimal charge level. For a heat load greater than Q_ cap , the MHP cannot operate effectively, irrespective of its charge level, since dryout, flooding, or the combination of both will occur. However, the MHP will operate effectively if the heat load Q_ is less than its b is confined heat transport capacity, provided its charge level M within Zone I of the operation-zone map for the given operating temperature (see Fig. 4), and is bounded by the intersections

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Fig. 4. An operation-zone map of a copper MHP operating at 60 °C. Depicted in the map are the various possible operation zones associated with the charge level and applied heat load of the MHP.

between the constant Q_ line and the curves cl and fl. If the actual b opt , the MHP is considered undercharged charge level is less than M for that operating temperature, and its performance is limited by the occurrence of dryout at the evaporator end; otherwise, the MHP is considered overcharged, and its performance is limited by the onset of flooding at the condenser end. The primary concern of this paper is the thermal performance of an overloaded MHP. For the purpose of illustration, we consider again an operating temperature equal to 60 °C. Then, the charge b Q_ Þ, would be located outside level and heat load of the MHP, ð M; Zone I of Fig. 4. We begin with Zone II, which is identified exclusively with an undercharged MHP. For any charge level within this zone, the evaporator section is plagued by the existence of a dryout region, the length of which increases with the heat load Q_ . For a charge level greater than a threshold value, a continual increase in Q_ will eventually result in the onset of flooding at the condenser b is smaller than the end, as indicated by the curve fl. However, if M threshold charge level, no flooding will occur; instead, the heat load is limited by the curve da, which indicates that the dryout region, having occupied the entire evaporator section, is about to intrude into the adiabatic section of the MHP. For an MHP to reside within Zone III, it must be overcharged. In this zone, the overloaded MHP is plagued by the existence of flooding in the condenser section. For a given charge level, the extent of flooding becomes greater as the heat load Q_ is increased. However, dryout will occur in the evaporator section before the entire condenser is flooded, if the MHP is not too severely overcharged. Otherwise, dryout can only occur after the whole condenser is flooded, as indicated by the line fa. From Fig. 4, it is clearly seen that an MHP residing in Zone IV is plagued by the simultaneous occurrence of dryout and flooding. However, both dryout and flooding are still confined within the evaporator section and the condenser section, respectively. Beyond the curves da and fa, i.e., in Zone V and Zone VI, dryout and flooding have, respectively, intruded into the adiabatic section of the MHP. This is a critical situation, because an MHP may stop functioning when its adiabatic section is entirely dried-out or flooded. From Fig. 4, we observe that the first critical zone, Zone V, can be reached by undercharged as well as overcharged MHPs, while the second, Zone VI, is reachable by overcharged MHPs only. Moreover, the charge level required by Zone VI is relatively high (above 30%). Consequently, failure of MHPs is usually attributed to dryout. To a large extent, the occurrence of dryout inside an overloaded MHP results from the failure of capillarity to overcome viscous dissipation and provide the circulation rate of the working fluid required by the heat load applied to the MHP. For capillarity to

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keep up with the increased circulation rate, viscous dissipation must be reduced, by a decrease in the effective length of the MHP. This can be accomplished in two ways: the occurrence of dryout at the evaporator section and the existence of flooded region in the condenser section. As stated earlier, the main concern of this paper is to investigate the effects of the thermal conductivity of the solid wall of an overloaded MHP. In accordance with this, we shall first examine the relation between the dryout/flooded length and the thermal conductivity of the solid wall. At first glance, it may seem irrelevant to investigate the roles of solid thermal conductivity, since the parameter which controls the working fluid’s ability to circulate, namely r=ll [1], is solely the property of the working fluid. However, as shown previously [13], the heat transport capacity of an MHP is partially determined by the solid thermal conductivity, through its ability to alter the axial solid temperature distribution and, hence, the rate of heat transport by phase change. It is, therefore, worthwhile to see if solid conductivity persists in playing a role in overloaded MHPs. For this purpose, we consider two MHPs made of two solid materials of significantly different thermal conductivities: copper and nickel. In Fig. 5, we show how these MHPs, each optimally charged for and operating at 60 °C, response, in terms of the lengths of dryout and flooded regions, to the applied heat load. Since each MHP is optimally charged, dryout and flooding set in simultaneously when the applied heat load reaches the critical value given by the heat transport capacity of the MHP. Beyond this critical load, which is a function of the solid thermal conductivity [13], the applied heat load forces the MHP to operate in Zone IV b map (see Fig. 4). We observe that both the dryout of the Q_  M and flooded lengths increase monotonically with the applied heat load, the former increasing with a greater rate than the latter. In Fig. 5, the increase in the applied heat load is terminated when the dryout region has occupied the entire evaporator section of the MHP. If the two MHPs of Fig. 5 are operating at 40 °C, which is below their design operating temperature, they become overcharged [1]. In this case, a continual increase in the applied heat load to an MHP will eventually result in the appearance of flooding in the condenser section when the applied heat load reaches a critical value, followed later by the appearance of dryout in the evaporator if the heat load is increased sufficiently further. The scenarios of the appearance of the flooded and dryout regions and their subsequent monotonic expansion with the applied heat load depicted in Fig. 6, in which the actual charge levels of the MHPs are fixed and greater than their respective optimal charge levels for 40 °C, correspond to b map (see traversing Zone I, Zone III, and Zone IV of the Q_  M Fig. 4). As in Fig. 5, the curves depicting the dryout and flooded lengths of each MHP are terminated when the dryout region has

Fig. 5. Dryout and flooded lengths vs. applied heat load. Both MHPs are optimally charged for and operating at 60 °C.

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observe that this order of performance superiority persists in overloaded MHPs. In each of the three cases, a copper MHP, compared to that of nickel, can sustain a higher heat load (or a higher overload, which is the difference between the actual and the critical heat loads) before the entire evaporator section dries out. In view of the fact that copper has a higher thermal conductivity than nickel, we conclude that a solid material of higher conductivity yields a better performance for an MHP, in terms of the heat transport capacity and the allowable overload before the occurrence of complete dryout of the evaporator section, and vice versa. To gain further insight into an overloaded MHP, we consider its effective length L ,

L ¼ L  ðLd þ Lf Þ; Fig. 6. Dryout and flooded lengths vs. applied heat load. Both MHPs are optimally charged for 60 °C but operating at 40 °C.

occupied the entire evaporator section. For each of the two MHPs in Fig. 6, we observe that the critical heat load is less than its counterpart in Fig. 5. This is a typical scenario for an MHP which is operating below the temperature for which it is optimally charged, as pointed out previously by Tio et al. [1]. If the two MHPs of Fig. 5 are operating at a temperature above their design operating temperature, say 80 °C, they become undercharged [1]. As depicted in Fig. 7, a continual increase in the applied heat load to an MHP will eventually result in the occurrence of dryout region in its evaporator section when the applied heat load reaches a critical value, with subsequent appearance of flooding in the condenser if the heat load is increased sufficiently further. For each of the two MHPs, we observe that the critical heat load is greater than its counterpart in Fig. 5, which is consistent with the observation by Tio et al. [1] for an MHP which is operating above its design operating temperature. Moreover, we also observe that, as in the previous two cases, the extent of the dryout and flooded regions increases monotonically with the applied heat load. Finally, comparison between Figs. 4 and 7 reveals that the two MHPs depicted in the latter are not severely undercharged, as the scenarios depicted in Fig. 7 correspond to traversing, with b Zone I, Zone II, and Zone IV, and stopping at curve da a fixed M, b map. of the Q_  M While the flooded lengths of Figs. 5–7 increase with the applied heat load, their rates of expansion are much smaller than those of the dryout lengths. Consequently, less than 20% of the condenser section of an MHP is flooded when its entire evaporator has already dried out. In view of this, we take note again that failure of MHPs is usually attributable to dryout. It was shown previously [13] that a copper MHP yields a higher heat transport capacity than a nickel MHP. From Figs. 5–7, we

Fig. 7. Dryout and flooded lengths vs. applied heat load. Both MHPs are optimally charged for 60 °C but operating at 80 °C.

ð33Þ

instead of the dryout and flooded lengths separately, as we have done previously. Thus, we plot the heat load Q_ that can be sustained as a function of the effective length L in Figs. 8–10, where we focus only on the overloaded regimes of, respectively, Figs. 5–7. We note that for a given effective length, the two MHPs of Fig. 8 may be considered as operating with their respective heat transport capacities associated with the given effective length, with onset of dryout at one end of the non-flooded wetted segment of the solid wall and onset of flooding at the other end. In Fig. 9, the two MHPs are overcharged for the operating temperature of 40 °C. An initial decrease in the effective length L from the actual physical length L, with the simultaneous increase in the heat load Q_ from the critical value, corresponds to the appearance and initial expansion of the flooded region. The heat load sustainable by the MHP increases rather rapidly with the decrease in the effective length until the onset of dryout. Beyond this point, which is marked by a cusp, the sustainable heat load increases less rapidly, and for a given effective length L , the MHP may be considered as operating with its heat transport capacity associated with this L . In Fig. 10, the two MHPs are undercharged for the operating temperature of 80 °C. With flooding commencing only after dryout has spread from the evaporator end of the MHP, the Q_ -vs.-L graphs do not exhibit the cusped nature of those in Fig. 9 and, in this sense, are qualitatively similar to those in Fig. 8. From Figs. 8–10, we observe that for a given effective length, a copper MHP can sustain a greater heat load than a nickel MHP. In view of the fact that copper has a greater thermal conductivity than nickel, this observation allows us to extend to overloaded regimes the observation of Hung and Tio [13]: Not only does an MHP made of a solid of greater thermal conductivity yield a greater heat transport capacity, it can, for a given effective length, also sustain a greater heat load in the overloaded regime. Moreover, the observation of Hung and Tio [13], that the heat transport capac-

Fig. 8. Applied heat load vs. effective length. Both MHPs, optimally charged for and operating at 60 °C, are overloaded. The effective length varies from the point of simultaneous onsets of dryout and flooding to the point of the dryout region starting to extend into the adiabatic section.

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For an MHP, we may write

Q_ ¼ Q_ p þ Q_ c ;

ð34Þ

where Q_ p is the portion of the applied heat load transported by phase change and Q_ c is the remaining portion transported by axial conduction in the solid wall. Obviously, the heat transport by phase change is equal to the heat transfer from the solid to the liquid over the axial segment of the MHP where evaporation takes place, and is also equal to the liquid-to-solid heat transfer over the remaining non-flooded segment. Thus, we can calculate the heat transport by phase change using either of the two formulas,

k 2 k s As Q_ p ¼ L Fig. 9. Applied heat load vs. effective length. Both MHPs, optimally charged for 60 °C but operating at 40 °C, are overloaded. The effective length varies from the point of onset of flooding to the point of the dryout region starting to extend into the adiabatic section.

ity of an MHP is greater if it is made of a solid of higher thermal conductivity, is also valid in the overloaded regimes where both dryout and flooding take place, provided that we consider the effective length (instead of the actual physical length of the MHP) and regard the sustainable heat load as the heat transport capacity. Finally, we note that the trend of sustainable heat load Q_ decreasing with increasing effective length L displayed by Figs. 8–10 is consistent with the observations of Peterson and Ma [22], Sugumar [23], and Hung and Seng [12] that the heat transport capacity of an MHP decreases as its total length is increased.  From Figs. 8–10, we also observe that the derivative dQ_ =dL is  a positive quantity which becomes larger as L is reduced further. Moreover, for a given effective length L , this derivative attains a larger value for copper than nickel. As pointed out earlier, the occurrence of dryout and/or flooding inside an overloaded MHP reduces its effective length, thus allowing capillarity to keep up with the required circulation rate of its working fluid to transport heat from the evaporator to the condenser by phase change. More over, that a larger dQ_ =dL is associated with a shorter effective length (and a greater heat load) means that a given reduction in the effective length of the MHP will allow a greater increase in the heat load at a higher overload. Therefore, the increase in  dQ_ =dL implies that the heat transport by conduction in the solid wall may have increased with the applied heat load and possibly taken up an increasing portion of the latter. In light of this, we shall now proceed to investigate the heat transport by conduction in the solid wall.

^x¼L0 =L

ðT s  T l Þd^x ¼

^x¼Ld =L

k2 ks As L

Z

^x¼1Lf =L

ðT l  T s Þd^x;

ð35Þ

^x¼L0 =L

x ¼ L0 being the axial position where the solid temperature is equal to that of the liquid. Once the heat transport by phase change is calculated, we can use Eq. (34) to determine the axial solid conduction. In Figs. 11 and 12, we plot the fractions np and nc ,

np 

Fig. 10. Applied heat load vs. effective length. Both MHPs, optimally charged for 60 °C but operating at 80 °C, are overloaded. The effective length varies from the point of onset of dryout to the point of the dryout region starting to extend into the adiabatic section.

Z

Q_ p ; Q_

nc 

Q_ c ; Q_

ð36Þ

as functions of the heat load, Q_ , applied to, respectively, copper and nickel MHPs optimally charged for and operating at 60 °C. In both cases, np dominates nc , and both are practically constant if the MHP is not overloaded. When the applied heat load exceeds a critical value, which in these cases is the heat transport capacity Q_ cap , with the subsequent appearance of dryout and flooding (see Fig. 5), np decreases while nc increases quite drastically with the heat load. However, it should be noted that nc is rather small, accounting for less than 2.5% and 0.03% of the heat load applied to, respectively, the copper and nickel MHPs of Figs. 11 and 12. While np decreases when the heat load exceeds the heat transport capacity Q_ cap , the rate of heat transport by phase change actually increases, as can be seen from Figs. 13 and 14, which depict the heat transport by phase change, Q_ p , and axial conduction, Q_ c , corresponding to the fractions np and nc of Figs. 11 and 12, respectively. For both copper and nickel MHPs, Q_ p dominates Q_ c and increases linearly with the applied heat load. While Q_ c also increases linearly with Q_ if the MHP is not overloaded, its rate of increase, i.e., dQ_ c =dQ_ , becomes larger and larger once the heat load exceeds the heat  transport capacity Q_ cap . It follows that the derivative dQ_ c =dL also becomes larger as the overload is increased, since   dQ_ c =dL ¼ ðdQ_ c =dQ_ Þ  ðdQ_ =dL Þ and, as discussed previously, so  _ does dQ =dL . Similarly, we can also conclude that the derivative  dQ_ p =dL becomes larger for a larger overload. To summarize: For

Fig. 11. Fractional heat transport by axial solid wall conduction, nc , and fractional heat transport by phase change, np , vs. applied heat load. The MHP is made of copper, optimally charged for and operating at 60 °C. Note that both nc and np are given as percentages.

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Fig. 12. Fractional heat transport by axial solid wall conduction, nc , and fractional heat transport by phase change, np , vs. applied heat load. The MHP is made of nickel, optimally charged for and operating at 60 °C. Note that both nc and np are given as percentages.

Fig. 13. Heat transport by axial solid wall conduction, Q_ c , and heat transport by phase change, Q_ p , vs. applied heat load. The MHP is made of copper, optimally charged for and operating at 60 °C.

Although axial solid conduction is rather insignificant compared to heat transport by phase change, as vividly illustrated by the fractions nc and np in Figs. 11 and 12, its effect on the latter may be significant. For, the temperature distribution associated with the axial solid conduction affects the solid-to-liquid heat transfer and, consequently, the heat transport by phase change, too [13]. In Fig. 15, axial solid temperature distributions are depicted for a copper MHP which is optimally charged for and operating at 60 °C under three different heat loads. They are: 0.444, 0.555, and 0.666 W, the second being the heat transport capacity of the MHP at 60 °C. At a given axial position apart from ^ x ¼ 0, ^ x ¼ 1, and the flooded region, we note that the temperature gradient dT s =d^ x increases as the heat load is increased. This implies that axial solid conduction also increases with the applied heat load, consistent with the trend of Q_ c with respect to Q_ in Fig. 13, from which we have previously observed that the heat transport by phase change, Q_ p , also increases with the applied heat load. The fact that both Q_ p and Q_ c have the same trend with respect to the applied heat load can be clearly understood by comparing the axial solid temperature profiles for the heat loads of 0.444 and 0.555 W. First, we note that larger temperature gradients dT s =d^ x do not only imply a greater Q_ c , they also result in a larger total axial temperature drop. In view of Eq. (35), it is then a simple matter to see that the heat transport by phase change associated with the heat load of 0.555 W is greater than that associated with 0.444 W, since the region bounded by the vertical axes, the isotherm T = 60 °C (the liquid temperature), and the T s curve takes up a larger area for the higher heat load. When the applied heat load exceeds the critical load, which is also the heat transport capacity Q_ cap in the case of Fig. 15, the solid temperature rises drastically from the operating temperature over the entire dryout region as well as a portion of the wetted evaporator section. The drastic rise of the solid temperature and the accompanying increase in its temperature gradient dT s =d^ x then result in the large increase in the heat transport by solid conduction, as illustrated by the trend of nc and Q_ c in the post-Q_ cap regime in Figs. 11 and 13. Although a substantial portion of the evaporator section is no longer directly involved in the solid–liquid heat transfer process, the loss of the solid–liquid heat transfer over the dryout region, 0 6 ^ x 6 0:1759, is apparently over-compensated by the higher solid temperature over the wetted portion of the evaporator section as well as the adjacent portion of the adiabatic section, culminating in an increase in the heat transport by phase change. In Fig. 16, we show the axial solid temperature distributions corresponding to the heat loads of 0.427, 0.534, and 0.641 W applied to a nickel MHP which is optimally charged for and operating at

Fig. 14. Heat transport by axial solid wall conduction, Q_ c , and heat transport by phase change, Q_ p , vs. applied heat load. The MHP is made of nickel, optimally charged for and operating at 60 °C.

a given overloaded MHP, a reduction in its effective length by a fixed amount allows its heat load Q_ , together with the two heat transport components Q_ p and Q_ c , to increase; moreover, the same amount of reduction in L at a greater overload will allow a greater increase in Q_ , Q_ p and Q_ c . Comparing the copper and nickel MHPs in Figs. 8, 13 and 14, we can also conclude that this increase in Q_ , Q_ p and Q_ c is greater for an MHP made of a solid material of greater thermal conductivity.

Fig. 15. Axial temperature profiles of the solid wall of a copper MHP optimally charged for and operating at 60 °C under three different heat loads. For the heat load of Q_ ¼ = 0.666 W, the MHP is overloaded, with dryout and flooded regions over 06^ x 6 0:1759 and 0:9788 6 ^ x 6 1, respectively.

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60 °C. Although quantitatively different, the two sets of temperature profiles of Figs. 15 and 16 are qualitatively similar and, therefore, all the previous qualitative observations for the former are also valid for the latter. In both figures, the dryout region is accompanied by high solid temperature and its large temperature gradient dT s =d^ x. Obviously, this is due to the absence of heat transfer by phase change, so that all the heat input to the MHP over its dryout region must be axially conducted through the small cross-sectional area of its solid wall. From Figs. 15 and 16, we also note that the degree of temperature rise over the dryout region is a function of the thermal conductivity of the solid wall, a larger conductivity yielding a lower temperature rise, and vice versa. That an overloaded MHP with dryout is accompanied by abrupt temperature rise over its dryout region may provide a practical means to detect the occurrence of dryout. In Figs. 17 and 18, we plot the total axial temperature drop DT as a function of the applied heat load Q_ . Here, the temperature drop is defined as

DT  T 0  T 1 ;

ð37Þ

where T 0 and T 1 are the solid temperatures at the evaporator and condenser ends, respectively. In Figs. 17 and 18, the temperature drop increases gradually with low heat loads. However, once the heat load reaches and then exceeds the critical value, which is the heat transport capacity in the two cases considered here, DT increases sharply with a steep gradient, corresponding to the simultaneous onsets of dryout and flooding and the subsequent expansion of the dryout and flooded regions (see Fig. 5). Although dryout and flooding take place concurrently, we can draw a conjecture from the axial temperature profiles in Figs. 15 and 16 that the steep gradient of DT is the result of dryout only. In fact, similar DTvs.-Q_ plots for the MHPs of Fig. 6 do not exhibit a sharp increase in DT until the onset of dryout, although the MHPs may have been overloaded. Therefore, observation of DT cannot serve as a tool to detect flooding. However, this does not pose a serious problem, since, as we noted previously, failure of MHPs is usually attributable to dryout. From Figs. 17 and 18, we observe that for a given heat load Q_ , the total axial temperature drop in a copper MHP is smaller than that in a nickel MHP. We note that this observation is also true for the cases of Figs. 6 and 7. We thus conclude that an MHP made of a solid of higher conductivity yields a better performance, in terms of the total axial temperature drop DT. The trend of DT increasing with the applied heat load depicted in Figs. 17 and 18 has also been observed experimentally by Peterson and Ma [3]. They use a silver trapezoidal MHP of 0.19 mm wall thickness, of the same cross-sectional area as the MHP of the pres-

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ent study, and filled with 32 mg of water. Their results and the temperature drops predicted by the present model are, for the purpose of comparison, juxtaposed in Fig. 19. The predicted values are obtained using a silver-water triangular MHP, of 0.19 mm (instead of the usual 0.14 mm) wall thickness, which is optimally charged for and operating at 50 °C. We observe from Fig. 19 that at the operating temperature of 50 °C, the theoretical predictions compare very favorably with the experimental results, in both overloaded and non-overloaded regimes. This excellent agreement between theoretical and experimental results thus further validates the mathematical model of the present study. In Figs. 15 and 16, we see how the axial temperature profile of the solid wall of an MHP responses to the applied heat load, thus changing the heat transport by phase change and by conduction in the solid wall. In order to see how the thermal conductivity of the solid wall affects these outcomes, we consider in Fig. 20 the axial temperature profiles of the copper and nickel MHPs of Fig. 10 which result from the heat loads of 1.2 and 1.1 W, respectively. Here, both MHPS are overloaded and operating at 80 °C with an effective length L equal to 0.0377 m. We observe from the temperature distribution of the nickel MHP that a substantial portion of its adiabatic section, 0:254 6 ^ x 6 0:746, participates in the solid–liquid heat transfer, but a large portion still remains dormant, approximately at the operating temperature of 80 °C. On the other hand, the more gradual axial temperature variation of the copper MHP practically leaves no dormant region in its adiabatic section, so that the whole effective length L is involved in the solid–liquid heat transfer. Moreover, it is clear from Fig. 20 that the area above the 80 °C isotherm and bounded by the temperature-distribution curve is larger for the copper MHP, yielding for

Fig. 17. Total axial temperature drop vs. applied heat load. The MHP is made of copper, optimally charged for and operating at 60 °C.

Fig. 16. Axial temperature profiles of the solid wall of a nickel MHP optimally charged for and operating at 60 °C under three different heat loads. For the heat load of Q_ = 0.641 W, the MHP is overloaded, with dryout and flooded regions over 06^ x 6 0:2064 and 0:9775 6 ^ x 6 1, respectively.

Fig. 18. Total axial temperature drop vs. applied heat load. The MHP is made of nickel, optimally charged for and operating at 60 °C.

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Fig. 19. Total axial temperature drop as a function of applied heat load of a silver– water MHP optimally charged for and operating at 50 °C. Experimental data of Peterson and Ma [3] are included for comparison.

thermal conductivity out-performs one with lower solid thermal conductivity. Apart from these performance indicators, this paper also examines how the heat transport by phase change and the heat transport by conduction in the solid wall vary with the applied heat load. In both copper and nickel MHPs, the trends are qualitatively the same. Finally, comparison between the axial temperature profile of overloaded copper and nickel MHPs, operating at the same operating temperature and with the same effective length, leads to the conclusion that a solid wall of higher thermal conductivity provides an axial solid temperature distribution which is more amicable to phase-change heat transport. In this paper, only one type of working fluid has been selected, i.e., water. However, as shown by Sugumar and Tio [24] as well as Chang and Hung [25], the thermo-physical properties of the working fluid of and MHP play an important role in its heat transport capacity. It would, therefore, be interesting to see if the importance of these properties persists in the case of overloaded MHPs. Conflict of interest None declared. Acknowledgments The first author, K.-K. Tio, would like to thank the Ministry of Education, Malaysia, for the support provided under the research grant FRGS/1/2013/TK01/MMU/02/03. The second author, Y.M. Hung, would like to thank the Ministry of Science, Technology and Innovation, Malaysia, for the support provided under the research grant ScienceFund 04-02-10-SF0113. Appendix A

Fig. 20. Axial temperature profile of the solid wall of copper and nickel MHPs optimally charged for 60 °C but operating at 80 °C. Both MHPs are overloaded and operating with an effective length L equal to 0.0377 m. For clarity, the dried out and flooded segments of the solid wall are excluded from the profiles.

this MHP a larger heat transport by phase change [see Eq. (35)]. We note that our observation here that a greater solid thermal conductivity renders an axial solid temperature distribution which is more conducive to phase-change heat transport is similar to that of Hung and Tio [13], who consider the heat transport capacity of MHPs made of copper, nickel, and monel. 4. Conclusion Based on the principles of mass, momentum, and energy conservations as well as the Young–Laplace capillary equation, a one-dimensional steady-state model of triangular MHPs has been developed to investigate their performance under overloaded conditions, with particular attention on the effects of the thermal conductivity of the solid wall. To this end, copper and nickel, which have significantly different thermal conductivities, have been selected for the wall material. A map delineating the various possible operation zones of a copper MHP was constructed, partly to serve as a guide for subsequent stages of investigation. To compare the performance of copper and nickel MHPs in the overloaded zones, attention was focused on various key indicators. They were: the dryout and flooded lengths, which increase with the applied heat load; the effective length, which decreases as the heat load is increased; and the total axial temperature drop, which increases rapidly with the applied heat load once it exceeds a critical value and, therefore, may serve as a tool for dryout detection. Comparison between copper and nickel MHPs on how these indicators response to the applied heat load shows that an MHP of solid wall of higher

Several geometrical parameters are involved in the governing equations derived in Section 2. The mathematical expressions of these parameters are given by Hung and Tio [13], but for ease of reference, they are reproduced here. At a typical axial position where phase change occurs, the cross section of the flow channel of an MHP is occupied by the liquid and vapor phases of the working fluid. The total area occupied by the liquid, Al , and that by the vapor, Av , are related to the volume fraction occupied by the liquid phase, s, by the formulas

Al ¼ As;

ð38Þ

Av ¼ Að1  sÞ;

ð39Þ

where A is the cross-sectional area of the flow channel. The radius of curvature, r, of the liquid–vapor interface is related to the liquid volume fraction by 2

s¼

~ rÞ ðx ; A

ð40Þ

~ is given by where x

~ ¼ N 1=2 x



cos h cosð/ þ hÞ p 1=2 þ ð/ þ hÞ  : sin / 2

ð41Þ

In Eq. (41), N is the number of vertices of the cross section, h is the contact angle of the liquid–vapor interface with the solid wall, and / is the half corner angle at the vertices. The total lengths of the solid–liquid, solid–vapor, and liquid–vapor interfaces are given by the following expressions:

Psl ¼ C sl ðAsÞ1=2 ;

ð42Þ

Psv ¼ Nw  C sl ðAsÞ1=2 ;

ð43Þ

K.-K. Tio, Y.M. Hung / International Journal of Heat and Mass Transfer 81 (2015) 737–749

Plv ¼ C lv ðAsÞ1=2 ;

ð44Þ

where w is the length of one side of the cross section, and

C sl ¼



2N1=2 cos h p=2  ð/ þ hÞ 1=2  ; cos2 ð/ þ hÞ sin / sin / cosð/ þ hÞ

C lv ¼ 2N1=2

hp 2

 ð/ þ hÞ

i cos h cosð/ þ hÞ sin /

ð45Þ

1=2

þ ð/ þ hÞ 

p 2

: ð46Þ

The hydraulic diameters of the liquid and vapor phases are, respectively, given by

DH;l 

4Al 4ðAsÞ1=2 ¼ ; Psl C sl

ð47Þ

DH;v 

4Av 4Að1  sÞ ¼ : Psv þ Plv Nw þ ðC lv  C sl ÞðAsÞ1=2

ð48Þ

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