Dielectric liquid pumping flow in optimally operated micro heat pipes

International Journal of Heat and Mass Transfer 108 (2017) 257–270

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Dielectric liquid pumping flow in optimally operated micro heat pipes Fun Liang Chang, Yew Mun Hung ⇑ School of Engineering, Monash University, 47500 Bandar Sunway, Malaysia

a r t i c l e

i n f o

Article history: Received 14 September 2016 Received in revised form 4 December 2016 Accepted 8 December 2016

Keywords: Circulation effectiveness Dielectric liquid pumping Heat transport capacity Micro heat pipe

a b s t r a c t Micro heat pipe is a micro-scale capillary-driven two-phase heat transfer device of which thermal performance is governed by the strength of evaporation and the circulation effectiveness of condensate from the condenser to the evaporator. By employing a mathematical model based on the conservation laws, this study demonstrates the application of dielectric pumping flow in enhancing the circulation effectiveness of condensate and hence the thermal performance of micro heat pipes. Through the application of a non-uniform electric field, the Maxwell pressure gradient is induced to drive the condensate flowing towards the evaporator. Two different dielectric pump configurations are compared and the micro heat pipe using planar electrodes is found performing better than that with the pin electrodes. The performance enhancement of different dielectric pump lengths where the total amount of electrical energy of the pump is conserved is analysed. The dielectric pump performs the best when it covers the entire length of micro heat pipe. Compared to the case without dielectric liquid pumping flow, significant enhancement in the heat transport capacity can be obtained where the maximum enhancement exceeds 220%. Even with a significant performance enhancement, the use of dielectric pump renders a sufficiently small solid wall temperature drop of a micro heat pipe, justifying the typical characteristic of a phasechange heat transfer device. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Towards the goal of improved thermal management in microelectronics, micro heat pipe renders a promising approach due to the fact that it can provide sufficient cooling effect even within a constrained space in the electronic device. Since the idea of micro heat pipe was proposed by Cotter [1] in 1984, this micro-scaled cooling device has been studied actively, either theoretically or experimentally, following the dramatic development in the electronic and micro-electronic industries in the recent years. Micro heat pipe possesses various advantages over other electronics cooling devices due to its benefit mostly from the phase-change and circulation effectiveness of working fluid. As shown in Fig. 1, micro heat pipe is a wickless capillary microchannel of which capillary pressure is induced by the sharp-angle corners. The heat loaded to the evaporator section is mainly absorbed as latent heat of evaporation by the liquid confined at the sharp corners. The remaining small fraction of the heat is conducted axially in the solid wall towards the condenser section [2,3]. The resultant vapour flows towards the condenser section through the adiabatic section, and condenses at the condenser section where the latent heat of ⇑ Corresponding author. E-mail address: [email protected] (Y.M. Hung). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.018 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

evaporation is dissipated to the surroundings. The capillary action induced at the sharp-angle corners due to the resultant liquid pressure drop from the condenser to the evaporator drives the condensate back to the evaporator to perpetuate the cycle of phase change and circulation of the working fluid. The maximum possible heat transport rate, denoted as the heat transport capacity, is obtained when the simultaneous onsets of dryout at the evaporator end and flooding at the condenser end take place [2,3]. When the micro heat pipe is overloaded (the heat input exceeds the heat transport capacity), dryout transpires at the evaporator section and the sharp-angled corners are depleted of liquid due to the fact that the capillary pressure is incapable of supplying an adequate amount of working fluid rapidly enough to the evaporator section to compensate for the liquid loss through the immense evaporation rate. The excessive condensate, which is not circulated to the evaporator section, accumulates in the condenser end and flooding occurs. A liquid block is formed, hindering the condensation heat transfer and thus reducing the effective condenser length of the micro heat pipe. The transport phenomena taking place in micro heat pipes are considerably complicated, encompassing the principles of fluid mechanics and phase-change heat transfer. The thermal performance of a micro heat pipe is essentially governed by its circulation effectiveness of working fluid [3]. External force induced by

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Nomenclature A BoP Bo C Ca DH d E F^e f f g hfg K k Lt M M opt _ m ^ m _ ref m _ m N Nu P p ^ p Po Q_ Q_ cap q_ w Re r T T op tw u V We w

cross-sectional area, m2 ponderomotive Bond number, defined in Eq. (26) Bond number, defined in Eq. (24) dimensionless geometrical parameter capillary number, defined in Eq. (25) hydraulic diameter, m distance between the tip of pin and planar electrodes, m electric-field vector, V/m dimensionless electric body force term, defined in Eq. (23) friction factor electric body force density, N=m3 gravitational acceleration, m=s2 latent heat of evaporation, J=kg integration constant thermal conductivity, W=m  K total length of micro heat pipe, m charge level optimal charge level mass flow rate, kg/s dimensionless mass flow rate reference mass flow rate, kg/s average mass flow rate, defined in Eq. (38), kg/s number of corners Nusselt number, defined in Eq. (40) contact length, m pressure, N=m2 dimensionless pressure Poiseuille number, defined in Eq. (15) heat transport rate, W heat transport capacity, W heat transfer rate per unit axial length, W/m Reynolds number, defined in Eq. (16) meniscus radius of curvature, m temperature, °C operating temperature of micro heat pipe, °C wall thickness, m velocity, m/s applied voltage, V Weber number, defined in Eq. (27) groove width, m

electric field is widely applied to generate fluid motion in a favourable direction for enhancing the circulation effectiveness. The incorporation of electrohydrodynamically-driven flow is a promising approach in enhancing the performance of a microfluidic device [4]. Electrohydrodynamic (EHD) or electrokinetic flows are induced by the application of an electric field to incorporate a net electrostatic (Maxwell) force in polarized surface regions in order to drive the fluid flow or to induce the motion of particles suspended within the liquid [4,5]. In practical applications, either a DC or an AC high-voltage low-current electric field is applied through a pair of charged and receiving electrodes [5]. Generally, attributed to the applied electric field, the electric body force density, f, which is more commonly known as Korteweg-Helmholtz force density, is given by [4]

1 f ¼ qe E  r 2



eq


 @ e

E  E; @ q
T

ð1Þ

where qe is the volume charge density and E being the electric-field vector. The first term on the right-hand side of Eq. (1) represents the Coulombic force, which arises due to the presence of the free space

x ^x

axial distance from evaporator end, m dimensionless x

Greek symbols aspect ratio, defined in Eq. (30) b angle of inclination, rad e dielectric permittivity, C=V  m U volume fraction occupied by liquid phase / half corner apex angle, rad C circulation parameter, defined in Eq. (37), kg=m  s c kinematic viscosity ratio, defined in Eq. (29) l dynamic viscosity, kg=s  m m kinematic viscosity, m2 =s density ratio, defined in Eq. (28) h contact angle, rad q density, kg=m3 qe volume charge density, C=m3 r surface tension, N/m s shear stress, N=m2 ! angular parameter, defined in Eq. (61)

a

Subscripts a adiabatic section c condenser section cl capillary limit dp dielectric liquid pump e evaporator section fl onset of flooding j phase j l liquid lv liquid-vapour interface m maximum s solid wall sl solid-liquid interface sv solid-vapour interface v vapour x x-direction y y-direction z z-direction 0 evaporator end 1 condenser end

charges while the second term consists of the ponderomotive force term and the electrostrictive term. The ponderomotive force term arises due to the inhomogeneity of the dielectric permittivity, e, which describes the permittivity change occurs at the liquidvapour interface. On the other hand, the electrostrictive term accounts for the compressibility of the media which is negligible for incompressible fluids [4]. In the absence of electrostrictive effects, two general principles of inducing electrohydrodynamic actuation are either through the generation of free space charges to engender Coulombic forces or through the polarization of induced charges by applying an electric field to trigger ponderomotive forces [4]. The Coulombic force acts on the free space charges within the fluid. A viable method of producing space charges in the fluid is by directly charging electrolyte solution to the working fluid such that electroosmotic flow can be induced because of the presence of free ions in the electrolyte solution [6,7]. A recent study by Chang and Hung [8] demonstrates the use of electroosmotic flow to increase the circulation rate of working fluids in micro heat pipes. Significant enhancement in thermal performance of micro heat pipe has been achieved.

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259

Fig. 1. Through the application of the non-uniform electric field, an electric (Maxwell) pressure gradient is established, which is responsible to enhance the liquid flow on top of the capillary action at the sharp-angle corners in a micro heat pipe.

Another way to induce electrohydrodynamic force is through the application of electric field within a dielectric fluid [9,10]. There are four different dielectric liquid flow mechanisms used to introduce electrohydrodynamic flow, i.e. ion injection, conduction pumping, induction pumping and Maxwell pressuregradient-driven flow [10]. The first two dielectric liquid flow mechanisms, i.e. ion injection and conduction pumping, could not be realized without the presence of free-space charges in a dielectric fluid. By applying ion injection, the mobile charges are directly injected into the dielectric fluid using an emitter tip [4]. On the other hand, in an application with conduction pumping, the freespace charges are generated within a dielectric fluid through the dissociation of ions. The bipolar charges which are formed due to the Faradaic dissociation of polar impurities or dielectric molecules, or a combination of both, provide a net flow of dielectric fluid through a channel [4]. In contrast to the first two mechanisms requiring the presence of free-space charges in a dielectric fluid, the induction pumping depends merely on the polarization of induced charges within the fluid. Even in the absence of freespace charges, owing to a conductivity or permittivity gradient across an interface, induction pumping is capable of generating electrohydrodynamic flow based on the polarization of the induced surface charges at a free surface which interact with the interfacial tangential field in the fluid [11]. In a more generalized configuration, the Maxwell pressure gradient can be induced even without the presence of a free surface to drive a dielectric fluid if an electric pressure gradient is established through the application of a nonuniform electric field [4]. The tangential or the normal component of the electric field or the combination of both could contribute to the normal component of the Maxwell stress, which is the driving force of fluid flow [4,10,12]. This type of dielectric liquid flow is denoted as Maxwell pressure-gradient-driven flow [4]. The first electrohydrodynamic heat pipe, classified as an induction-pumping-assisted heat pipe, was incepted in the 1970s [13]. A thin ribbon electrode extended in the axial direction is used to generate the polarization force which drives the dielectric liquid in a non-uniform electric field to establish the hydrostatic equilibrium. The surface tension induces a pressure differential across the liquid-vapour interface and its net effect drives the liquid flow. Later, the experiments successfully proved the concept of

electrohydrodynamic heat pipe [14]. However, the overall thermal conductance of the heat pipes was low and the temperature drops were large due to the hydrodynamic limit imposed by circumferential capillary distribution structures. Later on, improved thermal conductance with a smooth evaporator surface which was only 15% lower than the highest conductance measured for an evaporator surface with capillary grooves was obtained [15]. More recently, the induction pumping mechanism was applied on the evaporator section of capillary pumped-loop system to drive the dielectric fluid from condenser to evaporator for heat transfer enhancement [16–18]. Other dielectric pumping mechanisms such as ion injection and conduction pumping require the presence of free-space charges in a dielectric fluid. The feasibility and performance enhancement of wicked heat pipes were demonstrated with the use of an ion-drag pump to supplement the capillary pumping [19]. The operating characteristics and thermal performance of conventional heat pipes with the incorporation of conduction pumping was investigated [20,21]. The heat transport capacity was doubled and the polarization force was capable to recover the evaporator from dryout. In terms of microscale, very limited studies of dielectric-liquid-pumping-assisted micro heat pipes have been reported and only two such investigations employing induction pumping exist in the literature [22,23]. The electric field was applied to circulate the condensate from the condenser back to the evaporator of micro heat pipe arrays consisting of seven grooves of 1-mm width and 0.6-mm depth [22]. The inductionpumping-assisted micro heat pipe utilizes dielectrophoretic force resulted from application of an electric field at a liquid-vapour interface to enhance the liquid flow from the condenser to the evaporator. The force is attributed to the polarizing effect of the larger electric field in the liquid because of larger electrical permittivity. Another experimental study on micro heat pipe arrays assisted by induction pumping demonstrated that when the micro heat pipe is subjected to high heat load, dryout occurring at the evaporator restrains the role of the externally applied electric field [23]. A semi-empirical model of the system was developed for analytical investigation of the efficacy of an electric field in response to heat input variations. Up to date, the application of dielectric liquid pumping flow utilizing the normal component of the Maxwell stress in a microscale

260

F.L. Chang, Y.M. Hung / International Journal of Heat and Mass Transfer 108 (2017) 257–270 Table 1 Dimensions of the micro heat pipe with a square cross section. Evaporator length, Le Condenser length, Lc Adiabatic section length, La Total length, Lt Groove width, w Wall thickness, tw Half corner angle, /

12.7 mm 12.7 mm 24.6 mm 50.0 mm 0.684 mm 0.14 mm 45°

two-phase heat transfer device is absent in the existing literature. The objective of this study is to theoretically demonstrate how the basic principles of dielectric liquid pumping can be applied to a micro heat pipe. Fig. 1 depicts a dielectric liquid pump in a micro heat pipe of which the bulk liquid is driven by the normal component of Maxwell stress through the application of a non-uniform electric field. For a long and narrow channel in micro heat pipe, the flow is approximated as unidirectional in the longitudinal direction from the high-field region to the low-field region. In this study, we compare and investigate two different configurations of dielectric liquid pump incorporated in micro heat pipes. The effects of length of dielectric liquid pump on the circulation effectiveness and hence the thermal performance of a micro heat pipe is studied. The circulation of working fluid driven by dielectric liquid pump is scrutinized through the liquid volume fraction distribution inside the micro heat pipe and the circulation effectiveness is quantified by a properly defined parameter. By incorporating dielectric liquid pump that operates without moving parts in a micro heat pipe, we analyse the heat transport capacity of which the desirable values can be achieved by varying the applied potentials that control the strength of the dielectric liquid pump. 2. Mathematical formulation The mathematical model is derived based on the modification on the model developed previously by Hung and Tio [2], particularly to investigate the role of circulation effectiveness of working fluids in analysing the thermal performance of micro heat pipes. By applying energy conservations to the solid wall and the liquid phase, momentum and mass conservations to both liquid and vapour phases, and the Young-Laplace capillary equation to the

liquid-vapour interface, this model is extended to incorporate the electrohydrodynamic forces in the momentum equations, where the performance enhancement of a micro heat pipe under the influence of electrohydrodynamic forces can be examined. Unless otherwise stated, the micro heat pipe is operated at 60 °C, which is a typical operating temperature for electronic cooling devices. The thermo-physical properties and dielectric constant of water are obtained from Reay and Kew [24] and Dean [25] and the thermal conductivity and dielectric constant of silicon are acquired from Jacobsen et al. [26] and Sze and Ng [27], respectively. A silicon-water micro heat pipe with a square cross section is employed and Table 1 tabulates its dimensions. 2.1. Maxwell pressure-gradient-driven dielectric liquid actuation To proceed with the formulation of mathematical model, we engage Eq. (1) to incorporate the electric forces in the liquid and vapour momentum equations. After eliminating the electrostrictive term, Eq. (1) can be simplified as

1 f ¼ qe E  E  Erej : 2

ð2Þ

The second term on the right-hand side of Eq. (2) can be expanded and rearranged as [28]

1 1  E  Erej ¼ ej ½ðE  rÞE  rjEj2  ¼ ej ½ðr  EÞ  E: 2 2

ð3Þ

The first term on the right-hand side of Eq. (2), i.e. the Coulombic force acts on the free space charges within the dielectric fluid. In cases where the Coulombic force is eliminated due to the absence of free ions in the working fluid, the dielectric liquid pumping can still be actuated via the ponderomotive force, corresponding to the second term in Eq. (2). The free ions are absent in the working fluid. Therefore, the first term on the right-hand side of Eq. (2), corresponding to the Coulombic force, is omitted while the effect of the ponderomotive force, which is the remaining term in Eq. (2), is of primary consideration. The dielectric liquid pumping flow can be induced by configuring an electric circuit on a micro heat pipe. Fig. 2 depicts two different configurations of dielectric liquid pump instated on a micro heat pipe—Model 1 and Model 2. Model 1 consists of planar electrodes at the top and the bottom of entire length of micro heat pipe while Model 2 is installed with a pin electrode at the top and a planar electrode at the bottom. Through the application of a non-uniform electric field, Maxwell pressure gradient within the dielectric liquid is established and the ponderomotive force prevails. Unlike an induction pump, the free surface charges and hence a free surface are not necessarily required in a Maxwell pressure-gradient-driven dielectric pump [4]. Depending on the physical setup of the dielectric liquid pump, the tangential or the normal component of the electric field or the combination of both could contribute to the normal component of the Maxwell stress, which is the driving force in enhancing the circulation rate of fluid flow. The flow is directional from the high-electric-field region to the low-electric-field region, as illustrated in Fig. 2. By employing the identity expressed in Eq. (3), Eq. (2) can be simplified as

f ¼ hf x ; f y ; f z i ¼ ej ½ðr  EÞ  E:

ð4Þ

For a unidirectional flow along the axial direction inside a micro heat pipe, only the x-component of the electric body force density term is considered. From Eq. (4), the x-component of the electric body force density term can be written as

Fig. 2. A schematic diagram of a dielectric liquid pump in a micro heat pipe: (a) Model 1: Planar electrodes at the top and the bottom of entire length of micro heat pipe. (b) Model 2: Pin electrode at the top and planar electrode at the bottom. w is the groove width.

     @ @ @ @ f x ¼ ej Ez  ðEz Þ þ ðEx Þ  Ey ðEy Þ  ðEx Þ ; @x @z @x @y

ð5Þ

and it can be further simplified for a one-dimensional mathematical model as

F.L. Chang, Y.M. Hung / International Journal of Heat and Mass Transfer 108 (2017) 257–270

     @ @ f x ¼ ej Ez  ðEz Þ  Ey ðEy Þ : @x @x

261

ð6Þ

This x-component of the electric body force density term can be incorporated in the momentum equations of the liquid and the vapour phases to induce the dielectric liquid pumping flow in a micro heat pipe. 2.2. Conservation of momentum of working fluid The flow inside the micro heat pipe is assumed to be incompressible, laminar and fully developed with constant fluid thermo-physical properties. By applying a non-uniform electric field, the electric body force prevails. In accordance with the electrical circuit configuration as depicted in Fig. 2, the electric field acts in the normal direction to the electrodes (y-direction) only, and hence Ex ¼ Ez ¼ 0. For a unidirectional flow in the axial direction of a micro heat pipe, the x-component of electric body force density of Eq. (4) is given by

    @ ej @ 2 f x ¼ ej Ey ðEy Þ ¼  ðEy Þ : @x 2 @x

ð7Þ

For Model 1 with planar electrodes at the top and the bottom of micro heat pipe as shown in Fig. 2(a), its y-component electric field, Ey , is given by [4]

Ey ¼

V x ; w Lt

ð8Þ

while for Model 2 with a pin electrode at the top and planar electrode at the bottom as depicted in Fig. 2(b), its y-component electric field is expressed as [10]

2V Ey ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 ðLt  xÞ2 þ ðdÞ

ð9Þ

where d is the distance between the tip of the pin and planar electrodes, w being the groove width of micro heat pipe and x is the micro heat pipe’s axial distance from the evaporator end. Eqs. (8) and (9) are scaled using the following non-dimensional groups:

^ y ¼ Ey ; E V=w

^x ¼

x ; Lt

^¼ d: d Lt

ð10Þ

^y , for Following this, the dimensionless y-component electric field, E Model 1 and Model 2 can be written, respectively, as

^y ¼ ^x; E

ð11Þ

Fig. 3. Infinitesimal control volumes of a micro heat pipe: (a) Conservation of momentum for liquid and vapour phases. (b) Conservation of energy in the solid wall and working fluid.

end with respect to the horizontal of the micro heat pipe. The shear stress, sij at the solid-liquid (sl), liquid-vapour (lv) and solid-vapour (sv) interfaces are given by [30]

1 2

sij ¼ qj u2j f ;

ð12Þ

As depicted in Fig. 3(a), by incorporating the electric body force to the momentum balance of an infinitesimal control volume of liquid and vapour phases, respectively, the momentum equations for the liquid and vapour flows are, respectively, given by

d dp ql ðAl u2l Þ ¼ Al l þ ssl Psl þ slv Plv  g ql Al sin b þ f x;l Al ; dx dx

ð13Þ

and

qv

d dp ðAv u2v Þ ¼ Av v  ssv Psv  slv P lv  g qv Av sin b dx dx þ f x;v Av ;

ð14Þ

where Aj , uj , pj and qj are the cross-sectional area, velocity, pressure and mass density of phase j, respectively, g is the gravitational acceleration, and b is the angle of inclination at the evaporator

Poj ; Rej

ði ¼ s; l; j ¼ l; v; i–jÞ;

ð15Þ

where f is the Fanning friction factor. The liquid and vapour flows are assumed to be fully developed inside a micro heat pipe on account of negligible convective heat transport and small variation of the liquid and vapour cross-sectional areas in the axial direction. The Poiseuille number Poj ¼ f Rej is a constant for laminar flows and depends on the cross-sectional geometry only [30,31]. As the liquid phase is confined in the sharp corners, by neglecting the impact of the vapour flow on the liquid-vapour interface, Pol ¼ 14:2 corresponds to the value of a square. For the vapour phase, the cross section changes from a square at the evaporator end to a circle at the condenser end. Therefore, the value of Pov ¼ 15:1 is evaluated as the average of the values of a square and a circle. The Reynolds number, Rej , is defined based on the hydraulic diameter, DH;j (see Appendix B) as

and

2w ^y ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: E 2 ^ 2 Lt ð1  ^xÞ þ ðdÞ

f ¼

Rej ¼

qj uj DH;j ; j ¼ l; v; lj

ð16Þ

where lj is the dynamic viscosity of phase j. As no mass is accumulated through any cross section of the micro heat pipe during steady-state operation, the mass conservation is given by

_ ¼ ql ul Al ¼ qv uv Av ; m

ð17Þ

_ is the mass flow rate of liquid and vapour through a cross where m section. The mass flow rate profile can be obtained by solving the governing equation based on the energy conservation of working fluid as described in Appendix A. The pressures of the two phases are related by the capillary pressure and the Young-Laplace equation is given by

pv  pl ¼

r r

;

ð18Þ

where r is the radius of curvature of the liquid-vapour interface and

r is the surface tension. The radius of curvature r is related to the volume fraction occupied by the liquid phase, U, by

262

r¼

F.L. Chang, Y.M. Hung / International Journal of Heat and Mass Transfer 108 (2017) 257–270

A1=2 U1=2

!

:

ð19Þ

Differentiating both sides of Eq. (18) with respect to x yields

dpv dpl !r dU  ¼  1=2 3=2 ; dx dx dx 2A U

ð20Þ

where A is the cross-sectional area of the micro heat pipe, U is the volume fraction occupied by the liquid phase and ! is an angular parameter (see Appendix B). The governing equations are rendered dimensionless using the following non-dimensional groups:

^x ¼

x ; Lt

U¼

Al ; A

1U¼

Av ; A

^ ¼ m

_ m ; _ ref m

^¼ p

p

r=A

1 2

;

ð21Þ

_ ref ¼ Q_ Lt =hfg Le and hfg is the latent heat of evaporation of where m the working fluid. Subsequently, by combining Eqs. (13), (14), (17) and (20), together with the geometrical parameters defined in Appendix B, we obtain the following ordinary differential equation:

!#  2  ^ ^2 1 d 1 d m m  We -ð1  UÞ d^x 1  U U d^x U o ^  aBo sin b þ F^e : þaCa½Pol G1 ðUÞ þ cPov G2 ðUÞm

dU 2U3=2 ¼ d^x !

(

"

ð22Þ

The dimensionless electric body force term, F^e , in Eq. (22) is given by

1 @ ^ 2  E : F^e ¼  BoP 2 @x y

ð23Þ

The Bond number Bo, capillary number Ca, ponderomotive Bond number BoP , Weber number We, density ratio -, kinematic viscosity ratio c, and aspect ratio a are given, respectively, by

Bo ¼ Ca ¼

gðql  qv ÞA

r

ð24Þ

ll ðm_ ref =ql AÞ ; r

BoP ¼ We ¼

-¼

;

ðel  ev ÞðV=wÞ2 ðA1=2 Þ

r

ð25Þ ;

ð26Þ

ql ðm_ ref =ql AÞ2 A1=2 ; r

ð27Þ

qv ; ql

ð28Þ

m lq c¼ v ¼ v l; ml ll qv

ð29Þ

and

a¼

Lt A1=2

:

ð30Þ

The functions of G1 ðUÞ and G2 ðUÞ in Eq. (22) are, respectively, expressed a function of U as

G1 ðUÞ ¼

C 2sl 8U2

;

ð31Þ

and

G2 ðUÞ ¼

½NwA1=2 þ ðC lv  C sl ÞU1=2 ½NwA1=2 þ C lv U1=2  C sl U1=2  8ð1  UÞ3

:

ð32Þ

2.3. Conservation of energy of solid wall and working fluid In accordance with Fig. 3(b), the formulations based on conservation of energy pertaining to the solid wall and the working fluid are similar to those developed by Hung and Tio [2] for a triangular micro heat pipe. Here, we modify the governing equations to be applicable to a square micro heat pipe. In order to show a selfcontained picture of the model of this paper, a brief description of the formulations based on conservation of energy is presented in the Appendix A.

2.4. Solution technique To investigate the thermal performance of an optimally designed micro heat pipe subjected to dielectric liquid pumping, Eq. (22) is solved using the fourth order Runge-Kutta method with a sufficiently small step size of 0.0001. To achieve the optimal operating condition, the micro heat pipe has to fulfil two specified conditions simultaneously. The condition Uð0Þ ¼ Ucl ¼ 0:0001 corresponds to the onset of dryout at the evaporator end [2,3,32–36]. The condition of Uð^ xm Þ ¼ Ufl is specified as the onset of flooding where ^ xm is the axial position. The liquid saturation achieves its maximum value in the condenser section corresponding to the onset of flooding, Ufl , which is given by [2,3,33,35] 2

Ufl ¼

ð!w sin /Þ : 4A cos2 ð/ þ hÞ

ð33Þ

The solution steps are detailed as follows. (a) For an application of favourable electric body force (positively applied voltage where V > 0), the liquid saturation increases at the evaporator section and decreases at the condenser section with increasing applied voltage, compared to the case without influence from the dielectric liquid pump (V ¼ 0). The maximum point of the liquid saturation profile which prevails at the condenser end (^ x ¼ 1:0) for the micro heat pipe without a dielectric liquid pump, moves away from the condenser end (^ x < 1:0) when the voltage is applied. The existence of this maximum point is attributed to the counteraction between two opposite effects. While the electrokinetic effect tends to pull the liquid from the condenser section towards the evaporator section, the heat load moves the liquid towards the condenser section via evaporation. To locate the position of the maximum point at the condenser section, ^ xm , the gradient of liquid saturation along the axial direction must fulfil the condition as follows:


dU

¼ 0: d^x
^x¼^xm

ð34Þ

Hence, the right-hand side of Eq. (22) is equated to zero and the corresponding ^ xm value at a given operating temperature can be obtained. For a micro heat pipe not subjected to dielectric liquid pumping, its ^ xm is always located at the condenser end. (b) After ^ xm has been determined, the ‘‘initial” condition of the integration of Eq. (22) is set as Uð^ xm Þ ¼ Ufl . (c) The heat input for the solution of Eq. (22) is iterated in steps of DQ_ ¼ 1  104 W, until the condition Uð0Þ ¼ Ucl ¼ 0:0001 corresponding to the onset of dryout at the evaporator end, is satisfied [2,3,32–35]. The heat input corresponding to these conditions is regarded as the heat transport capacity, Q_ cap . The solution of the liquid volume fraction is considered convergent when the relative error between the new and old ~ at ^ x ¼ 0 satisfies the prescribed criterion of values of U

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263

Fig. 5. Dimensionless electric force variation under a positively applied voltage of 300 V by different dielectric liquid pump configurations, of an optimally charged silicon-water micro heat pipe.

Fig. 4. (a) Heat transport capacity, Q_ cap , as a function of the operating temperature, T op , for different contact angles of an optimally charged copper micro heat pipe filled with water. Numerical results are benchmarked with the experimental data by Babin et al. [37]. (b) Axial solid wall temperature difference, DT s , as a function of the applied heat load, Q_ in , of a silver-water micro heat pipe operated at 60 °C. The theoretical results satisfactorily agree with the experimental data by Peterson and Ma [38].

~ new  U ~ old j jU 6 107 : ~ jUnew j

ð35Þ

Furthermore, the deviation of Q_ cap between the runs with a step size of 1  104 and a step size of 8  105 is only 1:5  1010 %, justifying the accuracy of the current step size. The solution of the liquid volume fraction U is used to determine the charge level of micro heat pipe and its dimensionless form is expressed as

^ ¼ M ¼ - þ ð1  -Þ M ALt ql

Z

1

Uð^xÞd^x;

ð36Þ

h for a horizontally positioned micro heat pipe, without the external influence of the electrohydrodynamic and gravitational forces. We compare the numerical results with the experimental data by Babin et al. [37] for a copper micro heat pipe filled with 3.2 mg of water, with a trapezoidal cross-sectional area and a length equivalent to that of the present study. Both sets of results agree qualitatively showing an increasing trend of heat transport capacity with operating temperature. Quantitatively, they agree well on the order of magnitude of heat transport capacity. The heat transport capacity is a function of the contact angle between the liquid phase and the solid wall. The contact angle between the liquid and the solid wall is assumed to be a constant along the entire length of micro heat pipe. The heat transport capacity increases from zero contact angle to its maximum value of about 15° [2,33]. Beyond the threshold value, the heat transport capacity decreases with a further increase in the contact angle. Detailed study of the effect of contact angle is beyond the scope of the present study and hence a zero contact angle corresponding to complete wetting of the working fluid on the solid wall is assumed for the subsequent analyses. For another comparison in Fig. 4(b), the numerical results agree well with the experimental data of axial solid wall temperature difference, DT s , as a function of the applied heat load, Q_ in , for a silver-water micro heat pipe operating at 60 °C [38]. The numerical results are properly validated by the experimental results.

3.2. Analysis of different dielectric liquid pump configurations

0

where M is the mass of working fluid charged into micro heat pipe. Eq. (36) is integrated numerically using Simpson’s rule. When the micro heat pipe is loaded with its heat transport capacity, Q_ cap at specific operating conditions, the corresponding amount of working fluid is denoted as the optimal charge level, Mopt (in dimensionless ^ opt ) where the onsets of dryout and flooding take place form, M simultaneously [2,3,8,33–35]. 3. Results and discussion 3.1. Model validation We validate the present numerical results prior to investigating the role of dielectric pumping flow on the thermal performance of micro heat pipes. Fig. 4(a) depicts the heat transport capacity, Q_ cap , as a function of operating temperature for different contact angles

In the absence of free space charges in the dielectric liquid, an electric pressure gradient due to non-uniform electric field is used to drive the fluid flow. We compare the performance of two different dielectric liquid pump configurations in micro heat pipe (Model 1 and Model 2) as illustrated in Fig. 2. Under a positive voltage of 300 V, the axial profiles of the dimensionless electric force F^e of an optimally charged silicon-water micro heat pipe, as defined in Eq. (23), are plotted and compared in Fig. 5. Although the magnitude of electric force prevalent in Model 2 is larger than that of Model 1, the force distribution is more uniform in Model 1. For Model 1, F^e decreases linearly from the condenser end towards the evaporator end. For Model 2, a drastic spike of F^e is restricted in the condenser end and no effective electric force prevails across the evaporator and adiabatic sections. In addition to the capillary force, the electric (Maxwell) force assists to circulate the liquid flow from the condenser to the evaporator. The uniform Maxwell force distribution of Model 1 yields more effective circulation of

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working fluid over a practical range of applied voltage, as shown in Fig. 6(a). The circulation parameter, C, which is an indicator of the circulation effectiveness of working fluid in a micro heat pipe, is given by [3]





C ¼

Lt qv

Z 0

1

uv d^x þ ql

Z 1

0



ul d^x

:

ð37Þ

Model 1 manifests a significantly higher capability in circulating condensate back to the evaporator compared to Model 2 because of the uniform Maxwell force exerted across the entire length of micro heat pipe. Although no significant difference in the circulation parameter is observed between the two models at low voltages (V < 100 V), the increasing rate of C with the applied voltage of Model 1 is significantly larger than that of Model 2 at higher applied voltage. For Model 2, the pin electrode is located at the condenser end and its field strength is restricted within the confine of the condenser section. Although the amount of liquid is maximum in the condenser, the effect of the pumping force diminishes in the adiabatic section before the liquid reaches the evaporator section. Compared to Model 1 with the planar electrodes at the top and the bottom of entire length of micro heat pipe, the linearly decreasing effect of field strength from the condenser to the evaporator is inherently able to pump the liquid more effectively from the source (condenser) to the sink (evaporator). Identical trends are observed in Fig. 6(b) for the plots of heat transport capacity, Q_ cap , which is an intrinsic performance indicator of a micro heat pipe [34–36,39], versus the applied voltage. This shows that the thermal performance is enhanced due to the enhancement of working fluid’s circulation. Without the application of electric field, the heat transport capacity is evaluated as 0.17 W and its enhancement becomes more significant when the applied voltage increases, particularly for Model 1. In what follows, therefore, we only employ Model 1 in the subsequent analyses. 3.3. Optimal location of electrode

Fig. 6. (a) Circulation parameter, C, and (b) heat transport capacity, Q_ cap , as a function of the applied voltage, V, of an optimally charged silicon-water micro heat pipe for two different dielectric liquid pumping configurations.

The location of the planar electrode dictates the circulation effectiveness of dielectric liquid inside a channel [4]. To obtain an optimal layout of Model 1, we investigate the effect of location of the electrode (of same length) by comparing the performance of micro heat pipe with different electrode’s location. As depicted in Fig. 7(a), three cases are considered: electrode at the evaporator,

Fig. 7. (a) Schematic diagram of micro heat pipe with different electrode’s locations: electrode at the evaporator, electrode at the adiabatic section and electrode at the condenser. (b) Illustration of flows induced by the electrode located at different locations. (c) Heat transport capacity, Q_ cap , as a function of the applied voltage, V, of an optimally charged silicon-water micro heat pipe for two different electrode’s locations.

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electrode at the adiabatic section and electrode at the condenser. It should be noted that if the electrode is placed at the evaporator or the adiabatic section, as shown in Fig. 7(b), bidirectional flows would be induced [4]. The reversed flow (towards the condenser) counteracts the favourable flow in the opposite direction and hence the enhanced effect on circulation is eliminated. Similar bidirectional flows are engendered if the electrode is located at the evaporator section as shown in Fig. 7(b). Therefore, the most appropriate location for the planar electrode is at the condenser where the reversed flow is suppressed by the condenser end. To compare the performance of micro heat pipe for different electrode locations, we evaluate the heat transport capacity when the electrodes are located at the evaporator and the condenser. As depicted in Fig. 7(c), when the electrode is located at the evaporator section, the enhancement in heat transport capacity is less significant compared to its counterpart where the electrodes are placed at the condenser section—albeit without considering the effect of the unfavourable reversed flow. This is because the amount of liquid in the evaporator is very small and the pumping effect is not sufficient to induce effective circulation of the working fluid. On the other hand, the condenser contains larger amount of liquid compared to the other locations and the pumping efficiency and hence the heat transport capacity are the highest when the electrodes are installed here. Therefore, the dielectric pump performs the most efficiently when the planar electrodes are installed extending from the condenser end and spanning over the condenser section. The pumping efficiency is the highest attributed to the large amount of liquid on top of the suppression of flow reversal. In what follows, we only consider cases where the electrodes are extended from the condenser end. 3.4. Effects of electrode length To investigate the effects of different electrode lengths of dielectric liquid pump on the thermal performance of micro heat pipe, the dimensionless electric force term variation in the axial direction for three different electrode lengths is plotted in Fig. 8. The inset of Fig. 8 illustrates the micro heat pipes installed with electrodes of different length, Ldp . Three cases are considered here. First, the electrodes span the entire length of micro heat pipe, Ldp ¼ Lt . Second, the electrodes encompass the adiabatic and condenser sections, Ldp ¼ La þ Lc , while in the third case, the electrodes only span the condenser section, Ldp ¼ Lc . The optimally charged silicon-water micro heat pipe is operated under a positively applied voltage of 300 V. For all the three cases, the electric force

Fig. 8. The dimensionless electric force variation under a positively applied voltage of 300 V for different electrode lengths. The inset depicts a schematic diagram of electrodes of different lengths installed to an optimally charged silicon-water micro heat pipe: entire length of micro heat pipe (red), adiabatic and condenser sections (blue) and condenser section only (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

265

magnitude increases linearly from zero at the initial axial location of the pump and reaches its maximum value at the other end. The electric force is effective across the span of the length of electrode. The shorter the electrode length, the higher the gradient of electric force is. However, the area under the curve which manifests as the electrical energy delivered from the dielectric liquid pump is the same for the three cases. This implies that under a fixed applied voltage, the imposed electric energy remains constant despite the change of length of electrode. Fig. 9 shows the liquid saturation profiles in the axial direction of micro heat pipes with different electrode lengths, under a favourably applied voltage of 300 V. With the application of a favourable electric body force, the liquid volume fraction reaches its maximum value at ^ xm , where the corresponding liquid volume fraction value is equivalent to the flooding limit Ufl , as depicted in Eq. (33). For the case without the dielectric pumping effect (Ldp ¼ 0), ^ xm manifests at the condenser end for an optimally operated micro heat pipe. When the electric force prevails, this maximum point exists mainly due to the balance of the two competing opposite effects, which are the electric body force exerted by the dielectric liquid pump and the heat loaded to the micro heat pipe [8,35]. As the electric force assists in circulating the condensate back to the evaporator, the applied heat load moves the liquid in the opposite direction. Under the same applied voltage, we observe that ^ xm moves the furthest away from the condenser end in the third case, where the electrode of dielectric liquid pump only spans the condenser section (Ldp ¼ Lc ). This occurrence is attributed to the electric force as depicted in Fig. 8. As we utilize the shortest electrode in the third case, its effective electric force is the highest among the three cases. Although the applied voltage is the same, the highest electric force manages to push ^ xm the furthest away from the condenser end. The distance of ^ xm from the condenser end decreases with the electrode length in accordance with the fact that the electric force decreases with the increase in the dielectric pump length. To elucidate the actual thermal performance of micro heat pipe under the influence of dielectric liquid pumping, the heat transport capacity, Q_ cap , as a function of the favourably applied voltage, is plotted in Fig. 10. Different dielectric liquid pump lengths are applied. All the three cases show insignificant difference in the heat transport capacity at low voltages (V < 150 V). The heat transport capacity of the third case (Ldp ¼ Lc ) remains almost insensitive to the increase in the applied voltage while the other two cases show a discernible increasing trend of the heat transport capacity with the applied voltage. Although the electrode length of

Fig. 9. Liquid volume fraction variation in the axial direction for different dielectric liquid pump lengths of an optimally charged silicon-water micro heat pipe, operating under a positively applied voltage of 300 V.

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Fig. 10. Heat transport capacity, Q_ cap , as a function of the applied voltage, V, of an optimally charged silicon-water micro heat pipe subjected to different dielectric liquid pump lengths.

Fig. 11. Circulation parameter, C, as a function of the applied voltage, V, of an optimally charged silicon-water micro heat pipe subjected to different dielectric liquid pump lengths.

the first case is longer, the difference in heat transport capacity between them is not significant. For the first case (Ldp ¼ Lt ), although the electrode spans the evaporator, the enhancement in liquid circulation compared to the second case is not significant due to the fact that the amount of liquid in the evaporator is extremely small attributed to the intensive evaporation. Therefore, the heat transport capacity which is strongly dependent on the circulation effectiveness [3] is not enhanced pronouncedly in the first case (Ldp ¼ Lt ) compared to the second case (Ldp ¼ La þ Lc ). For the second case, further increasing the applied voltage beyond 6.7 kV will cause the ^ xm moves beyond the adiabatic-condenser section whereby the entire condenser section is flooded and the condensation process ceases, in which case the micro heat pipe malfunctions. On the other hand, for the third case where the electrode spans over only the condenser section, the heat transport capacity remains almost unchanged with the application of favourable electric force. Due to the absence of electric force in the adiabatic section, the strength of circulation of liquid deteriorates despite the fact that the electric force gradient spanning the condenser is the highest among the three cases. Without the assistance from the electric force, the liquid in the adiabatic section depends only on the capillary pressure induced by the sharp corners for circulation to the evaporator section. Besides having a relatively high amount of liquid in the adiabatic section, the electric force applied to the micro heat pipe in the adiabatic section exerts significant influence on the circulation of working fluid back to the evaporator. However, the dielectric pumping flow performs the best when the electrode spans the entire length of micro heat pipe. Compared to the case without dielectric pumping flow, significant enhancement in the heat transport capacity can be attained where the maximum enhancement amounts to 222% for the first case (Ldp ¼ Lt ). As discussed earlier, the aiding effect of electric force increases circulation rate of working fluid, which inherently resulting in an increment in the heat transport capacity. It is instructive to scrutinize the circulation rate of working fluid inside a micro heat pipe subject to dielectric liquid pumping. As defined in Eq. (37), a circulation parameter is used to quantify the circulation rate of working fluid in a micro heat pipe. Fig. 11 depicts the circulation parameter, C, as a function of the applied voltage of an optimally charged silicon-water micro heat pipe subject to different lengths of dielectric liquid pump. Similar to Fig. 10, the circulation parameter of the three cases show insignificant difference at low voltages. At higher voltages, the circulation parameter increases with the applied voltage. The increase rate of the circulation parameter increases with the length of the dielectric liquid pump. By comparing Figs. 10 and 11, the variation of C with the applied voltage is different from

that of Q_ cap . Referring to Eq. (17), the mass flow rate is a function of both cross-sectional areas and velocities of liquid and vapour, respectively. As we assume no mass accumulation through a cross section at any axial location within a micro heat pipe, the liquid’s cross-sectional area and velocity are inversely proportional to each other. As shown in Fig. 9, the liquid volume fraction, U, which is a function of the liquid’s cross-sectional area, exhibits a minimum value in the evaporator section. Therefore, the liquid velocity is the highest in the evaporator section. On the contrary, the liquid’s cross-sectional area is the highest in the condenser section, hence, its velocity is the minimum in the condenser section. According to Eq. (37), the circulation parameter involves only the velocity of both phases of the working fluid. As depicted in Fig. 8, the dielectric pump of the second case (Ldp ¼ La þ Lc ) exert no force in the evaporator section. In the third case (Ldp ¼ Lc ), the electrical force only prevails in the condenser. In the absence of electrical force, the liquid saturation profiles remain unchanged, resulting in an unaffected liquid and vapour velocities, at a particular section of a micro heat pipe. To provide a more justifiable comparison, we plot the mass flow rate profile to investigate the circulation rate of working fluid as it takes into account both cross-sectional area and velocity of the working fluid, as mentioned previously. Based on the principle of mass conservation, it is assumed that no mass accumulates anywhere along the axial direction. Therefore, the local axial mass flow rates of the liquid and vapour phases are equal in magnitude. Fig. 12 plots the mass flow rate variations

Fig. 12. Mass flow rate profiles of an optimally charged silicon-water micro heat pipe operating under a positively applied voltage of 300 V (black color lines) and 650 V (red color lines), for different dielectric liquid pump lengths. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

F.L. Chang, Y.M. Hung / International Journal of Heat and Mass Transfer 108 (2017) 257–270

267

in the axial direction of an optimally charged silicon-water micro heat pipe, subject to different length of dielectric pump and applied voltage. All the mass flow rate profiles manifest the same trend where they are equal to zero at the evaporator and condenser ends, and peak at the adiabatic section. This is because the evaporation and condensation processes transpire in the evaporator and condenser sections, respectively. For a particular dielectric pump length, the mass flow rate increases with the applied voltage. In _ which is evaluated from addition, the average mass flow rate, m, _ is plotted as a function of the applied voltits local counterpart, m, _ is expressed as age in Fig. 13. The average mass flow rate, m,

_ ¼ m

Z 0

1

_ ^x ¼ md

Z 1 Q_ Lt ^ ^x: md hfg Le 0

ð38Þ

_ as a function of the Fig. 13 depicts the average mass flow rate, m, applied voltage of an optimally charged silicon-water micro heat pipe subject to dielectric liquid pump of different lengths. Consistent with the observation in Fig. 10, both heat transport capacity and average mass flow rate possess the same trend. When the applied voltage is small, both heat transport capacity and average mass flow rate do not show significant difference for different lengths of dielectric pump. At higher applied voltage, the heat transport capacity and the average mass flow rate increase with the applied voltage for the first two cases. For the third case where the dielectric pump covers only the condenser section, both Q_ cap _ remain insensitive to the increase of the applied voltage. and m When comparing the mass flow rate profiles depicted in Fig. 12, a higher mass flow rate variation corresponds to a higher average mass flow rate in Fig. 13. The close resemblance between the heat transport capacity and the average mass flow rate plots indicates a strong relationship between the heat transport capacity and the circulation rate of working fluid. We conclude that the increase in working fluid’s circulation rate inherently leads to an increment in the heat transport capacity. Fig. 14 plots the axial solid wall temperature variations of an optimally charged silicon-water micro heat pipe, with different lengths of dielectric liquid pump for two different applied voltages: 300 V and 650 V. The solid wall temperature profiles exhibit a similar trend but differ in the temperature drop across the evaporator and condenser ends, for different applied voltages. For a fixed dielectric pump length, the temperature profile increases with the favourably applied voltage. Following this, in Fig. 15, the axial solid wall temperature drop across the evaporator and condenser ends, DT s , is plotted as a function of the applied voltage. The optimally charged silicon-water micro heat pipe is subject to different

Fig. 14. Axial solid wall temperature profiles of an optimally charged silicon-water micro heat pipe operating under a positively applied voltage of 300 V (black lines) and 650 V (red lines), for different dielectric liquid pump lengths. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Axial solid wall temperature difference, DT s , as a function of the applied voltage, V, of an optimally charged silicon-water micro heat pipe subjected to different dielectric liquid pump lengths.

dielectric liquid pump lengths. Similarly, by comparing Fig. 15 with Fig. 10, which plots the heat transport capacity as a function of the applied voltage, it is observed that both plots possess the same trend. We can deduce that the temperature drop is proportionately related to the corresponding heat transport capacity, which is attributed by the applied voltage. As shown in Fig. 15, at a high applied voltage (V > 650 V), a high temperature drop prevails in the micro heat pipe subject to the longest dielectric liquid pump, consistent with the solid wall temperature profile in Fig. 14. For the range of voltage being investigated, it is noteworthy that the maximum solid wall temperature drop across the evaporator and condenser ends is only about 1.5 °C. The low maximum temperature drop validates the commonly held notion that heat pipes are regarded as ‘‘isothermal” devices. However, in view of the necessity to apply such high voltage to achieve a relatively low heat transfer enhancement as compared to the electroosmoticallydriven micro heat pipe [8], this approach is deemed to be less practical in applications. 3.5. Performance comparison among different configurations and electrode lengths

_ as a function of the applied voltage, V, of an Fig. 13. Average mass flow rate, m, optimally charged silicon-water micro heat pipe subjected to different dielectric liquid pump lengths.

As discussed earlier, when the favourable electric body force is applied, the liquid volume profile exhibits its maximum value at ^ xm , where the maximum liquid volume fraction corresponds to

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4. Conclusions

Fig. 16. The maximum allowable applied voltage, V max , and its corresponding heat transport capacity, Q_ cap , of an optimally charged silicon-water micro heat pipe subjected to different dielectric liquid pump configurations and lengths. Here, the micro heat pipe is operating at 60 °C.

the flooding limit, Ufl , as defined in Eq. (33). This onset of flooding is attributed to the balance of the two competing opposite effects, which consist of the electric force and the heat loaded to the micro heat pipe [8,36]. As we apply higher voltage favourably, the ^ xm moves further away from the condenser end. The axial position of the onset of flooding, ^ xm , occurs in the condenser section where the liquid forms a throat that shrinks the passage of the vapour flow [8,36]. When the liquid coalesces and seals up the throat, the remaining condenser section is flooded and the flow of vapour will be blocked. Subsequently, the condensation heat transfer is impeded in the condenser section. The performance of a micro heat pipe deteriorates, or to the worst extent, it leads to a complete failure. In the present investigation, we always assume that ^ xm could only be located within the condenser section. If the ^ xm moves beyond the adiabatic-condenser border into the adiabatic section, we deem that the micro heat pipe malfunctions. Fig. 16 summarises the maximum allowable voltage that can be applied favourably to the micro heat pipe and its corresponding heat transport capacity which could be achieved, of an optimally charged silicon-water micro heat pipe, for different dielectric liquid pump configurations and lengths. Comparing Model 1 and Model 2, it is obvious that the range of the applied voltage of Model 2 is larger than that of Model 1. However, at the maximum allowable voltage, Model 1 possesses a higher heat transport capacity than Model 2, despite a much lower applied voltage. The thermal performance of micro heat pipe with different lengths of dielectric pump is also analysed. The maximum allowable voltage for the case where the dielectric pump covers the entire length of micro heat pipe is slightly higher than the second case where the pump covers the adiabatic and condenser sections. With a relatively higher maximum allowable applied voltage (19.4% increase), the former manifests a higher heat transport capacity (20.6% increase). The micro heat pipe subject to the shortest pump length (which covers only the condenser section) has the lowest heat transport capacity among others. Referring to Fig. 10, which depicts the variation of heat transport capacity with the applied voltage, for the case of the shortest pump length, its heat transport capacity remains insensitive to the increase in applied voltage, particularly at high applied voltage. This leads to a maximum allowable voltage which approaches an infinite value. Even with a very high voltage of 40 kV, the micro heat pipe fails to exhibit a significant increase in the heat transport capacity, as compared to the case without the dielectric liquid pumping effect. Therefore, it can be claimed that the dielectric pump performs the best when it covers the entire length of micro heat pipe (Model 1).

We investigate the effects of dielectric liquid pumping flow in enhancing the working fluid’s circulation effectiveness, and hence, the heat transport capacity of a micro heat pipe. Different configurations and lengths of dielectric liquid pump are compared and discussed. The performance of each dielectric pump depends solely on its effective electric field. Two different dielectric pump designs are compared: the planar electrodes and the pin electrodes. It is found that the micro heat pipe employing the planar electrodes performs better than that with the pin electrodes. The performance of each dielectric liquid pump depends merely on the electric field exerted. The favourably applied electric field enhances the circulation rate of condensate back to the evaporator, and inherently, increases the heat transport capacity of micro heat pipe. When the voltage is applied on the micro heat pipe, the axial position of the onset of flooding moves away from the condenser end towards the adiabatic section. At the onset of flooding, the liquid forms a throat that shrinks the passage of the vapour flow. Although the circulation of condensate is enhanced with the application of dielectric pumping, the risk of liquid coalescence and blocking the vapour flow in the condenser is also increased. Different dielectric pump length (with the planar electrodes) results in different enhancement rates of circulation and heat transport capacity of a micro heat pipe. This is mainly attributed to the effective electric fields exerted on different amount of condensate in different sections of a micro heat pipe. Nonetheless, at a particular applied voltage, regardless of the electrode length, the total amount of electrical energy of the pump is conserved. The role of the planar electrode located at the evaporator is of minor importance due to the fact that the amount of liquid in the evaporator is extremely small attributed to the intensive evaporation. Therefore, the difference in performance between the cases where the electrode spans the condenser and the adiabatic section and the electrode covers the entire length of micro heat pipe is not significant. However, the dielectric pump performs the best when it covers the entire length of micro heat pipe. Compared to the case without dielectric pumping flow, significant enhancement in the heat transport capacity can be attained where the maximum enhancement amounts to 222%. With the application of dielectric pump in a micro heat pipe, the solid wall temperature drop is sufficiently small, which is a typical phenomenon for a phase-change heat transfer device. Acknowledgment The authors acknowledge the support of the Ministry of Higher Education, Malaysia, through the Fundamental Research Grant Scheme (FRGS/1/2013/SG04/MUSM/02/1). Appendix A A.1. Conservation of energy of solid wall We apply the principle of energy conservation to the control volume depicted in Fig. 3(b) and the ordinary differential equation governing the axial temperature of the solid wall can be derived as 2

ks As

d Ts 2

dx

 hP sl ðT s  T l Þ þ q_ w ¼ 0;

ð39Þ

where 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, q_ w is the rate of heat transfer per unit axial length between the micro heat pipe and the ambient, and Psl is the length of the solid-liquid interface of the cross section at a given axial position x, which is measured from the evaporator end of the micro heat

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pipe. The mean heat transfer coefficient h between the solid wall and the liquid phase of the working fluid is related to the Nusselt number as

Nu ¼

hDH;l ; kl

ð40Þ

where kl is the thermal conductivity of the liquid phase. For a hydrodynamically and thermally fully developed laminar duct flow, the Nusselt number is generally a constant of order one dependent on the duct’s cross section and the thermal boundary conditions [29]. In the absence of theoretical and experimental results of the appropriate Nusselt number to be employed in the present study, we take the arithmetic mean of the two extreme Nusselt numbers for constant wall heat flux and constant wall temperature as 3.26 for a square cross section. In Eq. (40), the Nusselt number is a constant but the heat transfer coefficient h is evidently a function of the axial position, x, because the cross-sectional hydraulic diameter of the liquid phase, DH;l , varies with the axial position. By considering # ¼ T s  T l and ^ x ¼ x=Lt , Eq. (39) can be rewritten as 2

d #  k2 # þ H ¼ 0; d^x2

ð41Þ

where

k2 ¼

  2! C 2sl kl Lt Nu; 4 ks As

ð42Þ

and C sl is a geometrical parameter (see Appendix B). By assuming the heat fluxes are distributed uniformly over the evaporator section and the condenser section, respectively, the quantity H characterizing the heat load Q_ in Eq. (41), is given by

8 _ 2 > H ¼ Q Lt ; 0 6 ^x 6 LLet ; > > < e ks As Le Le < ^x < 1  LLct ; H ¼ 0; Lt > > 2 > _ : Hc ¼  ksQALstLc ; 1  LLct 6 ^x 6 1:



d#

d#

¼ ¼ 0: d^x
^x¼0 d^x
^x¼1

ð44Þ

The prescribed boundary conditions ensure that the heat rates entering and leaving the micro heat pipe are the same and equal to Q_ as a micro heat pipe is typically embedded in the semiconductor substrate where its ends are properly insulated and considered to be adiabatic. Eq. (41) can be integrated piecewise by imposing the continuity of temperature and heat flux at ^x ¼ Le =Lt and ^x ¼ 1  Lc =Lt . By applying the boundary conditions of Eq. (44), we obtain [2]

8 0 6 ^x 6 LLet ; C ðek^x þ ek^x Þ þ Hk2e ; > > < e Le 6 ^x 6 1  LLct ; # ¼ C a1 ek^x þ C a2 ek^x ; Lt > > : C ðekð2^xÞ þ ek^x Þ þ Hc ; 1  Lc 6 ^x 6 1; c Lt k2

C a2 ¼



2k 2k C a2 þ He ðe

kLe =Lt

2

C a1 ¼

ðek

e



kLe =Lt

2k2 2

Cc ¼



Hc ekLc =Lt  ekLc =Lt þ He ekð1þLe =Lt Þ  ekð1Le =Lt Þ 2



k C a2 e

kð1Lc =Lt Þ

ek Þ Þ

þ C a1 e

k2 ek ðekLc =Lt þ ekLc =Lt Þ

ð45Þ

;

ð46Þ

ð47Þ  Hc

k C a2 ekLe =Lt þ C a1 ekLe =Lt  He k2 ðekLe =Lt þ ekLe =Lt Þ

;

ð48Þ

:

ð49Þ

A.2. Conservation of energy of working fluid ^ in Eq. (22) can be obtained The dimensionless mass flow rate, m, by applying the energy conservation requirement in the liquid domain. As depicted in Fig. 3(b), the heat absorbed by the liquid from the solid wall is taken up as the latent heat of evaporation and the energy equation is given by

^ dm ¼ gðT s  T l Þ; d^x

ð50Þ

where

k2 k A

g¼ _ s s : mref hfg Lt

ð51Þ

^ With the boundary condition mð0Þ ¼ 0, Eq. (50) is solved to obtain the dimensionless mass flow rate profile as

8 

 g > C e ek^x  ek^x þ Hke ^x ; 0 6 ^x 6 LLet ; > k > > < Le ^ ¼ gk ðC a2 ek^x  C a1 ek^x þ K 1 Þ; 6 ^x 6 1  LLct ; m Lt > > > > : g ½C ðek^x  ekð2^xÞ Þ þ Hc ^x þ K ; 1  Lc 6 ^x 6 1; c 2 k k Lt

ð52Þ

where the constants K 1 and K 2 are given by

K 1 ¼ C e ðekLe =Lt  ekLe =Lt Þ þ C a1 ekLe =Lt  C a2 ekLe =Lt þ

He Le kLt

;

ð53Þ

and





K 2 ¼ C e ekLe =Lt  ekLe =Lt þ C a1 ekLe =Lt  ekð1Lc =Lt Þ



þ C a2 ekð1Lc =Lt Þ  ekLe =Lt þ C c ekð1þLc =Lt Þ  ekð1Lc =Lt Þ þ

He Le þ Hc ðLc  Lt Þ kLt

:

ð54Þ

Appendix B The geometrical parameters prevail in the governing equations essentially depend on the contact angle between the liquid phase and the solid wall h, the half corner angle /, the groove width w and the number of corners of the cross-section N. The crosssectional area A and the periphery of the square micro heat pipe P are given, respectively as

A ¼ w2 ;

ð55Þ

and

P ¼ Nw:

ð56Þ

On the other hand, the micro heat pipe’s hydraulic diameter is defined as

Dh ¼



4A 4w ¼ : P N

ð57Þ

The cross-sectional areas of the liquid and vapour phases are Al and Av , respectively, which are expressed as

Al ¼ AU;

;

kð1Lc =Lt Þ

Ce ¼

ð43Þ

Two boundary conditions are prescribed to solve Eq. (41) and they are given by

where

2

ð58Þ

and

Av ¼ Að1  UÞ;

ð59Þ

where the volume fraction occupied by the liquid phase U can be related to the meniscus radius of curvature r, by

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U¼

ð!rÞ2 : A

ð60Þ

The angular parameter ! is defined as



! ¼ N1=2

cos h cosð/ þ hÞ p þ ð/ þ hÞ  sin / 2

1=2 :

ð61Þ

The solid wall-liquid contact line length Psl , the solid wall-vapour contact line length Psv , and the liquid-vapour contact line length Plv , are respectively given by

Psl ¼ C sl ðAUÞ1=2 ;

ð62Þ

Psv ¼ Nw  C sl ðAUÞ1=2 ;

ð63Þ

and

Plv ¼ C lv ðAUÞ1=2 :

ð64Þ

The geometrical constants C sl and C lv are defined, respectively, as

C sl ¼

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

ð65Þ

and

C lv ¼ 2N1=2

hp 2

 ð/ þ hÞ

icos h cosð/ þ hÞ sin /

1=2

þ ð/ þ hÞ 

p 2

: ð66Þ

The hydraulic diameters of liquid and vapour phases are, DH;l and DH;v , respectively, given by

DH;l ¼

4Al 4ðAUÞ1=2 ¼ ; P sl C sl

ð67Þ

4Av 4Að1  UÞ ¼ : Psv þ Plv Nw þ ðC lv  C sl ÞðAUÞ1=2

ð68Þ

and

DH;v ¼

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