Thermal analysis of Al₂O₃/water nanofluid-filled micro heat pipes

Abstract

Micro heat pipe is an effective micro-scale cooling device and its operation is sustained by two-phase heat transfer and capillary suction. Based on the first-principles calculation to yield the heat and fluid flow characteristics, this work delineates the thermal performance of micro heat pipes utilizing a nanofluid as a working fluid. Heat transport capacity and thermal resistance are compared as the performance indicator of nanofluids at different nanoparticle concentration fractions in an optimally charged micro heat pipe. The performance of a nanofluid-filled micro heat pipe is deemed to decline if we use heat transport capacity as a performance indicator, which is contrary to the indication of thermal resistance. We elucidate the factors contributing to the contradictory results in the thermal performance of a nanofluid-filled micro heat pipe using different performance indicators to explain the underlying physical significance of the use of a nanofluid on the performance of micro heat pipes.

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Introduction

Efficient thermal management has been the focus of keen interest in the electronics cooling community in the light of a continued miniaturization of the electronic components, giving rise to a dramatic increase in heat generation per unit volume. High rates of heat generation provoke high operating temperatures and jeopardize the reliability and lifespan of electronic components. A micro heat pipe has been a promising micro-scale cooling device since its inception by Cotter in 1984.¹ Micro heat pipes rely on phase-change heat transfer, i.e. evaporation and condensation, and capillary suction to function as an effective cooling device. The applied heat at the evaporator section evaporates the working fluid, and the resultant vapour flows along the pipe to the condenser section, in which it condenses and the latent heat of vaporization is dissipated to the surroundings. Unlike conventional heat pipes, micro heat pipes are wickless; instead, the capillary pressure provided by their sharp-angled corners is utilized to drive the condensate back to the evaporator. The operation of a micro heat pipe is sustained by the perpetuation of the cycle of phase-change and circulation of its working fluid.

Because the operation of a heat pipe is governed by the transport processes of the working fluid, its overall thermal performance is predominantly dictated by the thermal properties of working fluids, and the choice of a working fluid has been a key issue.^2,3 The anomalous increase in the thermal conductivity of the dispersions of metal nanoparticles and metal oxide nanoparticles in fluids has stimulated substantial interests in the thermal transport mechanisms.^4–7 This type of a nano-colloidal fluid was coined as a “nanofluid.”⁸ In the light of their unique properties, nanofluids are utilized in an array of applications, including micro-scale cooling of electronics devices, chemical processing and biomedical research. Reviews on the investigations of the thermal application of nanofluids have been well documented.^8–10 Recently, relevant to micro-scale cooling, several studies employing nanofluids as working fluids in the thermal analysis of micro heat pipes have been reported.^11–13

In most of the existing experimental studies that use a nanofluid as the working fluid, the thermal performance of a micro heat pipe was quantified by thermal resistance as the performance indicator.^11,14,15 Thermal resistance can be obtained by evaluating the axial solid wall temperature drop between the evaporator and the condenser sections of a micro heat pipe. Based on this performance indicator, it was claimed that the use of a nanofluid in enhancing the heat transport capability of micro heat pipes is favorable. In addition, the only theoretical investigation that employed a nanofluid as a working fluid in a micro heat pipe also involved thermal resistance as the performance indicator.¹³ It was observed that the use of nanofluids led to a significant increase in heat transfer with a decrease in thermal resistance. However, in order to quantify the performance of a micro heat pipe, it is more appropriate to adopt the heat transport capacity, instead of the thermal resistance, as the performance indicator, following the common practice in most of the previous studies, either experimental or theoretical.^2,3,16–24 The heat transport capacity is the maximum possible heat load that a micro heat pipe can sustain without the occurrences of dryout and flooding.^2,3,16–22 Compared to the thermal resistance, which is simply dependent on the axial wall temperature of a micro heat pipe, the heat transport capacity is evaluated by solving the transport equations of the working fluid taking into account the geometry and dimensions of the micro heat pipe as well as the heat conduction in the solid wall.^3,19

The main focus of this paper is to investigate the effect of a nanofluid on the thermal transport performance of an optimally charged micro heat pipe. A mathematical model is developed based on the laws of conservation and Young–Laplace equation. A comprehensive analysis of a nanofluid-filled micro heat pipe is performed, emphasizing on the thermal performance based on the heat transport capacity at different nanoparticle concentration fractions. For comparison, the thermal performance based on the thermal resistance derived from the solid wall axial temperature drop is used as a benchmark. Other working fluid related performance indicators, such as figure of merit of working fluid, evaporation rate, liquid circulation rate and internal convective resistance, are also assessed. We delineate the factors contributing to the contradictory results in the thermal performance of a nanofluid-filled micro heat pipe using different performance indicators; thus, the underlying physical significance of the nanofluid on the thermal performance of a micro heat pipe is elucidated.

Thermo-physical properties of nanofluid

The addition of nanoparticles momentously changes the thermo-physical properties of the base fluid. Herein, the nanoparticles are assumed to be uniformly dispersed within a liquid pool; moreover, only changes in the thermo-physical properties of the liquid phase are involved, while those of the vapor phase are assumed to be same as those of the base fluid. The thermo-physical properties are also assumed to be constant at a given operating temperature. Al₂O₃/water nanofluid is selected as the working fluid. The effective thermal conductivity of the nanofluid is greater than that of its base fluid with increasing nanoparticle volume fraction, φ. By taking into account the Brownian motion-induced convection, the effective thermal conductivity can be expressed as follows:²⁵

k_nf = C_kk_f (1)

where C_k is the coefficient of enhancement, defined as

In eqn (2), κ is the thermal conductivity ratio, given as κ = k_p/k_f; Pr = c_p,fμ_f/k_f is the Prandtl number of the nanofluid; φ is the nanoparticle volume fraction; and Bi_nf = 2R_bk_f/d is the Biot number of the nanoparticle, where d is the diameter of the nanoparticle and R_b is the interfacial resistance. The Brownian–Reynolds number is given as with k_b as the Boltzmann constant and v_f as the fluid kinematic viscosity. For Al₂O₃/water nanofluid, the interfacial resistance R_b is estimated to be 0.77 × 10⁻⁸ km² W⁻¹, m = 2.5 and A = 40 000.²⁵ Due to the existence of slip velocity between nanoparticles and the base fluid, the conventional Einstein model is incapable of capturing these rheological features of a nanofluid. A modified Einstein model has been introduced and the effective viscosity is given as follows:²⁶

μ_l,nf = C_μμ_f (3)

where the coefficient C_μ is expressed as

For the suspension of Al₂O nanoparticles in water, the empirical constants are ξ = −1/4 and n = 280 with D being the inner diameter of the micro-channel. In the absence of theoretical formulas to adequately evaluate the viscosity and thermal conductivity without the loss of generality for Al₂O₃/water nanofluid, we employ the effective thermal conductivity and viscosity parameters suggested by Prasher et al.²⁵ and Jang et al.,²⁶ respectively. The coefficients C_k in eqn (2) and C_μ in eqn (4) can be easily modified to suit other semi-empirical models of thermal conductivity and empirical models of viscosity in the existing literature.

The effective liquid density of a nanofluid as a function of the volume fraction of a nanoparticle, φ, is given as follows:²⁷

ρ_l,nf = ρ_f(1 − φ) + ρ_pφ (5)

The latent heat, which is proportional to the volume fraction that is defined using a lever rule between the liquidus and solidus temperatures, is defined as follows:²⁸

Mathematical formulation

A steady-state one-dimensional mathematical model is developed by employing the laws of conservation and the Young–Laplace equation. For the current analysis, a micro heat pipe is positioned horizontally and optimally filled with a working fluid at its designated operating temperature. An optimally charged micro heat pipe is loaded with the maximum possible heat input such that it operates under the condition of simultaneous onsets of dryout at its evaporator end and flooding at its condenser end, as depicted in Fig. 1(a). The cross-sectional proportions of the liquid and vapor phases along the axial direction are also illustrated. Because the capillary limit dominates over other operating limits, we studied only the capillary limit in our formulation.²⁹

Fig. 1 (a) Schematic diagram of an optimally charged micro heat pipe. Infinitesimal control volume of micro heat pipe associated with (b) conservation of energy in solid wall and working fluid, and (c) conservation of momentum for liquid and vapor phases.

Energy conservation is applied to the control volume of solid wall, as shown in Fig. 1(a). The governing equation for the axial solid-wall temperature is derived as follows:

where k_s is the thermal conductivity of solid, A_s is the cross-sectional area of the solid wall, P_sl is the perimeter of solid–liquid interface, T_s and T_l are the solid and liquid temperature, respectively, and q̇ is the rate of heat transfer per unit axial length. By defining θ = T_s − T_l and x̂ = x/L_t, eqn (7) can be rewritten aswhere

In eqn (9), C_sl is a geometrical parameter, which is given in Appendix A. The Nusselt number Nu_D is specified as 2.68, which is an average value of two extreme cases of constant wall heat flux and constant wall temperature. Based on the axiom that uniform heat fluxes are applied at the evaporator and condenser sections, with the adiabatic section insulated, the parameter Θ can be expressed in terms of the heat load Q̇ as

By considering that the sealed ends of the micro heat pipe are properly insulated, the boundary conditions required to solve eqn (8) are given by

By imposing the requirement of no-temperature-jump condition and temperature gradient continuity at x̂ = L_e/L_t and x̂ = 1 − L_c/L_t, the temperature distribution in piecewise form is obtained as

where

C_a2={Θ_e[cosh(λ(−L_t + L_e)/L_t) + sinh(λ(−L_t + L_e)/L_t) − cosh(λ(L_t + L_e)/L_t) + sinh(λ(L_t + L_e)/L_t)] − 2Θ_c sinh(λL_c/L_t)}/4λ² sinh(λ) (13)

C_a1 = [λ²C_a2 + Θ_e sinh(λL_e/L_t)]/λ² (14)

C_c = {λ²[(C_a1 + C_a2)cosh(λ(−L_t + L_c)/L_t) + (C_a1 − C_a2)sinh(λ(−L_t + L_c)/L_t)] + Θ_c}/2λ²(cosh(λ) + sinh(λ))(cosh(λL_c/L_t)) (15)

C_e = {λ²[(C_a1 + C_a2)cosh(λL_e/L_t) + (C_a2 − C_a1)sinh(λL_e/L_t)] − Θ_e}/2λ² cosh(λL_e/L_t) (16)

The principle of momentum conservation is applied to a differential control volume of fully developed laminar incompressible fluid flow, as depicted in Fig. 1(b). The governing differential equation for liquid and vapor phases are given as

andrespectively, where A, u and p denotes the cross sectional area, velocity and pressure, respectively. The shear stress at the solid–liquid (τ_sl), solid–vapor (τ_sv) and liquid–vapor (τ_lv) interface, is given aswhere f = Po_i/Re_j is the fanning friction factor, and the Reynolds number Re_j is defined aswhere μ_j is the viscosity and D_H,j is the hydraulic diameter for phase j. For liquid phase (j = l), μ_l,nf in eqn (3) is applicable to the nanofluids. For laminar flow, the Poiseuille number, Po, is a constant and only depends on the geometry of the micro heat pipe. Because liquid flow is confined to the sharp edges of the wall, the cross section of the liquid flow could be assumed to be a triangle and the value of Poiseuille number is set as 13.3.³⁰ On the other hand, the cross section of the vapor flow varies from an equilateral triangle at the evaporator section to a circle at the end of the condenser due to the flooding limit. The Poiseuille number of the vapor flow is calculated by averaging the values of an equilateral triangle and a circle; in this case it is equal to a value of 14.7.

The pressure difference of the liquid and vapor phases is defined using the Young–Laplace equation as follows:

where p, σ, and r are the pressure, surface tension and radius of curvature of the liquid–vapor interface, respectively. The surface tension of the nanofluid in the convection process of boiling and condensation can be considered invariant with that of the base fluid.^31,32 The radius of curvature, r, is related to the liquid saturation by the following equation:

r = A_t^1/2s^3/2 (22)

The saturation, s, is the liquid volume fraction of the total volume of micro heat pipe given as

where A_l is cross-sectional area occupied by liquid phase and A_t is the total cross sectional area. By differentiating eqn (21) and substituting eqn (22) into it, we obtain

In eqn (24), ϖ is an angular parameter, as shown in Appendix A. By introducing the following non-dimensional variables

and by utilizing eqn (17) and (18), eqn (24) can be rewritten to yieldwhere the Weber number, We = [ρ_l,nf(ṁ_ref/ρ_l,nfA_t)²A_t^1/2]/σ; Capillary number, ; Bond number,; kinematic viscosity ratio, γ = (μ_vρ_l,nf)/(μ_l,nfρ_v); and density ratio, ε = ρ_v/ρ_l,nf. The functions F₁(s) and F₂(s) in eqn (26) are given bywhere N is the number of corners, w is the side length, and C_lv and C_sl are the geometrical parameters, as depicted in Appendix A.

From Fig. 1(a), the heat absorbed by the liquid from the solid wall is consumed as the latent heat of evaporation. The dimensionless mass flow rate m̂ is expressed as

where η is a parameter given by

With the “boundary” condition of m̂(0) = 0, eqn (29) can be integrated to yield

where

K₁ = 2C_e sinh(λL_e/L_t) + (C_a1 − C_a2)cosh(λL_e/L_t) − (C_a2 + C_a1)sinh(λL_e/L_t) + (Θ_eL_e)/(λL_t) (32)

K₂ = (2C_e − C_a1 − C_a2)sinh(λL_e/L_t) + (C_a1 − C_a2)cosh(λL_e/L_t) + (C_a2 − C_a1 − C_c)cosh(λ(−1 + L_c/L_t )) + (C_c − C_a2 − C_a1)sinh(λ(−1 + L_c/L_t)) + C_c[cosh([λ(L_t + L_c)]/L_t) + sinh([λ(L_t + L_c)]/L_t)] + [Θ_eL_e + Θ_c(L_t − L_c)]/λL_t (33)

The maximum heat load that a micro heat pipe can carry without the occurrence of dry-out and flooding is denoted as the heat transport capacity, Q_cap, where the onset of dry-out and the onset of flooding take place simultaneously at the evaporator end and condenser end, respectively.^3,18–22 To calculate the heat transport capacity, eqn (26) is integrated using the fourth order Runge–Kutta method with a step size of 0.0001 by fulfilling the onset conditions. The onset of dry-out at the evaporator end corresponds to s(0) = s_cl = 0.0001, while the onset of flooding at the condenser end, s(1) = s_fl is given as

By integrating eqn (26), the heat input Q̇ is iterated using the false position method.³³ When the onset condition of the condenser end is satisfied with a convergence criterion of 10⁻⁷, the heat input Q̇ is taken as the heat transport capacity, Q_cap.

Results and discussion

The figure of merit, Me = ρ_lh_fgσ/μ_l, which is merely a group of thermo-physical properties of fluid, is widely used to provide a quick and convenient approach for selecting the working fluid of a heat pipe.³⁴ It consists of two elements, namely, evaporation rate and liquid recirculation effectiveness. The ratio σ/μ_l with a dimension of velocity is considered to be a measure of recirculation rate, whereas the term ρ_lh_fg can be interpreted as the latent heat of vaporization per unit volume of the liquid phase.^3,35 Fig. 2(a) plots the figure of merit for both water and nanofluid with different volume fractions as a function of operating temperature. It is observed that water has the highest merit number. For nanofluids, the merit number decreases with increasing nanoparticle volume fraction. The recirculation rate and the latent heat of vaporization per unit volume are plotted as a function of operating temperature, as shown in Fig. 2(b) and (c), respectively. A nanofluid exhibits a lower σ/μ_l and ρ_lh_fg compared to water. The decrease in σ/μ_l is attributed to the increase in the viscosity of a nanofluid, while ρ_lh_fg decreases due to the decrease in the latent heat of vaporization of a nanofluid. The decrements of the circulation rate and the strength of evaporation lead to the decrease in the figure of merit of nanofluids. Based on a quick analysis on the figure of merit of the working fluid, we can observe that the performance of the nanofluid is inferior to that of its base fluid. This poses an ambiguous question because nanofluids have been claimed to enhance the performance of heat pipes.¹³ Therefore, a more comprehensive investigation by employing different types of performance indicators is motivated in the present study.

Fig. 2 (a) Figure of merit, (b) ratio of surface tension and liquid dynamic viscosity, and (c) latent heat of evaporation per unit volume as a function of operating temperature.

The figure of merit is a fast and a convenient method of evaluating the merits of working fluids based on their thermo-physical properties, whereas the heat transport capacity is obtained by solving the transport equations taking into account the dimensions and geometry of the heat pipe. The heat transport capacity can be employed to indicate the actual performance of a micro heat pipe. The present numerical results are compared with the experimental results by Babin et al.,³⁶ as depicted in Fig. 3. The theoretical results agree well with the experimental data in terms of the order of magnitude. Thus, they provide a solid validation of the present model. Fig. 4 plots the heat transport capacity of water and nanofluid with different volume fraction as a function of operating temperature. The heat transport capacity, Q_cap, increases with the operating temperature. The fact that water possesses the highest heat transport capacity, while the heat transport capacity of the nanofluid decreases with the nanoparticle volume fraction contradicts the previous findings that were obtained using thermal resistance.

Fig. 3 Heat transport capacity as a function of operating temperature at different contact angles for an optimally nanofluid-filled micro heat pipe. Experimental data from Babin et al.³⁶ are used for comparison.

Fig. 4 Heat transport capacity of the micro heat pipe filled with water and nanofluid with different volume fraction as a function of operating temperature.

The low thermal resistance of a nanofluid filled micro heat pipe is due to the decrease of solid wall temperature drop between the evaporator and the condenser section. The solid wall temperature profiles of a micro heat pipe filled with a nanofluid are plotted in Fig. 5. We observe that the temperature drop is smaller for the nanofluid of high nanoparticle volume fraction and vice versa. This is noteworthy because the solid wall temperature profile can be practically obtained through experimental investigation.

Fig. 5 Axial temperature profiles of the solid wall of the micro heat pipe filled with water and nanofluids.

In experiments, the temperature drop, as well as the thermal resistance, is commonly employed as a performance indicator of a heat pipe. Fig. 6 shows the temperature drop as a function of the applied heat load to provide a justification for the accuracy of the theoretical model. The experimental data reported by Peterson and Ma³⁷ for a silver trapezoidal micro heat pipe at a wall thickness of 0.19 mm filled with water is included for comparison. At the operating temperature of 60 °C, the theoretical results fit well with the experimental data. It provides a sound validation for the estimation of the temperature drop of solid wall.

Fig. 6 Longitudinal temperature drop of solid wall as a function of the applied heat load. The experimental data by Peterson and Ma³⁷ are included for comparison.³⁷

The overall thermal resistance, R_th = ΔT/Q̇_in, can be evaluated based on the temperature drop. Fig. 7 depicts the overall thermal resistance of the nanofluid-filled micro heat pipe operated at different operating temperatures. For an optimally operated micro heat pipe, the applied heat load, Q̇_in, is equal to the heat transport capacity, Q̇_cap. It is observed that the overall thermal resistance decreases with the operating temperature, showing an increase in the performance at a high operating temperature. The overall thermal resistance decreases with the nanoparticle volume fraction, indicating that the suspension of the nanoparticle in the working fluid enhances the performance of a micro heat pipe. This is contradictory to the finding in Fig. 4 when the heat transport capacity is employed as the performance indicator. Because the temperature drop of solid wall, ΔT, can be measured empirically, in most experimental investigations, the overall thermal resistance is used as the performance indicator. This explains why the use of nanofluids in a heat pipe is deemed to be favorable in experimental works.^9,38,39

Fig. 7 Overall thermal resistance of the nanofluid-filled micro heat pipe with different nanoparticle volume fractions operated at different operating temperatures.

To elucidate the role of the heat conduction of solid wall in the evaluation of the overall thermal resistance, we define the thermal resistance using two aspects: one based on heat transport in the solid wall and another based on heat transport in the liquid phase. The former parameter is given as^40,41

where Q̇_c is the net rate of the axial heat conduction in the solid wall. A small portion of the applied heat load is conducted in the solid wall to the condenser section, while the remaining portion is absorbed by the liquid phase and evaporated. Therefore, the heat transport capacity can be written as Q̇_cap = Q̇_c + Q̇_p,¹⁹ where Q̇_p is the portion of the applied heat load taken up as the latent heat of vaporization, which is denoted as heat transport by phase change, given as . The uniform heat flux over the evaporator and condenser of equal lengths leads to the upper limit of x = 0.5L_t in the integral. Another thermal resistance associated with the heat transport in the liquid phase is given as

This forms the resistance associated with heat transfer by convection between the internal wall surface and the liquid phase. The heat transport capacity comprises of two elements, namely, Q̇_c and Q̇_p. Fig. 8 depicts the ratios of Q̇_c/Q̇_cap and Q̇_p/Q̇_cap as a function of operating temperature. In Fig. 8(a), we observe that the axial heat conduction in the solid wall contributes less than 1% to the heat transport capacity, and in accordance with Fig. 8(b), we can ascertain that the heat transport capacity of the micro heat pipe is predominantly carried by the phase change heat transfer of the nanofluid, which increases with the nanoparticle volume fraction. A larger Q̇_p/Q̇_cap is associated with a greater effective thermal conductivity of the nanofluid and vice versa. However, as discussed earlier, the absolute value of Q̇_cap in Fig. 4 decreases with nanoparticle volume fraction, showing that the adverse effect of increased viscosity counteracts the favorable effect of increasing thermal conductivity, leading to a lower heat transport capacity.

Fig. 8 (a) The ratio of Q̇_c/Q̇_cap and (b) the ratio of Q̇_p/Q̇_cap of a micro heat pipe optimally filled with water and nanofluid with different nanoparticle volume fraction as a function of operating temperature.

Fig. 9 plots the thermal resistance of the micro heat pipe operated optimally at different operating temperatures. In Fig. 9(a), the thermal resistance associated with the net rate of axial heat conduction in the solid wall, ψ_s, increases with the nanoparticle volume fraction. As observed in Fig. 8(a), a smaller portion of the applied heat load is conducted in the solid wall from the evaporator section to the condenser section, leading to a smaller temperature drop when the nanoparticle volume fraction increases. However, by increasing the nanoparticle volume fraction, the degree of temperature drop, ΔT, is lower than that of Q̇_c, attributed to higher ψ_s, as deduced from eqn (35). Moreover, in Fig. 9(b), the thermal resistance associated with heat transfer by convection from the wall surface to the liquid phase, ψ_l, is also observed to be decreased with nanoparticle volume fraction. By referring to eqn (36), it can be seen that ψ_l is inversely proportional to the heat transfer coefficient, which is given as

Fig. 9 (a) Thermal resistance associated with the net rate of axial heat conduction in the solid wall, ψ_s, and (b) thermal resistance associated with heat transfer by convection from the wall surface to the liquid phase, ψ_l, of a micro heat pipe optimally filled with water and nanofluid with different nanoparticle volume fraction as a function of operating temperature.

From eqn (37), the heat transfer coefficient is proportional to the thermal conductivity of the liquid phase. Consequently, from eqn (36) and (37), we note that ψ_l is also inversely proportional to the liquid thermal conductivity. Therefore, decreased ψ_l is in accordance with a proportionate increase in liquid thermal conductivity when the nanoparticle volume fraction increases. Compared with the heat transport capacity, Q̇_cap, the thermal resistance associated convective heat transfer, ψ_l, only considers the liquid thermal conductivity, k_l, while the phase-change heat transfer and circulation of condensate are not considered. Thus, the variations in the latent heat of vaporization and the viscosity induced by the nanoparticle suspensions in the working fluid impart no effect on ψ_l. However, the heat transport capacity incorporates the effects of the change of all the thermo-physical properties, i.e. thermal conductivity, latent heat of vaporization and viscosity, which are closely associated with the variation in nanoparticle volume fraction in the nanofluid. This makes the contradictory results between the heat transport capacity and the thermal resistance understandable. Therefore, the performance evaluation of the application of a nanofluid in micro heat pipes is susceptible to the choice of a performance indicator. From a theoretical point of view, the performance of a nanofluid-filled micro heat pipe degenerates using heat transport capacity as a performance indicator. For empirical evaluation, the application of a nanofluid in a micro heat pipe enhances the performance by employing thermal resistance as a performance indicator.

Conclusions

A comprehensive analysis emphasizing the thermal performance evaluation based on the heat transport capacity and the thermal resistance is performed at different nanoparticle concentration fractions of an optimally charged micro heat pipe. From a theoretical aspect, the heat transport capacity, which indicates the actual performance of a micro heat pipe, incorporates the effects of the change of all the thermo-physical properties, i.e. thermal conductivity, latent heat of vaporization and viscosity, which are closely associated with the variation of nanoparticle volume fraction in the nanofluid. The performance of a nanofluid-filled micro heat pipe is deemed to decline when heat transport capacity is used as a performance indicator. This is consistent with the evaluation based on the figure of merit, a group of thermo-physical properties that provide a convenient approach in selecting a working fluid for heat pipe application. However, the use of nanofluids enhances the performance of a micro heat pipe when the thermal resistance is employed as a performance indicator, which is commonly used in experimental investigations. The factor contributing to the contradictory results in the thermal performance of a nanofluid-filled micro heat pipe using different performance indicators is elucidated. Compared with the heat transport capacity, the thermal resistance associated convective heat transfer only considers the liquid thermal conductivity, while other thermo-physical properties, such as the latent heat of vaporization and viscosity, are not considered. We reveal that the performance enhancement of nanofluid-filled micro heat pipes associated with thermal resistance is susceptible to a benchmark against heat transport capacity for a more rigorous evaluation.

Appendix A

The geometrical parameters appearing in the governing equation are associated with the contact angle between solid wall and liquid, θ, half corner angle, ϕ, groove width, w, and the number of corners, N, of the micro heat pipe. The cross sectional area and perimeter of the micro heat pipe are given as

A_t = w² sin(ϕ)cos(ϕ) (38)

and

P = Nw, (39)

respectively. The cross-sectional area of the liquid and vapor phases are given as

A_l = A_ts (40)

and

A_v = A_t(1 − s), (41)

respectively, where s is the liquid to total area fraction, which is related to the radius of curvature aswhere ω is the angular parameter defined as

The solid–liquid contact perimeter, P_sl, the vapor-wall contact perimeter, P_sv, and the liquid–vapor contact perimeter, P_lv, are given as

P_sl = C_sl(A_ts)^1/2 (44)

P_lv = C_lv(A_ts)^1/2 (45)

P_sv = Nw − C_sl(A_ts)^1/2, (46)

respectively. The geometrical constant C_sl and C_lv are defined asand

The hydraulic diameter of both liquid phase, D_H,l and vapor phase, D_H,v are given as

respectively.

Acknowledgements

This work was supported by the eScienceFund (04-02-10-SF0113), Ministry of Science, Technology and Innovation (MOSTI), Malaysia.

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