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Effects of a surface-tension gradient on the performance of a micro-grooved heat pipe: an analytical study

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Abstract

We investigate effects of surface-tension gradients on the performance of a micro-grooved heat pipe in this work. The surface-tension gradient force is accounted for in the present model, and expressions for radius of curvature, liquid pressure, liquid velocity, and maximum heat throughput are found analytically using a regular perturbation technique. With a favorable surface-tension gradient, the liquid pressure drop across the heat pipe can be decreased by ∼90%, and the maximum heat throughput can be increased by ∼20%. In contrast, using an unfavorable surface-tension gradient, the liquid pressure drop increases by ∼150%, and the maximum heat throughput decreases by ∼15%. For the same values of the favorable and unfavorable surface-tension gradients, the unfavorable effect is more pronounced than the favorable one. The effects of the surface-tension gradients are found to be increasing with the corner angle of a polygonal heat pipe. Adverse effects of the surface-tension gradient could be due to the variations in the liquid temperature and/or surfactant concentration. Nevertheless, a favorable situation where the surface-tension gradient can facilitate the liquid flow in a heat pipe can also be obtained using a suitable surfactant, surface charge, etc., and then the performance of a micro heat pipe can be improved.

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Abbreviations

a :

smaller side length/side length of a heat pipe polygon, m

b :

larger side length of a heat pipe polygon, m

A l :

liquid flow cross-section area, m2

B, B 1, B 2, C 1, C 11 , C 3, C 4 :

constant

F s :

force due to surface-tension gradient, N

f r :

friction factor

f 1 :

dimensionless length of a condensing section

f 2 :

dimensionless length of condensing and adiabatic sections

g :

acceleration due to gravity, m s−2

K′:

constant in the expression for B 2

La:

length of an adiabatic section, m (Fig. 1)

Lc:

length of a condensing section, m (Fig. 1)

Le:

length of an evaporative section, m (Fig. 1)

L :

length of a heat pipe, m

\( L_{\rm h}^\prime \) :

half of wetted length per corner, m

L h :

half of total wetted length, m

n :

number of corners in a heat pipe polygon

N Re :

Reynolds number

P l :

liquid pressure, N m−2

P vo :

constant vapor pressure, N m−2

\( \Updelta{p}_{\rm l}^\ast\) :

dimensionless liquid pressure difference between hot and cold ends

ΔP c :

capillary pressure, N m−2

ΔP g :

pressure loss due to gravity, N m−2

ΔP l :

pressure loss due to liquid flow, N m−2

ΔP v :

pressure loss due to vapor flow, N m−2

Q :

heat supplied to coolant liquid from substrate, W

Q cr :

maximum heat throughput, W

\( Q_{\rm c}^{\prime\prime}\) :

heat flux in a condensing section, W m−2

\( Q_{\rm e}^{\prime\prime}\) :

heat flux in an evaporative section, W m−2

R :

radius of curvature, m

R R :

reference radius of curvature, m

R*:

nondimensional radius of curvature

\( R_{\rm c}^{\ast}\) :

nondimensional radius of curvature at a cold end

\( R_{\rm h}^{\ast}\) :

nondimensional radius of curvature at a hot end

\( R_0^{\ast}\) :

nondimensional radius of curvature without surface-tension gradient

\( R_1^{\ast}\) :

nondimensional radius of curvature with the first-order perturbation variable

\( R_{01}^{\ast}\) :

\( R_0^{\ast}\) at f 1

\( R_{02}^{\ast}\) :

\( R_0^{\ast}\) at f 2

\( R_{11}^{\ast}\) :

\( R_{1}^{\ast}\) at f 1

\( R_{12}^{\ast}\) :

\( R_{1}^{\ast}\) at f 2

V l :

axial liquid velocity, m s−1

\( V_{l}^{\ast}\) :

nondimensional liquid velocity

V R :

reference liquid velocity, m s−1

W b :

perimeter of a heat pipe polygon, m

x :

coordinate along the heat pipe length starting from a cold end, m

X*:

nondimensional coordinate along a heat pipe length starting from a cold end

α:

half of the corner angle of a heat pipe polygon, rad

β:

inclination of a substrate with horizontal, rad

ɛ:

perturbation variable

\(\Uppsi,\xi, \acute{\eta}\) :

used as an integration variable

γ:

contact angle, rad

ϕ:

curvature, rad

λ:

latent heat of vaporization of a coolant liquid, J kg−1

μl :

viscosity of a coolant liquid, kg m−1 s−1

ρl :

density of a coolant liquid, kg m−3

σ:

surface tension of a coolant liquid, N m−1

σ*:

dimensionless surface tension of a coolant liquid

σm :

mean surface tension of a coolant liquid, N m−1

σh :

surface tension of a coolant liquid at the hot end, N m−1

σc :

surface tension of a coolant liquid at the cold end, N m−1

τw :

wall shear stress, N m−2

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Acknowledgments

The valuable suggestions from Dr. Manohar S. Sohal, Idaho National Laboratory, USA, are gratefully acknowledged. Insightful comments from the reviewers helped improve this manuscript.

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  1. Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue, SE, Minneapolis, 55455, MN, USA

    Balram Suman

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  1. Balram Suman
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Suman, B. Effects of a surface-tension gradient on the performance of a micro-grooved heat pipe: an analytical study. Microfluid Nanofluid 5, 655–667 (2008). https://doi.org/10.1007/s10404-008-0282-8

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