### Analytical and Numerical Solutions of Vapor Flow in a Flat Plate Heat Pipe

#### Abstract

In this paper, the optimal homotopy analysis method (OHAM) and differential transform method (DTM) were applied to solve the problem of 2D vapor flow in flat plate heat pipes. The governing partial differential equations for this problem were reduced to a non-linear ordinary differential equation, and then non-dimensional velocity profiles and axial pressure distributions along the entire length of the heat pipe were obtained using homotopy analysis, differential transform, and numerical fourth-order Runge-Kutta methods. The reliability of the two analytical methods was examined by comparing the analytical results with numerical ones. A brief discussion about the advantages of the two applied analytical methods relative to each other is presented. Furthermore, the effects of the Reynolds number and the ratio of condenser to evaporator lengths on the flow variables were discussed.**Graphical abstract**

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A Basiulis, H Tanzer and S McCabe. Thermal management of high power PWB’s through the use of heat pipe substrates. In: Proceedings of the 6th Annual International Electronics Packaging Conference, San Diego, California. 1986, p. 501.

MJ Rightley, CP Tigges, RC Givler, CV Robino, JJ Mulhall and PM Smith. Innovative wick design for multi-source flat plate heat pipes. Microelectron. J. 2003; 34, 187-94.

M Thomson, C Ruel and M Donato. Characterization of a flat plate heat pipe for electronic cooling in a space environment. In: Proceedings of the 1989 National Heat Transfer Conference, Heat Transfer in Electronics, Philadelphia, Pennsylvania. 1989, p. 59-65.

H Ooijen and CJ Hoogendoorn. Vapor flow calculations in a flat-plate heat pipe. AIAA J. 1979; 17, 1251-9.

K Vafai, N Zhu and W Wang. Analysis of flow and heat transfer characteristics of an asymmetrical flat plate heat pipe. Int. J. Heat Mass Tran. 1992; 35, 2087-99.

K Vafai, N Zhu and W Wang. Analysis of asymmetrical disk-shaped and flat-plate heat pipes. ASME J. Heat Tran. 1995; 117, 209-18.

N Zhu and K Vafai. Vapor and liquid flow in an asymmetrical flat plate heat pipe: a three dimensional analytical and numerical investigation. Int. J. Heat Mass Tran. 1998; 41, 159-74.

N Zhu and K Vafai. Analytical modeling of the startup characteristics of asymmetrical flat plate and disk-shaped heat pipes. Int. J. Heat Mass Tran. 1998; 41, 2619-37.

Y Wang and K Vafai. Transient characterization of flat plate heat pipes during startup and shutdown operations. Int. J. Heat Mass Tran. 2000; 43, 2641-55.

EK Levy. Theoretical investigation of heat pipes operating at low vapor pressures. J. Eng. Ind. 1968; 90, 547-52.

YM Brovalsky, PI Bystrov and MV Melkinov. The method of calculation and investigation of high temperature heat pipe characteristics taking into account the vapor flow compressibility, friction, and velocity profile. In: Proceedings of the 2nd International Heat Pipe Conference, Bologna, Italy. 1976, p. 113-22.

A Faghri. Performance characteristics of a concentric annular heat pipe: part II-vapor flow analysis. ASME J. Heat Tran. 1989; 111, 851-7.

JH Jang. 1988, An Analysis of Startup from the Frozen State and Transient Performance of Heat Pipes, Ph. D. Dissertation. Georgia Institute of Technology, USA.

Y Cao, A Faghri and ET Mahefkey. The thermal performance of heat pipes with localized heat input. Int. J. Heat Mass Tran. 1989; 32, 1279-87.

N Zhu and K Vafai. The effects of liquid vapor coupling and non-Darcian transport on asymmetrical diskshaped heat pipes. Int. J. Heat Mass Tran. 1996; 39, 2095-113.

MT North and CT Avedisian. Heat pipe for cooling high flux/high power semiconductor chips. J. Electro. Packag. 1993; 115, 112-7.

N Zhu and K Vafai. Numerical and analytical investigation of vapor flow in a disk-shaped heat pipe incorporating secondary flow. Int. J. Heat Mass Tran. 1997; 40, 2887-900.

YS Chen, KH Chien, CC Wang, TC Hung and BS Pei. A Simplified transient three-dimensional model for estimating the thermal performance of the vapor chambers. Appl. Therm. Eng. 2006; 26, 2087-94.

Y Xuan, Y Hong and Q Li. Investigation on transient behaviors of flat plate heat pipes. Exp. Therm. Fluid Sci. 2004; 28, 249-55.

R Sonan, S Harmand, J Pellé, D Leger and M Fakès. Transient thermal and hydrodynamics model of flat heat pipe for the cooling of electronics components. Int. J. Heat Mass Tran. 2008; 51, 6006-17.

Y Koito, H Imura, M Mochizuki, Y Saito and S Torii. Numerical analysis and experimental verification on thermal fluid phenomena in vapor chamber. Appl. Therm. Eng. 2006; 26, 1669-76.

R Boukhanouf, A Haddad, MT North and C Buffone. Experimental investigation of a flat plate heat pipe performance using IR thermal imaging camera. Appl. Therm. Eng. 2006; 26, 2148-56.

AH Nayfeh. Perturbation Methods. Wiley, New York, 2000.

SJ Liao. 1992, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph. D. Thesis: Shanghai Jiao Tong University, China.

SJ Liao. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton, 2003.

G Domairry and M Fazeli. Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature dependent thermal conductivity. Commun. Nonlinear Sci. Numer. Simul. 2009; 14, 489-99.

MM Rashidi, G Domairry and S Dinarvand. Approximate solutions for the Burger and regularized long wave equations by means of the Homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 2009; 14, 708-17.

Z Ziabakhsh, G Domairry and M Esmaeilpour. Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the Homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 2009; 14, 1284-94.

MM Rashidi and S Dinarvand. Purely analytic approximate solutions for steady three-dimensional problem of condensation film on inclined rotating disk by Homotopy analysis method. Nonlinear Anal. R. World Appl. 2009; 10, 2346-56.

Z Ziabakhsh and G Domairry. Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using Homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 2009; 14, 1868-80.

JH He. Homotopy perturbation technique. Comput. Meth. Appl. Mech. Eng. 1999; 178, 257-62.

JK Zhou. Differential Transformation and Its Applications for Eelectrical Circuits. Huazhong University Press, Wuhan, China, 1986.

CK Chen and SH Ho. Solving partial differential equations by two dimensional differential transform method. Appl. Math. Comput. 1999; 106, 171-9.

F Ayaz. Solutions of the systems of differential equations by differential transform method. Appl. Math. Comput. 2004; 147, 547-67.

IH Abdel-Halim Hassan. Comparison differential transformation technique with Adomain decomposition method for linear and nonlinear initial value problems. Chaos Solitons & Fractals 2008; 36, 53-65.

A Arikoglu and I Ozkol. Solution of differential-difference equations by using differential transform method. Int. J. Appl. Math. Comput. 2006; 181, 154.

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