# Three-dimensional numerical analysis of heat and mass transfer in heat pipes

## Abstract

A three-dimensional finite-element numerical model is presented for simulation of the steady-state performance characteristics of heat pipes. The mass, momentum and energy conservation equations are solved for the liquid and vapor flow in the entire heat pipe domain. The calculated outer wall temperature profiles are in good agreement with the experimental data. The estimations of the liquid and vapor pressure distributions and velocity profiles are also presented and discussed. It is shown that the vapor flow field remains nearly symmetrical about the heat pipe centerline, even under a non-uniform heat load. The analytical method used to predict the heat pipe capillary limit is found to be conservative.

### List of symbols

*A*_{c}total outer surface area of condenser (m

^{2})*A*_{e}heated outer surface area of evaporator (m

^{2})*C*_{F}dimensionless form-drag constant

*c*_{p}specific heat at constant pressure (J kg

^{−1}K^{−1})*h*_{fg}latent heat of evaporation (J kg

^{−1})*k*thermal conductivity (W m

^{−1}K^{−1})*K*permeability (m

^{2})*L*length of the heat pipe (m)

*L*_{h}distance from the evaporator end cap to the heated region (m)

- \({\dot{m}}\)
mass flow rate (m s

^{−1})*p*pressure (Pa)

- Δ
*p*_{cap} capillary pressure (Pa)

- Pe
Peclet number

*q*heat flux (W m

^{−2})*Q*_{in}applied heat load (W)

*r*radial coordinate (m)

*r*_{p}effective pore radius of the wick (m)

*R*_{p}heat pipe wall outer radius (m)

*R*_{v}vapor core radius (m)

*R*_{w}heat pipe wall inner radius (m)

- ℜ
gas constant (J kg

^{−1}K^{−1})*T*temperature (K)

**u**velocity vector (m s

^{−1})*U*average velocity (m s

^{−1})*u*velocity component in

*x*direction (m s^{−1})*v*velocity component in

*y*direction (m s^{−1})*w*velocity component in

*z*direction (m s^{−1})*x*axial horizontal coordinate (m)

*y*lateral horizontal coordinate (m)

*z*lateral vertical coordinate (m)

### Greek symbols

- ∂/∂
*n* differential operator along the normal vector to a boundary

- μ
viscosity (Pa s)

- ρ
density (kg m

^{−3})- \(\varphi \)
porosity

- σ
surface tension of the working fluid (N m

^{−1})

### Subscripts

- a
adiabatic section

- c
condenser section

- e
evaporator section

- eff
effective

- int
interface

- l
liquid

- p
heat pipe outer wall

- ref
reference

- v
vapor

- w
wick

### References

- 1.Chen MM, Faghri A (1990) An analysis of the vapor flow and the heat conduction through the liquid-wick and pipe wall in a heat pipe with single or multiple heat sources. Int J Heat Mass Transfer 33:1945–1955CrossRefGoogle Scholar
- 2.Faghri A, Buchko M (1991) Experimental and numerical analysis of low-temperature heat pipes with multiple heat sources. J Heat Transfer 113:728–734Google Scholar
- 3.Schmalhofer J, Faghri A (1993) A study of circumferentially-heated and block-heated heat pipes-II. Three-dimensional modeling as a conjugate problem. Int J Heat Mass Transfer 36:213–226CrossRefGoogle Scholar
- 4.Zhu N, Vafai K (1999) Analysis of cylindrical heat pipes incorporating the effects of liquid-vapor coupling and non-Darcian transport-a closed form solution. Int J Heat Mass Transfer 42:3405–3418MATHCrossRefGoogle Scholar
- 5.Tournier JM, El-Genk MS (1994) A heat pipe transient analysis model. Int J Heat Mass Transfer 37:753–762MATHCrossRefGoogle Scholar
- 6.Layeghi M, Nouri-Borujerdi A (2004) Vapor flow analysis in partially-heated concentric annular heat pipes. Int J Comp Eng Sci 5:235–244CrossRefGoogle Scholar
- 7.Cao Y, Faghri A (1993) A numerical analysis of high temperature heat pipe startup from the frozen state. J Heat Transfer 115:247–254CrossRefGoogle Scholar
- 8.Tournier JM, El-Genk MS (2003) Startup of a horizontal lithium-molybdenum heat pipe from a frozen state. Int J Heat Mass Transfer 46:671–685CrossRefGoogle Scholar
- 9.Ergun S (1952) Fluid flow through packed columns. Chem Eng Prog 48:89–94Google Scholar
- 10.Chi SW (1976) Heat pipe theory and practice. McGraw-Hill, New YorkGoogle Scholar
- 11.Bai L (2004) The FEM analysis of a heat pipe. MASc dissertation, Carleton University, OttawaGoogle Scholar
- 12.Kuznetzov GV, Sitnikov AE (2002) Numerical modeling of heat and mass transfer in a low-temperature heat pipe. J Eng Phys Thermophysics 75:840–848CrossRefGoogle Scholar
- 13.Cao Y, Faghri A (1990) Transient two-dimensional compressible analysis for high-temperature heat pipes with pulsed heat input. Num Heat Transfer A18:483–502Google Scholar
- 14.Schmalhofer J, Faghri A (1993) A study of circumferentially-heated and block-heated heat pipes-I. Experimental analysis and generalized analytical prediction of capillary limits. Int J Heat Mass Transfer 36:201–212CrossRefGoogle Scholar