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A boundary element solution approach for the conjugate heat transfer problem in thermally developing region of a thick walled pipe

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Abstract

This paper presents a sole application of boundary element method to the conjugate heat transfer problem of thermally developing laminar flow in a thick walled pipe when the fluid velocities are fully developed. Due to the coupled mechanism of heat conduction in the solid region and heat convection in the fluid region, two separate solutions in the solid and fluid regions are sought to match the solid-fluid interface continuity condition. In this method, the dual reciprocity boundary element method (DRBEM) with the axial direction marching scheme is used to solve the heat convection problem and the conventional boundary element method (BEM) of axisymmetric model is applied to solve the heat conduction problem. An iterative and numerically stable BEM solution algorithm is presented, which uses the coupled interface conditions explicitly instead of uncoupled conditions. Both the local convective heat transfer coefficient at solid-fluid interface and the local mean fluid temperature are initially guessed and updated as the unknown interface thermal conditions in the iterative solution procedure. Two examples imposing uniform temperature and heat flux boundary conditions are tested in thermally developing region and compared with analytic solutions where available. The benchmark test results are shown to be in good agreement with the analytic solutions for both examples with different boundary conditions.

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Abbreviations

A,B,m :

Parameters for the complete elliptic integral of the 1st kind

b :

Heat source-like term

E(m) :

Complete elliptic integral of the 1st kind of modulusm

f :

Interpolating function

F :

Matrix of its elementf

G :

Coefficient matrix involvingT* oru*

Gz :

Graetz number

g :

Boundary integrals involvingT* oru*

H :

Coefficient matrix involvingq*

h :

Convective heat transfer coefficient or boundary integrals involvingq*

K:

Ratio of thermal conductivity of pipe wall to that of the fluid considered

k :

Thermal conductivity

L :

Number of internal points or length of thermal boundary section

N :

Number of boundary elements

Nu :

Nusselt number

n :

Normal unit vector

p :

Pressure

Pe:

Peclet number

Pr:

Prandtl number

q :

Normal derivative ofT oru

r,z :

Cylindrical coordinates

T :

Temperature

u :

General dependent variable

w :

Axial flow velocity

x,y,z :

Cartesian coordinates

α:

Thermal diffusivity

β:

Initially unknown coefficient

Г:

Boundary

δ:

Kronecker delta

ε:

Relative error

θ:

Angle

ζ:

Parameter taking the values between 0 and 1

λ:

Relaxation factor

μ:

Coefficient of viscosity

φ:

Linear interpolation function

Ω:

Solution domain

e:

Pipe entrance of heating section

f:

Fluid region

i:

Source point or inner boundary

j:

Collocation point

o:

Outer boundary

s:

Solid region

w:

Wall surface

∞:

Ambient condition

m:

Time level

-:

Specified value

*:

Fundamental solution

∧:

Particular solution

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Authors and Affiliations

  1. Department of Mechanical & Automotive Engineering, Jeonju University, Hyojadong, 3 Ga 1200, 560-759, Jeonju, Korea

    Chang-Yong Choi

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  1. Chang-Yong Choi
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Correspondence to Chang-Yong Choi.

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Choi, CY. A boundary element solution approach for the conjugate heat transfer problem in thermally developing region of a thick walled pipe. J Mech Sci Technol 20, 2230–2241 (2006). https://doi.org/10.1007/BF02916340

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  • DOI: https://doi.org/10.1007/BF02916340

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