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Ch 5: Conduction Shape Factor

Two-D, Steady State Case:For a 2 - D, steady state situation, the heat equation is simplified to

∂2T∂x2 + ∂

2T∂ y2=0

it needs two boundary conditions in each direction.

There are three approaches to solve this equation:

• Analytical Method: The mathematical equation can be solved using techniques like the method of separation of variables.

• Graphical Method: Limited use. However, the conduction shape factor concept derived under this concept can be useful for specific configurations. (See Table below for selected configurations)

• Numerical Method: Finite difference or finite element schemes usually will be solved using computers.Analytical Method:

Chapter 5: Conduction Shape Factor

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Some problems can be solved analytically, but the solution procedure is so complex and the resulting solution expressions so complicated that it is not worth all that effort.

With the exception of steady one-dimensional or transient lumped system problems, all heat conduction problems result in partial differential equations. Solving such equations usually requires mathematical sophistication, such as orthogonality, eigenvalues, Fourier and Laplace transforms, Bessel and Legendre functions, and infinite series. In such cases, the evaluation of the solution, which often involves double or triple summations of infinite series at a specified point, is a challenge in itself.

Radial Fins with rectangular shape

Chapter 5: Conduction Shape Factor

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Chapter 5: Conduction Shape Factor

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The expressions for the temperature distribution, rate of heat transfer, and fin efficiency are

I n ( x ) = Modified Bessel Function of the First kind of order ‘n’Kn ( x )=Modified Bessel Function of Second kind of order ‘n’

Chapter 5: Conduction Shape Factor

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Chapter 5: Conduction Shape Factor

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Numerical Methods:

Use computer to solve the governing equations.

Graphical approach applied to 2-D conduction involving two isothermal surfaces, with all other surfaces being adiabatic.

The heat transfer from one surface (at a temperature T1) to the other surface (at T2) can be expressed as:

q̇=Sk (T2−T 1 )

S is the conduction shape factor. The shape S can be related to the thermal resistance.

q̇=Sk (T2−T 1 )=(T 2−T 1 )

( 1ks )

=(T 2−T 1 )Rt

Where Rt=1/ (kS ) is the thermal resistance in 2-D

Chapter 5: Conduction Shape Factor

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Chapter 5: Conduction Shape Factor

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Chapter 5: Conduction Shape Factor

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Chapter 5: Conduction Shape Factor

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Chapter 5: Conduction Shape Factor