Page 18 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

INTRODUCTION
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In the above equation, the first term is derived from the heat diffusion (conduction)
within the material, the second term is due to convection from the material surface to
ambient, the third term represents the heat transport due to the motion of the material and
finally the last term is added to account for heat generation within the body.
1.5 Heat Conduction Equation

The determination of temperature distribution in a medium (solid, liquid, gas or combination
of phases) is the main objective of a conduction analysis, that is, to know the temperature
in the medium as a function of space at steady state and as a function of time during
the transient state. Once this temperature distribution is known, the heat flux at any point
within the medium, or on its surface, may be computed from Fourier’s law, Equation 1.1.
A knowledge of the temperature distribution within a solid can be used to determine the
structural integrity via a determination of the thermal stresses and distortion. The optimiza-
tion of the thickness of an insulating material and the compatibility of any special coatings
or adhesives used on the material can be studied by knowing the temperature distribution
and the appropriate heat transfer characteristics.
We shall now derive the conduction equation in Cartesian coordinates by applying
the energy conservation law to a differential control volume as shown in Figure 1.4. The
solution of the resulting differential equation, with prescribed boundary conditions, gives
the temperature distribution in the medium.

Q y +∆y Q z +∆z

∆y

Q x
Q x +∆x

∆z
y
z

∆x
x
Q
z
Q y
Figure 1.4 A differential control volume for heat conduction analysis