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EX1: Infinite Slab

Suppose an infinite slab of material, bounded by planes and , has a constant thermal conductivity . If the faces are maintained for a long time at temperatures and respectively, what is the temperature distribution inside the slab? What are the direction and magnitude of the heat flux at each point?

Since the boundary (face) temperatures do not vary with time, it is reasonable to assume that the temperature within the slab is also independent of time (a steady state). Furthermore, since on each face the temperature is independent of the other space variables, and , we assume that within the slab is independent of and . In other words, , so that



Equation 4

and from equation (2.1), the heat flux vector is also in the x-direction (perpendicular to the faces). Therefore



Equation 5

where is the -component (and only component) of the heat flux vector.

Since depends only on so do and . Thus we may write:



Equation 6

In fact, it may be shown that:



Equation 7

throughout the slab.



Click here for the proof!

The relations and give



Equation 10

While is constant, its value is not yet known.

 
 
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