To determine the effect of the temperature dependence of the thermal conductivity on the temperature distribution in a solid, consider a material for which this dependence may be represented as k = ko + at where ko is a positive constant and a is a coefficient that may be positive or negative. sketch the steady-state temperature distribution associated with heat transfer in a plane wall for three cases corresponding to a > 0, a = 0, and a < 0.

Given that

k = ko + aT

From Fourier law

Heat transfer per unit volume given as

x measured from left hand side of wall.

By integrating

Where C is constant

We can that temperature of wall is varying as parabolically with distance x

When a = 0 :

This is become straight line.

When a > 0  :

K decrease when x is increases.

When a < 0  :

K increases when x is increases.

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