Question

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.

Answer 1

Answer:

Given that

k = ko + aT

From Fourier law

Heat transfer per unit volume given as

$q=-\dfrac{KdT}{dx}$

x measured from left hand side of wall.

$q=-\dfrac{(K_o+aT)dT}{dx}$

$q{dx}=-{(K_o+aT)dT}$

By integrating

$\int q{dx}=\int -{(K_o+aT)dT}$

$q{x}= -\left(K_oT+\dfrac{aT^2}{2}\right)+C$

Where C is constant

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

When a = 0 :

$q{x}= -\left(K_oT\right)+C$

This is become straight line.

$\dfrac{dT}{dx}=Constnat$

When a > 0  :

$\dfrac{dT}{dx}=increases$

K decrease when x is increases.

When a < 0  :

$\dfrac{dT}{dx}=decrease$

K increases when x is increases.