Chapter 3, Problem 1CP

Consider heat conduction through a wall of thickness L and area A. Under what conditions will the temperature distributions in the wall be a straight line?

To determine

The condition under which the temperature distribution in the wall is a straight line.

Explanation of Solution

The temperature distribution of a wall with thickness L and area A is shown in the figure below:

  

The one-dimensional steady state heat conduction equation without heat generation for the wall is $\frac{d^{2}T}{d{x}^2}=0\_\_\_\_\_\_(1)$

Here, T is the temperature and x is the direction of flow of heat.

Integrate equation (1) with respect to x.

  $\frac{dT}{dx}=c_1\_\_\_\_\_\_(2)$

Where, c₁ is the constant of integration.

Further integrate the equation (2) with respect to x.

  $T=c_{1}x+c_2\_\_\_\_\_\_(3)$

Now, calculate the constant of integration.

At x = 0, T =T₁

Substitute the value in equation (3);

  $\begin{array}{l}{T}_1=c_1(0)+c_2\\ c_2=T_1\end{array}$

Substitute the value of c₂ in equation (3);

  $\begin{array}{l}T=c_{1}x+T_1\\ Atx=L,T=T_2\\ T_2=c_1(L)+T_1\\ c_1=\frac{T_2-T_1}{L}\end{array}$

Substitute the value of c₁ and c₂ in equation (3);

  $T=\frac{T_2-T_1}{L}x+T_1$

The above equation is in the form of a straight line equation y = mx+c. Thus, the distribution of temperature in the plane wall will be a straight line during steady-state one dimensional heat transfer.

Thus, the following are the conditions during which the temperature distribution in the wall is a straight line.

1. Heat conduction in one-dimensional steady state direction.
2. Thermal conductivity should be constant.
3. There is no any internal heat generation in the wall.