Chapter 2, Problem 2.1P

Assume steady-state, one-dimensional heat conduction through the axisymmetric shape shown below.

Assuming constant properties and no internal heat generation, sketch the temperature distribution on $T-x$ coordinates. Briefly explain the shape of your curve.

To determine

To sketch: The temperature distribution on T-x coordinates and explain the shape.

Explanation of Solution

Draw the axisymmetric shape as shown below:

  

Write the expression as per Fourier law.

  $q_x=-k{A}_x\frac{dT}{dx}$

Here, $q_x$ is the heat conducted through the axisymmetric shape, $k$ is the thermal conductivity of shape, $A_x$ is the area of shape at position $x$ , $dT$ is change in temperature and $dx$ is the change in position.

Since the heat transfer through the body remains constant and thermal conductivity of the body remains constant, the Fourier law can be explained as:

  $\begin{array}{c}{A}_x\frac{dT}{dx}=\text{constant}\\ \frac{dT}{dx}\propto \frac{1}{A_x}\end{array}$

The above expression indicated that temperature and thickness is inversely proportional to one another.

On T-x curve the independent variable is x and the dependent variable is T .

Draw the T-x curve for axisymmetric shape as shown below:

  

The T-xcurve for axisymmetric shape is shown above and the slope $\frac{dT}{dx}$ decreases as distance increases.