Question

The temperature of a plane wall during steady onedimensional heat conduction varies linearly when the thermal conductivity is constant. Is this still the case when the thermal conductivity varies linearly with temperature?

Short answer

Answer: No, the temperature of a plane wall during steady one-dimensional heat conduction does not vary linearly when the thermal conductivity varies linearly with temperature. Instead, it results in a nonlinear temperature distribution.

Step-by-step solution

Understand the given information

We are given that the temperature of a plane wall during steady one-dimensional heat conduction varies linearly when the thermal conductivity is constant. Now we need to find out if this is still the case when the thermal conductivity varies linearly with temperature. Let 'T' be temperature, 'x' be the coordinate parallel to the wall, and 'k' be the thermal conductivity.

Formulate the linear relationship for thermal conductivity

Since thermal conductivity varies linearly with temperature, we can write: k = k_0 + b*T where k_0 is the initial thermal conductivity, b is the slope, and T is the temperature.

Apply the heat conduction equation

For one-dimensional heat conduction, the heat conduction equation is: \frac{dq}{dx} = -k \frac{dT}{dx} where dq/dx is the heat flux and dT/dx is the temperature gradient.

Substitute the linear relationship for thermal conductivity

Now, we will substitute the linear relationship for thermal conductivity from Step 2 into the heat conduction equation from Step 3: \frac{dq}{dx} = -(k_0 + b*T) \frac{dT}{dx}

Integrate the differential equation

In order to find the temperature distribution, we need to integrate the above equation with respect to 'x': \int \frac{dq}{dx} dx = -\int (k_0 + b*T) \frac{dT}{dx} dx By integrating and rearranging, we obtain a nonlinear temperature distribution. Therefore, the temperature of a plane wall during steady one-dimensional heat conduction does not vary linearly when the thermal conductivity varies linearly with temperature.