Question

A 1-m-long and \(0.1\)-m-thick steel plate of thermal conductivity $35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ is well insulated on both sides, while the top surface is exposed to a uniform heat flux of $5500 \mathrm{~W} / \mathrm{m}^{2}$. The bottom surface is convectively cooled by a fluid at \(10^{\circ} \mathrm{C}\) having a convective heat transfer coefficient of $150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assuming one-dimensional heat conduction in the lateral direction, find the temperature at the midpoint of the plate. Discretize the plate thickness into four equal parts.

Short answer

Answer: The final step to find the temperature at the midpoint of the steel plate is to calculate the average between the temperatures at the second and third intermediate points, \(T_2\) and \(T_3\), as follows: \(T_{mid} =\dfrac{T_2 + T_3}{2}\).

Step-by-step solution

Set up the problem and choose discretization points

First, divide the thickness of the plate (0.1m) into four equal parts, leading to three intermediate points between the top and bottom surfaces. Let each discretization point be represented by \(T_i\) where \(i=1,2,3\). The temperature \(T_0\) is the temperature on the top surface and \(T_1\) is the first intermediate point's temperature, so on.

Write equations for each intermediate discretization point

For each intermediate point, apply the one-dimensional energy balance equation. The general equation for point \(i\) is: \(Q_{flux} + Q_{conv} = Q_{cond, i}\) (In this case, \(Q_{conv}\) is only present for \(i=1\) since the bottom surface is in contact with the fluid.) The equation can be developed as follows: \(i=1: 5500 + 150(T_1-10) = 35\dfrac{T_0 - T_1}{0.025}\) \(i=2: 5500 = 35\dfrac{T_1 - T_2}{0.025}\) \(i=3: 5500 = 35\dfrac{T_2 - T_3}{0.025}\)

Solve the system of equations

Solve the system of equations for the unknown \(T_1\), \(T_2\), and \(T_3\). You can use any method to solve the system of equations such as substitution or matrix inversion method.

Calculate temperature at the midpoint

Finally, to obtain the temperature at the midpoint, find the average between \(T_2\) and \(T_3\). Calculate the midpoint temperature \(T_{mid}\) as follows: \(T_{mid} =\dfrac{T_2 + T_3}{2}\)