Question

Consider a \(1.5-\mathrm{m}\)-high and \(0.6-\mathrm{m}\)-wide plate whose thickness is \(0.15 \mathrm{~m}\). One side of the plate is maintained at a constant temperature of \(500 \mathrm{~K}\), while the other side is maintained at \(350 \mathrm{~K}\). The thermal conductivity of the plate can be assumed to vary linearly in that temperature range as \(k(T)=k_{0}(1+\beta T)\) where \(k_{0}=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and $\beta=8.7 \times 10^{-4} \mathrm{~K}^{-1}$. Disregarding the edge effects and assuming steady onedimensional heat transfer, determine the rate of heat conduction through the plate.

Short answer

Answer: To find the rate of heat conduction through the plate, follow these steps: 1. Determine the average temperature of the plate. 2. Calculate the temperature-dependent thermal conductivity. 3. Apply Fourier's law of heat conduction. 4. Calculate the rate of heat conduction using the results from the previous steps.

Step-by-step solution

Determine the average temperature of the plate

As the thermal conductivity varies with temperature, we need to find the average temperature of the plate. To do this, simply add the temperatures of both sides and divide by 2: $$ T_{avg} = \frac{T_1 + T_2}{2} $$

Calculate the temperature-dependent thermal conductivity

Using the given equation for the temperature-dependent thermal conductivity, plug the average temperature found in Step 1 into the equation: $$ k(T) = k_0(1 + \beta T_{avg}) $$

Apply Fourier's law of heat conduction

Fourier's law of heat conduction states that the rate of heat transfer through a material is proportional to the negative gradient of temperature and the material's thermal conductivity: $$ q = -k(T) A \frac{\Delta T}{d} $$ where \(q\) is the heat transfer rate, \(k(T)\) is the temperature-dependent thermal conductivity, \(A\) is the cross-sectional area, \(\Delta T\) is the difference in temperature across the material, and \(d\) is its thickness.

Calculate the rate of heat conduction

Using the results from the previous steps, we can calculate the rate of heat conduction: $$ q = -k(T) A \frac{T_1 - T_2}{d} $$ Plug in the values for \(k(T)\), \(A\), \(T_1\), \(T_2\), and \(d\) to get the final answer.