Question

Consider steady one-dimensional heat transfer through a plane wall exposed to convection from both sides to environments at known temperatures $T_{\infty 1}\( and \)T_{\infty 2}\( with known heat transfer coefficients \)h_{1}$ and \(h_{2}\). Once the rate of heat transfer \(\dot{Q}\) has been evaluated, explain how you would determine the temperature of each surface.

Short answer

Answer: To find the heat transfer rate, first calculate the overall thermal resistance in the system, then use energy balance and temperature difference to get the heat transfer rate. Next, use convective heat transfer formulas to find the temperatures of each surface.

Step-by-step solution

Understand the problem components and variables

In this problem, consider the heat transfer through a plane wall as steady one-dimensional. The wall is exposed to convection from both sides, with known environment temperatures \(T_{\infty 1}\) and \(T_{\infty 2}\), and known heat transfer coefficients \(h_1\) and \(h_2\). Also, let the temperature of the wall surfaces be \(T_1\) and \(T_2\). We need to determine the rate of heat transfer \(\dot{Q}\) and explain how to find the temperature of each surface.

Apply the formula for the heat transfer rate

In order to determine the rate of heat transfer, we can apply the formula for the overall thermal resistance (\(R_{total}\)) in a one-dimensional conduction problem: \[R_{total} = \frac{1}{h_{1}A} + \frac{L}{kA} + \frac{1}{h_{2}A}\] Where \(A\) is the surface area, \(L\) is the thickness of the wall, and \(k\) is the wall's thermal conductivity. Now, using the energy balance and given \(\Delta T = T_{\infty 1} - T_{\infty 2}\), we can determine the heat transfer rate \(\dot{Q}\) as follows: \[\dot{Q} = \frac{\Delta T}{R_{total}}\]

Determine surface temperatures using the convective heat transfer formula

Now that we know the heat transfer rate, we can determine the temperatures of both surfaces. Let's examine the convective heat transfer from the surface at \(T_1\) to the environment with temperature \(T_{\infty 1}\). We can write the heat transfer rate as follows: \[\dot{Q} = h_{1}A(T_{1} - T_{\infty 1})\] Solving for \(T_1\): \[T_1 = \frac{\dot{Q}}{h_{1}A} + T_{\infty 1}\] Similarly, let's examine the convective heat transfer from the surface at \(T_2\) to the environment with temperature \(T_{\infty 2}\). We can write the heat transfer rate as follows: \[\dot{Q} = h_{2}A(T_{\infty 2} - T_{2})\] Solving for \(T_2\): \[T_2 = T_{\infty 2} - \frac{\dot{Q}}{h_{2}A}\] By first evaluating the heat transfer rate and then using convective heat transfer formulas, we are able to find the temperature of each surface, \(T_1\) and \(T_2\), in this steady one-dimensional heat transfer problem.