Question

Consider two walls, \(A\) and \(B\), with the same surface areas and the same temperature drops across their thicknesses. The ratio of thermal conductivities is \(k_{A} / k_{B}=4\) and the ratio of the wall thicknesses is \(L_{A} / L_{B}=2\). The ratio of heat transfer rates through the walls \(\dot{Q}_{A} / \dot{Q}_{B}\) is (a) \(0.5\) (b) 1 (c) 2 (d) 4 (e) 8

Short answer

Answer: The ratio of heat transfer rates through the walls A and B is 2:1.

Step-by-step solution

Recall the formula for heat transfer rate

The formula for the heat transfer rate (Q) through a wall is given by: $$ \dot{Q} = \frac{k A \Delta T}{L} $$ Where \(\dot{Q}\) is the heat transfer rate, k is the thermal conductivity, A is the surface area, \(\Delta T\) is the temperature drop across the wall, and L is the wall thickness.

Write the equation for heat transfer rate ratio

We want to find the ratio of the heat transfer rates for wall A and wall B. Using the formula from Step 1, we can write the equation as follows: $$ \frac{\dot{Q}_{A}}{\dot{Q}_{B}} = \frac{\frac{k_{A} A \Delta T_{A}}{L_{A}}}{\frac{k_{B} A \Delta T_{B}}{L_{B}}} $$

Simplify the equation and plug the given ratios

Notice that the surface areas and temperature drops across both walls are the same, so those will cancel out. We can simplify the equation and insert the given ratios into the equation, such as: $$ \frac{\dot{Q}_{A}}{\dot{Q}_{B}} = \frac{k_{A} L_{B}}{k_{B} L_{A}} = \frac{\frac{k_{A}}{k_{B}}}{\frac{L_{A}}{L_{B}}} $$ We are given that \(\frac{k_{A}}{k_{B}} = 4\) and \(\frac{L_{A}}{L_{B}} = 2\). Plugging these values into the equation, we get: $$ \frac{\dot{Q}_{A}}{\dot{Q}_{B}} = \frac{4}{2} $$

Calculate the heat transfer rate ratio

Now we can simply calculate the heat transfer rate ratio from the equation: $$ \frac{\dot{Q}_{A}}{\dot{Q}_{B}} = \frac{4}{2} = 2 $$ So, the ratio of heat transfer rates through the walls is 2, which corresponds to option (c).