Question

Consider steady one-dimensional heat transfer through a multilayer medium. If the rate of heat transfer \(\dot{Q}\) is known, explain how you would determine the temperature drop across each layer.

Short answer

Question: Given the rate of heat transfer, \(\dot{Q}\), the thickness and thermal conductivity of each layer in a multilayer medium, determine the temperature drop across each layer using the concept of thermal resistance and Fourier's Law of heat conduction. Answer: To find the temperature drop across each layer in a multilayer medium, follow these steps: 1. Identify the layers, their thicknesses, and their thermal conductivity. 2. Calculate the thermal resistance of each layer using the formula \(R = \frac{d}{kA}\) where \(d\) is the thickness, \(k\) is the thermal conductivity, and \(A\) is the area through which heat is being transferred. 3. Find the total thermal resistance of the system by adding the individual thermal resistances: \(R_{total}=R_1+R_2+\cdots+R_n\). 4. Apply Fourier's Law to the entire system and calculate the total temperature drop using ΔT = \(\frac{\dot{Q}R_{total}}{A}\). 5. Determine the temperature drop across each layer by multiplying the individual thermal resistance of each layer by the known rate of heat transfer: \(\Delta T_i = \frac{\dot{Q}R_{i}}{A}\) where \(\Delta T_i\) is the temperature drop across layer \(i\).

Step-by-step solution

To solve this problem, we first need to identify the layers in the multilayer medium, their thicknesses and their thermal conductivity. The thermal resistance of each layer, \(R\), can be calculated using the formula \(R = \frac{d}{kA}\) where \(d\) is the thickness of the layer, \(k\) is the thermal conductivity, and \(A\) is the area through which heat is being transferred. #Step 2: Find the total thermal resistance of the system#

Once we know the thermal resistance of each layer, we can calculate the total thermal resistance of the system by adding the individual thermal resistances: \(R_{total}=R_1+R_2+\cdots+R_n\). #Step 3: Apply Fourier's Law to the entire system#

Fourier’s law of heat conduction states that \(\delta{Q} = -kA \frac{\delta{T}}{\delta{x}}\), which means \(\dot{Q} = -kA \frac{dT}{dx}\). In this scenario, we have a known rate of heat transfer (\(\dot{Q}\)) and we want to determine the temperature drop across each layer. Rewriting the formula in terms of the temperature difference, we get \(\Delta T = \frac{\dot{Q}R_{total}}{A}\). #Step 4: Determine the temperature drop across each layer#

Now that we have calculated the total temperature drop across the medium, we can determine the temperature drop across each layer by multiplying the individual thermal resistance of each layer by the known rate of heat transfer: \(\Delta T_i = \frac{\dot{Q}R_{i}}{A}\) where \(\Delta T_i\) is the temperature drop across layer \(i\). Remember to follow these steps for each layer in the multilayer medium to determine the temperature drop across each layer.