Assume the muscle is 37C and is separated from the outside air by layers of fat and skin. The layer of fat, at a particular location on the skin, is 2 mm thick and has a conductivity of 0.53 W/mK. Finally, the outermost epidermal layer is 0.1 mm thick with a thermal conductivity of 0.21 W/mK. How much heat is lost per unit area and unit time if the ambient air temperature is 0C?
To calculate the heat lost per unit area and unit time, we need to use the formula for heat transfer known as Fourier's Law, which states that the heat transfer rate is directly proportional to the temperature difference and inversely proportional to the thermal resistance:
Q/At = (ΔT) / (ΣR)
Where:
Q/At is the heat lost per unit area and unit time.
ΔT is the temperature difference between the muscle (37C) and the ambient air (0C).
ΣR is the sum of the thermal resistances.
To calculate the heat lost through each layer, we need to calculate the thermal resistance (R) of each layer using the formula:
R = (thickness) / (conductivity x area)
Let's break down the calculation step by step:
1. Calculate the thermal resistance of the fat layer:
R_fat = (2 mm) / (0.53 W/mK x 1 m^2) = 3.77 m^2K/W
2. Calculate the thermal resistance of the epidermal layer:
R_epidermal = (0.1 mm) / (0.21 W/mK x 1 m^2) = 0.476 m^2K/W
3. Calculate the total thermal resistance:
ΣR = R_fat + R_epidermal = 3.77 m^2K/W + 0.476 m^2K/W = 4.246 m^2K/W
4. Calculate the temperature difference:
ΔT = (37C) - (0C) = 37C
5. Calculate the heat transfer rate per unit area and unit time using Fourier's Law:
Q/At = (ΔT) / (ΣR)
Q/At = 37C / 4.246 m^2K/W
Q/At ≈ 8.70 C/m^2s
So, the heat lost per unit area and unit time, assuming the ambient air temperature is 0C, is approximately 8.70 degrees Celsius per square meter per second.