A down jacket has a surface area of 0.82 m2 and is filled with a 3.8 cm thick layer of down, which has a thermal conductivity of 6.0 10-6 kcal/(s · m · C°). The coat is worn by a person whose skin temperature is maintained at 27° C. The outer surface of the coat is at -23° C. At what rate is heat conducted through the coat?
6• 10^-6 kcal/(s • m • C°) =0.025 J/(s • m • C°)
Fourier's Law of Conduction
ΔQ/ Δt = λ•A•( ΔT/ Δx) =0.025• 0.82•(50/0.038) =0.27 J/s
To find the rate at which heat is conducted through the coat, we can use Fourier's law of heat conduction:
Q = (k * A * ΔT) / d
Where:
Q is the rate of heat conducted through the coat,
k is the thermal conductivity of the material (6.0 * 10^-6 kcal/(s · m · C°)),
A is the surface area of the jacket (0.82 m^2),
ΔT is the temperature difference across the jacket (27°C - (-23°C) = 50°C),
d is the thickness of the down layer (3.8 cm = 0.038 m).
Let's substitute the given values into the equation:
Q = (6.0 * 10^-6 kcal/(s · m · C°)) * (0.82 m^2) * (50°C) / (0.038 m)
Now, we can calculate the rate of heat conducted through the coat:
To calculate the rate at which heat is conducted through the coat, we can use Fourier's law of heat conduction, which states that the rate of heat conduction is directly proportional to the surface area, temperature difference, and inversely proportional to the thickness and thermal conductivity of the material.
The formula for heat conduction is:
Q = (k * A * ΔT) / d
Where:
Q is the rate of heat conduction (in kcal/s)
k is the thermal conductivity of the material (in kcal/(s · m · C°))
A is the surface area of the coat (in m2)
ΔT is the temperature difference between the outside and inside of the coat (in °C)
d is the thickness of the down layer (in meters)
Using the given values:
k = 6.0 * 10^(-6) kcal/(s · m · C°)
A = 0.82 m2
ΔT = (-23 - 27) °C = -50 °C
d = 3.8 cm = 0.038 m
Substituting the values into the formula, we get:
Q = (6.0 * 10^(-6) * 0.82 * -50) / 0.038
Calculating this, we find:
Q ≈ -0.51 kcal/s
Therefore, the rate at which heat is conducted through the coat is approximately -0.51 kcal/s. The negative sign indicates that heat is being conducted from the outside (-23°C) to the inside (27°C) of the coat.