A copper pipe with an outer radius of 0.010 m runs from an outdoor wall faucet into the interior of a house. The temperature of the faucet is 3.3°C, and the temperature of the pipe, 3.0 m from the faucet, is 25°C. In fifteen minutes, the pipe conducts a total of 270 J of heat to the outdoor faucet from the house interior. Find the inner radius of the pipe. Ignore any water inside the pipe
To find the inner radius of the copper pipe, we need to use the formula for heat conduction:
Q = k * A * ΔT / d
Where:
Q is the amount of heat conducted (270 J in this case)
k is the thermal conductivity of copper (we'll use 401 W/m*K)
A is the surface area of the pipe (we'll calculate it using the formula for the area of a ring)
ΔT is the temperature difference between the pipe and the ambient temperature (25°C - 3.3°C)
d is the thickness of the pipe (the difference between the outer radius and the inner radius)
Let's calculate each of these steps:
1. Calculate the surface area of the pipe:
The formula for the area of a ring is A = π(R2 - R1), where R2 is the outer radius and R1 is the inner radius.
Given:
Outer radius (R2) = 0.010 m
Substituting the values into the formula, we get:
A = π(0.010^2 - R1^2)
2. Calculate the temperature difference (ΔT):
Given:
Temperature at the faucet (T1) = 3.3°C
Temperature 3.0 m from the faucet (T2) = 25°C
Substituting the values into the formula, we get:
ΔT = T2 - T1 = 25°C - 3.3°C
3. Rearrange the formula for heat conduction to solve for the inner radius (R1):
Q = k * A * ΔT / d
Rearranging the formula, we have:
R1 = sqrt((k * A * ΔT) / (Q * π))
Now, substitute the known values into the formula and solve for R1:
R1 = sqrt((401 * A * ΔT) / (Q * π))
By plugging in the values for A, ΔT, and Q, you should be able to calculate the inner radius (R1) of the pipe.