An iron cylinder with radius 1.0 cm and length 25 cm has one end held at 250°C and the other at 20°C.
Find the heat-flow rate through the cylinder
To find the heat-flow rate through the cylinder, we can use the formula for heat conduction:
Q = (k * A * ΔT) / d
Where:
Q is the heat-flow rate (in Watts),
k is the thermal conductivity of the material (in Watts per meter per Kelvin),
A is the cross-sectional area of the cylinder (in square meters),
ΔT is the temperature difference across the cylinder (in Kelvin),
and d is the thickness or length of the cylinder (in meters).
First, let's find the cross-sectional area of the cylinder:
A = π * r^2
A = π * (0.01 m)^2
A = 0.000314159 m^2
Next, let's calculate the temperature difference across the cylinder:
ΔT = T2 - T1
ΔT = 20°C - 250°C
ΔT = -230°C
Note that we converted the temperatures to Kelvins by adding 273.15 because the formula requires temperatures in Kelvin.
Now, we need to find the thermal conductivity of the material. The thermal conductivity of iron is approximately 80 W/mK.
Substituting these values into the formula, we have:
Q = (80 W/mK * 0.000314159 m^2 * -230 K) / 0.25 m
Simplifying the calculation, we get:
Q = -18.84 W
Since heat is flowing from the higher temperature end to the lower temperature end, the value is negative, indicating the heat flow direction.
Therefore, the heat-flow rate through the cylinder is approximately -18.84 Watts.