Question:
A metal bar of length 1 meter and cross-sectional area 0.02 square meters is initially at a temperature of 200°C. It is then placed in contact with a heat reservoir at a temperature of 25°C. The thermal conductivity of the metal is 50 W/(m•°C) and its specific heat capacity is 200 J/(kg•°C). Assuming no heat escapes to the surroundings, calculate:
a) The rate of heat conduction through the bar. b) The final temperature reached by the bar.
Answer:
a) The rate of heat conduction through the bar can be calculated using the formula:
Q = k * A * (ΔT / L)
where Q is the rate of heat conduction, k is the thermal conductivity of the material, A is the cross-sectional area of the bar, ΔT is the temperature difference between the ends of the bar, and L is the length of the bar.
Given:
Substituting the values into the formula:
Q = 50 * 0.02 * (175 / 1)
Q = 175 W
Therefore, the rate of heat conduction through the bar is 175 Watts.
b) To calculate the final temperature reached by the bar, we can use the heat equation:
Q = m * c * ΔT
where Q is the heat absorbed or released by the bar, m is the mass of the bar, c is the specific heat capacity of the material, and ΔT is the change in temperature.
The heat absorbed by the bar can be given by:
Q = m * c * (Tf - Ti)
Since no heat escapes to the surroundings, the heat absorbed by the bar is equal to the heat conducted through it:
Q = 175 W (from part a)
Next, we need to calculate the mass of the bar. The mass can be determined using the formula:
m = (density) * A * L
Given that the density of the bar is 8000 kg/m^3, substituting the values:
m = 8000 * 0.02 * 1
m = 160 kg
Substituting the values of Q and m into the heat equation:
175 = 160 * 200 * (Tf - 200)
(Tf - 200) = 0.00109375
Finally, solving for Tf:
Tf = 0.00109375 + 200
Tf ≈ 200.001°C
Therefore, the final temperature reached by the bar is approximately 200.001°C.
Hence, the answers are: a) The rate of heat conduction through the bar is 175 Watts. b) The final temperature reached by the bar is approximately 200.001°C.