A bar of aluminum (bar A) is in thermal contact with a bar of iron (bar B) of the same length and area. One end of the compound bar is maintained at Th = 75.5°C while the opposite end is at 30.0°C. Find the temperature at the junction when the energy flow reaches a steady state.

conductivity iron = FE

conductivity al = AL

temp difference over iron = 75.5 - T
temp difference over Al = T - 30

(75-T) (FE) proportional to Q
(T-30) (AL) same proportional to q because areas and lengths the same
so same Q through each
(75-T)(FE) = (T-30) (AL)

by the way the ratio of AL/FE is about 136/46

(you don't need units here because all you need is the ratio)

Thanks for your help! The only thing is that the AL and FE need to be switched around and then I got the correct answer. There is a picture that I forgot to post that has Aluminum by the 75.5 and 30 next to the Iron. Thanks again!

Thanks for the help

To find the temperature at the junction when the energy flow reaches a steady state, we can use the principle of thermal conductivity and temperature gradient.

The principle of thermal conductivity states that heat flows from a higher temperature to a lower temperature. It can be quantified by Fourier's Law, which states that the rate of heat conduction through a substance is directly proportional to the temperature gradient and the cross-sectional area, and inversely proportional to the length.

In this case, we have a compound bar with two materials, aluminum (bar A) and iron (bar B), in thermal contact. Since the bars are in contact, they will reach a steady state where the energy flow is constant.

To find the temperature at the junction, we need to first calculate the heat flow through each bar separately using Fourier's Law:

Q_A = k_A * A * (Th - T_junction) / L_A
Q_B = k_B * A * (T_junction - Tc) / L_B

Where:
Q_A: Heat flow through bar A
k_A: Thermal conductivity of aluminum
A: Cross-sectional area of the bars
Th: Temperature at the hot end of the compound bar
T_junction: Temperature at the junction (unknown)
L_A: Length of bar A
Q_B: Heat flow through bar B
k_B: Thermal conductivity of iron
Tc: Temperature at the cold end of the compound bar
L_B: Length of bar B

Since the compound bar is in thermal contact, the heat flow through both bars will be equal at the junction when the steady state is reached. Therefore:

Q_A = Q_B

k_A * A * (Th - T_junction) / L_A = k_B * A * (T_junction - Tc) / L_B

By rearranging the equation, we can solve for T_junction:

(T_junction - Tc) / (Th - T_junction) = (k_A * L_B) / (k_B * L_A)

T_junction - Tc = (Th - T_junction) * (k_A * L_B) / (k_B * L_A)

T_junction - Tc = (Th * k_A * L_B) / (k_B * L_A) - (T_junction * k_A * L_B) / (k_B * L_A)

T_junction - T_junction * (k_A * L_B) / (k_B * L_A) = (Th * k_A * L_B) / (k_B * L_A) - Tc

T_junction * (1 - (k_A * L_B) / (k_B * L_A)) = (Th * k_A * L_B) / (k_B * L_A) - Tc

T_junction * ( (k_B * L_A - k_A * L_B) / (k_B * L_A)) = (Th * k_A * L_B) / (k_B * L_A) - Tc

T_junction = ((Th * k_A * L_B) / (k_B * L_A) - Tc) / ( (k_B * L_A - k_A * L_B) / (k_B * L_A))

By substituting the known values of Th, Tc, k_A, k_B, L_A, and L_B into the equation, we can calculate T_junction.