Calculating Temperature at Aluminum-Copper Junction in Heat Conducting Rod

[preluderacer]

Homework Statement

A heat conducting rod, 0.90 m long, is made of an aluminum section, 0.10 m long, and a copper section, 0.80m long. Both sections have a cross-sectional area of 0.0004m^2 The aluminum end and the copper end are maintained at temperatures of 40C and 150C respectively. The thermal conductivities of aluminum and copper are 205 and 385 W/m ∙ K, respectively. The temperature of the aluminum-copper junction in the rod, in is closest in C to.

The Attempt at a Solution

I don't have the slightest clue even how to solve this.

 

[anathema]

Hello,

Thank you for posting this question. I can help you solve this problem.

First, let's review the information given in the problem. We have a heat conducting rod that is made up of two sections - aluminum and copper. The lengths and cross-sectional areas of each section are also provided. The temperatures of the aluminum and copper ends are given, as well as the thermal conductivities of each material.

To solve this problem, we need to use the equation for thermal conductivity: Q = kAΔT/L, where Q is the heat transferred, k is the thermal conductivity, A is the cross-sectional area, ΔT is the temperature difference, and L is the length.

We can start by calculating the heat transferred through each section of the rod. For the aluminum section, we have Q_al = (205 W/m ∙ K)(0.0004m^2)(150C - T_junc)/0.1m, where T_junc is the temperature at the aluminum-copper junction. Similarly, for the copper section, we have Q_cu = (385 W/m ∙ K)(0.0004m^2)(T_junc - 40C)/0.8m.

Since the heat transferred through each section must be equal, we can set these equations equal to each other and solve for T_junc. This gives us:

(205 W/m ∙ K)(0.0004m^2)(150C - T_junc)/0.1m = (385 W/m ∙ K)(0.0004m^2)(T_junc - 40C)/0.8m

Solving for T_junc gives us a temperature of 91.4C. Therefore, the temperature at the aluminum-copper junction in the rod is closest to 91.4C.

I hope this helps you understand how to approach and solve this problem. Let me know if you have any further questions. Good luck!