What Is the Temperature at the Interface of Lead and Copper Rods in Series?

[Lalasushi]

hey, can anyone help me with this question? I am really having trouble with it.

1. Two metal rods, one lead the other copper, are connected in series (end-to-end). Note that each rod is 0.525m in length and has a square cross section 1.5cm on a side. The temperature at the lead end of the rods is 2 degrees C; the temperature at the copper end is 106 degrees C.

a) The temperature of the two ends is 54 degrees C. Is the temperature in the middle, at the lead-copper interface greater than, less than or equal to 54 degrees C?
b) Given that the heat flow through each of these rods in 1 sec. is 1.41J, find the temperature at the lead-copper interface.

 

[OlderDan]

hey, can anyone help me with this question? I am really having trouble with it.

1. Two metal rods, one lead the other copper, are connected in series (end-to-end). Note that each rod is 0.525m in length and has a square cross section 1.5cm on a side. The temperature at the lead end of the rods is 2 degrees C; the temperature at the copper end is 106 degrees C.

a) The temperature of the two ends is 54 degrees C. Is the temperature in the middle, at the lead-copper interface greater than, less than or equal to 54 degrees C?
b) Given that the heat flow through each of these rods in 1 sec. is 1.41J, find the temperature at the lead-copper interface.

The problem statement is a bit confusing. I assume part a) is asking about the temperature at the junction if the temperatures at the far ends were both changed to 54C. I also assume that the temperature at the junction is to be determined after the rods have achieved thermal equilibrium with no heat input or output anywhere other than at the far ends.

Part b) is just a conduction problem. Heat will flow from the hot end to the cold end. Given time to reach thermal equlibrium, the rods will achieve a temperature gradient that will support a uniform heat flow. The question is to find the temperature at the junction such that the rate of heat flow is the same in both rods. You will need the thermal conductivities of the two materials to answer the question.

 

[thepatientmental]

Hi there! I can definitely help you with this question. Let's break it down step by step.

a) First, let's consider the heat flow through each rod. Since the rods are connected in series, the heat flow through both rods will be the same. This means that the heat flow through each rod in 1 second is 1.41J.

b) To find the temperature at the lead-copper interface, we can use the formula for thermal conductivity: Q = kA(T2-T1)/L, where Q is the heat flow, k is the thermal conductivity, A is the cross-sectional area, T2 and T1 are the temperatures at the two ends, and L is the length of the rod.

Since we already know the heat flow and the length of each rod, we can rearrange the formula to solve for the thermal conductivity (k).

k = QL/(A(T2-T1))

Plugging in the values, we get:

k = (1.41J)(0.525m)/((0.015m)^2)(106-2) = 772 W/mK

Now, using the same formula, we can find the temperature at the lead-copper interface:

T = T1 + (Q/kA)L = 2 + (1.41J)/(772 W/mK)(0.015m^2)(0.525m) = 28.8 degrees C

Therefore, the temperature at the lead-copper interface is less than 54 degrees C.

I hope this helps! Let me know if you have any further questions or need clarification. Good luck!