Thermal Conductivity and Temperature Drop Across Metal Walls

Thermal conductivity is a measure of a material's ability to conduct heat. It is defined as the rate of heat transfer through a unit area of a material per unit time per unit temperature gradient. The higher the thermal conductivity of a material, the better it is at conducting heat.

In this problem, we are given three metal walls of the same thickness and cross-sectional area but with different thermal conductivities: k, 2k, and 3k. We need to find the ratio of the temperature drop across these walls for the same heat transfer.

Assumptions

- The metal walls are in steady-state heat transfer.
- The heat transfer is one-dimensional and uniform.
- The thermal conductivities of the metal walls are constant.

Solution

Let us consider a metal wall of thickness L and cross-sectional area A. The heat transfer rate through the wall can be expressed as:

Q = kA(T1 - T2)/L

where Q is the heat transfer rate, k is the thermal conductivity, T1 is the temperature on one side of the wall, and T2 is the temperature on the other side of the wall.

For the same heat transfer rate, the temperature drop across the wall can be expressed as:

(T1 - T2) = QL/kA

Now, let us consider three metal walls with thermal conductivities k, 2k, and 3k. For the same heat transfer rate, the temperature drop across these walls can be expressed as:

(T1 - T2)1 = QL/kA
(T1 - T2)2 = QL/2kA
(T1 - T2)3 = QL/3kA

Dividing the second equation by the first equation, we get:

(T1 - T2)2/(T1 - T2)1 = 1/2

Dividing the third equation by the second equation, we get:

(T1 - T2)3/(T1 - T2)2 = 3/2

Multiplying these two equations, we get:

(T1 - T2)3/(T1 - T2)1 = 3/4

Therefore, the temperature drop across the metal walls will be in the ratio of 3:2:1 (or 3/2:2/2:1/2) for thermal conductivities of 3k, 2k, and k respectively. This means that the metal wall with the highest thermal conductivity will have the lowest temperature drop and vice versa.

3:2:1

δ1 = δ2 = δ3 and cross sectional areas are same i.e. temperature drop varies inversely with thermal conductivity.