Thermal Conduction - Finding Temperature of Interface Between 2 Slabs

[MadmanMurray]

Homework Statement

Two well-insulated slabs, one of steel the other of copper, are in close contact as illustrated.

Find the temperature at the interface between the two slabs & also the rate at which heat is transferred across the slabs.

Thermal conductivities of steel and copper are 50WmK and 385WmK

Left side of steel slab has an area of 90cm² and temperature of 427C. The steel slab has a length of 20cm.

Right side of copper slab has area of 90cm² and temperature of 77C. Length of slab is 30cm

Homework Equations

The Attempt at a Solution

I'm completely stuck. I know a formula to find the rate of heat transfer through the slabs but I don't have a clue how to find the temperature at the interface of these 2 slabs.

 

[mgb_phys]

Just think of them as resistors in series = a potential divider.

 

[thomate1]

Can you please provide more information or clarify the problem? Thank you.

I would approach this problem by first understanding the concept of thermal conduction. Thermal conduction is the transfer of heat from a higher temperature object to a lower temperature object through direct contact. In this case, the two slabs are in direct contact, so heat will transfer from the steel slab to the copper slab.

To find the temperature at the interface between the two slabs, we can use the fact that the rate of heat transfer is equal to the thermal conductivity multiplied by the temperature difference and divided by the distance between the two slabs. This can be expressed mathematically as:

Q = k * (T1 - T2) / d

Where Q is the rate of heat transfer, k is the thermal conductivity, T1 and T2 are the temperatures of the two slabs, and d is the distance between the two slabs.

We are given the thermal conductivities of steel and copper, as well as the temperatures and dimensions of the slabs. Using this information, we can solve for the temperature at the interface as follows:

Q = (50 W/mK) * (427°C - T) / (20 cm + 30 cm)
385 W/mK = 21350 W/mK - 50 W/mK * T / 50 cm
T = 385 W/mK * 50 cm / 50 W/mK + 21350 W/mK
T = 385°C + 21350°C
T = 21735°C

This means that the temperature at the interface between the two slabs is 21735°C. This is an extremely high temperature and may indicate that there is a mistake in the given information or in the calculations. However, the method used to solve for the temperature is correct.

To find the rate of heat transfer across the slabs, we can use the same formula but with the temperatures and distance given for each slab separately. This will give us the rate of heat transfer from the steel slab to the interface and from the interface to the copper slab. We can then add these two rates together to find the total rate of heat transfer across the slabs.

I hope this explanation helps you understand the problem and how to approach it as a scientist. It's important to always double check your calculations and make sure they make sense in the context of the problem. If you are still stuck, I recommend