# Heat transfer through a cylindrical shell

• Ian Baughman
In summary: The solution will give you the temperature profile as a function of radius. Once you have that you can integrate to find the heat flow per unit length.In summary, the conversation discusses the calculation of the rate of heat flow per unit length between the inner and outer surfaces of an infinitely long cylindrical shell with inner radius a, outer radius b, and thermal conductivity k. The formula used for heat transfer through a flat surface is not applicable in this scenario, and instead, the heat equation in cylinder coordinates must be solved to find the temperature profile as a function of radius. Integration can then be used to determine the heat flow per unit length.
Ian Baughman

## Homework Statement

An infinitely long cylindrical shell has an inner radius a and outer radius b. If the inside is maintained at a temperature Ta and the outside at a temperature Tb, determine the rate of heat flow per unit length between inner and outer surfaces assuming the shell has a thermal conductivity k.

## Homework Equations

[/B]
H = -KA((TH-TC)/L)

## The Attempt at a Solution

[/B]
1) I said let TH = Ta and TC = Tb
2) I let L = b-a so my new expression is:
H = -KA((Ta-Tb)/(b-a))​
3) My issue here is I can not figure out what to use for the cross sectional area A. In the example I was using as reference the heat was flowing through the pipe not the outer shell of it so the cross sectional area was easy to calculate.
4) My idea was to use 2πr and multiply it by the length of the shell but since it is an infinitely long cylindrical shell that wouldn't make sense to do.

Your formula is only applicable to heat transfer through a flat surface. To find the stationary state in a different geometry you have to solve the Laplace equation for that geometry.

Also note that you are asked for the transfer per unit length of the cylinder.

Orodruin said:
Your formula is only applicable to heat transfer through a flat surface. To find the stationary state in a different geometry you have to solve the Laplace equation for that geometry.

Also note that you are asked for the transfer per unit length of the cylinder.
Do you mean like starting with the definition of heat transfer, H = (dQ/dT) or H = -KA(dT/dx), and solving from there using Laplace?

Ian Baughman said:
Do you mean like starting with the definition of heat transfer, H = (dQ/dT) or H = -KA(dT/dx), and solving from there using Laplace?
That differential equation is only valid for heat transfer in one dimension (flat geometry). You cannot apply it here unless your shell is very thin. In general you need to solve the heat equation in cylinder coordinates - which for a stationary situation is equivalent to laplace equation.

## 1. How does the temperature of a cylindrical shell affect heat transfer?

The temperature of a cylindrical shell has a direct impact on the rate of heat transfer. As the temperature difference between the inner and outer surfaces of the shell increases, the rate of heat transfer also increases. This is because a larger temperature difference creates a greater driving force for heat to flow from the hotter surface to the cooler surface.

## 2. What is the role of the material in heat transfer through a cylindrical shell?

The material of the cylindrical shell plays a significant role in heat transfer. Different materials have different thermal conductivities, which determine how easily heat can pass through them. Materials with higher thermal conductivity will allow for faster heat transfer compared to those with lower thermal conductivity.

## 3. How does the thickness of a cylindrical shell impact heat transfer?

The thickness of a cylindrical shell affects heat transfer in two ways. Firstly, a thicker shell will result in a longer distance for heat to travel, which can slow down the rate of heat transfer. Secondly, a thicker shell has a larger surface area, which can increase the amount of heat that can be transferred. Therefore, the thickness of a cylindrical shell must be considered when analyzing heat transfer.

## 4. Can the shape of a cylindrical shell affect heat transfer?

Yes, the shape of a cylindrical shell can impact heat transfer. A cylindrical shell with a larger diameter will have a larger surface area, allowing for more heat to be transferred compared to a thinner shell. Additionally, the shape of the ends of the shell can also affect heat transfer. A flat end will have a larger surface area compared to a dome-shaped end, resulting in faster heat transfer.

## 5. How do boundary conditions affect heat transfer through a cylindrical shell?

Boundary conditions, such as the temperature and material of the surrounding environment, can greatly impact heat transfer through a cylindrical shell. For example, a colder surrounding environment will result in a larger temperature difference and faster heat transfer. Additionally, a material with a higher thermal conductivity in contact with the shell will also increase the rate of heat transfer.

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