# Heat transfer on a cylinder (doubt)

In summary, the speakers discuss a past university exam on partial differential equations and heat transfer in a cylinder. The conditions of the problem included an insulated lateral surface, a heat source on the bottom face, and a heat transfer coefficient on the top face based on a digit in the student's ID number. The speakers discuss the correct approach for solving the problem, with one speaker questioning if their initial assumption of ∂u/∂r = 0 was correct. The other speaker explains that this assumption was only correct if the lateral surfaces were perfectly insulated. They suggest looking up "Transient Heat Conduction" for more information on the correct approach.
Hi, there.

I remember when I was in the University (mech. engineering), I had an exam on partial differential equations about heat transfer in a cylinder. We had to determine the temperature distribution. I remember the conditions were that the cylinder was insulated in the side area, had a heat source in the bottom face at 9°C and the top face had a heat transfer coefficient that was taken from a digit in the student's ID number. For my particular case, this digit was 0, so I knew that the result was going to be that the cylinder ended up at 9º in all of its volume, or that the limit of the temperature function when t (time) tends to infinity equaled 9, independent of any other parameter.

Now, I don't remember the procedure, but I remember that I assumed that the rate of heat transfer in the radial direction or ∂u/∂r was going to be 0 because there's no heat being transfer in that direction and proceded from there. I got the result right: Lim(T) when t tends to infinity = 9 and wrote the reasoning. The teacher gave me all points in the problem because of the reasoning but said that the procedure was wrong.

My question is, was I correct in making that assumption? If anyone could maybe solve this problem here, I'd appreciated.This happened 10 years ago, but I never got the answer. It's been bugging me ever since and I forgot about D.E.

This is a 1D transient heat transfer problem in the axial z direction. Do you remember the partial differential equation describing transient heat conduction in 1D?

Chet

Q = kA(T1-T2), right?

Edit: no, nevermind. That's not a DE. I don't remember!

Since you say this is a 1D transient heat transfer problem, I take it that my initial assumption was correct. Am I right?

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Your approach was correct only if the cylinder is perfectly insulated on its lateral surfaces. You solve the steady heat equation in 2-D r-z cylindrical geometry, and apply the boundary condition of zero heat flow at the outer radius. This will prove that there is no temperature gradient in the radial direction at steady-state. If the top is also perfectly insulated, the cylinder will reach equilibrium with a spatially uniform temperature.

Yes, that was one of the boundary conditions: the cylinder was insultated on its lateral surface. And, because the digit in my ID number was 0, which corresponded to HT coefficient of the top face, the top face was also insulated, so the cylinder was completely insulated except for the bottom face where the heat source was. I'd love to see what I did, because I don't remember. I do remember just assuming that ∂u/∂r = 0.

Look up "Transient Heat Conduction" on Google.

## 1. What is heat transfer on a cylinder?

Heat transfer on a cylinder refers to the process of transferring thermal energy between the surface of a cylinder and its surroundings. This can occur through conduction, convection, or radiation.

## 2. How does heat transfer occur on a cylinder?

Heat transfer on a cylinder occurs through conduction, which is the transfer of heat through direct contact between the cylinder and another object or surface. It can also occur through convection, which is the transfer of heat through the movement of fluids or gases around the cylinder, or through radiation, which is the transfer of heat through electromagnetic waves.

## 3. What factors affect heat transfer on a cylinder?

The factors that affect heat transfer on a cylinder include the material of the cylinder, its surface area, the temperature gradient between the cylinder and its surroundings, and the properties of the surrounding medium (such as air or water).

## 4. How does the shape of a cylinder affect heat transfer?

The shape of a cylinder can affect heat transfer in several ways. A larger surface area will allow for more heat transfer, while a smaller surface area will restrict it. The shape can also affect the direction of heat transfer, as a cylindrical shape may promote more efficient convection compared to a flat surface.

## 5. What is the importance of understanding heat transfer on a cylinder?

Understanding heat transfer on a cylinder is important for many engineering and scientific applications. This knowledge can be used to design more efficient cooling systems, optimize heat transfer in power generation and manufacturing processes, and predict and prevent thermal damage to cylinders and surrounding equipment. It also allows for a better understanding of heat transfer in general and its impact on the environment.

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