Heat conduction in hollow cilinder.

In summary, the problem involves a hollow cylinder with inner radius R1 and outer radius R2, with a heat conductivity of k and length l. The inner and outer temperatures are T1 and T2 respectively. The task is to determine the temperature gradient and heat flow in the cylinder. The equation used is Heat flux = k*A(r)(dT/dr), where A(r) is the area of the cylinder at radius r and dT/dr is the temperature gradient. After integrating and solving for T, the final equation is T = T1 + heat flux/(k*2*pi*l)*ln(r/R1). The heat flux can be found by plugging in the known temperatures at the inner and outer radii.
  • #1
aranud
20
0

Homework Statement


Given is a hollow cilinder with inner radius R1 and outer radius R2. The heat conductivity of the material is k. The cilinder has length l and an inner temperature of T1 and outer temperature T2. Determine the temperature gradient in the cilinder and the heat flow that leaks away radially.

Homework Equations


Power = k*Area(Temperatureinside-Temperatureoutside)/(thickness)
Area = 2*pi*R*l

The Attempt at a Solution


The reason I can't solve this problem is the changing area when going from inside to outside the cilinder. Obviously the heat flow going from the inside to the outside should be minus the heat flow from outside to inside, since the areas are different I think the only way this is possible if the temperature gradient falls off quicker going inside causing the heat flow to be the equal despite the smaller area.
So much for intuition though since I've tried fruitlessly to write this down mathematically.
I also think the trick is to use an intermediate temperature T and equate the powers going through an area between R2 and R1 since we've used that one before.
 
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  • #2
aranud said:
Power = k*Area(Temperatureinside-Temperatureoutside)/(thickness)

This equation assumes that the temperature profile is linear, which isn't the case here. A more general equation is [itex]\mathrm{Heat~flux}=kA(x)(dT/dx)[/itex], or since you're working with radius r, [itex]\mathrm{Heat~flux}=kA(r)(dT/dr)[/itex]. You should be able to find T(r) through integration.
 
  • #3
When I integrate the formula you've given me I get T = (heat flux)/(k*2pi*l)*ln(R)
To find the heat flux should i simply fill in T1 and R1? I feel like I've missed something since the problem seems a bit simple this way.
 
  • #4
aranud said:
When I integrate the formula you've given me I get T = (heat flux)/(k*2pi*l)*ln(R)
To find the heat flux should i simply fill in T1 and R1? I feel like I've missed something since the problem seems a bit simple this way.

Check your integration. A lone ln(R) can't be correct; you can't take the logarithm of a parameter with units.
 
  • #5
You're right, is this better? T = T1+ heatflux/(k*2*pi*l)*ln(r/R1)
It seems to describe the right heat gradient but how do I get the heatflux from this?
 
  • #6
You know the temperature at the outer radius too.
 
  • #7
Ah yes of course:blushing: My brain is all but melted right now because of all the studying I guess. Thanks for the help.
 

FAQ: Heat conduction in hollow cilinder.

What is heat conduction?

Heat conduction is the transfer of thermal energy from one region of a material to another due to a temperature difference. It occurs through direct contact between the particles of the material.

How does heat conduction work in a hollow cylinder?

In a hollow cylinder, heat conduction occurs through the walls of the cylinder. The inner wall of the cylinder, which is in contact with the hot material, absorbs the thermal energy and transfers it to the outer wall. The thermal energy then dissipates to the surrounding environment.

What factors affect heat conduction in a hollow cylinder?

The rate of heat conduction in a hollow cylinder is affected by several factors, including the temperature difference between the inside and outside of the cylinder, the thermal conductivity of the material, the thickness of the walls, and the surface area of the walls.

How can heat conduction in a hollow cylinder be calculated?

The rate of heat conduction in a hollow cylinder can be calculated using the Fourier's Law of Heat Conduction, which states that the rate of heat conduction is directly proportional to the temperature difference, thermal conductivity, and surface area, and inversely proportional to the thickness of the walls.

What are some practical applications of heat conduction in hollow cylinders?

Heat conduction in hollow cylinders is commonly used in household appliances such as refrigerators, air conditioners, and water heaters. It also plays a crucial role in industrial processes such as heat exchangers and thermal insulation in buildings.

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