Solving Heat Transfer Problems with Rods: Tips & Tricks

[JSBeckton]

My book is fairly explicit when it comes to solving problems with rods of uniform temp, and fins that have a heat source at one end but I have a couple of HW problems that deal with rods.

1) A rod connected to 1 hot wall

given:

Wall temp
environment temp
dia. and length of rod
heat reansfer coef. between rod and wall

find:
heat transfer

2) Rod connected between two hot walls of different temps

given:
temp of both walls
dia and length of rod
heat transfer coef.

find:
heat loss in watts

Can anyone get me pointed in the right direction? I really don't know where to start since the rods ar not of uniform temp.

Thanks

 

[OlderDan]

These could be very complex problems, depending on the assumptions you are allowed to make. If the rod is an excellent heat conductor the assumptions could be constant temperature throughout the rod for the first case, and a constant temperature gradient in the second case.

For the first problem you have heat transfer into the rod at the wall over an area equal to the end of the rod, and heat transfer out of the rod into the environment at all other surfaces of the rod. You can perhaps ignore any temperature variation within the rod; if not things will be more complicated.

For the second case you have the additional heat transfer out of the rod into the lower temperature wall, and a uniform temperature variation over the length of the rod. More heat will be lost to the environment at the hot end of the rod than at the cooler end. If the rod is a very good conductor, you may be able to ignore any temperature variation over its length.

I think this should get you started.

 

[JSBeckton]

I don't think that we can assume uniform temp over the rods length. Is there any kind of a formula or sequence of formulas for these situations?

 

[OlderDan]

I don't think that we can assume uniform temp over the rods length. Is there any kind of a formula or sequence of formulas for these situations?

I can see why you can't make that assumption in the second case, but not the first. I perhaps should not have suggested it for the rod between two walls.

You may know more about this topic than I do, but I'll tell you what I am thinking and maybe it will get you going. I'll look at the second case. If you can do the second problem, I'm confident you can do the first.

I'm assuming the following:
The heat transfer coefficient is a measure of the heat transfer per unit area per unit time for a unit temperature difference between two objects. The heat transferred is assumed directly proportional to the temperature difference and the surface area.

Heat will flow from the hot wall into the rod at some rate.

Heat will flow out of the rod at the cold wall at some rate.

If the cylindrical surface were well insulated, the flow rate into the rod at one wall would have to equal the flow rate out of the rod at the other end (once steady state is achieved; the heat needed to warm up the rod is another story). If the coefficients and areas are the same at the two walls, then the temperature differences would have to be the same to equalize these flow rates. If the rod is a perfect heat conductor, its temperature would have to be midway between the two wall temperatures. If the rod is not a great conductor, then you would need some conductivity coefficient that relates heat flow to the temperature gradient in the rod. You did not mention any such coefficient in your origanal post, but if you have that data it can be incorporated. You would need to establish a temperature gradient that supports the same flow rate as you have at the walls.

With heat loss to the environment added to the mix, you would need a greater flow rate at the source wall than at the sink wall. I assume the coefficient for this is much less than the rod/wall coefficient, but the area exposed to the environment would probably be much larger. You would need a bigger temperature difference between the source wall and the rod than the difference between the sink wall and the rod. If the conductor is less than perfect, the gradient within the rod would be more complicated with heat escaping through the walls because less and less heat would flow through each successive elemental length of the rod. It would be a calculus problem, but I'm pretty sure it can be done.

I'll wait to har if any of this makes sense to you before going any further.

 

[JSBeckton]

It makes sense but i am not sure how to go about doing it. I don't have my book on me but I believe that the problem involved both convection and conduction from the rod, I believe that this has something to do with solving a differential equation since I have heat moving across the rod and also away from the rod due to convection.

 

[OlderDan]

It makes sense but i am not sure how to go about doing it. I don't have my book on me but I believe that the problem involved both convection and conduction from the rod, I believe that this has something to do with solving a differential equation since I have heat moving across the rod and also away from the rod due to convection.

The complete solution surely does involve a DE. Take a look at this site. http://imartinez.etsin.upm.es/bk3/c11/Heat conduction.htm I think Exercise 2 probably is the approach you are looking for to solve your first problem. See if any of that makes sense to you.

 

[JSBeckton]

Thanks Dan, but I just can't follow, I will have to see the TA.

 

[OlderDan]

Thanks Dan, but I just can't follow, I will have to see the TA.

Good idea- and good luck with it.