JSBeckton said:
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.
