Consider a rod that is 1m long whose radius changes from 1cm at one end to 4cm at the other end in a uniform fashion. Assume that the end with r=1cm i kept at 0°C and that the end with r=4cm is kept at 100° C. Determine the temperature profile along the rod. One can assume that at any position along the rod the temperature is uniform across its cross section. Hint: Determine an expression for the incrimental change in temperature, dT, at any position x along the rod and integrate to get a general expression for the total temperature change between the ends of the rod. Remember that the heat flowing through any cross section of the rod is constant.
The Attempt at a Solution
I think what I have done so far is wrong, as I have trouble with integral problems. (My problem is with setting them up, not solving them). I tried to follow the hint given, but i am stuck. Here is what I have so far:
(Hx)/(kA) = ΔT
ΔT=∫H/[k∏(dr)^2]dx (This is supposed to be aDefinite integral from 0 to 1-couldn't figure out the format).
ΔT= H/[k∏(dr)^2] and here is where I get stuck, and think I set up the equation wrong, as 1/dr doesn't exist. Any help would be greatly appreciated. :)