# Thermo, Temperature profile of rod

• digipony
In summary, the rod's temperature changes uniformly along its length, but the temperature at any given point changes depending on the radius at that point. The heat entering the rod at a given point is proportional to the radius at that point, and the heat leaving the rod at a given point is proportional to the radius squared.f

## Homework Statement

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.

H=(kaΔT)/x
A=∏r^2

## 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
dΔT=[Hdx]/[k∏(dr)^2]
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. :)

Last edited:
Write an expression for the area of the rod radius as a function of its length, x. Use that expression in formulating the differential equation that equates the heat entering a slice of length dx to the heat leaving at x+dx. The derivation of the ODE is similar to the case for 'standard' 1-D conduction except the area (pi*radius^2) is within the derivative operator that you get from using the first term of a Taylor series expansion to evaluate the heat leaving at x+dx. Integrate the differential equation and evaluate the two constants of integration by the boundary conditions.

Thanks!