Steady state conduction in a hollow sphere

[rcummings89]

Hello,

I'm having trouble with a conduction problem, I have access to the answer but not the solution. I did it on my own and my value is half of what the answer is. Now, my calculus is a little rusty, but I don't know where I am going wrong. So the dimensions and temperatures of the sphere are:

r_i=0.01 m
r_o=0.02 m
T_i=70 ^oC
T_o=100 ^oC
k=0.58 W/m K

Attempt:

q/A = -k dT/dr

q/A dr = -k dT

Where A = 4[itex]\pi[/itex]r² then

q ∫ dr/r² evaluated from r_i to r_o = -4 [itex]\pi[/itex] k ∫ dT from T_i to T_o yields...

-q (r_o^-1 - r_i^-1) = -4 [itex]\pi[/itex] k (T_o - T_i)

Substituting in the values, I get q = 4 [itex]\pi[/itex] [(0.58)(100-70)](0.02-0.01) = 2.18 W

The actual answer is double that and negative, -4.37 W. Any idea what I'm doing wrong?

Thanks in advance!

 

[Chestermiller]

Your algebraic solution is correct, but you substituted into it incorrectly. The problem is with the r_o and r_i. You should have (1/r_i-1/r_o) in the denominator.