Homework Help: Heat Flow across cylindrical surface

1. May 5, 2012

jerzey101

1. The problem statement, all variables and given/known data
The temperatur at the point (x,y,z) in a substance with conductivity K=6.5 is u(x,y,z)=2y2+2z2. Find the rate of heat flow inward across the cylindrical surface y2+z2=6, 0≤x≤4.

2. Relevant equations
F=-k∇u
-k∫∫s∇u*ds

3. The attempt at a solution
So F=-6.5(0,4y,4z)
I get lost with how to find n because it is not in the usual cylindrical form x2+y2=6. Please give me some guidance. Thank you.

2. May 5, 2012

sharks

Plot $y^2+z^2=6,\, 0≤x≤4$. It is a cylinder with center origin, radius=√6 and height=4, from plane x=0 to plane x=4. It's faster and easier if you use Gauss Divergence Theorem in this case, as using the sum of (3) surface integrals will require some work.

Since $\vec F=-k(∇u)$, $\vec F= -26y\hat j -26z\hat k$.

Last edited: May 5, 2012
3. May 5, 2012

jerzey101

Thank you sharks. I really appreciate it.
∫∫∫52r drdθdz = 1248∏ which is the correct answer.

How do you know right off the bat to use the divergence theorem? Just from practice?

4. May 5, 2012

sharks

Since your formula for $\vec F$ involves -k, your answer should be negative. Unless you mistyped your relevant equation for $\vec F$?

The divergence theorem is just a 'shortcut' to evaluate the flux through a smooth completely enclosed surface. You don't have to evaluate the triple integral using cylindrical coordinates. Just use the formula for finding the volume of a cylinder: $\pi r^2h$, where $r=\sqrt6$ and h=4.

5. May 5, 2012

jerzey101

my answer should be negative, i just forgot to type the -. I see about just using the volume formulas. I'm just studying for my calc class so I wanted to learn the integral way because that's what I will have to do on the exam. Thanks again.

6. May 6, 2012

cjc0117

Actually, the answer is positive, since the heat flow is defined by the vector field F=-K∇u=-6.5<0,4y,4z>, and we want to find the rate of heat flow inward. The divergence is divF=-6.5(8). The volume of the cylinder is $π(\sqrt{6})^{2}4=24π$. So the INWARD flux through the cylinder with closed tops is $-[-6.5(8)(24π)]=1248π$. Then you'd have to show that the rate of heat flow through the tops of the cylinder is zero, which is easy since the x-component of F is zero.

Last edited: May 6, 2012