Heat Flow across cylindrical surface

[jerzey101]

Homework Statement

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.

Homework Equations

F=-k∇u
-k∫∫_s∇u*ds

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.

 

[DryRun]

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.

 

[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?

 

[DryRun]

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?

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.

 

[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.

 

[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.