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Homework Help: Heat Flow across cylindrical surface

  1. May 5, 2012 #1
    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

    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. jcsd
  3. May 5, 2012 #2


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    Plot [itex]y^2+z^2=6,\, 0≤x≤4[/itex]. 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 [itex]\vec F=-k(∇u)[/itex], [itex]\vec F= -26y\hat j -26z\hat k[/itex].
    Last edited: May 5, 2012
  4. May 5, 2012 #3
    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?
  5. May 5, 2012 #4


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

    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: [itex]\pi r^2h[/itex], where [itex]r=\sqrt6[/itex] and h=4.
  6. May 5, 2012 #5
    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.
  7. May 6, 2012 #6
    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 [itex]π(\sqrt{6})^{2}4=24π[/itex]. So the INWARD flux through the cylinder with closed tops is [itex]-[-6.5(8)(24π)]=1248π[/itex]. 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
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