1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    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. jcsd
  3. May 5, 2012 #2

    sharks

    User Avatar
    Gold Member

    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

    sharks

    User Avatar
    Gold Member

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Heat Flow across cylindrical surface
Loading...