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Transient heat transfer in a cylinder with internal heating

  1. Jan 11, 2018 #1
    hi, I met a problem about heat transfer in cylinder, if you can help, I will appreciate it.

    The question is simple. I want to know the transient heat distribution in a cylinder with internal heating(constant temperature not constant flux). The boundary conditions comprises two constant temperature, inner and outer surface. we assume the cylinder radius is infinite. therefore, we have

    the governing equation:
    dT/dt=k(d2T/dr2+(1/r)*dT/dr)
    rw<r<∞
    i.c. : r(r,0)=t0
    b.c.: r(∞, t) = t0
    r(rw ,t) = t1



    It looks like a hollow cylinder problem. But unfortunately, with surprise, I failed to find a solution for the heat transfer with two constant temperature in literatures. In Carslaw and Jaeger, I only found one solution for constant heat flux internal heating. So I derived the solution by Laplace Transform and that gives me a solution with Bessel function , here, I just want to find a example to verify my solution.

    Thank you very much in advance !!

    Bruce
     
    Last edited: Jan 11, 2018
  2. jcsd
  3. Jan 11, 2018 #2
    I think i’ve Seen the solution to this somewhere. I”lol check it out tomorrow.
     
  4. Jan 12, 2018 #3
    See Practical Aspects of Groundwater Modeling by William C. Walton, National Water Well Association, 1984, Section 5.3.1

    Jacob, C. E., and S. W. Lohman, 1952, Nonsteady flow to a well of a constant drawdown in an extensive aquifer, Trans. Am. Geophys. Union, Vol. 33, No. 4

    Hantush, M. S., 1964, Hydraulics of wells., In Advances in Hydroscience, Vol 1., Academic Press Inc., New York.

    Hantush, M.s., 1954, Drawdown around wells of variable discharge, Jour. of Geophys. Res., 69 (20)
     
  5. Jan 12, 2018 #4
    hi, Chestermiller, thank you very much for the reference. Actually, I derived my solution using the method from groundwater pumping. But I am just curious if there is a solution from the aspect of heat transfer, which is more straightforward. still appreciate your help !! By adjusting the boundary value, it is also possible to make a verification.
     
  6. Jan 12, 2018 #5
    The equations are exactly the same. So, in place of hydraulic diffusivity, just use thermal diffusivity, and in place of head, just use temperature.
     
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