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Heat transfer via conduction from hollow cylinder to Bitumen

  1. Oct 4, 2012 #1
    Hey all, not sure if this is where I'm supposed to post this question, but it is all about heat transfer. So this is part of my design project for my 4th year of chemical engineering.

    Problem Statement:
    A hollow cylinder 400 m long has hot combustion gases T = 500 Celsius flowing through it. The bitumen surrounding the cylinder is initially at 10 Celsius. How much time and how much heat is required to heat the bitumen 5 metres away from the outer edge of the cylinder to 85 celsius. (Assume the bitumen being heated is concentric with respect to the cylinder).

    Known Variables:
    Tr = 10 C
    Ts = 500 C (for now just assuming the surface of the cylinder in contact with the bitumen is at the combustion gas temperature, will need to iterate until I get convergence though I imagine)
    L = 400 m
    OD = 10" (25.4 cm)
    ID = 8"
    OD of Bitumen being heated = 10.254

    Unknown Variables:
    time required for surrounding bitumen 5 m thick to be heated from 10 to 85 Celsius
    heat required to do so

    Attempt:

    So far I've grabbed the PDE I think I would need to solve this.
    Keep in mind that since this isn't a textbook problem I have to solve this differently than usual. How I want to solve this (assuming zero free convection at boundaries for simplicity) is neglect the cylinder at first and just use the surface temperature condition (the outer edge of cylinder assuming it's temp is the same as the combustion gases) as well as the initial and final temperature conditions given above to just solve for the required heat that would need to leave the cylinder in order to heat the bitumen 5m away.
    Once solved I would use the value of heat required along with various efficiency estimates (essentially sensitivity analysis) to calculate the amount of heat that the gases would need to transfer internally to the cylinder walls to reach that condition, and as a result the mass flow rate of gases that would also be required.

    The PDE I grabbed is Trr + (1/r)Tr = (1/α)Tt (subscripts denote orders of differentials)
    Where r is the radius
    α is the thermal diffusivity
    t is the time

    The boundary conditions I have: (possibly incorrect)

    Initial Cond. (R3 is OD of bitumen)
    T(R3,0) = 10 C

    Boundary Cond. (R2 is OD of cylinder, aka ID of Bitumen)
    T(R2, t) = Ts = 500 C
    T(R3, t) = 85 C

    Essentially what I'm wondering is the best way I can go about doing this? (Solving for time and heat requirements that is) Also any suggestions as to how I can solve this problem differently (or better) are very welcome.

    Let me know if I can clarify anything, I realize I might have put this question confusingly. Also let me know if my formatting is wrong too.

    Thanks for any help guys
     
  2. jcsd
  3. Oct 10, 2012 #2
    Your doping out of the problem looks pretty good. I wouldn't be so quick to discount solutions in books. This problem is essentially transient heat transfer from a cylinder to an infinite medium. Why don't you check out Conduction of Heat in Solids by Carslaw and Jaeger. They have lots of problems solved, and this sounds like the type of problem you can find there.
     
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