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Temperature hollow sphere - Temperature varying inside

  1. Sep 4, 2009 #1
    Hi. Just for illustration purposes I am trying to model the temperature in the walls of a hollow sphere. The purpose is to (very crudly) approximate the heating effects of a combustion process (running engine). It is assumed that there is gas inside of the sphere.

    Assumptions:
    A. Inner and Outer radii are constants - Ri and Ro, respectively.
    B. The temperature outside of the sphere is constant - Tw (w stands for "water")
    C. The temperature of the gas (Tg) is assumed to be uniform and either
    a) Tg(t) = C (constant) or
    b) Tg(t) = C1 + C2*sin(w*t)

    where "t" naturally denotes time. I want to study mainly the long term solution, but for fun also the transients.

    So far so good. Now to the problematic part. How to model and solve this? I understand the following:
    I. Because of symmetry it can be modeled as a one-dimensional problem (T(r,t))
    II. The Outer surface temperature of the sphere (T(Ro,t)) should be modeled as T(Ro,t)=Tw. Or maybe it shouldn't?

    After that I am lost! Is it a good idea to model the Inner surface temperature (T(Ri,t)) of the sphere as T(Ri,t)=Tg(t)? I don't think it corresponds very well to reality, or does it? Maybe it would be better to model the flux across the surface in some way, but how?

    Please help! I really would like to model and solve this in an appropriate manner.
     
    Last edited: Sep 4, 2009
  2. jcsd
  3. Sep 4, 2009 #2
    You should also account for the inside and outside convection coefficients. Especially the outside one to take in account the outside airflow over the sphere.

    Thanks
    Matt D.
     
  4. Sep 4, 2009 #3
    Depending on the heat source the system can reach a steady state where the power of the heater becomes equal to the heat loss per second through the walls of the sphere.To find the latter you can use the equation of thermal conductivity(P=kAdT/dx)
    K= thermal conductivity of sphere material dT= temperature difference across walls of sphere,dx= thickness of sphere,A= effective surface area of sphere
     
  5. Sep 4, 2009 #4
    A more simplistic example would be a good place to start.

    In Dennis Zill's book "A First Course in Differential Equations" Eight Edition there is a problem that could get you started in the right direction. It is not one you would use to model your problem but it could get you started.

    The problem goes like this;

    Two concentric spheres of radius r=a and r=b, a < b the temperature u(r) in the region between the spheres is determined from the boundary value problem

    r*d^2u/dr^2 + 2*du/dr = 0 u(a) = u(0) and u(b) = u(1)

    u(0) and u(1) are constants

    This type of ODE is considered a boundary value problem.

    I haven't solved this one yet but I will work on it.

    Thanks
    Matt
     
  6. Sep 4, 2009 #5
    Thank you guys. This problem seems to be more involved than I thought. I understand it is a one-dimensional PDE, but the problem is how to model it adequately and to get the initial and boundary conditions right. I am pretty skilled with ODE's, but PDE's are MUCH harder, :( Even setting up the right framework to solve is not easy.
     
  7. Sep 4, 2009 #6
    You can always use an FEA program such as ANSYS or NASTRAN.

    Thanks
    Matt
     
  8. Sep 4, 2009 #7

    jambaugh

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    As a start I'd suggest you first work out the heat flow through the metal sphere for given inner and outer temperature. Unless your walls are extremely thick relative to the inner radius I would suppose a linear heat flow approximation would work best.

    Beyond that modeling the gas temp. depends alot on what simplifying assumptions you'll make. If the gas becomes incandescent (even in the infra-red) you'll get a substantial amount of heat transport via radiation. This also depends on the opacity of the gas in the relevant temperature scale. You'll also find the emissivity (blackness) of the surface of your chamber is a factor determining the rate at which radiative inner heat is transmitted through the surface of the metal.

    You've picked a complex problem even for the steady-state case. Have fun with it!
     
  9. Sep 4, 2009 #8
    jambaugh:
    I thought I had simplified it so much it would be pretty easy to handle, :) But no.
     
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