Math578 - Alexiades
Axially Symmetric Heat Transfer - Algorithm development
(to be done ON PAPER )
1. State precisely the mathematical problem describing axially symmetric heat conduction
in a hollow cylinder Ω: Rin ≤ r ≤ Rout, 0 ≤ z ≤ Z,
with imposed temperature boundary conditions on all boundaries,
starting at a uniform temperature uinit.
2. Set up a 2-dimensional (r,z) mesh for the axially symmetric heat transfer problem above,
by subdividing [Rin, Rout] into Mr subintervals and [0,Z] into Mz subintervals.
Determine (formulas for) the location of nodes: ri , zj , of faces: ri-½ , zj-½ ,
areas of radial faces: Ari-½,j , areas of axial faces: Azi,j-½ , and control volumes: ΔVij.
Make sure you specify the range of indices for each.
3. Derive the discrete heat balance for axially symmetric heat conduction in a
(hollow) cylinder using the above mesh, for any (implicitness parameter) θ in [0,1].
[This is the conservation law applied on each control volume, in terms of U and fluxes, where
fluxes=...., areas=...., volume=... as it will be coded up, do NOT plug these into the expression.
Do NOT write code, only formulas needed for update.]
4. Outline the explicit Finite Volume algorithm (formulas, NOT code)
for the problem with convective BC at r=Rin, imposed flux at r=Rout,
and insulated top and bottom (at z=0, z=Z), starting at uniform initial temperature uinit.
For simplicity, assume constant properties and mesh uniform in each direction.
5. Find the CFL condition and the stable Δtexpl for the explicit scheme (for internal nodes only).
[For this you DO need to plug in formulas for fluxes, areas, vol, to see the coefficients]