Determine the difference equations applicable to the centerline and at the surface of an axisymmetric cylindrical geometry with volumetric heat generation and convection boundary condition. Assume steady-state conditions.
GIVEN
Axisymmetric, steady, cylindrical geometry with volumetric heat generation and surface
convection boundary condition
FIND
(a) Difference equations for the centerline and surface
SKETCH
The solution to this problem completes the formulation of the cylindrical geometry presented in
Section 3.5, with the added constraints of steady state conditions and symmetry.
As the radius is given by
Let the convection coefficient be h and ambient temperature be T?. The inner surface area per unit
length of the shaded control volume is
and the outer surface area is
The volume of the control volume per unit length is
The energy balance on the control volume gives
The energy balance on the control volume gives
For the control volume for the centerline node, i = 1, the volume per unit length is
and the surface area per unit length is
The energy balance gives
Simplifying and putting into the tridiagonal form
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