Page 303 - Numerical Methods for Chemical Engineering

292 6 Boundary value problems

drt 1
2 2

2 2
n n

drt 1 drt 1 2
2 2

2 2
n 2121 n 22
Figure 6.17 Effect of drop tolerance on the sparsity of incomplete Cholesky factors.

Example. 3-D heat transfer in a stove top element
Consider the following “kitchen” transport problem. We have an electric stove top compris-
ing several heating elements (Figure 6.18). Each element contains within a ceramic matrix
two annular regions in which heat is generated by electrical resistance at a volumetric rate
S. The thermal conductivities of the ceramic matrix and heat generation material have a
constant value λ. The dimensions of an individual element are defined in Figure 6.18.
We wish to calculate the temperature profile within a single element, assuming that it
is part of a periodic array (the same boundary conditions apply if it is insulated on all
sides). We assume that a metal pot containing boiling water has been placed on top of the
heating element. If the thermal conductivity of the metal pot is much higher than the thermal
conductivity of the ceramic matrix, we expect that at steady state, the temperature of the
upper surface will be equal uniformly to that of boiling water, T b . On the bottom surface,
we assume that there is an underlying insulator layer, so that the heat flux out of the bottom
is zero, implying a zero gradient there.
We define a dimensionless temperature and dimensionless coordinates,

T − T b x 1 y 1 z
θ = χ = − η = − ζ = (6.168)
T b L 2 L 2 L
and convert the governing heat conduction equation to dimensionless form
2
2
2
∂ θ ∂ θ ∂ θ
− − − = σ H(χ, η, ζ) (6.169)
∂χ 2 ∂η 2 ∂ζ 2
The following function “switches on” the heat generation only within the specified annular