Page 304 - Numerical Methods for Chemical Engineering

Discretized PDEs with more than two spatial dimensions 293

(b)
(a) BC6 T(x, y, z = 0) = T b
z = 0
z = −d 1

z = −D
BC5 ∂T/∂z = 0 heat generation material
ceramic material
BC2 ∂T/∂y = 0
(c) ctatina y = L
setr
dain
t 1
BC3 r 1 BC4
∂T/∂x = 0 ∂T/∂x = 0
r 2

setr
t 2
y = 0
x = 0 BC1 ∂T/∂y = 0 x = L
Figure 6.18 Stove top geometry: (a) 2 × 2 grid of heating elements; (b) side view of an individual
element in a periodic array; (c) top view of an individual element.
regions,

1, if (χ, η, ζ) is within an annular region
H(χ, η, ζ) = (6.170)
0, otherwise
This reduces the number of independent parameters to seven,
SL 2
r 1 /L t 1 /L r 2 /L t 2 /L a ≡ L/D b ≡ d 1 /L σ ≡ (6.171)
T b λ
As we expect the solution to possess the symmetry

θ(χ, η, ζ) = θ(−χ, η, ζ) θ(χ, η, ζ) = θ(χ, −η, ζ) (6.172)

we restrict the domain to 0 ≤ χ ≤ 1/2, 0 ≤ η ≤ 1/2 (dashed lines in Figure 6.18), and
enforce symmetry at χ = 0,η = 0, to yield the boundary conditions:
1 1 ∂θ
BC 1 η = 0 0 ≤ χ ≤ − ≤ ζ ≤ 0 = 0
2 a ∂η
1 1 1 ∂θ
BC 2 η = 0 ≤ χ ≤ − ≤ ζ ≤ 0 = 0
2 2 a ∂η
1 1 ∂θ
BC 3 χ = 0 0 ≤ η ≤ − ≤ ζ ≤ 0 = 0
2 a ∂χ (6.173)
1 1 1 ∂θ
BC 4 χ = 0 ≤ η ≤ − ≤ ζ ≤ 0 = 0
2 2 a ∂χ
1 1 1 ∂θ
BC 5 ζ =− 0 ≤ χ ≤ 0 ≤ η ≤ = 0
a 2 2 ∂ζ
1 1
BC 6 ζ = 0 0 ≤ χ ≤ 0 ≤ η ≤ θ = 0
2 2