BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 71
2.2. Numerical Approach
The heat transfer equation (2), transposed in the finite difference form,
becomes (Horbaniuc et al., 2003), (Horbaniuc & Dumitraşcu, 2006)
2
∂ θ 1∂θ 1 ⎡⎛ h ⎞ ⎛ h ⎞
+ ≅ 1 θ 2θ 1 θ
2
r r
2 ⎢⎜ − ⎟ − + ⎜ + ⎟
∂ r ∂ h ⎢⎣⎝ 2rm
⎠ ⎝ 2r
⎠
m− 1 m m+1
m
⎤
⎥, (7)
⎥⎦
where h is the grid step of the domain (wall, or SSM) and subscript m represents
the current node number. After a series of manipulations, Eq (7) becomes
p p p
im , − 1 im , im , m im , + 1 im ,
− θ + σ θ − γ θ = β , (8)
where superscript p accounts for the current time step (time step number p).
The coefficients σ, γ, and β are
σ
i,m
⎡ Ri
⎤
2⎢Ni
+ m⎥
⎡
( ) 2 ⎤
Δ h
i ⎢ i ⎥
=
⎣ ⎦
2
R
⎢
+
i
ai
τ ⎥ , (9)
⎡ ⎤ Δ
2⎢Ni
+ m⎥−1⎢
⎣
⎥
Δ
⎦
⎣ i ⎦
γ
m
⎡ Ri
⎤
2⎢Ni
+ m⎥
+ 1
Δi
=
⎣ ⎦
, (10)
⎡ Ri
⎤
2⎢Ni
+ m⎥
−1
⎣ Δi
⎦
β
m
⎡ Ri
⎤
2⎢Ni
+ m⎥
2
Δi
hi
=
⎣ ⎦
⎡ Ri
⎤ aiΔτ
2⎢Ni
+ m⎥−1
⎣ Δi
⎦
, (11)
where Δ represents the depth of region i which is W or S, h is the grid step (see
also Fig. 1.), Δτ is the time step, N is the total number of nodes of a domain, and
m is the current node for which the equation has been written.
For a time step, all of the finite difference equations of type (8), along
with the heat balance and the boundary conditions equations written in terms of
finite differences, form a set that has to be simultaneously solved. Since the
matrix of this set is tridiagonal (Croft & Lilley, 1977), (LeVeque, 2007), the set