⎛ ( ∆z )2 ⎞
⎛ kthj ,i+1
⎞
α j,i
⎟ , γ j,i = α j,i - 2, β j,i = 2 ⎜
= -2 ⎜
⎟,
⎜ kth ( ∆t ) ⎟
⎜ kth
⎟
⎝ j ,i
⎠
⎝ j ,i
⎠
(4.15)
⎛ c p,w ∆z ⎞
l ρ
⎟ , j,i = α j,i fus i ( ∆θi ) j,i .
δ j,i
= 2v j,i ⎜
⎜ c p kth ⎟
c p ρw
⎝ j ,i j ,i ⎠
Equation (4.14) is solved for Tj+1 using a
Newton-Raphson technique so that the final
matrix equation becomes
⎡-γ j,1 -1
⎤ ⎡∆T j+1,1 ⎤ ⎡ B1 ⎤
⎢-1
⎥ ⎢∆T
⎥ ⎢B ⎥
-γ j,2 -1
⎢
⎥⎢
j+1,2 ⎥
⎢ 2 ⎥
⎢
⎥⎢
⎥=⎢
⎥
(4.16)
⎢
⎥⎢
⎥
-1 ⎥ ⎢∆T j+1,n-1 ⎥ ⎢ Bn-1 ⎥
-1 -γ j,n-1
⎢
⎢
⎥
⎢
-γ j,n ⎥ ⎢ ∆T j+1,n ⎥ ⎢ Bn ⎥
-1
⎦ ⎣
⎦
⎣
⎦⎣
where
Bi = -T j+1,i-1 - γ j,iT j+1,i - T j+1,i+1 + φ j,i
2 ≤ i ≤ n - 1.
(4.17)
T1 must satisfy the boundary condition set forth in Equation (4.2). Following the
technique of Kahle (1977), Equation (4.2) may be rewritten in the form
T 4 + c1T + c2 = 0
(4.18)
with
1 ⎡
1 ⎤
∂qs
⎢( e + ρac p,aCDW ) + ρaCDW ' lM g
+ U pc p + κ1
c1 =
h
e
⎥
ε1σ ⎢ 0
∂T T =Tj-1,1
∆z1 ⎥
⎣
⎦
⎡
T ⎤
(1 - α ) I s ↓ +ε Ii↓ + ( e0 + ρac p,aCDW ) Ta + U pc pTa + κ1 j-1,2 ⎥ .
h
(4.19)
⎢
r
∆z1
-1 ⎢
⎥
c2 =
ε1σ ⎢
⎥
⎛
⎞
⎛
⎞
∂qs
⎢- ρaCDW ' l ⋅ ⎜ M g ⎜ qs (T j-1,1 ) -
⎥
⋅ T j-1,1 ⎟ - qa ⎟
e
⎜
⎟
⎜
⎟
∂T T =Tj-1,1
⎢
⎥
⎝
⎠
⎝
⎠
⎣
⎦
The above assumes that κ dT / dz = κ1 (T2 - T1 ) / ∆z1 and that the saturated mixing ratio
may be represented as
∂q
qs = qs (Tj-1,1 ) + s (Tj -1,1 ) ⎡Tj,1 - Tj-1,1 ⎤ .
(4.20)
⎣
⎦
∂T
Equation (4.18) is solved for T using the Newton-Raphson technique.
To ensure numerical stability when solving Equation (4.16), the following criterion holds
on the time step for each node:
⎛ ∆t ⎞ 1
⎟< .
kth,i ⎜
(4.21)
⎜ ( ∆z )2 ⎟ 2
⎝
⎠
i
34
Previous Page
