Published in : Proceedings Institut Teknologi Bandung, Lembaga Penelitian ITB,
Volume 32, No. 1, pp. 425-433 r
T D
r r
r r
X D
r r
r 1
t X
p p
eff p
2 2
p eff
p 2
2 p
∂ ∂
ρ ∂
∂ δ
+ ∂
∂ ρ
∂ ∂
= ∂
ρ ∂
3
t X
H r
T r
r r
t T
Cp
v p
p p
2 2
eff p
p p
p
∂ ∂
∆ ε
ζρ −
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂
∂ ∂
λ =
∂ ∂
ρ 4
For this study, model parameters such as λ
az eff
, λ
ar eff
, D
z eff
, D
r eff
taken as constant parameters along the bed, also for U
z
, ρ
a
and ρ
p
, while U
r
=0 will be considered. Note that symbol
ζ in this equation describes phase conversion factor. However, the effect of momentum balance for grain drying process is not yet considered.
Computation of the equilibrium moisture content of the drying gas having direct contact with the grain surface the following relations can be used [Zahed et al 1992] :
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
⎟ ⎠
⎞ ⎜
⎝ ⎛
− ⎟⎟⎠
⎞ ⎜⎜⎝
⎛ −
= 16
. 273
T ln
31 .
5 T
6887 0214
. 27
exp 100
p
b b
w
5 6
w s
w
p p
Φ =
s w
s w
p 760
p 622
. Y
− =
7
Thompson et al. 1968 proposed Φ parameter or equilibrium moisture content relation
for corn grain drying:
for 8
⎟⎠ ⎞
⎜⎝ ⎛
+ −
− −
= Φ
49.81 16
. 273
T X
100 8.6541x10
exp 1
b 8634
. 1
5 -
cr s
X X
≤
The boundary and initial conditions for all equation developed in above are given elsewhere [Sitompul et al. 2000, Sitompul 1994].
3. NUMERICAL SOLUTION
The heat and mass transfer equations in the fluid and particle phases have been developed in the previous section are coupled, non-linear, unsteady, and involve
variable transport properties. The equations are solved numerically by using finite difference method with alternating direction implicit method algorithm applied to two-
dimensional cylindrical coordinate together with boundary and initial conditions. The most attractive feature of this method is that the solution converges fast.
3.1. Domain Computation and Boundary Condition
By employing the discretization scheme, the computed domain are divided into a number of control volumes with uniform grid spacing applied on both column dryer and
grain section as shown in Fig.1 and Fig.2 respectively. 3.2. Discretization Scheme
The second order discretization was employed to reduce discretization error of differential approximation. The discretization of fluid-phase differential terms is in
implicit and explicit at alternate time intervals for the axial and radial direction. The discretization of grain-phase differential terms is in implicit and explicit at alternate
time intervals in accordance with fluid phase algorithm. Discretization for time derivative is made use of backward difference method, while spatial derivative are
Published in : Proceedings Institut Teknologi Bandung, Lembaga Penelitian ITB,
Volume 32, No. 1, pp. 425-433 discretized by central difference one. The implicit discretization for all governing
equations will give set of linear algebraic equations that can solved using tridiagonal matrix routine TDMA algorithm [Davis 1984, Sitompul et al. 1999, Hoffmann et
al.
1993].
Fig.1. Domain computation and boundary conditions of cylindrical column dryer
Fig.2. Domain computation and boundary conditions of spherical grain
r=0 r=RC
z=0 z=L
∆r ∆z
z Y
r ,
L ,
t
= ∂
∂ z
T
r ,
L ,
t b
= ∂
∂
Yt,0,r=Y
i
T
b
t,0,r=T
bi
z Y
RC ,
z ,
t
= ∂
∂
∞ −
− ⎟
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎜ ⎝
⎛ +
λ ∆
= −
T T
ho 1
x 1
T T
hi
JM ,
z ,
t b
Wall JM
, z
, t
b 1
JM ,
z ,
t b
r Y
, z
, t
= ∂
∂ r
T
, z
, t
b
= ∂
∂
r X
, t
= ∂
∂ r
T
, t
P
= ∂
∂ Y
Y k
r X
D
m RP
t, eff
p p
− =
∂ ∂
ρ −
Y Y
H k
T T
h r
T
v m
b p
RP t,
p eff
p
− ∆
+ −
= ∂
∂ λ
−
∆
r
Generally, the discretization of differential terms are as follows, in which φ
expresses dependent variables such as drying air temperature T
b
and absolute humidity Y, grain temperature T
p
and moisture content X:
t t
1 n
j ,
i n
j ,
i
∆ φ
− φ
= ∂
φ ∂
−
;
z 2
z
n j
, 1
i n
j ,
1 i
∆ φ
− φ
= ∂
φ ∂
− +
;
2 n
j ,
i n
j ,
1 i
n j
, 1
i 2
2
z 2
z ∆
φ −
φ +
φ =
∂ φ
∂
− +
;
r 2
r
n 1
j ,
i n
1 j
, i
∆ φ
− φ
= ∂
φ ∂
− +
;
2 n
j ,
i n
1 j
, i
n 1
j ,
i 2
2
r 2
r ∆
φ −
φ +
φ =
∂ φ
∂
− +
9 After rearrangement of the equations above, the discretization equations for two-
dimensional model drying gas phase only can be written as follows:
1 -
n j
i, n
1 j
, i
1 j
, i
n 1
j ,
i 1
j ,
i n
j ,
1 i
j ,
1 i
n j
, i
j ,
i n
j ,
1 i
j ,
1 i
S A
A A
A A
= φ
+ φ
+ φ
− φ
+ φ
−
+ +
− −
+ +
− −
10 However, discretization for one-dimensional model grain phase only can be written as
follows:
1 -
n k
j, i,
n 1
k ,
j ,
i 1
k ,
j ,
i n
k ,
j ,
i k
, j
, i
n 1
k ,
j ,
i 1
k ,
j ,
i
S A
A A
= φ
− φ
+ φ
−
+ +
− −
11
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Volume 32, No. 1, pp. 425-433
3.3. Solution of Algebraic Equations
