Published in: Proceedings Institut Teknologi Bandung, Lembaga Penelitian ITB, Volume 32, No. 1, pp. 425-433
In view of numerical studies, researchers have been developed numerical method for solving the coupled-partial differential equations. Numerical simulation of deep-bed
grain drying has been developed by researchers that is considering drying air phase and grain phase, for malt drying [Lopez et al. 1997], barley drying [Sun et al. 1997],
hazelnut drying [Lopez et al. 1998], corn drying [Thompson et al. 1968, Courtois 1997], wheat drying [Giner et al. 1996], and others [Franca et al.1994]. However,
these models have been developed were not a complete model, only one-dimensional, and solve them by finite-difference scheme.
The aim of this research is numerical studies of two-dimensional grain drying models. The second order discretization scheme was developed to predict differential
terms by finite-difference method. The two-dimensional mass and heat transfer equations have been developed in the drying gas phase in which consider convective
and diffusive mechanism, and the coupled of heat and mass transfers within intraparticle is considered in the grain phase. In order to verify the mathematical models, an
experimental prototype of deep-bed grain dryer was built, and the characteristics of the dryers were studied. In order to simulate the drying process using this model, a
computer code has been developed using the set of equations proposed in the following section.
2. MATHEMATICAL MODEL
In this paper, we proposed two-phase model for deep-bed dryer by taking into account the conservation of mass and energy within the bed and the spherical grains.
The assumptions were made in order to simplify the mathematical modelling and computation, i.e. a no shrinkage during drying, b uniform grain kernels are in size
and internally homogeneousisotropic spheres, c moisture migration path within each particle is in the radial direction only, d no conduction between close grains, and e
moisture transfer within the corn grain is controlled by liquid diffusion only [Sitompul et al. 1999].
The governing equations of moisture and heat balance for describing humidity and temperature of the drying gas are written in a two-dimensional cylindrical coordinate r,
z system. The moisture balance can be written as follows:
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
ε ε
− −
+ ∂
ρ ∂
− ∂
ρ ∂
− ⎟
⎠ ⎞
⎜ ⎝
⎛ ∂
ρ ∂
∂ ∂
+ ⎟
⎠ ⎞
⎜ ⎝
⎛ ∂
ρ ∂
∂ ∂
= ∂
ρ ∂
b b
m r
a z
a a
eff r
a eff
z a
1 a
Y Y
k r
Y U
r r
1 z
Y U
r Y
D r
r r
1 z
Y D
z t
Y
1
Heat balance can be written as follows:
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
ε ε
− −
+ −
∆ −
+ ρ
∂ ∂
− +
ρ ∂
∂ −
⎟ ⎠
⎞ ⎜
⎝ ⎛
∂ ∂
λ ∂
∂ +
⎟ ⎠
⎞ ⎜
⎝ ⎛
∂ ∂
λ ∂
∂ =
∂ +
ρ ∂
b b
p b
v m
b r
v a
a b
z v
a a
b eff
ar b
eff z
a b
v a
a
1 a
T T
h Y
Y H
k T
U YCp
Cp r
r r
1 T
U YCp
Cp z
r T
r r
r 1
z T
z t
T YCp
Cp
2 The governing equations of moisture and heat balance for describing moisture content
and temperature of the grains are written in a one-dimensional spherical coordinate r system are as follows:
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
