CHAPTER 4 MATHEMATICAL MODEL A moving boundary layer model is used to carry out drying kinetics analysis of the proposed system and illustrated in this chapter.Figure 4.1 shows a schema
Trang 1CHAPTER 4 MATHEMATICAL MODEL
A moving boundary layer model is used to carry out drying kinetics analysis of the
proposed system and illustrated in this chapter.Figure 4.1 shows a schematic of the
one-dimensional physical model of the AFD process The mathematical model consists
of the applicable conservation equations of energy and mass It is based on the work of
(Jaakko and Impola, 1995) who considered drying of wood slab
y
X
P va
Q
Ice Layer
………
………
…
………
(f)
(s)
p vwa
Dry Layer
T a
Cold
dry air
Figure 4.1 Physical model of atmospheric freeze drying (f- interface; s-surface; Q-heat
transfer; m-mass transfer; Ta-temperature gradient; Pva-partial pressure gradient; ps-
partial pressure of water vapor around the product surface; pvwa- partial pressure of
water in the drying chamber)
With references to Figure 4.1, the ice interface (f) recedes to the centre line as heat of
sublimation (Q) flows from the surface (s) to the interface to a temperature gradient
(Ta) represented by the dotted curve Simultaneously, water vapor flows through the
Trang 2dry layer in response to the water vapor pressure (Pva) gradient represented by the firm
line curve Following mechanism is considered in the model: convective heat transfer
from the carrier gas to the surface of the solid mass; radiant heat transfer from the IR
radiation heater to the material surface and conductive heat transfer within the solid
mass
4.1 Assumptions
• One dimensional heat and mass transfer (thin slab)
• There is equilibrium between ice and water vapor at the interface
• Supplied energy is used to remove only ice at the sublimation front
• The frozen region is considered to have homogeneous and uniform thermal
conductivity, density and specific heat
• The shape of the product remains constant during the drying period considered
Shrinkage and deformation are neglected
4.2 Math Model
The conservation equation of energy for dry layer is:
ρp CP
t
T
∂
∂
= Kp 2
2
x
T
∂
∂
-
x
T C
∂
∂
′′
The conservation equation of water vapor inside the dry layer is:
ρg
t
Y
∂
∂
=ρg 2
2
x
Y
D p
∂
∂
-
x
Y m
∂
∂
′′
Here, Lv<x<L
The boundary conditions for heat and mass transfer
Trang 3s p s
e
e
x
T K T
T
⎠
⎞
⎜
⎝
⎛
∂
∂
=
− )
s g p s
g
d
x
Y D
Y
Y
⎠
⎞
⎜
⎝
⎛
∂
∂
=
The effective temperature of the atmosphere air is as follows:
) /(
) ( c g r r c r
e h T h T h h
The effective heat transfer coefficient including both convection and radiation
of the atmosphere air is as follows:
r
c
e h h
The radiation and convective heat transfer co-efficient are as follows:
s r s r
1 exp
//
//
−
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
•
co s
co s
co
c
h C
h C
h
h
(8)
Here,
L Nu
Re Pr
6
0
=
The mass transfer co-efficient is:
1 exp
/
//
//
−
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
•
do s
do s
do
d
h m
h m
h
h
(11)
Here,
L Sh
D
Trang 4Re 6
0
2 Sc 3
Analytical solutions of equations (1) and (2) subject to boundary condition equations
(3) and (4) are given by (Jaakko and Impola, 1995) as follows:
1 )]
, 1 ( exp[
) / 1
(
1 )]
, ( exp[
−
″ +
−
=
−
−
Γ
Γ
v c
e s
v c
v
e
v
Z g K h
C
Z z g K T
T
T
T
1 )]
, 1 ( exp[
) / 1
(
1 )]
, ( exp[
−
″ +
−
=
−
−
Γ
Γ
v d
d s
v d
v
g
v
Z g K h
m
Z z g K Y
Y
Y
Y
&
For thin slab,
Γ = 0,
O
z
z
v
Z z
Z
z
g
ds S Z
z
g
v
−
=
= ∫ Γ
Γ
,
1 ,
(16)
Therefore, equations (14) and (15) can be rearranged using equation (16) as follows:
1 )]
1 ( exp[
) / 1
(
1 )]
( exp[
−
−
″ +
−
−
=
−
−
v c e
s
v c
v
e
v
Z K h
C
Z z K T
T
T
T
1 )]
1 ( exp[
) / 1
(
1 )]
( exp[
−
−
″ +
−
−
=
−
−
v d d
s
v d
v
g
v
Z K h
m
Z z K Y
Y
Y
Y
Here,K c = &C s″L/K p, ,
p g s
d m L D
//
•
= z= x/L, Z L v L
v = The vaporization rate is related to the heat flux at the vaporization front
Lv
x
p
v
s
v
v
v
v e
v
v
dx
dT
L
L
m
l
m
l
T T
G
q
=
Γ
•
•
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
=
−
=
λ
//
//
//
//
(19)
Trang 5By using equation (17) and (19), the vaporization temperature is obtained as follows:
g v
c e
s v
e
v v
v
e
v
c Z
K h
C
l
T
G m
l
T
T
/ } )]
1 ( exp[
) / 1
{(
/ //
//
−
−
″ +
−
=
−
Where the effective heat conduction between the atmosphere and the drying front
exp /
1
/
//
//
//
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
=
Γ
•
• Γ
•
v c
s s
v s
v
Z g K h
C
L L C
The boundary condition for the moisture content at the vaporization front is
v v
L l v L x p
g
dx
dY D
•
=
•
−
⎟
⎠
⎞
⎜
⎝
⎛
=
The moisture mass fraction at the receding front is given by using equations (18) and
(22)
d s
v d g
v
h m
Z K Y
Y
/ 1
)]
1 ( exp[
) 1
(
1
& ′′
+
−
−
−
−
The evaporation rate can be obtained from
p
g v
sat v t
sat g t g
g
p
g v v g g
g
s
D
ShD Z g
p p
p p R
Sh
D
D
ShD Z Y Y R
SH
D
m
2 , 1 1
ln
2
2 1
1 1
1 ln 2
, ,
Γ
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
=
− +
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
=
′′
ϕ ρ
ρ
&
(24)
The thermophysical and transport properties of the carrier gas, potato and carrot used
in the simulations are summarized in Table 4.1(a) and Table 4.1(b)
Trang 6Table 4.1(a) Thermodynamic and transport properties of potato and carrot
Table 4.1(b) Thermophysical and transport properties of subzero air (Oosthuizen and
Naylor, 1999)
Properties of Air
Temperature of air -11oC -6oC
Saturated vapor pressure at tair (0.000000003)*exp(0.0957
*tair)
(0.000000003)*exp(0.09 57*tair)
Drying air relative humidity 0.0 0.0
Diffusivity of gas 0.000019 m2/sec 0.000019 m2/sec
Total pressure 101000.0 Pa 101000.0 Pa
Density of air 1.352 kg/m3 1.326 kg/m3
Specific heat of moist gas 1004.713 J/kg-k 1004.718 J/kg-k
Viscosity of air 0.000016 kg/ms 0.000017 kg/ms
Heat conductivity of gas 0.023 W/mk 0.024 W/mk
Slip velocity of surface 2.5 m/sec 2.5 m/sec
Molecular weight of air 29 29
Effective thermal
conductivity of dry
Product
0.552 W/m K 0.564 W/m K Saravacos and
Maroulis, (2001)
Diffusivity of
product
Sablani et al., (2000) 5.2e-6 m2/sec 7.8e-6 m2/sec
Mean density of dry
product
1.526 Kg/m3 1.253 Kg/m3 Senadeera et al.,
(2000)
Effective Sp heat of
dry product
7616 J/Kg K 3780 J/Kg K Oosthuizen and
Naylor, (1999
105*(2.29e-10*TV ^3-4.06e-6*TV ^2+1.9e-3*TV+2.612)
105 *(2.29e-10*TV ^3-4.06e-6*TV ^2+1.9e-3*TV+2.612)
Keey, (1972) Latent heat of
evaporation
Trang 74.3 Simulations
A MATLAB computer code was written to solve the analytical equations with the
relevant boundary conditions The solution procedure was initiated with a guess of the
initial evaporating temperature Iterations were continued until it matches with the
calculated value through satisfying the preset condition Subsequently, mass fluxes,
locations of evaporation front, temperature and moisture distribution inside the dry
layer were calculated
4.4 Summary
A two-layer moving boundary model was developed to simulate the variation of the
transient moisture content distribution in the drying product in the form of a thin slab,
sublimation front temperature, location of the sublimation front and moisture content
as well as temperature distribution inside the dry layer of a novel fixed bed
atmospheric freeze dryer using a vortex tube to generate cryogenic temperature air for
drying and multi-mode heat supply Model is also capable to capture the drying
kinetics phenomena under different range of operating condition as well as effect of
different size of the model heat sensitive products