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Nonlinear Driving Force Approximations of Intraparticle Mass Transfer in Adsorption Processes The Effect of Pellet Shape.

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Short Research Communication
Nonlinear Driving Force Approximations of
Intraparticle Mass Transfer in Adsorption
Processes: The Effect of Pellet Shape
A. Georgiou and K. Kupiec*
Institute of Chemical Engineering and Physical Chemistry
Technical University of Cracow, 31-155 Krakbw, ul. Warszawsku 24
POLAND
A general nonlinear drivingforce (NLDF) approximation of adsorbate uptake rate in
terms of pellet geometry is presented. The results show that the accuracy of the
proposed NLDF model is very good over the entire time fractional uptake) domain,
significantly improving over both the Linear Driving Force (LDF) and the Quadratic
Driving Force (QDF) models.
Introduction
Substantial computational simplification is achieved by the use of driving force
approximations in difisiodadsorption or diffusionheaction modelling and has
motivated numerous studies whose objective was to develop such expressions. In
most cases the methodology of intraparticle concentration profile approximation has
been applied (Buzanowski and Yang (1987); Do et al. (1986, 1987, 1988, 1995);
* Authorfor correspondence.
235
A. Georgiou and K. Kupiec
Georgiou and Kupiec (1995); Goto et a]. (1990,1991,1993); Liaw et al. (1979);
Tomida and McCoy (1987)). A different approach was recently illustrated in the case
of diffusion and adsorption (Georgiou and Kupiec, 1996) and in the case of diffusion
and reaction
(Georgiou and Tabice, 1995) resulting in expressions of excellent
accuracy.
In this communication an extension of the results obtained by Georgiou and
Kupiec (1 996) to non-spherical pellets is presented.
Approximate Expressions for the Case of Negligible External
Mass-Transfer Resistances.
The transient diffusion and adsorption in an initially uniformly loaded particle (at?=
0) to a unit step change of the concentration at its surface is described by:
,..(l)
with the boundary conditions
=o,
x<l;
A=O
...(la)
220,
x=l;
A, = 1
...( 1b)
2
x=O;
-dA- - 0
ax
...(1c )
where A = (Q - Q)/ (Qs- Q,)
The exact solution (Crank, 1956) is given as:
...(2)
236
Nonlinear drivingforce approximations of intraparticle mass transfer in adsorptionprocesses:
At short times, upon a step change, the particle can be treated as a semi-infinite
medium and this assumption leads to the following approximations (Crank, 1956):
I---
..(3a)
...(3b)
Georgiou and Kupiec (1 996) define the function:
dz
F(z)= 1-1
...(4)
for which it can be shown that:
...(5)
lim F(T) = a
r+m
and the function:
...(6 )
From Equations (3a) and (3b), it can be shown that:
-
G(0) = lim G@)= lim A
7-0
1-0
a2
2m2
- = __
dz
7~
The function G( 2) normalized with respect to G(0) for the cases of a slab, a cylinder
and a sphere is shown in Fig. 1.
Any approximation of the function F(r) or
C( A ) leads to an equivalent driving
force model as follows.
237
A. Georgiou and K . Kupiec
1
.o
G (A)
m
0.8
0.6
0.4
0.2
-
0 .o
-
0 .o
0.2
0.4
0.6
0.8
A
1 .O
Figure 1. Thefunction G(A)/G(O) for various pellet shape.
(I) The approximation:
F(T)z m(m + 2 )
...(8)
leads to the Linear Driving Force (LDF) model resulting from the parabolic
approximation of the intraparticle concentration profile.
(11) The approximation G @ ) = a: /2 leads to a model similar in form to Quadratic
Driving Force (QDF) approximation (Vermeulen, 1953):
a4
dT
-
+A2
-=a,
...(9)
2A
(111) The approximation:
G@)z -(12m2
2)
...(10)
x
yields the following nonlinear driving force (NLDF) model:
dz
238
(1 - 2)
...(11)
Nonlinear drivingforce approximations of intraparticle mass transfer in adsorption processes:
h"
1.2
1.o
0.8
0.6 -
0.4
1
\
1 -slab
2 -cylinder
3 -sphere
a - L D F model
b - Q D F model
c - N L D F model
,
0.2
i 1
3
0.0
0.o
0.2
0.4
0.6
0.8
A
1.o
Figure 2. A comparison of various approximations.
For small fkactional uptakes this equation reduces to Equation 3b, while for large
fractional uptakes it reduces to a LDF approximation.
In dimensional terms, Equation 11 can be rewritten as:
The results of the integration of Equation 11 are shown in Fig. 2 where
-
-
A = Aqp 1A,,
was adopted as an accuracy measure of the model. For comparison
purposes, analogous results are shown for the QDF model (see Equation 9):
-..(12)
239
A. Georgiou and K. Kupiec
...(12a)
AQDF(0)=&%
2m
and the LDF model:
ALDF
= 1- exp[-m(m + 2)t
]
...(13)
The results show that the accuracy of the proposed NLDF model (Equation 11) is
very good over the entire time (fiactional uptake) domain, significantly improving
upon both the LDF and the QDF model results.
1 .o
A
0.8
0.6
0.4
0.2
0.0
0
20
40
60
80
1O C
t [min]
Figure 3. A comparison of experimental and theoretical uptake curves (m=I).
In Fig. 3, the uptake curve for cyclohexane in silicalite (Cavalcante and Ruthven,
1995) and the predictions based on the exact model for slab-shaped particles and the
240
Nonlinear drivingforce approximationsof intraparticle mass transfer in adsorption processes:
NLDF model are shown. The experiments were carried out at following conditions:
2260 Pa, 3OO0C, crystal size 66x66~223pm. Excellent agreement is observed which
shows that the NLDF model can be used in parameter estimation from experimental
data.
Applications
The application of the NLDF model in batch adsorber modelling is illustrated for the
case of adsorption with finite external mass transfer resistances. Consider the case of
an initially adsorbate-free pellet plunged at T=O into an infinite medium of
concentration C, . The dimensionless model equations are:
...(14)
=o,
x<l;
z20,
x=l;
- = Bi(1- A,)
x = 0;
-aA= o
z
A=O
dA
ax
ax
...(14a)
...( 14b)
..(1 4c)
where A = Q / (b C,,)
The exact solution (Crank, 1956) is:
W
2=1-2mBi2z
P:[Pf,
exp (-P 3)
-t Bi2 + Bi(2-m)]
...(15)
From Equations (14) and (14b), it can be shown that:
a
- = m Bi(1- As)
dz
...(16)
241
A. Georgiou and K. Kupiec
Rewriting Equation 1 1 in dimensionless form:
...( 17)
From Equations 16 and 17:
I
(As - 2)=m Bi(1- A S )
This equation was solved to give Asas a function of
2 = X(T).
was integrated to obtain
2,and subsequently Equation 17
The results of these computations for a few
values of the Biot number are shown in Fig. 4.
1.1
8
A
1 .o
0.9
B i = 0.05
1
0.8
0.7
1
Figure 4. A vs
242
I
T
...( 18)
I
(NDLF model) with Biot number as aparameter.
Nonlinear driving force approximations of intraparticle mass transfer in adsorption processes:
Conclusions
A nonlinear driving force approximation of intraparticle mass transfer in adsorption
processes has been presented. The main virtues of the model are:
(a) a unified form with respect to pellet geometry;
(b) very good accuracy over the entire time domain;
(c) direct relation with the exact results known from the theory of diffusion.
Furthermore, the method applied for the derivation of this model can be a useful
alternative to the method based on intraparticle concentration profile approximations.
Nomenclature
a
A
b
Bi
C
D
J, (a>
kin
K
m
q
Q
R
t
E,+K for pore diffusion model
=1
for solid diffision model
Fractional uptake
= 1 for pore diffision model
= K for solid diffusion model
Biot number (LR / bD)
Gas phase concentration (kg/nf)
= D, (pore diffisivity) for pore diffusion model (m2/s)
= D, (solid diffusivity) for solid diffusion model (m2/s)
Bessel function of the first kind of order I
Film mass transfer coefficient (m/s)
Linear equilibrium adsorption constant
Geometric factor; m= 1 for slab
m=2 for cylinder
m=3 for sphere
Solid phase concentration (kg/d)
= C for pore diffusion model
= q for solid diffusion model
Characteristic geometric dimension equal to the cylinder
and sphere radius or to the half of the slab thickness (m)
Time (s)
=
Greek letters
a.
= (2n-l)x/2
=
=
Pn
=
=
=
nth non-zero root of JO(a)=O,
(a,=2.4048)
nx
nth non-zero root of ptan(p) - Bi = 0
nth non-zero root of pJ,(p) - Bi J@) = 0
nth non-zero root of Pcot(p) + Bi - 1 = 0
for slab
for cylinder
for sphere
for slab
for cylinder
for sphere
243
A. Georgiou and K. Kupiec
A
Model accuracy measure A = ZqF I A,
Ep
T
Pellet porosity
Dimensionless time ( = Dt / R2a)
Superscripts
average value
Subscripts
0
Value at T = 0
appr
Approximate value
b
Refers to bulk
ex
Exact value
Refers to the pellet surface
S
References
Buzanowski, M.A., and Yang, R.T. 1989. Extended linear driving- force approximation for intraparticle
diffusion rate including short times, Chem. Eng. Sci.,44(1 I), 2683-2689.
Cavalcante, C.L. Jr, and Ruthven, D.M.1995. Adsorption of branched and cyclic paraffins in silicalite.
Z.Kinetics, Ind. Eng. Chem.,34(1), 185-191.
Crank, J. 1956. The Mathematics of Diffusion, Oxford University Press, Oxford, UK.
Do, D.D.,and Rice, R.G. 1986. Validity of the parabolic profile assumption in adsorption studies, AlChE
J., 32(1), 149-154.
Do, D.D.,
and Mayfeld, P.L.J. 1987. A new simplified model for adsorption in a single particle, AIChE J.,
33(8), 1397-1400.
Do, D.D.,and Nguyen, T.S. 1988. A power law adsorption model and its significance, Chem. Eng.
Comun., 72, 171-185.
Do, D.D.,and Rice, R.G.1995. Revisiting approximate solutions for batch adsorbers: explicit half time,
AlChE J., 41(2), 426-429.
Georgiou, A., and Kupiec, K. 1995. The kinetics of adsorption in a spherical pellet, Inz Chem. Proc.,
16(1), 75-94 (in polish).
Georgiou, A. and Kupiec, K. 1996. Nonlinear driving force approximations of intraparticle mass transfer
in adsorption processes, Int. Comm. in Heat Mass T m f e r , 23(3), 367-376.
Georgiou, A,, and Tabis, B. 1995. An approximation of unsteady state diffusion and reaction process in a
catalyst pellet, Inz. Chem. Proc.,16(3), 379-391 (in polish).
Goto, M., Smith, J.M., and Mc Coy, B.J. 1990. Parabolic profile approximation (Linear driving force
model) for chemical reactions, Chem. Eng. Sci.,45(2), 443448.
Goto M.,and Hirose, T. 1991. Modified parabolic profile approximation of intraparticie concentration for
catalytic chemical reaction and adsorption, J. Chem. Eng. Japan,24(4), 538-542.
Goto, M., and Hirose, T. 1993. Approximate rate equation for intraparticle diffusion with or without
reaction, Chem. Eng. Sci., 48(10), 1912-1915.
Liaw, C.H., Wan& J.S.P., Greencorn, R.A., and Chao., K.C. 1979. Kinetics of fixed-bed adsorption: A
new solution, AiChE J.,25(2), 376-381.
Tomida, T., and McCoy, B.J. 1987. Polynomial profile approximation for intraparticle diffusion, AIChE
J.,33(11), 1908-1911.
Vermeulen, T. 1953. Theory of irreversible and constant-pattern solid diffusion, Ind. Eng. Chem., 45(8),
1664-1670.
Received: 30 October 1995; Accepted after revision: 5 May 1996.
244
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