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Development of Solutions to Two-Resistance Mass Transport Models Based on External and Pore Diffusion. Part II Experimental Results

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Development of Solutions to
Two-Resistance Mass Transport
Models Based on External and Pore
Diffusion. Part II: Experimental Results
G. McKay', 9. Al-Duri and Stephen McKee
Department of Chemical Engineering, The Queen's University
of Belfast, Belfast, BT9 5AG, Northern lrland
Introduction
This paper applies the theoretical mass transport models developed in Part I to
experimental results for several adsorption systems. The nomenclature has been
presented in Part I. All Figures, Tables and References presented are in the present
paper.
Discussion of Results
Porosity is the main characteristic of all adsorbents since the porous structure
determines the surface area and accessible adsorption sites in any sorbent. This
makes the film-pore diffusion mechanism worthy of investigation as an adsorption
mechanism. In general adsorption is considered a three-step process:
(1) mass transport across the boundary layer surrounding the particle;
(2) intraparticle diffusion through the liquid filled pores expressed by the effective
pore diffusivity, (Deff);
(3) adsorption at a site.
Step (3) is very rapid and steps (1) and (2) are the rate controlling processes.
The transport process is driven by the concentration gradient in the liquid
filled pores and hence intra-particle diffusion is the main step that needs thorough
investigation. First, molecules transfer from one site to another by desorption in
the liquid- filled pores, then adsorption again on the new site. External mass
transfer is controlling in the first few minutes of the process when sorbate
molecules are diffusing through the boundary layer to the external surface.
The film-pore diffusion model, according to the present solutions, assumes
irreversible adsorption. Irreversibility is indicated when the adsorption and
desorption data do not coincide, i.e. some of the adsorbed material is still in the
solid phase even after desorption. This occurs in the cases of chemisorption when
the surface functional groups on the sorbent form complexes with the sorbate
molecules.' The fraction of adsorption which is reversible is due to the Van der
Waal forces that are formed at some of the sorption sites. Complete irreversibility
is characterized by a rectangular shaped isotherm where Y* is constant regardless
of the fluid phase concentration value (Cs). The experimental isotherm shown in
* To whom correspondence should be addressed.
Developments in Chemical Engineering and Mineral Processing, Vol. 1, No. 2/3,Page 146
Two-Resistance Mass Transport Models
147
Adsorption
isotherm
Concn. in liquid, C
Cz
Concn. in liquid, C
Figure 1 Relationship between bulk and su@ace concentration during the
adsorption in batch tests.
Table 1 Values of Langmuir equilibrium constants.
Adsorbent
Solute
K
(dm3 g-1 )
Chitin
Chitin
Activated carbon
Activated carbon
Silica
Acid Blue 25
Acid Blue 158
Phenol
p-chlorophenol
Astrazone Blue
22
4.1
11.3
48
500
b
(dm3 mg-')
0.12
0.019
0.050
0.11
0.016
Figure 2 is typical of the systems investigated in this work. The curve shows a
degree of reversibility and a saturation monolayer which could best be described
mathematically by the Langmuir equation. However, for liquid-phase equilibrium
concentrations greater than 200 mg dm-3, all the points have a constant monolayer
saturation value typical of the rectangular isotherm. The saturation monolayer
value (Yh*)is used for the analytical solution to the adsorption problem when the
operating line, and all tie lines terminate on the monolayer. For all other operating
conditions, the appropriate value of Y* is used as predicted by the isotherm
equation.
Model I : Pseudo-analytical batch model (PAM)
This is a film-pore diffusion model with the saturation isotherm constraint, i.e. it
assumes complete irreversibility. This assumption enabled an analytical solution
G. McKay, B. Al-Duri and S. McKee
I48
Table 2 External liquid mass-transfer coeflcients (PL) and effective difisivities
(Defl)for the analytical solution represented by Model 1.
~~
Adsorbent
Solute
PL
(cm s-')
Sorbsil silica A
Sorbsil silica B
Activated carbon
Chitin
Chitin
Activated carbon
Astrazone Blue
Astrazone Blue
Phenol
Acid Blue 25
Acid Blue 158
p-Chlorophenol
8.of1.0
2.ofo.5 lo4
2.Mo.5
4.W1.0
5.Ok1.5
1. M . 4
1.8M.5
8.M1.0
2.5f1.0
1 3 0 . 5 10"
3.0k1.0 10"
1.0M.3 lo4
Figure 2 Operating and tie lines for the adsorption of Astrazone Blue on silica.
to be obtained which yields the concentration decay curve. Application of this
model is restricted to system conditions where the operating line terminates on the
'plateau' of the isotherm. Equilibrium is described by the Langrnuir isotherm.
Table 1 shows all isotherm constants for the systems under investigation.
In the present work, results have been obtained by using a Fortran 77 program
that carries out the solution method outlined in Part I. The input to the program
comprises the following eight parameters: initial solute concentration, adsorbent
mass, adsorbent particle diameter, adsorbent voidage and density, the external
Two-Resistance Mass Transport Models
149
mass transfer coefficient, effective diffusivity, and a capacity based on the
monolayer saturation capacity (Yh*).
This saturation capacity is obtained by either
reading the monolayer value from isotherm curve Figure 2, or evaluating it as the
ratio Kib.
Figure 2 shows a typical output for the adsorption of Astrazone Blue on silica
type B. For a number of systems whose conditions terminate on the monolayer, it
was possible to correlate theoretical and experimental concentration decay curves
using a single constant external mass tIansfer coefficient and a single constant
effective diffusion coefficient. The results for the adsor tion of Astrazone Blue on
Sorbsil silica type A have been reported previously! the values of the mass
transfer coefficients for this and other adsorption systems are compared in
Table 2. This table shows the values of PL and Dew which can be used
successfully as fitting paramete values.
Reported literature3 values of Deff are 2.70 x lo-’, 2.80 x lo-’, 1.70 x lo4
and 1.00 x 10“ cm2 s-l for phenylacetic acid, benzoic acid, acetophenone,
alizarin and crystal violet respectively, all on carbon B10. The above values have
been obtained using the graphical method. Applying the present method to other
systems has yielded Defi values of 18.0 x lo-’ and 2.50 x lo-’ cm2 s-l for
cm2 s-l f o r a
BB69Isilica and phenol/carbon
and 2.50 x
BB69/woodmeal system.6 Comparing these values to the present values, it is
evident that the present Defi values are lower than literature values. This is due to
two major factors. First, the molecular size of adsorbates. Dye molecules are
considered as large organic molecules compared to phenols, erc., and hence are
expected to diffuse much slower than other organics. Second, the type of
adsorbent. Activated carbon is a polar adsorbent with a porous structure and
controlled active sites, hence it has a high capacity or strong affinity to many
organic substances. Therefore, diffusivities of these organics are expected to be
strong in carbon systems, more so than other adsorbents.
However, despite the model constraint that restricts the variation in system
conditions, it is an easy-to-use practical model for industrial purpose^,^ and once
applied it gives good agreement between experiment and theory.
Model 2: Polynomial curve-fit model (POLYFIT)
Typical outputs are shown in Figures 3,4 and 5 for the adsorption of Acid Blue 25
on chitin, phenol on activated carbon, and p-chlorophenol on activated carbon,
respectively. The data for the adsorption of phenol on carbon are poorly correlated
mainly due to the poor curve fits obtained when the concentration-time curves
decay rapidly initially and then almost instantaneously level out to a plateau.
Therefore, polynomial curve fits were difficult to achieve for this system. The
results produced by the model were heavily dependent on the order of the
polynomial chosen to describe the experimental concentration decay curve. Too
small an order tended to flatten out the concentration gradient, whereas too high
an order introduced fluctuations (undulations) in the theoretical concentration
decay curves yielding very erroneous gradients. However, the program execution
time was very short and many runs could be analysed without the need for a
powerful computer. The program can be run for a small number of experimental
points. The external film mass-transfer coefficients and effective diffusion
coefficients are listed in Table 3.
In general, the results were acceptable but it was apparent that a better curve
fitting equation(s) was required. Furthermore, the mathematical forms of the
equilibrium isotherms were becoming more important now that systems, whose
G. McKay, B. Al-Duri and S. McKee
I50
Table 3 Values of
Adsorbent
PL and Defifor Model 2.
Solute
~
D
4 -I
(cm s )
3.0f1.0 lo4
9.of3.0
5.of1.0
1.ofo.3
2.ofl.O lo4
1.3rto.3 lo4
1.5rto0>5
PL
~~~
Silica B
Chitin
Chitin
Activated Carbon
(cm s-'1
~
~~
Astrazone Blue
Acid Blue 25
Acid Blue 158
Phenol
1.~o.2
Table 4 Values of PL and Degfor Model 3.
Adsorbent
Solute
PL
(cm s-')
Carbon
Silica
Chitin
Chitin
Phenol
Astrazone Blue
Acid Blue 158
Acid Blue 25
2.Mo.5
4.W.O lo4
1.Mo.5
1.Mo.3
D
qf -I
(cm s 1
1.4rto.4
1.0rto.2
3.0k1.5 lo4
1.5H.5 lod
operating and tie lines did not terminate on the monolayer, were being analysed.
Table 3 shows the values of the Langmuir isotherm equilibrium constants.
Model 3: Exponential curve-fit Model (EXPOFIT)
As explained in Part I, this is a film pore-diffusion model with a more advanced
solution method, such that the saturated isotherm constraint is no longer required,
. ~ technique
and it provides more accurate results than the graphical s ~ l u t i o nThis
has widened the range of experimental conditions over which the model can be
applied, leading to further investigation of intraparticle diffusion. In the present
model, ranges of PL and D,ff values are input to the Fortran 77 program, which
responds with the best five combinations of PL and Deff based on the minimum
residual R'.
Figures 6, 7 and 8 show the quality of the curve-fit for the adsorption of
phenol on carbon, Astrazone Blue on silica, and Acid Blue 158 on chitin,
respectively. The values of
and Deff for the experimental systems are given in
Table 4. The variations in PL and D,ff are shown in Table 5 for the adsorption of
phenol. The data listed are those based on the minimum residual for each
particular analysis. This program requires a longer execution time than Model 2,
largely due to the detailed procedures in the program. The program also requires
more data points (AO) than Model 2 to fit the exponential series equations.
A closer inspection of Tables 1, 2 and 4 allows comparison of the D,ff values
from the present systems, obtained by the three different solution methods.
Table 6 shows that generally, D,ff values obtained by PAM and EXPOFIT are
nearly equal, whereas those of POLYFIT are much lower values for AB158/chitin
PL
Two-Resistance Mass Transport Models
I51
systems compared to PAM and EXPOFIT. This can be explained in terms of the
relatively poor fit of the curves obtained by the POLYFIT program, and makes
EXPOFIT the preferred method rather than POLYFIT. However, further
investigations on a wider range of systems is needed before drawing final
conclusions. The time of application is also worth considering. Table 7 shows that
EXPOFIT has been used to fit data up to 100 minutes, i.e. 1-2 hours. For this
period it has been possible to describe the whole range of conditions by a single
.-C
E
Sherwood number
3
0
m
a
0
c
0
0
Sherwood rumber
Figure 3 Plots of Sherwood number versus time curves for the adsorption of Acid
Blue 25 dye on chitin.
152
G. McKay, B. Al-Duri and S. McKee
D e value
~
nearly equal to the Deff used in PAM. However, previous studies’ with
other systems have shown that when the kinetic data supplied is for 24 hours, Deff
varies with C,. Since the present data does not exceed 120 minutes, this point is
beyond the scope of this work.
Model 4: ‘Tes@t’ program
Program ‘Testfit’ was developed to be run prior to using the pseudo-analytical
Models 2 and 3. The program reads in the time-concentration coordinates and uses
three methods for fitting curves and hence determining the concentration
gradients. The previous two methods for polynomial and exponential series ;re
Sherwood number
Sherwood number
Figure 4 Plots of Shewood number versus time curves f o r the adsorption of
phenol on activated carbon.
I53
Two-ResistanceMass Transport Models
used, and also a finite difference program. The finite difference technique assumes
that the concentration gradient at a point is the average of the straight line
gradients before and after that point, as defined by:
(60)
(dC/dt) = [(Cj - Ci-I)/(tj - tj-1) + (Ci+1 - Ci)/($+l - $)]I2
The results produced by this program will enable the user to compare the
exponential and polynomial curve-fit data with the gradients predicted by the
finite difference method. In this way the number of possibilities can be reduced
before trying the selected curve-fit equations in the main programs.
Typical outputs from the testfit program are shown in Table 7 for the
adsorption of Acid Blue 25 on chitin. Only three sets of results are presented and
0
Sherwood number
In
Sherwood number
Figure 5 Plots of Shenvood number versus time for the adsorption of
p-chlorophenol on activated carbon.
I
G.McKay, B. Al-Duri and S.McKee
I54
Table 5 Values of PL and D,,,
for phenol adsorption onto carbon.
Initial
concentration
(mg dm-3)
Adsorbent
mass
(g)
PL
(cm
s-')
400
400
400
300
300
300
300
300
200
200
1.70
2.66
3.20
0.85
1.70
2.81
6.31
7.77
0.85
1.70
0.15
0.15
0.15
0.2
0.2
0.2
0.2
0.2
0.2
0.2
D
105)
(cm$ff-l
s
0.8
1.1
1.4
1.2
1.8
1.3
1.3
2.0
3.5
3.4
11
l0C
- Theoretical
0
C,= 200mg dm-3
3
Time (minsl
Figure 6 Plots of Shenvood number versus time for the adsorption of phenol on
activated carbon.
it can be seen that the curve-fit equations for the exponential and polynomial
series are in agreement with experimental results. However, when the gradients
are compared, the exponential function gives much closer agreement to the finite
difference method. Furthermore, after the first seven points in Table 7,the slope
of the gradient changes in the wrong direction obviously due to a fluctuation. This
was characteristic of polynomials with more than four terms, and often with only
four terms. These equations, therefore, were applicable only for a short period of
time, but by reducing the order of the polynomial the gradients exhibited greater
deviation.
Two-Resistance Mass Transport Models
155
36.
32 -
Vol ( u n 3 1 = 1700
PL (cmls) 3300 x 10-3
28 .
A5 2 c -c
- Theoretical
-
1i: -
I
n
0
C,= 300 mg dm-3
C,= lOOmg dm-3
-
12
84
L.
z
0
3
Time (mins)
Figure 7 Plots of Sherwood number versus time f o r the adrorption of Astrazone
Blue Dye on silica.
R (cm) = 0 0302
V o l . ( c d I = 1700
PL (cm/s 1 = o 900 x 10-2
- Theoretical
C0=50mgdm-3
In
10
0
10
20
30
LO
50
60
70
80
90
Time Imins)
Figure 8 Plot of Shenvood number versus time for the adsorption of Acid Blue
158 dye on chitin.
1
0
G. McKay, B. Al-Duriand S. McKee
156
Table 6 Mean diffusion coeficients (cm2 s-l lo7)for the systems under
investigation.
BB69/Silica B
AB 1WChitin
AB25IChitin
PAM
POLYFIT
EXPOFIT
0.8
30
15
1.o
13
20
1.o
30
15
Table 7 Comparison between theoretical concentration predicted by Model 4 and
experimental results.
Time
(min)
0.0
1.o
2.0
4.0
9.0
16
25
36
49
64
100
Experimental
concentration
(mg dms)
100
96
94
92
87
83
79
75
71
68
63
Theoretical
concentration
(mg d ~ n - ~ )
100
96
94
92
87
83
79
75
71
68
63
Gradient
(mg dm-3 min-')
-4.00
-3.00
-1.50
-1.00
-0.785
-0.507
-4.404
-0.335
-0.253
-0.169
-0.138
Conclusions and Significance
Model 1 is useful for predicting mass transfer coefficients and effective
diffusivities for adsorption systems in which the operating lines and tie lines
terminate on the monolayer of a pseudo-irreversible isotherm. The analysis is
based on external liquid mass transfer and internal pore-diffusion mass transport,
and the model has been applied successfully to several systems.
Models 2, 3 and 4 were developed to extend the use of the above basic model
for adsorption systems in which operating and tie lines terminated anywhere on
the equilibrium isotherm. The value of the concentration gradient with respect to
time, (dC/dt),, is needed and had been determined graphically in previous work.3
Consequently, these models tested curve fitting equations and gradient
determination by polynomial series equations (model 2), exponential series
equations, and finite difference techniques. It was decided that exponential series
equations are most useful for this procedure after comparing the outputs from a
number of tests on the computer. Both curve-fit equations were applied
successfully to a number of systems.
.
Two-ResistanceMass Transport Models
157
Future work will involve the application of the exponential solution for:
(i) longer time periods; (ii) other single component systems; (iii) multicomponent
adsorption systems.
References
1 Mattson, J.S.,Mark, H.B., Malbin. M.D., Weber, W.J. and Crittenden, J.C. 1969. Surface
chemistry of active carbon: specific adsorption of phenols. J. Colloid Interface Sci.,
31, 116-121.
2 McKay, G. 1984. Analytical solution using a pore diffusion model for a
pseudo-irreversible isotherm for the adsorption of basic dye on silica. AIChEJ, 30(4),
692-697.
3 Spahn, H. and Schlunder, E.U. 1975. The scale-up of activated carbon columns for water
purification, based on results from batch tests (PartI). Chem. Eng. Sci., 30,529-537.
4 Bino, M.J. 1983. Mass Transfer Processes in the Adsorption of Industrial Pollutants onto
Activated Carbon. Ph.D Thesis, The Queen's University of Belfast.
5 McKay, G.and Bino, M. 1985. Application of two resistance mass transfer model to
adsorption systems. Chem. Eng. Res. Des., 63, 168-178.
6 McConvey, I.F. and McKay, G. 1985. Mass transfer model for the adsorption of basic
dyes on woodmeal in agitated batch adsorbers. Chem.Eng. Proc., 19,267-277.
7 Neretnieks, I. 1974. Adsorption of components having a saturation isotherm. Swedish
Paper J., 77,4074 13.
8 Al-Dun, B. and McKay, G. 1990. Comparison in theory and application of several
mathematical models to predict kinetics of single component batch adsorption
systems. Proc. Safety Environ. Protn. (Trans. I. Chem E.), 68B,254-268.
Received: 27 February 1991; Accepted: 10 April 1992.
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