Ind. Eng. Chem. Process Des. Dev. 1984, 23, 221-226
221
External Mass Transfer and Homogeneous Solid-Phase Diffusion Effects during the Adsorption of Dyestuffs Gordon McKay,' Stephen J. Allen, Ian
F. McConvey, and James H. R. Wallerst
Department of Chemical .Engineering, The Queen's Universlty of Belfast, Belfast BT9 5DL, Northern Ireland
A mass-transfer model has been developed to explain the adsorption of dyestuffs (Acid Blue 25 and Basic Blue 69) onto various adsorbents in agitated batch adsorbers. The model is based on two resistances, namely, extemal mass transfer and homogeneous solid-phasediffusion. The basis of the model has been proposed previously, but
on applying this prior result over a wide range of our operating conditions the model was found to go unstable. Consequently, an analytical solution to replace a previously employed numerical solution is presented In this paper, which has been utilized with complete success over a wMe range of operating conditions, provided that a suitable time interval for analytical iteration is adopted.
Introduction The removal of dye from dilute aqueous solutions consists of a number of steps: (a) transport of dye from the bulk of the solution to the solution/adsorbent interface; (b) mass transport across the boundary layer (film diffusion); (c) adsorption onto external surface sites; and (d) internal mass transport of dye within the particle using either a pore diffusion or homogeneous surface diffusion model. Whether film or internal mass transport offers the most resistance depends to a great extent on the method of contacting. In agitated batch adsorbers film resistance will only be rate limiting for a short initial period of time, whereas film diffusion will exert more influence in continuous fixed bed adsorbers. In the homogeneous model, although the adsorbent particles may be rather porous, there are no sinks for the adsorbate since it diffuses into them in the adsorbed state. The entire process might be pictured as an adsorption at the outer surface of the adsorbent particles followed by a sponge-like absorption of the adsorbate into them. In the pore diffusion model there is an adsorption of the adsorbate into the pores with a concurrent adsorption distributed all along the pore walls. The aim of this paper is to propose an adsorption model based on film and homogeneous solid phase mass transport to explain the adsorption mechanism of dyestuffs onto adsorbents. Literature Review and Homogeneous Models Adsorption models should include: (a) a description of the diffusion processes within an adsorbent particle; (b) a mass balance for the dye in the fluid phase in the external voids; (c) a coupling equation between the fluid and the surface of an adsorbent pellet to account for the mass transfer resistance of the film between the bulk of the fluid and the fluid at the surface of a pellet. The homogeneous surface diffusion equation for the general case (Rosen, 1952,1954; Weber and Chakravorti, 1974; Nertenieks, 1977) is given by
For the case in which diffusivity is concentration independent, eq 1 becomes +Departmentof Applied Mathematics and Theoretical Physics, The Queen's University of Belfast, Belfast BT7 lNN, Northern Ireland. 0196-4305l8411123-0221$01.50/0
The film resistance equation for the homogeneous model is given by (3)
This equation relates the rate of mass transfer between the bulk of the fluid and the outer surface of the solid and acts as the coupling equation. The mass balance for a finite batch adsorber is given by (4) The homogeneous model has been solved analytically without a film resistance and assuming a linear isotherm for a number of liquid-olid systems using batch adsorbers (Eagle and Scott, 1950, Dryden and Kay, 1954). Solutions are available for a nonlinear isotherm but film resistance is neglected (Tien, 1960, 1961; Miller and Clump, 1970; Colwell and Dranoff, 1966) once again. A n unsteady-state homogeneous model incorporating film mass transfer and a nonlinear isotherm has been developed (Mathews and Weber, 1966) for the adsorption of phenol and bromophenol on carbon. Experimental Section The adsorption of dyestuffs onto various adsorbents has been studied. The dyestuffs selected were Acid Blue 25 (Telon Blue ANL) supplied by Bayer Ltd. and Basic Blue 69 (Astrazone Blue FRR) supplied by Ciba-Geigy Ltd. Measurements of all dye concentrations were made by a Perkin-Elmer 550s absorption spectrophotometer and with a wavelength corresponding to the wavelength for maximum absorption. Two cheap adsorbents, namely, peat and wood, were selected as materials; a few results are reported using Sorbsil silica supplied by J. Crossfield and Sons, Widnes, and activated carbon supplied by Chemviron Ltd. Details of the apparatus and experimental results have been reported in previous papers (Poots et al., 1976; McKay and McConvey, 1981; McKay and Allen, 1980; McKay et al., 1978, 1980; HMSO, 1961). Theoretical Section The theoretical model is that of Mathews and Weber (1976), in which the adsorbent particles are assumed to be identical spheres of radius R. The concentration of 0 1984 American Chemlcai Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 2, 1984
-
T h e l i n e s for two resistance model
I
C
( ng d ~ n - ~ )
Figure 1. Wood particles, of constant size range, contacted with Basic Blue 69 dye solution of initial concentration 50 mg dm-3 at room temperature.
solute inside the typical particle at distance r from the center and at time t is denoted by qi(r, t). The variation of qi with distance and time is governed by the diffusion equation
The final equation required in the model is that which described conservation of solute, i.e., any solute which is not in the fluid must be inside the adsorbent particles. Thus at any time t
(5)
subject to the boundary conditions qi(r, 0) = 0 (6) qi(R, t ) = qs(t) (7) In ( 5 ) it is assumed that the diffusion coefficient D is constant. The boundary condition (6) represents the initial (t = 0) state of the particle, while (7) expresses the fact that the concentration at the surface of the particle varies with time. The surface concentration, q,, is related to C,, the equilibrium concentration in the liquid phase at the surface of the particle, by the adsorption isotherm. In the present work it is assumed that the isotherm has the form acs = 1 bC:
+
(16)
(8)
where a, b, and p are constants. However, it should be noted that the model and mathematical techniques are not restricted to the particular form (8). The movement of solute from the liquid phase into the particle is described by the mass-transfer equation
where S,, V,, p , and e are respectively the surface area, volume, density, and porosity of the particle; k f is the mass transfer coefficient and C is the concentration of solute in the liquid phase. The quantity Q is the average concentration of solute in the particle Q(t) =
where C, is the initial concentration of solute in the liquid phase, W is the total weight of adsorbent particles in the system, and V is the total volume of fluid. Introducing dimensionless variables T = Dt/R2, x = r / R and defining u xqi, eq 5 to 13 become
R3 J R q i ( r , t)r2dr
Using S , = 4aR2 and V, = (4/3)aR3, (9) may be reduced to
c=co--W q -
V The mathematical problem presented by the model is the simultaneous solution of eq 13 to 19. Mathews and Weber (1976) have proposed a scheme whereby eq 13 is solved numerically by the method of Crank-Nicholson (Ambramovitz and Stegun, 1964) the other equations, of course, also being handled numerically. At first, we adopted their technique and verified that i t worked for the typea of system they had studied. However, on applying their methods to the experimental situations reported in this paper, it was found that the computer program ran into problems. It was not difficult to isolate the source of this trouble; it was the numerical solution of (13). An alternative procedure which avoided the direct numerical solution of (13)was therefore sought. The starting point is a semi-analytic solution of (13) which satisfies the
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984
The numerical procedure for solving eq 13 to 19 is then as follows. (i) Taking c,(70) = Q ( T ~ )= 0 and c(70) = Co it is possible to calculate Q(TJ from (32). (ii) Using this result in (28) (with n = 1) and in (19) f , and UT1) are obtained, i.e.
boundary conditions (14), i.e.
The function f(+) in (20) is determined by the boundary condition (15) and, on physical grounds, is expected to be a fairly smooth function of its argument. Consider now the series of times T ~ T, ~ T, ~ ..., , where 70 = 0; 71 > Ti-1 (21)
fi
Ti-1
W
a (71)
(7,
{e-(~-1)’/4(~,t-f)
(22) It is next assumed that the time intervals T,-T~-~ are sufficiently short that f ( # ) is effectively constant in each interval. I t is then a good approximation to write n
CfiGi(x,
7,)
is1
(23)
where f i is the average value off(#) in the interval ~ i tori and Gib, 7,) = 1 {e-(X-1)’/4(Tn-f)- e-(X+1)*/4(Tn+))(24) ( T n - 7’)1/2
1::d.’
where
and erf ( x ) is the error function
In the approximation (23) eq 17 becomes n Q(7n)
3EfiGi(7n) i=1
(34)
(35)
(iv) The result (35) enables one to invert (16) using Newton’s rule (Abramovitz and Stegun, 1964) as recommended by Mathews and Weber, to get CS(T1). These four steps give all the relevant quantities at time T ~ i.e., , Q(sl),C(T~), q8(sl),C,(T~).Upon using these values the cycle can now be repeated and the relevant quantities at time 72 can be calculated. Thus, from (32) Q ( T ~ )is obtained (19), c(72), from (28) 1 (36) f 2 = -@(72) - 3fi01(7z)) 3Gdrz)
- e-(X+1)2/4(Tr,-f)}
- r’)1/2
u(X, 7,)
(33)
(iii) Employing (33) in (23) and (23) in (15) gives ~ ~ ( 7=1 fiGi(1, ) 71)
$ STid7’f(7’)
&=1
= a(Ti)/@i(71)
co -
Then, trivially u ( ~7,,) = 1
223
~
from (23) and (15) qs(nJ, and, finally, from (16) C,(T~). Clearly, by going through the above cycle enough times one can generate the values of Q, C, qs, and C, at any time 7,.
We have programmed the above procedure on an ICL 1906s computer and found that it yields reliable results, even in those cases where we could not get the Mathews and Weber technique to work. We hope to publish a more precise comparison between our method and that of Mathews and Weber at a future date. It should be noted that we require to know the error function (27). A scientific subroutine to calculate this function should be available on most computers; we have used that provided by the Numerical Algorithms Group (U.K.) Ltd. (1959). Although, depending upon the circumstances, it may not be desirable to choose the time intervals AT to be of equal length, it is still worthwhile to point out that the numerical cycle outlined above is particularly easy and efficient to program when the time intervals are all of equal length; in fact, all of the calculations reported in this present paper were carried out with length equal intervals.
Discussion Adsorption Isotherms. The adsorption isotherms determined experimentally were analyzed mathematically according to the modified isotherm, eq 8.
where
The values of the constants a and b and the exponent /3 are given in Table I for the various dyeladsorbent sys-
and 4 3x3
2 3x3
K ( x )= - + - ( x 2 - 2)e-x’
+3
In a similar manner (18) is approximated by
tems reported in this work. Figure 1 shows an isotherm plot for the adsorption of Acid Blue 25 onto wood. The operating line for one batch contact time run is shown, connecting initial (C, = 100, qs = 0) to final condition (C, = 38, qs = 6.4). Tie lines have been drawn on Figure 1depicting various conditions. Tie lines for a single resistance model based on external mass transfer only are presented and then two cases of Yworesistance” models based on external mass transfer and homogeneous solid phase diffusion. In the latter cases the same internal diffusivity has been used, 2.0 X lo4 cm2s-’, but different external coefficients were selected. A value
224
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984
Table I. Modified Isotherm Parameters
A
I
I
h
I 1.0 I r
I I4 m
1
18
Figure 2. Two resistance homogeneous model for Acid Blue 25 adsorption on peat.
a , dm3 b, (dm3 g-' mg-')p
adsorbent
dye
carbon peat wood silica
Acid Blue 25 Acid Blue 25 Acid Blue 25 Basic Blue 69
18.5 0.62 0.26 0.38
0.12 0.021 0.036 0.016
p
0.99 0.99 0.76 0.99
cm s-l was closest to experimentally of kf = 3.0 X observed results. Kinetic Studies The proposed model has been tested using a number of experimental systems. Figures 2 and 3 show the effect of initial dye concentration for the adsorption of Telon Blue ANL (Acid Blue 25) dye onto peat and activated carbon, respectively. Experimental data are in good agreement with the theoretical-solidus curves predicted by the tworesistance model. The film mass transfer coefficients are cm s-l for carbon and peat, 2.0 X cm s-' and 4.5 X respectively, based on an impeller speed of 400 rpm. A particle size range of (500-710) X lo4 m was used in both instances and the homogeneous solid-phase diffusion coefficients were 2.0 X lo4 cm2 s-' and 1.5 X lo4 cm2 s-l for peat and carbon, respectively. The concentration decay curves, for single film resistance only, are readily obtained using the equation
_0 c/
co
2W k d -.. .-I-
exPR,V(l
-t)
(37)
Single and two-resistance cases are considered in Figure 4 using experimental data for the adsorption of Telon Blue
Figure 3. Effect of initial dye concentration for the adsorption of Telon Blue on carbon.
on wood. The initial dye concentrations for these wood runs were 25 and 100 mg dm-3, respectively. Initially the theoretical decay curves are in good agreement with experimental data, but after a limited period of time the single film resistance curves begin to fall too sharply for the mechanism to be solely one of film mass transfer. The versatility of the model may be demonstrated by considering other variables. Figure 5 shows the effect of varying the mass of wood in a batch adsorber system for an initially dye concentration of 100 mg dm-3 and the same film mass transfer coefficient and homogeneous solid phase diffusion coefficient used previously for Figure 4. The effect of initial dye concentration for the adsorption of Astrazone Blue FRR (Basic Blue 69) dye on silica is shown in Figure 6. The film mass transfer coefficient was
A
Figure 4. Wood particles, of constant size range, contacted with Acid Blue 25 dye solution of various initial concentrations at room temperature.
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984
225
1.00
.80
:I;: -
D
.60
.40
, '
,
60
90
,
i) 200
x IO-'
cnZ 3
\ i = 0 300 x 10- 'cm2
,
,
,
120
S
I80
5-
.20 30
0
the
(mlnn)
Figure 5. Wood particles, of constant size range, contacted with Acid Blue 25 dye solution for various masses at room temperature.
.00
1%
I
60
I 80
I 120
I 163
time (mins)
Figure 7. Dimensionless liquid phase concentration, 4, in bulk liqC
=
uid and at particle surface against time.
150 "6 dm-3
100
C
RUZ 2
50
Solidus
-
RUN 1
Theoretical
0
I
I
50
100
t i m e (mins)
Figure 6. Effect of initial dye concentration for the adsorption of basic blue 69 on silica.
8.0 X cm s-l and the homogeneous solid-phase diffusion coefficient was 1.8 x lo+' cm2 s-l. The adaptability of the film and homogeneous diffusion model has been demonstrated successfully for different variables and various systems. The model has also been applied over a wide range of operating conditions on the adsorption isotherm. The application of the model to operating lines terminating on the monolayer is difficult and requires extensive computer time to maintain stability within the program. However, if the incremental time interval for integration is sufficiently small, the model is suitable for such a system. If the operating line terminates below the saturation capacity of the isotherm, application of the model becomes progressively easier, and as 4, 0 the time increment for integration becomes larger and the model is extremely stable. Figure 7 points out another interesting feature of this model; a curve of C, the bulk liquid concentration, vs. time is shown. In addition, the theoretical plot of liquid concentration at the particle surface, C,t, against time can be predicted. This plot indicates a rapid rise in concentration at the particle surface, reaching a maximum C,,value; this value then decreases steadily as the bulk liquid concen-
-
tration decreases. A possible explanation is the changing mechanism of the adsorption process. Initially, the mass transfer to the external surface sites is rapid but then the slower intraparticle diffusion becomes predominant and the dye at the surface is distributed into the internal structure of the adsorbent particle. Conclusion A mathematical model and computer program have been presented to predict the performance of batch adsorbers in which film mass transfer and homogeneous solid phase diffusion are rate controlling. The model has been tested for the adsorption of dyestuffs from aqueous effluents onto various adsorbents. Furthermore, the effects of initial dye concentration and adsorbent mass have been investigated and in all instances theoretically predicted decay curves and experimentally obtained data were in good agreement. Acknowledgment The authors wish to thank Ciba-Geigy Ltd. and Bayer Ltd. for the provision of dyestuffs, J. Crossfield Ltd. (Widnes) for the provision of Sorbsil silica, and Chemviron Ltd. for the provision of activated carbon. Nomenclature a = isotherm constant b = isotherm constant B = mathematical function defined in formula 26 C = concentration of solute in liquid phase, g Co = initial concentration of solute in liquid phase, g cm-3 C, = equilibrium concentration at the surface in liquid phase, g
D = diffusion coefficient for particle, cm2s-l f = mathematical function, first appears in formula 20 G = mathematical function defined in formula 24 Gi= mathematical function defined in formula 29 K = mathematical function defined in formula 31 kf = liquid phase mass transfer coefficient, cm s-l q = concentration of solute per unit volume in particle, g cm-3 qi = concentration of solute at any point in the particle, solute per gram of adsorbent q, = equilibrium concentration at the surface in solid phase, solute per adsorbent q = average concentration of solute in the particle, g of solute per g of adsorbent r = radial coordinate, cm R = radius of particle, cm
Ind. Eng. Chem. Process Des. Dev. 1084, 23, 226-234
228
S, = surface area of particle, cm2 t = time, s
U = transformed solid phase concentration V = reactor volume, cm3 V = volume of particle, cm3 d = total weight of adsorbent material, g x = dimensionless radial co-ordinate Creek Letters p = isotherm constant e = particle porosity p = particle density, g cm-3 = dimensionless time Literature Cited Abramowltz, M.; Stegun, 1. A. “Handbook of Mathematical Functions”; National Bureau of Standards: Washington, DC, 1964. Colwell, C. J.; Dranoff, J. S. AIChE J . 1866, 72, 304. Dryden, C. E.; Kay, W. B. Ind. Eng. Chem. 1854, 46, 2294. Eagle, S.; Scott,J. W. Ind. Eng. Chem. 1850. 42, 1287. H.M.S.O. “Notes on Applied Science No. 16: Modern Computing Methods”.
Mathews, A. P.; Weber, W. J. AICMSymp. Ser. 1876, 7 3 , 91. McKay, G.; McConvey, I . F. J. Chem. Technol. Blotechnol. 1881, 3 7 , 401. McKay, G.; Allen, S. J. Can. J . Chem. Eng. 1880, 58, 521. McKay, G.; Alexander, F.; Poots, V. J. P. Ind. Eng. Chem. Process D e s . Dev. 1878. 77, 406. McKay, G.; Otterburn, M. S.; Sweeney, A. G. Water Res. 1880, 74, 15. Miller, C. 0.; Clump, C. W. A I C M J. 1870, 76, 169. Natlonal Physical Laboratory Publlcation: Teddington. Middlesex, 1961. Neretnlelts, I . I . Ch. E . Symp. Ser. No. 54 1877, 49. Numerical Algorithms Group (U.K.) Ltd.; Central Office: 7 Banbury Road, Oxford OX2 6NN, England, 1959. Pwts, V. J. P.; McKay, G.; Healy. J. J. Water Res. 1976, 70, 1071. Rosen, J. B. J. Chem. Phys. 1852. 2 0 , 387. Rosen, J. B. I n d . Eng. Chem. 1854, 4 6 , 1590. Smith, S. 6.; Hlltgen, A. X.; Juhola, A. J. AIChE Symp. Ser. No. 24 1858, 55, 25. Tien, C. Can. J. Chem. Eng. 1860, 38, 25. Tien, C. AIChE J. 1881, 7 , 410. Weber, T. W.; Chakravorti, R. K. AIChE J. 1874, 2 0 , 224.
Received for review August 9, 1982 Revised manuscript received March 24, 1983 Accepted June 6, 1983
SolubHies of Heavy FogSil Fuels in Compressed Gases. Calculation of Dew Points in Tar-Containing Gas Streams Agustlne Mongot and John M. Prausnltr’ Chemical Engheering Department and Meterials and Mokcular Research Division, Lawrence Berkeley Laboratory, Universlty of Callfornia, Bericeky, Callfornk 94720
A molecular-thermodynamic model is used to establish a correlation for solubilities of heavy fossil fuels in dense gases (such as those from a coal gasifier) In the region ambient to 100 bar and 600 K. This model is then applied to calculate dew points in tarcontaining gas streams. The heavy fuel is fractionated in a spinning-band column at high reflux; each fraction is conskiered to be a pseudocomponent. Each fraction is characterized by one vaporpresswe datum (obtainedduring fractionation),elemental analysis, and proton NMR spectra (to determine aromaticity). Uqukl-phaseproperties are obtained from the SWAP equation for vapor pressure and from a density correlation. Vapor-phase properties are obtained with the virial equation of state with virlal coefficients from Kaul’s correlation. Experimental solubWy measurements have been made for two Lurgi coal-tar fractions in dry and moist methane. Calculated and experime~ntalsotubllities agree well. The correlation is used to establish a design-orbnted computer program for calculating isobaric condensation as a function of temperature as required for design of a continuous-flow heat exchanger.
Introduction The high cost of energy has stimulated new process technologies toward more efficient utilization of energy resources; therefore, there has been growing interest in upgrading coal, heavy petroleum fractions, tar sands, shale oil, etc. Design of “downstream units” for emerging processes requires quantitative information for equilibrium properties of heavy fossil fuels at elevated temperatures and pressures. This work is concerned with the solubility of a heavy fossil fuel mixture in a compressed gas. This solubility is of interest in process design: e.g., for petroleum-reservoir pressurization with light gases, toward removal of high-molecular-weight hydrocarbons remaining after primary recovery; for design of coal-liquefaction reactors; for design of coal-gasification process steps (condensation and quenching) where product gas streams often contain high-boiling coal tars; and for design of heavyfossil-fuel/lighbgasseparation operations. Solubilities may
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be of particular interest for calculating dew points as required in heat-exchanger design. Fossil fuel mixtures typically contain very many components. The wide range of properties and the analytical problem to identify these components makes phaseequilibrium predictions difficult. A common procedure is to separate the mixture into fractions with fewer components and smaller ranges of properties and to characterize each fraction. Phase-equlibrium predictions and process design are then based on average or effective (pseudocomponent) properties of the characterized fractions. In this work, we separate a heavy fuel by distillation into narrow-boiling fractions and we characterize each fraction. Characterization data are used with previously established physical-property correlations to determine required parameters in our molecular-thermodynamic model. Fossil Fuel Fractionation and Characterization To obtain narrow-boiling fractions, we use a PerkinElmer Model 251 annular still, operated at high reflux, as described by Macknick (1979) and Alexander and Prausnitz (1981). Each fraction has a boiling-point range of 0 1984 American Chemical Society
