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Branched-pore model applied to the adsorption of basic dyes on carbon
1
Branched-Pore Model Applied to the Adsorption of Basic Dyes on Carbon GORDON Department (Received
MCKAY*
and BUSHRA
AL DURI
of Chemical Engineering, The Queen’s University of Belfast, Belfast BT9 SDL (U.K.)
June 9, 1987; in final form February
2, 1988)
Abstract The branched-pore adsorption model, expressed by an external mass transfer coefficient kr, a solid diffusivity D,, a lumped micropore diffusion rate parameter k,, and the fraction of macroporesf, describes kinetic data from initial contact of adsorbent-adsorbate to the long-term ( > 24 hours) adsorption stages with reasonable accuracy. In this work the model is applied for three basic dye systems, namely Basic Red 22, Basic Yellow 21 and Basic Blue 69, all on carbon. A single value of each parameter describes each dye system. The kr values are 0.18 x lo-‘+ 28%, 0.3 x lO-2 * 17% and 0.2 x lOA2 + 20% cm s-l, the D, values are 0.33 x 10e9 + 21%, 0.72 x lop9 + 9% and 0.72 x 10d9 f 9% cm2 s-‘, the k,, values are 0.65 x 1OV f 7.7%, 1.8 x 1OV + 17% and 0.2 x lop6 + 1% s-‘, while thefvalues are 0.55 f 9%, 0.60 + 10% and 0.18 + ll%, each for Basic Red 22, Basic Yellow 21 and Basic Blue 69 respectively. The model is based on the internal structure of the carbon particle being divided into a macropore and a micropore region. The latter has an upper-bound capacity of 241, 245 and 656 mg gg’ for Basic Red 22, Basic Yellow 21 and Basic Blue 69 respectively. A sensitivity analysis for each parameter has been carried out.
Kurzfassung Das Adsorptionsmodell auf der Basis verzweigter Poren (‘branched-pore model’) beschreibt kinetische Daten vom Zeitpunkt des ersten Kontaktes Adsorbent-Adsorbat bis zu langen Adsorptionszeiten ( >24 h) mit angemessener Genauigkeit. Die Modellparameter sind ein ZiuBerer Stofftransportkoefhzient kr, ein Feststoffdiffusionskoeffizient D,, ein Gesamtstofftransportparameter kb fur die Mikroporen und der MakroporenanteilJ Das Model1 wird in dieser Arbeit auf die drei Farbstoffsysteme Basic Red 22, Basic Yellow 21 und Basic Blue 69 (jeweils auf Aktivkohle) angewendet. Ein Farbstoffsystem wird dabei durch jeweils einen Wert der obengenannten Parameter beschrieben. Die Werte fur k, sind 0,18 x lo-’ + 28%, 0,3 x 1O-2 + 17% und 0,2 x 10d2 + 20% cm s-l; die Werte fur D, sind 0,33 x 10u9 f 21%, 0,72 x 10d9 f 9% und 0,72 x 1O-9 + 9% cm s-l; die Werte fiir k, sind 0,65 x lop6 f 7,7%, 1,8 x 10e6 + 17% und 0,2 x lop6 + 1% s-‘; schliel3lich &d die Werte fur f 0,55 + 9%, 0,6 &- 10% und 0,18 f 11% jeweils & Basic Red 22, Basic Yellow 21 und Basic Blue 69. Grundlegend fur das Model1 ist die innere Struktur der Aktivkohlepartikel die in Makroporen und Mikroporen unterteilt wird. Die Obergrenze der Kapazitiit der Mikroporen ist 241 mg g-i fur Basic Red 22,245 mg g-l fiir Basic Yellow 21 und 656 mg gg’ fur Basic Blue 69. Eine Sensitivitiitsanalyse wurde fur jeden Parameter durchgeftihrt.
Synapse Gegenstand der Untersuchung war die Adsorption der drei Farbstofle Basic Red 22, Basic Yellow 21 und Basic Blue 69 auf Aktivkohle (Filtrasorb 400). Ein Model1 mit drei Stofltransportwiderstiinden ist entwickelt worden, urn den Konzentrationsverlauf in der Aktivkohle zu beschreiben. Das Adsorptionsmodel auf der Basis verzweigter Poren (branched-pore model)
*Author to whom correspondence
0255-2701/88/$3.50
should be addressed.
beschreibt kinetische Daten vom Zeitpunkt des ersten Kontaktes Ahorbent-Adsorbat bis zu langen Adsorptionszeiten (> 24 h) mit angemessener Genauigkeit. Die Modellparameter sind ein 2iuJerer Stofftransportkoefizient kr, ein Feststofldtrusionskoefizient D,, ein Gesamtstofltransportparameter kb fir die Mikroporen und der Makroporenanteil fDie Kapazitiit der Aktivkohle wurde fGr jeden der drei Farbstoffe durch Messung der Gleichgewichtsisothermen bestimmt. Zur Messung der Isothermen wura’e eine bekannte Menge Aktivkohle zusammen mit einem festen Volumen Farbstoflosung in einem
Chem. Eng. Process., 24 (1988) 1-13
0 Elsevier Sequoia/Printed
in The Netherlands
2
geschlossenen GeftiJ sechs Wochen bei 20 “C aujbewahrt. Variiert wurde die Farbstoflonzentration in der Losung. Die MeQwerte wurden mit der FreundlichGleichung korreliert: ae = afCeb Die Freundlich-Konstanten a, und b sind in Tabelle 2 zusammengefa&. In der Isothermen in Bild 4(a) und (b) kann man, der inneren Struktur der Aktivkohle-Partikel entsprechend, eine Makroporen-Adsorption und eine Adsorption in den Mikroporen unterscheiden. Die der Kapazitiit der Mikroporen ist Obergrenze 241 mg g - ’ fir Basic Red 22, 245mg g - ’ fur Basic Yellow 21 und 656 mg g ~ ’ fur Basic Blue 69. Bild 1 zeigt den diskontinuierlich betriebenen, durchmischten Adsorber, in dem die Untersuchungen der Kinetik durchgefiihrt wurden. Die Entnahme von Proben aus dem Adsorber zur Analyse erfolgte mit einer Spritze in regelm$igen Zeitabstiinden wiihrend der Versuchsdauer von 24 Stunden. Mit einem Computerprogramm wurden aus dem Konzentrationsverlaufskurven Stofltransportmodell berechnet. Die Werte fur kr, D,, kb undf wurden durch Anpassung an die experimentellen Ergebnisse bestimmt. Die experimentellen Variablen fur jeden Farbsto# waren die Menge der Aktivkohle und die Anfangskonzentration des Farbstoffes. Die gilder 5-8 zeigen typische experimentelle Ergebnisse sowie berechnete Kurven. Ein Farbstoffsystem wird durch jeweils einen Wert der Parameter beschrieben. Die Werte fiir k, sind 0,18 x iUp2 k 28%, 0,3 x IO- 2 f 17% und 0,2 x 10e2 f 20% cm s-‘; die Werte fur D, sind 0,33 x IUp9 + 21%, 0,72 x 10m9 + 9% und 0,72 x IUp9 + die Werte fur k, sind 0,65 x IUp6 + 9% c?&-‘; 7,7%, 1,8x 10p6+ 17% und 0,2x 10-6+ I%s-‘; SchlieJich sind die Werte fir f 0,55 + 9%, 0,60 + 10% und 0,lS & 11% jeweils fir Basic Red 22, Basic Yellow 21 und Basic Blue 69. Die Ergebnisse einer Sensitivitiitsanalyse fur jeden Stofftransportparameter sind in Bild 911 dargestellt.
1. Introduction Activated carbon has been highly effective as an adsorbent in batch, fixed and moving bed adsorption systems for water treatment owing to its efficiency in producing high quality water, meeting the requirements of environmental pollution legislation, at a price comparable with secondary biological treatment! The design of adsorption systems has been aided by mathematical models that predicted kinetic data, based on a single effective intraparticle diffusion parameter, throughout adsorption time, assuming identical adsorption sites on the carbon particle. These are solid, pore and solid-pore diffusion models. Widely used at present, a solid diffusion model [l] assumes a homogeneous adsorbent surface and is applied for porous adsorbents. It produces reasonable agreement between experimental and theoretical data for a wide range of systems [14], but for a limited period of time (4
hours). The pore diffusion model, though assuming porous adsorbents, has a narrow range of application because it does not account for surface migration on the pore walls and this leads to inaccurate pore diffusivities with misleading results [ 5,6]. Experimentally, kinetic data start with a rapid solute uptake followed by a slower adsorption rate that levels off as equilibrium is attained [7, 81. In the above models, a high diffusion rate parameter leads to rapid attainment of equilibrium and a low diffusion rate parameter leads to slow attainment of equilibrium. Therefore these previous models succeeded in predicting the rapid uptake period but, owing to the high diffusivity implied, they led to equilibrium being attained at earlier stages than in experiments. This discrepancy suggested that there is more than one intraparticle diffusion parameter controlling the different stages of the system kinetics and related to the internal structure of the adsorbent, namely carbon in the present work, whose effect has been neglected in the single intraparticle diffusion rate parameter models. Carbon is an adsorbent with a heterogeneous surface and a polydisperse porous structure, that is, pores of a wide size range uniformly dispersed throughout the granules. Beck and Schultz [9] and Satterfield et al. [lo] have proved a strong relation between the diffusion rate and the ratio of solute molecular diameter to adsorbent pore diameter. Hence the wide pore size range implies a wide range of diffusion rates making a single diffusion rate either an average approximate value or an accurate value for a limited time. Peel et al. [ 111 have approximated the microscopic description of the adsorption mechanism by dividing a carbon particle into two regions of two pore size ranges and therefore two diffusion rates: a macropore region of pores of radius larger than 2 nm, where rapid uptake occurs, providing access to the micropore region of pores of radius smaller than 2 nm, where a slow adsorption rate is expected. Based on the above, Peel et al. proposed a threeresistance model: a film resistance expressed by the external mass transfer coefficient kr (cm s-l), a resistance in the macropore region given by the solid diffusivity D, (cm2 s-l), a resistance in the micropore region given by a lumped diffusion coefficient k,, (s-l), and finally a factor f which is the fraction of macropores. This work utilizes Peel’s model in a wide range of experiments and applies it to an agitated batch adsorption system for three basic dyes, namely Basic Blue 69 (Astrazone Blue), Basic Red 22 (Maxillon Red) and Basic Yellow 21 (Astrazone Yellow).
Experimental Kinetic
studies
Kinetic studies were carried out in an agitated batch adsorption system which consisted of a 2 dm’
3
A constant carbon particle size range, diameters 35&500 pm, was used in each experiment. The general characteristics of the Filtrasorb 400 are shown in Table 1. About 70% of the pore volume lies between pore sizes of 10 A and 1000 A, with 25% of the overall ore volume in the pores of diameters from 20 A to 30 x .
Variable Speed Motor
Rubber Support Ring a
Isotherms
A set of dye solutions was prepared for each dye system in predetermined concentrations. These were brought into contact with pre-weighed amounts of carbon in jars. The latter were sealed and shaken in a water bath at room temperature (20 “C) for three weeks until equilibrium was attained. The solution was tested to ensure no biological degradation had taken place during this period. Solutions were then analysed for the fluid-phase concentrations C,, while solid-phase concentrations qe were obtained by a material balance on the system. Equilibrium data were correlated by the Freundlich formula,
-
-4
Fig. 1. Schematic representation of batch adsorber.
q= = arc,6
glass vessel, of diameter 0.13 m, filled with 1.7 dm3 of solution, giving a solution height of 0.13 m. The solution was agitated by a six-flat-blade impeller driven by a Heidolph electric motor with a speed adjustable from 100 to 600 rev min-‘. The width of the impeller blades was 0.013 m. Complete mixing was facilitated by eight baffles, each of width 0.01 m, distributed at 45” around the circumference of the beaker and held in position by a polystyrene baffle holder. Figure 1 illustrates the system. In all experiments where the initial dye concentration or the carbon mass was varied, the agitation speed was maintained constant at 400 rev min-‘. Sorbate-sorbent
system
Carbon Filtrasorb F400, supplied by Chemviron Ltd., was used as the adsorbent. It was crushed by a hammer mill, screened to a series of particle sizes by sieve analysis and then washed thoroughly in distilled water to remove fines. It was dried in an oven at 120 “C for 24 hours. The sorbate system consisted of three basic dyes: (i) Basic Blue 69 (Astrazone Blue), BB 69; (ii) Basic Yellow 21 (Astrazone Yellow), BY 21; and (iii) Basic Red 22 (Maxillon Red), BR 22. The first two were supplied by Bayer and the third by Ciba Geigy. TABLE 1. Characteristics of Filtrasorb 400 carbon Total surface area, N, BET method (m* kg-‘) Bed density, backwashed and drained (kg m-j) Particle density, wetted in water (kg rnm3) Pore volume (m3 kg- ‘) Iodine number Methylene Blue number
because of its applicability for heterogeneous surfaces. Freundlich constants for the three dye systems are given in Table 2. Equilibrium is between the solid surfaces and the liquid phase at all adsorption sites. Analysis
Kinetic and equilibrium data were analysed using a Perkin Elmer SS05 spectrophotometer. The optical densities of the coloured solutions were measured at certain wavelengths corresponding to the maximum absorbance. These wavelengths, termed Iz,,,, were 585, 537 and 417 nm for BB 69, BR 22 and BY 21 respectively. The dye concentration is related to the optical density by a predetermined calibration graph. Theoretical
analysis
Figure 2 illustrates the approximate structure of a carbon particle. Macropores are the main passages that lead to the interior of the particle. Micropores branch off from the macropores forming a network which is dispersed uniformly throughout the particle. The adsorption mechanism is described in this model by three resistances in series: (i) the external resistance across the liquid film, (ii) the intraparticle
TABLE 2. Freundlich (temperature = 20 “C) 1.10 x IV 425 x 10’ 1350 0.90 x IO-’ 1050 280
constants
for three basic dye isotherms
Dye
ar(dm3g-‘)b
b
BR 22 BY 21 BB 69
299.0 330.3 337.0
0.130 0.133 0.226
I
qs: Cs
qm
Fig. 3. Conceptual
The following
diagram
structure
of carbon
system.
set is obtained:
3Qtn a2Qm x =4q - at12 + 4 2 Fig. 2. Approximation of the internal posed by the branched-pore model.
of the adsorption
‘c
- Bi,(Qm - Q,,)
(3)
as pro-
aQb = Bi p ;f(Qm
de
resistance in the macropores and (iii) the intraparticle resistance in the micropores. Owing to the complexity of mass transfer in this region, caused by multidirectional interactions, a lumped parameter kb has been used to describe the rate of mass transfer from the macropore to the micropore region. Bidisperse adsorbents like molecular sieves have been modelled relatively easily by Furusawa and Smith [6], Ruckenstein et al. [ 121, and Shah and Ruthven [ 131, since the relative proportions and locations of different regions are defined. In the case of carbon, modelling is complex owing to carbon’s polydisperse nature, that is, it has a continuous range of pore sizes, distributed uniformly throughout the particles. Therefore, a factor f has been selected to describe the fraciton of macropores in the particle, 1 -f being the fraction of micropores. Figure 3 is the conceptual diagram of the system. The mathematical description of the model is based on the material balance backed with mass transfer principles. Peel et al. [ 1l] explain the theory extensively, therefore to avoid duplication only the dimensionless form of the equations is stated briefly below. Details on the conversion of the equations to dimensionless form are given in the Appendix. Given the dimensionless terms
- Qd
(4)
(5)
(6) In the above, Bi
=
f
kKoR
(7)
Pc_m90
kbR2 Bi, = ~
(8)
fDs
are modified Biot numbers, where Bi, represents the ratio of the internal to the external resistance and Sir represents the ratio of the macropore to the micropore resistance;
F=YLO w90 is the separation factor. The initial and boundary
Q&L 0) = 0 Qdrl. 0) = 0 C(0) = 1 Qm( 1, t) = Q.(t)
(9) conditions
become:
(10) (11) (12)
(13) Equations (3)-( 13), together with the equilibrium data, are solved by the Crank-Nicolson finite difference method. Owing to the nonlinearity of the isotherm, the coupling eqn. (6) is not linear and the
solution is iterative at each point in time. Initial guesses of the future concentrations are calculated by linearly extrapolating the change over the previous time interval and the value used in the first iteration. The newly calculated values are then substituted directly and the solution repeated until a converged solution is obtained. A computer program based on this method has been developed by McKee [ 141 and extended by Al Duri [ 151. It is fed with the system variables, namely the mass, particle size and particle density of carbon, the initial sorbate concentration, volume of the solution, equilibrium constants, and kinetic data, together with time and space intervals required for integration. In addition, the four rate parameters of the system, k,-, D,, kb and f, are supplied. The program yields the liquid-phase concentration with time (theoretical kinetic data), the surface concentration, the dimensionless distance q across the particles, and the macropore and micropore solid-phase concentrations. Discussion The branched-pore model is based on the internal structure of the carbon particles. The duality in the adsorption mechanism makes analysis of the kinetics very complex. Several mechanisms act in parallel, with contributions depending on each particular system. In order to simplify things, the pore regions in a carbon particle were subdivided into macropores and micropores [ 113. This adsorption model, with its four parameters, namely, the mass transfer coefficient k, the solid diffusivity D,, the lumped micropore diffusion parameter k,, and the fraction of macropores f, is sufficient to describe experimental kinetic data from initial sorbate-sorbent contact to equilibrium. Peel et al. [ 1 I] developed the branched-pore model and applied it to the carbon-phenol adsorption system. Table 3 summarizes their results; they obtained the parameters by regression analysis of the data. kr was evaluated by a single resistance assumption and the variation in k, with carbon mass was attributed to varying hydraulic flow patterns with different amounts of carbon. McKee [ 141 found kf to be 0.28 x 10W2 cm s-’ for the chitin/Acid Blue 25 system and 0.47 x lop2 cm S-I for the chitin/Acid Blue 158 system. In the present work kf was evaluated by the single resistance assumption as an initial estimate to the best curve fit. The values obtained were 0.18 x 10e2 f 28%, 0.30 x lop2 f 17% and TABLE
0.20 x lop2 f 20% for BR 22, BY 21 and BB 69 respectively. Compared with the values in Table 3, these external mass transfer coefficients are low. This is attributed to the molecular sizes of these dyes which are relatively large compared with the size of the phenol molecule, therefore the mass transfer rates are slower. The effective diameters of the dye molecules are 27, 28 and 30 8, for Basic Red 22, Basic Blue 69 and Basic Yellow 21 respectively. In the experimental section the characteristics of Filtrasorb 400 activated carbon show that 25% of the pore volume is in the range 2&30 A, therefore molecular sieve effects may be occurring. However, since the dye molecules are so similar in size and Basic Blue 69, having the highest adsorptive capacity, falls between the other two dyes in terms of molecular diameter, it is considered that molecular sieving is not important in determining the relative adsorption capacities. The pH of the dye solutions at a concentration of 300 mg dmd3 were 5.9,6.5 and 5.0 for Basic Yellow 21, Basic Red 22 and Basic Blue 69. The low pH for Basic Blue 69 may have an influence on the high uptake of this dye on carbon. The adsorption capacities of the other two dyes do not follow this trend and therefore pH is again not the criterion for adsorption capacity. Intraparticle diffusion in the macropores is described by a solid diffusion mechanism, characterized by the solid diffusivity D, [ 11, 141. Peel et al. [ 111 obtained D, values comparable with those obtained from single intraparticle parameter models [ 3, 16, 171. This fact is related to the small molecular size of phenol molecules which results in a high solid diffusivity and consequently a rapid uptake, making all macropores saturated before diffusion into the micropores starts. This implies that mass transfer is solely in macropores at this stage of the adsorption process, that is, in the macropore mass balance equation, f
ak _fD, dt
r2
a
ar
cr -bR 2
a9,
ar
b
f = 1 and Rb = 0, making eqn. (2) similar to the intraparticle diffusion mass balance in the single solid diffusion parameter model, implying a similar D, to yield the same kinetic data. Other D, values in the literature include 1.0 x lop8 cm2 s-’ for the chitin/ Acid Blue 25 system and 7.5 x lo-’ cm2 s-’ for the chitin/Acid Blue 158 [ 141. In the present work D, was evaluated by a best curve fit technique and regression analysis and a single constant ( k 10%) value described each dye system for a set of masses and concentrations. Table 4 gives D, values to be 0.33 x 10p9k 21%, 0.72 x 10p9f9% and
3. Parameters of phenol kinetic rum by Peel et al. [ 1 I]
w cd
C,(rngd~~-~)
k,x
0.9898 0.7216 0.5393
96.3 96.3 94.7
1.33 f 5% 1.82 f 5% 2.01 f. 5%
10-2(cms-‘)
D, x lo-*
(cm2s-‘)
7.78 f 10% 9.01 f 9% 7.15 + 7%
k, x 106(sr’)
f
l&*27% 1.87 + 18% 1.80 + 16%
0.63 + 2% 0.66 + 2% 0.68 i 2%
6 TABLE 4. Data of kinetic runs for three basic dye systems Run
Dye
W(g)
C,, (mg dme3)
kf x 102 (cm s-l)
II, x lo~(cm*s-‘)
k, x 106 (s-l)
f
0.60
No.”
1 2 3 4 5 6
BR BR BR BR BR BR
22 22 22 22 22 22
1.27 0.850 0.638 0.425 0.638 0.638
98.5 102.0 101.0 100.0 49.0 75.6
0.15
0.20 0.20 0.20 0.23 0.20
0.27 0.35 0.35 0.40 0.32 0.35
0.70 0.70 0.70 0.70 0.60 0.70
7 8 9 10 11 12 13 14 I5 16 17
BY BY BY BY BY BY BY BY BY BY BY
21 21 21 21 21 21 21 21 21 21 21
0.850 0.638 0.425 0.319 0.213 0.107 0.425 0.425 0.425 0.425 0.425
103.0 99.4 99.1 98.7 101.0 102.0 25.0 49.8 74.6 154.0 206.0
0.28 0.26 0.25 0.25 0.25 0.26 0.35 0.35 0.30 0.25 0.25
0.68 0.68 0.65 0.79 0.70 0.65 0.79 0.79 0.79 0.79 0.75
1.5 1.8 1.5 2.1 1.5 1.5 2.0 2.1 1.5 2.0 2.0
0.60
18 19
BB 69 BB 69
1.70 1.27 0.850 0.638 0.213 0.850 0.850
89.6 90.5 85.2 91.1 92.8 68.5 106.0
0.25 0.20 0.15 0.15 0.20 0.18 0.15
0.66 0.80 0.70 0.78 0.78 0.79 0.66
0.20 0.15 0.20 0.20 0.20 0.20 0.20
0.16 0.18 0.18 0.18 0.20 0.18 0.16
20
21 22 23 24
BB 69 BB 69 BB 69 BB 69 BB 69
0.55 0.58 0.55 0.50 0.52
0.56 0.55 0.65 0.57 0.55 0.65 0.65 0.58 0.66 0.66
“For all runs: R = 0.0214 cm, V = 1700cm3 and pC= 1.26 g cnm3.
0.72 x lop9 f 9% for BR 22, BY 21 and BB 69 respectively. Compared with the values in Table 3 the carbondye solid diffusivities are low, and again this is related to the difference in molecular size of the sorbates (i.e. the diameter of the molecules) in the systems listed in Tables 3 and 4. Table 5 compares D, values in the present work with those obtained from the conventional film-solid diffusion parameter model (FSDPM) [ 151. For the
same set of masses and concentrations, the branchedpore diffusion model (BPDM) D, values in the dyecarbon systems are higher because the conventional FSDPM includes only two resistances. As mentioned earlier, macropores provide the only access to the interior of particles, therefore diffusion from macropores to micropores reduces the sorbate concentration in the former by a fractionf. Therefore a higher
D, is needed to maintain the intraparticle diffusion rate equal to that in the single solid diffusion parameter model. Thus the comparison between D, values in the present model (BPDM) and the FSDPM in Table 5 results in smaller D, values for the FSDPM because it only includes two resistances. Mathematically: Macropore + Micropore Intraparticle diffusion = diffusion diffusion i.e.
dq --+(l-/)d$ f dqm
dt-
(15)
dt
-_R
b
+R
b
(16)
which yields TABLE 5. Comparison of solid diffusivities obtained by the filmsolid diffusion parameter model (FSDPM) and the branched-pore diffusion model (BPDM) for three basic dye systems
Dye
BR 22 BY21 BB 69
D, x lo9 (cm s-‘) FSDPM
BDPM
0.085 + 5.9% 0.23 k 17% 0.02 * 20%
0.33 * 21% 0.72 + 9.7% 0.72 * 9.7%
(17) In the single diffusion parameter model Fritz et al. [ 31 used
(18) Comparing eqns. (17) and (18), they differ byfand qm, which implies a higher D, value in eqn. (17) to yield the same kinetic performance as eqn. ( 18).
7 The lumped diffusion parameter kb approximates the mass transfer at the later stages of the adsorption mechanism. Peel et al. [ 111 evaluated k, by regression analysis of their data, obtaining the values in Table 3. McKee [ 141 obtained k, values of 10.0 x 10d6cm S-I for chitin/Acid Blue 25 and 500 x IO-“cm s-’ for chitin/Acid Blue 158. In the present work a constant k,, was obtained for each dye system. The values of k, obtained were 0.65 x lop6 f 7.7%, 1.8 x lop6 * 17% and 0.20 x 10e6 f 1% for BR 22, BY 21 and BB 69 respectively. The adsorption mechanism in the micropores is strongly dependent on the pore geometry. Their small diameters (up to 2 nm in Filtrasorb F400) make the molecular sizes of the sorbates comparable with the pore sizes of the micropores, causing multidirectional sorbate-sorbent interactions [ 181. In addition, decreased pore radii raise the energy barrier in the surface migration process, greatly reducing its rate. On the other hand, the rate of diffusion is a strong function of the ratio of solute diameter to pore diameter [9, lo]. A ratio as low as 1: 10 reduces the diffusion rate by 40%. All the above observations suggested that the diffusion mechanism
in the micropores is by solid phase diffusion characterized by sorbate-sorbent multidirectional interactions. These are responsible for the irreversibility in the isothermal systems. Model1 et al. [ 191, in adsorption-desorption studies, succeeded in desorbing 70% of the phenol adsorbed on Filtrasorb F300 carbon, using CO, as a regenerant. They attributed the lost capacity to irreversible sorbate-sorbent interactions and obtained a better recovery in desorbing phenol after a short adsorption period. This indicated that irreversibility is caused by interactions during the later slow adsorption period. Snoeyink [20] was able to desorb 50% of adsorbed phenol from a Columbia LC carbon after a 5 month contact. Furthermore, owing to multidirectional interactions, sorbents acquire a strong affinity for the sorbate the therefore the micropores tend to be saturated at lower solution concentrations than the macropores. They reach an upper-bound capacity independent of equilibrium concentrations. This is given by qe(1 -f) and is shown in Fig. 4. Micropore capacities were found to be 241, 245 and 656 mg g-’ for BR 22, BY 21 and BB 69 respectively. The macropore fraction f evaluated by Peel et al. for the carbon-phenol system is given in Table 3.
Capacity
i : E
500
d ~I_
Capacitv
Cs,
m9
dm-'
(a)
t---f-__.
541
_.__i________.~___ _______ ____ ________ __-__________-___._ I____________ _______ ___----__I 541
541 I
0
50
100
(b) Fig. 4. Postulated distribution for (a) the BB 69 isotherm and (b) the BR 22 isotherm.
1
.
150 C,.
mg
dm
-3
0
3611
1080
tine,
istn
Ds =
0.35 x
lo-’ cm%-’
kf = 0.18 x lo-* cm s-' kb = 0.7 x (10-7, 10-6, 10-5, 10-4) f = 0.58
1
0
2
3
4
5
6
7
8 time, hrs
9
Fig. 10. Sensitivity analysis of the micropore diffusion rate parameter k,.
Ds = 0.35 x lo-’ cm’s_’ kf = 0.18 x 1O-2 cm s-' kb = f =
0.7 x 1O-6 cm s -’ 0.65, 0.75, 0.85, 0.999
, 0
1
2
3
4
5
6
7
8 time,
9 hrs
Fig. 1I Sensitivity analysis of parameter f, the fraction of macropores.
controlling influence to terminate after 4 hours owing to early salination of the macropore region for the reasons explained earlier. Figure 10 illustrates the effect of I&,, which becomes evident at late stages where micropore diffusion is the feature of adsorption. The influence off, shown in Fig. 11, alters the whole adsorption process. Furthermore, the extent of the contribution of each parameter depends totally on the particular sorption system, the physical and chemical properties of the sorbate.
Computer simulation and the running time of the model depend strongly on the combination of values of the system variables supplied. For example, run No. 9 took 8 minutes to yield kinetic data for 9 hours, while it took 11 hours to produce kinetic data for a contact time of 24 hours. This restricts the conditions under which this model can be run, particularly with respect to computational costs. However, the branched-pore model, applied on a wide experimental basis, proved to be theoretically sound and could reduce difficulties encountered with
11 the internal structure neous systems.
of carbon
and other heteroge-
kf
Conclusions From the present work the following points can be made. (i) The branched-pore model proposed by Peel et al. [ 1 I] proves to be applicable over a wide range of experimental conditions concerning three basic dye adsorption systems, namely BR 22, BY 21 and BB 69, all on carbon. (ii) Single values for the diffusion parameters kr, D,, k,, and f describe each respective stage of the adsorption mechanism for a whole range of masses and concentrations of each dye system. (iii) The chemical properties of the sorbate (like the molecular structure, molecular weight, radius and polar properties) have a strong effect on the adsorption mechanism and the contribution of each step in it. (iv) The solid diffusivity D, evaluated in the branched-pore model is higher than that evaluated in the single solid diffusion model. This is attributed to the contribution of the micropores since it reduces the carbon loading in the macropores, implying a higher D, to keep the macropores saturated. (v) The micropore region has an upper-bound capacity, calculated by (1 -f )qe, independent of the equilibrium data. It was found to be 241, 245 and 656 mg g-i for BR 22, BY 21 and BB 69 respectively. (vi) D, controls the first few hours of the adsorption process, k, controls the late micropore adsorption, while f affects the whole process. (vii) Extensive computational running times for certain system conditions, in order to obtain longterm data, are a disadvantage of this model.
Nomenclature
af A b
Bi, Bi, C
Freundlich constant, dm3 g-’ surface area, m2 Freundlich constant = kfRC,/pJD,qo, modified Biot number = kbR2/fD,, modified Biot number = C,/C,, dimensionless fluid-phase concentration equilibrium fluid-phase concentration, mg drnp3 fluid-phase concentration at particle’s surface, mg drne3 = C,,/C,, dimensionless fluid-phase concentration at particle’s surface fluid-phase concentration at any time t, mg drne3 initial fluid-phase concentration at particle’s surface, mg dm-3 solid diffusivity, cm2 s-i fraction of macropores in particles = COV/q, W, separation factor lumped diffusion parameter, cm SK’
%
Q r
R R, t V W B ;I PC
external mass transfer coefficient, cm s-’ micropore solid-phase concentration, mg gg’ equilibrium solid-phase concentration, mg g-i macropore solid-phase concentration, mg g-i solid-phase concentration at particle surface, mg g-i maximum solid-phase concentration, mg g- ’ = q/go, dimensionless solid-phase concentration radial distance from centre of carbon particle, cm radius of carbon particle, cm = (1 -f) dqb/dt, micropore diffusion rate, mg g-i mini time variable, min fluid-phase volume weight of carbon, g =rJR, dimensionless distance carbon particle = 8’ =D, tIR2, dimensionless time particle density, g crnV3
from
centre
of
References 1 G. McKay, The adsorption of basic dye onto silica from aqueous solutions-solid diffusion model, Chem. Eng. Sci., 39 (1984) 129. 2 A. P. Mathews and W. J. Weber, Jr., Effects of external mass transfer and intraparticle diffusion on adsorption rates in slurry reactors, AIChE Symp. Ser., 73 (1976) 166. 3 W. Fritz, W. Merk and E. U. Schlilnder, Competitive adsorption of two dissolved organ& onto activated Carbon-II, Chem. Eng. Sci., 36 (1981) 731. 4 G. McKay and I. F. McConvey, Adsorption of acid dye onto woodmeal by solid diffusional mass transfer, Chem. Eng. Process., 19 (1985) 287. 5 T. Furusawa and J. M Smith, Fluid particle and intraparticle mass transport rates in slurries, Ind. Eng. Chem., Fun&m., 12 (1973) 197. 6 T. Furusawa and J. M. Smith, Diffusivities from dynamic data, AIChE J., 19(1973) 401. 7 J. S. Zogorski, S. D. Faust and J. H. Haas, The kinetics of adsorption of phenols by granular activated carbon, J. Colloid Interface Sci., 55 ( 1976) 329. 8 V. L. Snoeyink, W. J. W&r and H. B. Mark, Sorption of phenol and nitrophenol by activated carbon, Environ. Sci. Technol., 3 ( 1969) 9 18. 9 R. E. Beck and J. S. Schultz, Hindered diffusion in microporous membranes with known pore structure, Science, I70 (1970) 1302. 10 C. N. Sattertield, C. K. Colton and W. H. Pitcher, Restricted diffusion in liquids within fine pores, AZChE J., 19 (1973) 628. 11 R. G. Peel, A. Benedek and C. M. Crowe, A branched pore kinetic model for activated carbon adsorption, AZChE J., 27 (1981) 26. 12 E. Ruckenstein, A. S. Vaidyanathan and G. R. Youngquist, Sorption by solids with bidispersc pore structures, Chem. Eng. Sci., 29 (1971) 1305. 13 D. B. Shah and D. M. Ruthven, Measurement of xeolitic diffusivities and equilibrium isotherms by chromatography, AIChE J., 23 (1977) 804. 14 S. McKee, The development of computer programs for the solution of batch adsorption problems, M.Sc. Thesis, Queen’s Univ. Belfast, 1984.
12 15 B. Al-Duri, Ph.D. Thesis, Queen’s Univ. Belfast, in preparation. 16 J. C. Crittenden and W. J. Weber, Predictive model for design of fixed-bed adsorbers: parameter estimation and model development, ASCE J. Environ. Eng. Div., 104 (EE2) (1978) 185. 17 Y. Sudo, C. M. Misic and M. Suyuki, Concentration dependence of effective surface diffusion coefficients in aqueous phase adsorption on activated carbon, Chem. Eng. Sci., 33 (1978) 1287. 18 M. M. Dubinin, Porous structure and adsorption properties of active carbons, in Chemistry and Physics qf Carbon, Vol. 2, Marcel Dekker, New York, 1966. 19 M. Modell, R. P. de Fillipi, V. Krukonis and A. D. Little, Regeneration of activated carbon with supercritical CO,, Am. Chem. Sm. Dill. Environ. Chem., Prepr., Miami, 1978. 20 V. Snoeyink, Adsorption of strong acids, phenol and 4-nitrophenol from aqueous solution by active carbon in agitated nonRow systems, Ph.D Thesis, Univ. Michigan, 1968.
From q=$
aq=$2r& therefore ar =
a,$
and
atj2$
ar2= Thus,
$!!$
=fD,
(3
$
+j
%$-kb(Qm
Rearrangement Reduction of the mass balance equations dimensionless form
Macropore
Starting tion
equa-
Bi,
=
!%I!? Dsf
we obtain the macropore dimensionless form:
WE49
at =FgF
(
_ R
) ‘a,> +;ar
-RR,
%
to introduce dimensionless terms without altering the equation we multiply the left-hand side by
(1 -f)
= k,(q,
we multiply
qoR2D
--2
qoR2Ds
respectively.
>
Transferring ‘6&m - qb)
q,, to the right-hand for
atD,/R’
fDE aqm/qO = fD, R2 at D,lR2
sides by
f f
Grouping
the variables
we obtain
=k,-!-(q,_qb)
R2
and after rearrangement,
side and substituting
&
equa-
- q,J
aqb/q0@(1 -f) -Rb
mass balance
the left- and right-hand and
qo R2 D,
+f%
in
region mass transfer
from the micropore
qoR2D, which yields
equation
a’Q
Starting tion
b
region transfer
-?+4%-Bi,(Q,-Q,) a12
Micropore z aq,
fDS a
-kb(Q,-Qb)
As
aQm
mass balance
gives
=
region mass transfer
from the macropore
of this equation
to
This has two main advantages: (i) reduction of the number of equations, thus simplifying the solution procedure; (ii) relations for the system variables in terms of useful chemical engineering dimensionless groups like Bi and F.
f aq,
- Qt.)
>
Appendix
(1 -f)
a&
x
kbR2
= mf(Qm
-
Qb)
Introducing
Bip=kbR2 Df
Inserting
Q, = qm/qO. Qb = qb/qO, and 80 = at D,/R2,
aQm RZae =fR
fD,
~+~%$)-kdQm-Qb)
we have the micropore region mass transfer equation in dimensionless form:
aQb -= ae &f
Bi,
13 Liquid-phase mass transfer Starting from the fluid-phase tion,
mass balance
equa-
%=k$(C,-C,,)
g
Following the same principle the left-hand side by
= ?!$f
(C - C,)
as above, we multiply The coupling equation Starting from the mass balance liquid interface,
CoD,R2 --CoD,R2 to obtain dC, COD, R2
and group the parameters depending on the equations for Bi, and F (with W = pc V). This leads to the equation For the liquid-phase mass transfer in dimensionless form:
W(Ct - G, J =fD&c
k,A
~~~jii=~(C*-Cs,,)
0 s which, after rearrangement, and insertion of C = C,/C,, C, = C,,, - Co and 9 = D,t/R2, becomes
;g;yD2 (C -
hn
(dr > =f~(ct-cs.*) ,IR
qoR2 -C,)
,=R
that is,
We multiply =
across the solid-
the left- and right-hand
sides by
CO
and
qo R2
c,
s
We wish to obtain this equation where Bi
_
WoR
’ - pcfD,q, So we multiply qof ---- COPGV qof Co&V
and
in terms of Bi, and F,
h&o
to obtain
qoR2
--=f~cc,-G,3 ar R2
F=s!r
qow the right-hand
respectively,
k&o
SE 0
Inserting side by
k&OR
Bi, =fa9cqo
and
leads to the coupling
r2 q = z equation
in dimensionless
form:
Branched-Pore Model Applied to the Adsorption of Basic Dyes on Carbon GORDON Department (Received
MCKAY*
and BUSHRA
AL DURI
of Chemical Engineering, The Queen’s University of Belfast, Belfast BT9 SDL (U.K.)
June 9, 1987; in final form February
2, 1988)
Abstract The branched-pore adsorption model, expressed by an external mass transfer coefficient kr, a solid diffusivity D,, a lumped micropore diffusion rate parameter k,, and the fraction of macroporesf, describes kinetic data from initial contact of adsorbent-adsorbate to the long-term ( > 24 hours) adsorption stages with reasonable accuracy. In this work the model is applied for three basic dye systems, namely Basic Red 22, Basic Yellow 21 and Basic Blue 69, all on carbon. A single value of each parameter describes each dye system. The kr values are 0.18 x lo-‘+ 28%, 0.3 x lO-2 * 17% and 0.2 x lOA2 + 20% cm s-l, the D, values are 0.33 x 10e9 + 21%, 0.72 x lop9 + 9% and 0.72 x 10d9 f 9% cm2 s-‘, the k,, values are 0.65 x 1OV f 7.7%, 1.8 x 1OV + 17% and 0.2 x lop6 + 1% s-‘, while thefvalues are 0.55 f 9%, 0.60 + 10% and 0.18 + ll%, each for Basic Red 22, Basic Yellow 21 and Basic Blue 69 respectively. The model is based on the internal structure of the carbon particle being divided into a macropore and a micropore region. The latter has an upper-bound capacity of 241, 245 and 656 mg gg’ for Basic Red 22, Basic Yellow 21 and Basic Blue 69 respectively. A sensitivity analysis for each parameter has been carried out.
Kurzfassung Das Adsorptionsmodell auf der Basis verzweigter Poren (‘branched-pore model’) beschreibt kinetische Daten vom Zeitpunkt des ersten Kontaktes Adsorbent-Adsorbat bis zu langen Adsorptionszeiten ( >24 h) mit angemessener Genauigkeit. Die Modellparameter sind ein ZiuBerer Stofftransportkoefhzient kr, ein Feststoffdiffusionskoeffizient D,, ein Gesamtstofftransportparameter kb fur die Mikroporen und der MakroporenanteilJ Das Model1 wird in dieser Arbeit auf die drei Farbstoffsysteme Basic Red 22, Basic Yellow 21 und Basic Blue 69 (jeweils auf Aktivkohle) angewendet. Ein Farbstoffsystem wird dabei durch jeweils einen Wert der obengenannten Parameter beschrieben. Die Werte fur k, sind 0,18 x lo-’ + 28%, 0,3 x 1O-2 + 17% und 0,2 x 10d2 + 20% cm s-l; die Werte fur D, sind 0,33 x 10u9 f 21%, 0,72 x 10d9 f 9% und 0,72 x 1O-9 + 9% cm s-l; die Werte fiir k, sind 0,65 x lop6 f 7,7%, 1,8 x 10e6 + 17% und 0,2 x lop6 + 1% s-‘; schliel3lich &d die Werte fur f 0,55 + 9%, 0,6 &- 10% und 0,18 f 11% jeweils & Basic Red 22, Basic Yellow 21 und Basic Blue 69. Grundlegend fur das Model1 ist die innere Struktur der Aktivkohlepartikel die in Makroporen und Mikroporen unterteilt wird. Die Obergrenze der Kapazitiit der Mikroporen ist 241 mg g-i fur Basic Red 22,245 mg g-l fiir Basic Yellow 21 und 656 mg gg’ fur Basic Blue 69. Eine Sensitivitiitsanalyse wurde fur jeden Parameter durchgeftihrt.
Synapse Gegenstand der Untersuchung war die Adsorption der drei Farbstofle Basic Red 22, Basic Yellow 21 und Basic Blue 69 auf Aktivkohle (Filtrasorb 400). Ein Model1 mit drei Stofltransportwiderstiinden ist entwickelt worden, urn den Konzentrationsverlauf in der Aktivkohle zu beschreiben. Das Adsorptionsmodel auf der Basis verzweigter Poren (branched-pore model)
*Author to whom correspondence
0255-2701/88/$3.50
should be addressed.
beschreibt kinetische Daten vom Zeitpunkt des ersten Kontaktes Ahorbent-Adsorbat bis zu langen Adsorptionszeiten (> 24 h) mit angemessener Genauigkeit. Die Modellparameter sind ein 2iuJerer Stofftransportkoefizient kr, ein Feststofldtrusionskoefizient D,, ein Gesamtstofltransportparameter kb fir die Mikroporen und der Makroporenanteil fDie Kapazitiit der Aktivkohle wurde fGr jeden der drei Farbstoffe durch Messung der Gleichgewichtsisothermen bestimmt. Zur Messung der Isothermen wura’e eine bekannte Menge Aktivkohle zusammen mit einem festen Volumen Farbstoflosung in einem
Chem. Eng. Process., 24 (1988) 1-13
0 Elsevier Sequoia/Printed
in The Netherlands
2
geschlossenen GeftiJ sechs Wochen bei 20 “C aujbewahrt. Variiert wurde die Farbstoflonzentration in der Losung. Die MeQwerte wurden mit der FreundlichGleichung korreliert: ae = afCeb Die Freundlich-Konstanten a, und b sind in Tabelle 2 zusammengefa&. In der Isothermen in Bild 4(a) und (b) kann man, der inneren Struktur der Aktivkohle-Partikel entsprechend, eine Makroporen-Adsorption und eine Adsorption in den Mikroporen unterscheiden. Die der Kapazitiit der Mikroporen ist Obergrenze 241 mg g - ’ fir Basic Red 22, 245mg g - ’ fur Basic Yellow 21 und 656 mg g ~ ’ fur Basic Blue 69. Bild 1 zeigt den diskontinuierlich betriebenen, durchmischten Adsorber, in dem die Untersuchungen der Kinetik durchgefiihrt wurden. Die Entnahme von Proben aus dem Adsorber zur Analyse erfolgte mit einer Spritze in regelm$igen Zeitabstiinden wiihrend der Versuchsdauer von 24 Stunden. Mit einem Computerprogramm wurden aus dem Konzentrationsverlaufskurven Stofltransportmodell berechnet. Die Werte fur kr, D,, kb undf wurden durch Anpassung an die experimentellen Ergebnisse bestimmt. Die experimentellen Variablen fur jeden Farbsto# waren die Menge der Aktivkohle und die Anfangskonzentration des Farbstoffes. Die gilder 5-8 zeigen typische experimentelle Ergebnisse sowie berechnete Kurven. Ein Farbstoffsystem wird durch jeweils einen Wert der Parameter beschrieben. Die Werte fiir k, sind 0,18 x iUp2 k 28%, 0,3 x IO- 2 f 17% und 0,2 x 10e2 f 20% cm s-‘; die Werte fur D, sind 0,33 x IUp9 + 21%, 0,72 x 10m9 + 9% und 0,72 x IUp9 + die Werte fur k, sind 0,65 x IUp6 + 9% c?&-‘; 7,7%, 1,8x 10p6+ 17% und 0,2x 10-6+ I%s-‘; SchlieJich sind die Werte fir f 0,55 + 9%, 0,60 + 10% und 0,lS & 11% jeweils fir Basic Red 22, Basic Yellow 21 und Basic Blue 69. Die Ergebnisse einer Sensitivitiitsanalyse fur jeden Stofftransportparameter sind in Bild 911 dargestellt.
1. Introduction Activated carbon has been highly effective as an adsorbent in batch, fixed and moving bed adsorption systems for water treatment owing to its efficiency in producing high quality water, meeting the requirements of environmental pollution legislation, at a price comparable with secondary biological treatment! The design of adsorption systems has been aided by mathematical models that predicted kinetic data, based on a single effective intraparticle diffusion parameter, throughout adsorption time, assuming identical adsorption sites on the carbon particle. These are solid, pore and solid-pore diffusion models. Widely used at present, a solid diffusion model [l] assumes a homogeneous adsorbent surface and is applied for porous adsorbents. It produces reasonable agreement between experimental and theoretical data for a wide range of systems [14], but for a limited period of time (4
hours). The pore diffusion model, though assuming porous adsorbents, has a narrow range of application because it does not account for surface migration on the pore walls and this leads to inaccurate pore diffusivities with misleading results [ 5,6]. Experimentally, kinetic data start with a rapid solute uptake followed by a slower adsorption rate that levels off as equilibrium is attained [7, 81. In the above models, a high diffusion rate parameter leads to rapid attainment of equilibrium and a low diffusion rate parameter leads to slow attainment of equilibrium. Therefore these previous models succeeded in predicting the rapid uptake period but, owing to the high diffusivity implied, they led to equilibrium being attained at earlier stages than in experiments. This discrepancy suggested that there is more than one intraparticle diffusion parameter controlling the different stages of the system kinetics and related to the internal structure of the adsorbent, namely carbon in the present work, whose effect has been neglected in the single intraparticle diffusion rate parameter models. Carbon is an adsorbent with a heterogeneous surface and a polydisperse porous structure, that is, pores of a wide size range uniformly dispersed throughout the granules. Beck and Schultz [9] and Satterfield et al. [lo] have proved a strong relation between the diffusion rate and the ratio of solute molecular diameter to adsorbent pore diameter. Hence the wide pore size range implies a wide range of diffusion rates making a single diffusion rate either an average approximate value or an accurate value for a limited time. Peel et al. [ 111 have approximated the microscopic description of the adsorption mechanism by dividing a carbon particle into two regions of two pore size ranges and therefore two diffusion rates: a macropore region of pores of radius larger than 2 nm, where rapid uptake occurs, providing access to the micropore region of pores of radius smaller than 2 nm, where a slow adsorption rate is expected. Based on the above, Peel et al. proposed a threeresistance model: a film resistance expressed by the external mass transfer coefficient kr (cm s-l), a resistance in the macropore region given by the solid diffusivity D, (cm2 s-l), a resistance in the micropore region given by a lumped diffusion coefficient k,, (s-l), and finally a factor f which is the fraction of macropores. This work utilizes Peel’s model in a wide range of experiments and applies it to an agitated batch adsorption system for three basic dyes, namely Basic Blue 69 (Astrazone Blue), Basic Red 22 (Maxillon Red) and Basic Yellow 21 (Astrazone Yellow).
Experimental Kinetic
studies
Kinetic studies were carried out in an agitated batch adsorption system which consisted of a 2 dm’
3
A constant carbon particle size range, diameters 35&500 pm, was used in each experiment. The general characteristics of the Filtrasorb 400 are shown in Table 1. About 70% of the pore volume lies between pore sizes of 10 A and 1000 A, with 25% of the overall ore volume in the pores of diameters from 20 A to 30 x .
Variable Speed Motor
Rubber Support Ring a
Isotherms
A set of dye solutions was prepared for each dye system in predetermined concentrations. These were brought into contact with pre-weighed amounts of carbon in jars. The latter were sealed and shaken in a water bath at room temperature (20 “C) for three weeks until equilibrium was attained. The solution was tested to ensure no biological degradation had taken place during this period. Solutions were then analysed for the fluid-phase concentrations C,, while solid-phase concentrations qe were obtained by a material balance on the system. Equilibrium data were correlated by the Freundlich formula,
-
-4
Fig. 1. Schematic representation of batch adsorber.
q= = arc,6
glass vessel, of diameter 0.13 m, filled with 1.7 dm3 of solution, giving a solution height of 0.13 m. The solution was agitated by a six-flat-blade impeller driven by a Heidolph electric motor with a speed adjustable from 100 to 600 rev min-‘. The width of the impeller blades was 0.013 m. Complete mixing was facilitated by eight baffles, each of width 0.01 m, distributed at 45” around the circumference of the beaker and held in position by a polystyrene baffle holder. Figure 1 illustrates the system. In all experiments where the initial dye concentration or the carbon mass was varied, the agitation speed was maintained constant at 400 rev min-‘. Sorbate-sorbent
system
Carbon Filtrasorb F400, supplied by Chemviron Ltd., was used as the adsorbent. It was crushed by a hammer mill, screened to a series of particle sizes by sieve analysis and then washed thoroughly in distilled water to remove fines. It was dried in an oven at 120 “C for 24 hours. The sorbate system consisted of three basic dyes: (i) Basic Blue 69 (Astrazone Blue), BB 69; (ii) Basic Yellow 21 (Astrazone Yellow), BY 21; and (iii) Basic Red 22 (Maxillon Red), BR 22. The first two were supplied by Bayer and the third by Ciba Geigy. TABLE 1. Characteristics of Filtrasorb 400 carbon Total surface area, N, BET method (m* kg-‘) Bed density, backwashed and drained (kg m-j) Particle density, wetted in water (kg rnm3) Pore volume (m3 kg- ‘) Iodine number Methylene Blue number
because of its applicability for heterogeneous surfaces. Freundlich constants for the three dye systems are given in Table 2. Equilibrium is between the solid surfaces and the liquid phase at all adsorption sites. Analysis
Kinetic and equilibrium data were analysed using a Perkin Elmer SS05 spectrophotometer. The optical densities of the coloured solutions were measured at certain wavelengths corresponding to the maximum absorbance. These wavelengths, termed Iz,,,, were 585, 537 and 417 nm for BB 69, BR 22 and BY 21 respectively. The dye concentration is related to the optical density by a predetermined calibration graph. Theoretical
analysis
Figure 2 illustrates the approximate structure of a carbon particle. Macropores are the main passages that lead to the interior of the particle. Micropores branch off from the macropores forming a network which is dispersed uniformly throughout the particle. The adsorption mechanism is described in this model by three resistances in series: (i) the external resistance across the liquid film, (ii) the intraparticle
TABLE 2. Freundlich (temperature = 20 “C) 1.10 x IV 425 x 10’ 1350 0.90 x IO-’ 1050 280
constants
for three basic dye isotherms
Dye
ar(dm3g-‘)b
b
BR 22 BY 21 BB 69
299.0 330.3 337.0
0.130 0.133 0.226
I
qs: Cs
qm
Fig. 3. Conceptual
The following
diagram
structure
of carbon
system.
set is obtained:
3Qtn a2Qm x =4q - at12 + 4 2 Fig. 2. Approximation of the internal posed by the branched-pore model.
of the adsorption
‘c
- Bi,(Qm - Q,,)
(3)
as pro-
aQb = Bi p ;f(Qm
de
resistance in the macropores and (iii) the intraparticle resistance in the micropores. Owing to the complexity of mass transfer in this region, caused by multidirectional interactions, a lumped parameter kb has been used to describe the rate of mass transfer from the macropore to the micropore region. Bidisperse adsorbents like molecular sieves have been modelled relatively easily by Furusawa and Smith [6], Ruckenstein et al. [ 121, and Shah and Ruthven [ 131, since the relative proportions and locations of different regions are defined. In the case of carbon, modelling is complex owing to carbon’s polydisperse nature, that is, it has a continuous range of pore sizes, distributed uniformly throughout the particles. Therefore, a factor f has been selected to describe the fraciton of macropores in the particle, 1 -f being the fraction of micropores. Figure 3 is the conceptual diagram of the system. The mathematical description of the model is based on the material balance backed with mass transfer principles. Peel et al. [ 1l] explain the theory extensively, therefore to avoid duplication only the dimensionless form of the equations is stated briefly below. Details on the conversion of the equations to dimensionless form are given in the Appendix. Given the dimensionless terms
- Qd
(4)
(5)
(6) In the above, Bi
=
f
kKoR
(7)
Pc_m90
kbR2 Bi, = ~
(8)
fDs
are modified Biot numbers, where Bi, represents the ratio of the internal to the external resistance and Sir represents the ratio of the macropore to the micropore resistance;
F=YLO w90 is the separation factor. The initial and boundary
Q&L 0) = 0 Qdrl. 0) = 0 C(0) = 1 Qm( 1, t) = Q.(t)
(9) conditions
become:
(10) (11) (12)
(13) Equations (3)-( 13), together with the equilibrium data, are solved by the Crank-Nicolson finite difference method. Owing to the nonlinearity of the isotherm, the coupling eqn. (6) is not linear and the
solution is iterative at each point in time. Initial guesses of the future concentrations are calculated by linearly extrapolating the change over the previous time interval and the value used in the first iteration. The newly calculated values are then substituted directly and the solution repeated until a converged solution is obtained. A computer program based on this method has been developed by McKee [ 141 and extended by Al Duri [ 151. It is fed with the system variables, namely the mass, particle size and particle density of carbon, the initial sorbate concentration, volume of the solution, equilibrium constants, and kinetic data, together with time and space intervals required for integration. In addition, the four rate parameters of the system, k,-, D,, kb and f, are supplied. The program yields the liquid-phase concentration with time (theoretical kinetic data), the surface concentration, the dimensionless distance q across the particles, and the macropore and micropore solid-phase concentrations. Discussion The branched-pore model is based on the internal structure of the carbon particles. The duality in the adsorption mechanism makes analysis of the kinetics very complex. Several mechanisms act in parallel, with contributions depending on each particular system. In order to simplify things, the pore regions in a carbon particle were subdivided into macropores and micropores [ 113. This adsorption model, with its four parameters, namely, the mass transfer coefficient k, the solid diffusivity D,, the lumped micropore diffusion parameter k,, and the fraction of macropores f, is sufficient to describe experimental kinetic data from initial sorbate-sorbent contact to equilibrium. Peel et al. [ 1 I] developed the branched-pore model and applied it to the carbon-phenol adsorption system. Table 3 summarizes their results; they obtained the parameters by regression analysis of the data. kr was evaluated by a single resistance assumption and the variation in k, with carbon mass was attributed to varying hydraulic flow patterns with different amounts of carbon. McKee [ 141 found kf to be 0.28 x 10W2 cm s-’ for the chitin/Acid Blue 25 system and 0.47 x lop2 cm S-I for the chitin/Acid Blue 158 system. In the present work kf was evaluated by the single resistance assumption as an initial estimate to the best curve fit. The values obtained were 0.18 x 10e2 f 28%, 0.30 x lop2 f 17% and TABLE
0.20 x lop2 f 20% for BR 22, BY 21 and BB 69 respectively. Compared with the values in Table 3, these external mass transfer coefficients are low. This is attributed to the molecular sizes of these dyes which are relatively large compared with the size of the phenol molecule, therefore the mass transfer rates are slower. The effective diameters of the dye molecules are 27, 28 and 30 8, for Basic Red 22, Basic Blue 69 and Basic Yellow 21 respectively. In the experimental section the characteristics of Filtrasorb 400 activated carbon show that 25% of the pore volume is in the range 2&30 A, therefore molecular sieve effects may be occurring. However, since the dye molecules are so similar in size and Basic Blue 69, having the highest adsorptive capacity, falls between the other two dyes in terms of molecular diameter, it is considered that molecular sieving is not important in determining the relative adsorption capacities. The pH of the dye solutions at a concentration of 300 mg dmd3 were 5.9,6.5 and 5.0 for Basic Yellow 21, Basic Red 22 and Basic Blue 69. The low pH for Basic Blue 69 may have an influence on the high uptake of this dye on carbon. The adsorption capacities of the other two dyes do not follow this trend and therefore pH is again not the criterion for adsorption capacity. Intraparticle diffusion in the macropores is described by a solid diffusion mechanism, characterized by the solid diffusivity D, [ 11, 141. Peel et al. [ 111 obtained D, values comparable with those obtained from single intraparticle parameter models [ 3, 16, 171. This fact is related to the small molecular size of phenol molecules which results in a high solid diffusivity and consequently a rapid uptake, making all macropores saturated before diffusion into the micropores starts. This implies that mass transfer is solely in macropores at this stage of the adsorption process, that is, in the macropore mass balance equation, f
ak _fD, dt
r2
a
ar
cr -bR 2
a9,
ar
b
f = 1 and Rb = 0, making eqn. (2) similar to the intraparticle diffusion mass balance in the single solid diffusion parameter model, implying a similar D, to yield the same kinetic data. Other D, values in the literature include 1.0 x lop8 cm2 s-’ for the chitin/ Acid Blue 25 system and 7.5 x lo-’ cm2 s-’ for the chitin/Acid Blue 158 [ 141. In the present work D, was evaluated by a best curve fit technique and regression analysis and a single constant ( k 10%) value described each dye system for a set of masses and concentrations. Table 4 gives D, values to be 0.33 x 10p9k 21%, 0.72 x 10p9f9% and
3. Parameters of phenol kinetic rum by Peel et al. [ 1 I]
w cd
C,(rngd~~-~)
k,x
0.9898 0.7216 0.5393
96.3 96.3 94.7
1.33 f 5% 1.82 f 5% 2.01 f. 5%
10-2(cms-‘)
D, x lo-*
(cm2s-‘)
7.78 f 10% 9.01 f 9% 7.15 + 7%
k, x 106(sr’)
f
l&*27% 1.87 + 18% 1.80 + 16%
0.63 + 2% 0.66 + 2% 0.68 i 2%
6 TABLE 4. Data of kinetic runs for three basic dye systems Run
Dye
W(g)
C,, (mg dme3)
kf x 102 (cm s-l)
II, x lo~(cm*s-‘)
k, x 106 (s-l)
f
0.60
No.”
1 2 3 4 5 6
BR BR BR BR BR BR
22 22 22 22 22 22
1.27 0.850 0.638 0.425 0.638 0.638
98.5 102.0 101.0 100.0 49.0 75.6
0.15
0.20 0.20 0.20 0.23 0.20
0.27 0.35 0.35 0.40 0.32 0.35
0.70 0.70 0.70 0.70 0.60 0.70
7 8 9 10 11 12 13 14 I5 16 17
BY BY BY BY BY BY BY BY BY BY BY
21 21 21 21 21 21 21 21 21 21 21
0.850 0.638 0.425 0.319 0.213 0.107 0.425 0.425 0.425 0.425 0.425
103.0 99.4 99.1 98.7 101.0 102.0 25.0 49.8 74.6 154.0 206.0
0.28 0.26 0.25 0.25 0.25 0.26 0.35 0.35 0.30 0.25 0.25
0.68 0.68 0.65 0.79 0.70 0.65 0.79 0.79 0.79 0.79 0.75
1.5 1.8 1.5 2.1 1.5 1.5 2.0 2.1 1.5 2.0 2.0
0.60
18 19
BB 69 BB 69
1.70 1.27 0.850 0.638 0.213 0.850 0.850
89.6 90.5 85.2 91.1 92.8 68.5 106.0
0.25 0.20 0.15 0.15 0.20 0.18 0.15
0.66 0.80 0.70 0.78 0.78 0.79 0.66
0.20 0.15 0.20 0.20 0.20 0.20 0.20
0.16 0.18 0.18 0.18 0.20 0.18 0.16
20
21 22 23 24
BB 69 BB 69 BB 69 BB 69 BB 69
0.55 0.58 0.55 0.50 0.52
0.56 0.55 0.65 0.57 0.55 0.65 0.65 0.58 0.66 0.66
“For all runs: R = 0.0214 cm, V = 1700cm3 and pC= 1.26 g cnm3.
0.72 x lop9 f 9% for BR 22, BY 21 and BB 69 respectively. Compared with the values in Table 3 the carbondye solid diffusivities are low, and again this is related to the difference in molecular size of the sorbates (i.e. the diameter of the molecules) in the systems listed in Tables 3 and 4. Table 5 compares D, values in the present work with those obtained from the conventional film-solid diffusion parameter model (FSDPM) [ 151. For the
same set of masses and concentrations, the branchedpore diffusion model (BPDM) D, values in the dyecarbon systems are higher because the conventional FSDPM includes only two resistances. As mentioned earlier, macropores provide the only access to the interior of particles, therefore diffusion from macropores to micropores reduces the sorbate concentration in the former by a fractionf. Therefore a higher
D, is needed to maintain the intraparticle diffusion rate equal to that in the single solid diffusion parameter model. Thus the comparison between D, values in the present model (BPDM) and the FSDPM in Table 5 results in smaller D, values for the FSDPM because it only includes two resistances. Mathematically: Macropore + Micropore Intraparticle diffusion = diffusion diffusion i.e.
dq --+(l-/)d$ f dqm
dt-
(15)
dt
-_R
b
+R
b
(16)
which yields TABLE 5. Comparison of solid diffusivities obtained by the filmsolid diffusion parameter model (FSDPM) and the branched-pore diffusion model (BPDM) for three basic dye systems
Dye
BR 22 BY21 BB 69
D, x lo9 (cm s-‘) FSDPM
BDPM
0.085 + 5.9% 0.23 k 17% 0.02 * 20%
0.33 * 21% 0.72 + 9.7% 0.72 * 9.7%
(17) In the single diffusion parameter model Fritz et al. [ 31 used
(18) Comparing eqns. (17) and (18), they differ byfand qm, which implies a higher D, value in eqn. (17) to yield the same kinetic performance as eqn. ( 18).
7 The lumped diffusion parameter kb approximates the mass transfer at the later stages of the adsorption mechanism. Peel et al. [ 111 evaluated k, by regression analysis of their data, obtaining the values in Table 3. McKee [ 141 obtained k, values of 10.0 x 10d6cm S-I for chitin/Acid Blue 25 and 500 x IO-“cm s-’ for chitin/Acid Blue 158. In the present work a constant k,, was obtained for each dye system. The values of k, obtained were 0.65 x lop6 f 7.7%, 1.8 x lop6 * 17% and 0.20 x 10e6 f 1% for BR 22, BY 21 and BB 69 respectively. The adsorption mechanism in the micropores is strongly dependent on the pore geometry. Their small diameters (up to 2 nm in Filtrasorb F400) make the molecular sizes of the sorbates comparable with the pore sizes of the micropores, causing multidirectional sorbate-sorbent interactions [ 181. In addition, decreased pore radii raise the energy barrier in the surface migration process, greatly reducing its rate. On the other hand, the rate of diffusion is a strong function of the ratio of solute diameter to pore diameter [9, lo]. A ratio as low as 1: 10 reduces the diffusion rate by 40%. All the above observations suggested that the diffusion mechanism
in the micropores is by solid phase diffusion characterized by sorbate-sorbent multidirectional interactions. These are responsible for the irreversibility in the isothermal systems. Model1 et al. [ 191, in adsorption-desorption studies, succeeded in desorbing 70% of the phenol adsorbed on Filtrasorb F300 carbon, using CO, as a regenerant. They attributed the lost capacity to irreversible sorbate-sorbent interactions and obtained a better recovery in desorbing phenol after a short adsorption period. This indicated that irreversibility is caused by interactions during the later slow adsorption period. Snoeyink [20] was able to desorb 50% of adsorbed phenol from a Columbia LC carbon after a 5 month contact. Furthermore, owing to multidirectional interactions, sorbents acquire a strong affinity for the sorbate the therefore the micropores tend to be saturated at lower solution concentrations than the macropores. They reach an upper-bound capacity independent of equilibrium concentrations. This is given by qe(1 -f) and is shown in Fig. 4. Micropore capacities were found to be 241, 245 and 656 mg g-’ for BR 22, BY 21 and BB 69 respectively. The macropore fraction f evaluated by Peel et al. for the carbon-phenol system is given in Table 3.
Capacity
i : E
500
d ~I_
Capacitv
Cs,
m9
dm-'
(a)
t---f-__.
541
_.__i________.~___ _______ ____ ________ __-__________-___._ I____________ _______ ___----__I 541
541 I
0
50
100
(b) Fig. 4. Postulated distribution for (a) the BB 69 isotherm and (b) the BR 22 isotherm.
1
.
150 C,.
mg
dm
-3
0
3611
1080
tine,
istn
Ds =
0.35 x
lo-’ cm%-’
kf = 0.18 x lo-* cm s-' kb = 0.7 x (10-7, 10-6, 10-5, 10-4) f = 0.58
1
0
2
3
4
5
6
7
8 time, hrs
9
Fig. 10. Sensitivity analysis of the micropore diffusion rate parameter k,.
Ds = 0.35 x lo-’ cm’s_’ kf = 0.18 x 1O-2 cm s-' kb = f =
0.7 x 1O-6 cm s -’ 0.65, 0.75, 0.85, 0.999
, 0
1
2
3
4
5
6
7
8 time,
9 hrs
Fig. 1I Sensitivity analysis of parameter f, the fraction of macropores.
controlling influence to terminate after 4 hours owing to early salination of the macropore region for the reasons explained earlier. Figure 10 illustrates the effect of I&,, which becomes evident at late stages where micropore diffusion is the feature of adsorption. The influence off, shown in Fig. 11, alters the whole adsorption process. Furthermore, the extent of the contribution of each parameter depends totally on the particular sorption system, the physical and chemical properties of the sorbate.
Computer simulation and the running time of the model depend strongly on the combination of values of the system variables supplied. For example, run No. 9 took 8 minutes to yield kinetic data for 9 hours, while it took 11 hours to produce kinetic data for a contact time of 24 hours. This restricts the conditions under which this model can be run, particularly with respect to computational costs. However, the branched-pore model, applied on a wide experimental basis, proved to be theoretically sound and could reduce difficulties encountered with
11 the internal structure neous systems.
of carbon
and other heteroge-
kf
Conclusions From the present work the following points can be made. (i) The branched-pore model proposed by Peel et al. [ 1 I] proves to be applicable over a wide range of experimental conditions concerning three basic dye adsorption systems, namely BR 22, BY 21 and BB 69, all on carbon. (ii) Single values for the diffusion parameters kr, D,, k,, and f describe each respective stage of the adsorption mechanism for a whole range of masses and concentrations of each dye system. (iii) The chemical properties of the sorbate (like the molecular structure, molecular weight, radius and polar properties) have a strong effect on the adsorption mechanism and the contribution of each step in it. (iv) The solid diffusivity D, evaluated in the branched-pore model is higher than that evaluated in the single solid diffusion model. This is attributed to the contribution of the micropores since it reduces the carbon loading in the macropores, implying a higher D, to keep the macropores saturated. (v) The micropore region has an upper-bound capacity, calculated by (1 -f )qe, independent of the equilibrium data. It was found to be 241, 245 and 656 mg g-i for BR 22, BY 21 and BB 69 respectively. (vi) D, controls the first few hours of the adsorption process, k, controls the late micropore adsorption, while f affects the whole process. (vii) Extensive computational running times for certain system conditions, in order to obtain longterm data, are a disadvantage of this model.
Nomenclature
af A b
Bi, Bi, C
Freundlich constant, dm3 g-’ surface area, m2 Freundlich constant = kfRC,/pJD,qo, modified Biot number = kbR2/fD,, modified Biot number = C,/C,, dimensionless fluid-phase concentration equilibrium fluid-phase concentration, mg drnp3 fluid-phase concentration at particle’s surface, mg drne3 = C,,/C,, dimensionless fluid-phase concentration at particle’s surface fluid-phase concentration at any time t, mg drne3 initial fluid-phase concentration at particle’s surface, mg dm-3 solid diffusivity, cm2 s-i fraction of macropores in particles = COV/q, W, separation factor lumped diffusion parameter, cm SK’
%
Q r
R R, t V W B ;I PC
external mass transfer coefficient, cm s-’ micropore solid-phase concentration, mg gg’ equilibrium solid-phase concentration, mg g-i macropore solid-phase concentration, mg g-i solid-phase concentration at particle surface, mg g-i maximum solid-phase concentration, mg g- ’ = q/go, dimensionless solid-phase concentration radial distance from centre of carbon particle, cm radius of carbon particle, cm = (1 -f) dqb/dt, micropore diffusion rate, mg g-i mini time variable, min fluid-phase volume weight of carbon, g =rJR, dimensionless distance carbon particle = 8’ =D, tIR2, dimensionless time particle density, g crnV3
from
centre
of
References 1 G. McKay, The adsorption of basic dye onto silica from aqueous solutions-solid diffusion model, Chem. Eng. Sci., 39 (1984) 129. 2 A. P. Mathews and W. J. Weber, Jr., Effects of external mass transfer and intraparticle diffusion on adsorption rates in slurry reactors, AIChE Symp. Ser., 73 (1976) 166. 3 W. Fritz, W. Merk and E. U. Schlilnder, Competitive adsorption of two dissolved organ& onto activated Carbon-II, Chem. Eng. Sci., 36 (1981) 731. 4 G. McKay and I. F. McConvey, Adsorption of acid dye onto woodmeal by solid diffusional mass transfer, Chem. Eng. Process., 19 (1985) 287. 5 T. Furusawa and J. M Smith, Fluid particle and intraparticle mass transport rates in slurries, Ind. Eng. Chem., Fun&m., 12 (1973) 197. 6 T. Furusawa and J. M. Smith, Diffusivities from dynamic data, AIChE J., 19(1973) 401. 7 J. S. Zogorski, S. D. Faust and J. H. Haas, The kinetics of adsorption of phenols by granular activated carbon, J. Colloid Interface Sci., 55 ( 1976) 329. 8 V. L. Snoeyink, W. J. W&r and H. B. Mark, Sorption of phenol and nitrophenol by activated carbon, Environ. Sci. Technol., 3 ( 1969) 9 18. 9 R. E. Beck and J. S. Schultz, Hindered diffusion in microporous membranes with known pore structure, Science, I70 (1970) 1302. 10 C. N. Sattertield, C. K. Colton and W. H. Pitcher, Restricted diffusion in liquids within fine pores, AZChE J., 19 (1973) 628. 11 R. G. Peel, A. Benedek and C. M. Crowe, A branched pore kinetic model for activated carbon adsorption, AZChE J., 27 (1981) 26. 12 E. Ruckenstein, A. S. Vaidyanathan and G. R. Youngquist, Sorption by solids with bidispersc pore structures, Chem. Eng. Sci., 29 (1971) 1305. 13 D. B. Shah and D. M. Ruthven, Measurement of xeolitic diffusivities and equilibrium isotherms by chromatography, AIChE J., 23 (1977) 804. 14 S. McKee, The development of computer programs for the solution of batch adsorption problems, M.Sc. Thesis, Queen’s Univ. Belfast, 1984.
12 15 B. Al-Duri, Ph.D. Thesis, Queen’s Univ. Belfast, in preparation. 16 J. C. Crittenden and W. J. Weber, Predictive model for design of fixed-bed adsorbers: parameter estimation and model development, ASCE J. Environ. Eng. Div., 104 (EE2) (1978) 185. 17 Y. Sudo, C. M. Misic and M. Suyuki, Concentration dependence of effective surface diffusion coefficients in aqueous phase adsorption on activated carbon, Chem. Eng. Sci., 33 (1978) 1287. 18 M. M. Dubinin, Porous structure and adsorption properties of active carbons, in Chemistry and Physics qf Carbon, Vol. 2, Marcel Dekker, New York, 1966. 19 M. Modell, R. P. de Fillipi, V. Krukonis and A. D. Little, Regeneration of activated carbon with supercritical CO,, Am. Chem. Sm. Dill. Environ. Chem., Prepr., Miami, 1978. 20 V. Snoeyink, Adsorption of strong acids, phenol and 4-nitrophenol from aqueous solution by active carbon in agitated nonRow systems, Ph.D Thesis, Univ. Michigan, 1968.
From q=$
aq=$2r& therefore ar =
a,$
and
atj2$
ar2= Thus,
$!!$
=fD,
(3
$
+j
%$-kb(Qm
Rearrangement Reduction of the mass balance equations dimensionless form
Macropore
Starting tion
equa-
Bi,
=
!%I!? Dsf
we obtain the macropore dimensionless form:
WE49
at =FgF
(
_ R
) ‘a,> +;ar
-RR,
%
to introduce dimensionless terms without altering the equation we multiply the left-hand side by
(1 -f)
= k,(q,
we multiply
qoR2D
--2
qoR2Ds
respectively.
>
Transferring ‘6&m - qb)
q,, to the right-hand for
atD,/R’
fDE aqm/qO = fD, R2 at D,lR2
sides by
f f
Grouping
the variables
we obtain
=k,-!-(q,_qb)
R2
and after rearrangement,
side and substituting
&
equa-
- q,J
aqb/q0@(1 -f) -Rb
mass balance
the left- and right-hand and
qo R2 D,
+f%
in
region mass transfer
from the micropore
qoR2D, which yields
equation
a’Q
Starting tion
b
region transfer
-?+4%-Bi,(Q,-Q,) a12
Micropore z aq,
fDS a
-kb(Q,-Qb)
As
aQm
mass balance
gives
=
region mass transfer
from the macropore
of this equation
to
This has two main advantages: (i) reduction of the number of equations, thus simplifying the solution procedure; (ii) relations for the system variables in terms of useful chemical engineering dimensionless groups like Bi and F.
f aq,
- Qt.)
>
Appendix
(1 -f)
a&
x
kbR2
= mf(Qm
-
Qb)
Introducing
Bip=kbR2 Df
Inserting
Q, = qm/qO. Qb = qb/qO, and 80 = at D,/R2,
aQm RZae =fR
fD,
~+~%$)-kdQm-Qb)
we have the micropore region mass transfer equation in dimensionless form:
aQb -= ae &f
Bi,
13 Liquid-phase mass transfer Starting from the fluid-phase tion,
mass balance
equa-
%=k$(C,-C,,)
g
Following the same principle the left-hand side by
= ?!$f
(C - C,)
as above, we multiply The coupling equation Starting from the mass balance liquid interface,
CoD,R2 --CoD,R2 to obtain dC, COD, R2
and group the parameters depending on the equations for Bi, and F (with W = pc V). This leads to the equation For the liquid-phase mass transfer in dimensionless form:
W(Ct - G, J =fD&c
k,A
~~~jii=~(C*-Cs,,)
0 s which, after rearrangement, and insertion of C = C,/C,, C, = C,,, - Co and 9 = D,t/R2, becomes
;g;yD2 (C -
hn
(dr > =f~(ct-cs.*) ,IR
qoR2 -C,)
,=R
that is,
We multiply =
across the solid-
the left- and right-hand
sides by
CO
and
qo R2
c,
s
We wish to obtain this equation where Bi
_
WoR
’ - pcfD,q, So we multiply qof ---- COPGV qof Co&V
and
in terms of Bi, and F,
h&o
to obtain
qoR2
--=f~cc,-G,3 ar R2
F=s!r
qow the right-hand
respectively,
k&o
SE 0
Inserting side by
k&OR
Bi, =fa9cqo
and
leads to the coupling
r2 q = z equation
in dimensionless
form: