Ind. Eng. Chem. Res. 1997, 36, 407-413
407
Determination of Multicomponent Adsorption Equilibria by Liquid Chromatography Parangusam K. Muralidharan and Chi B. Ching* Department of Chemical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore
Equilibrium relations for the glucose-sucrose-sorbitol-water system (a quaternary system) on 13X zeolite molecular sieves have been determined by an extension of the method suggested for a ternary system. The quaternary and the constituent ternary adsorption equilibrium relations derived from chromatographic retention time measurements of binary systems are verified by comparing experimental retention times with that obtained from simulations and the equilibrium theory of bisolute systems respectively. The accuracy of the adsorption equilibria is further verified by the adsorbed phase concentrations predicted by the ideal adsorbed solution theory and potential theory. Introduction The adsorption equilibrium isotherm quantitatively describes the equilibrium distribution of a solute between two phases involved in the chromatographic process over a wide concentration range. Traditionally, isotherms have been measured by the batch method (Fritz and Schluender, 1974; Jossens et al., 1978; James and Do, 1991; Hatanaka and Ishida, 1992). The primary drawbacks of this technique are the uncertainty in reaching equilibrium, the large amounts of solute and adsorbent required for accurate measurements, and the long period of time required for attainment of equilibrium. These drawbacks usually limit the application of the batch technique to the measurement of singlecomponent isotherms only. In order to circumvent the problems associated with the batch method, various chromatographic methods have been developed, which are well suited for the measurement of single-component as well as the more complicated multicomponent isotherms. Chromatography, being a dynamic method, offers some clear advantages in terms of accuracy, simplicity, and rapidity in obtaining experimental data. The application of the chromatographic technique to the measurement of binary adsorption equilibria in both gas and liquid systems is fairly well established (van der Vlist and van der Meijden, 1973; Ruthven and Kumar, 1980; Hyun and Danner, 1982; Barker and Thawait, 1984; Ruthven, 1986). Ching et al. (1990) demonstrated the feasibility of determining equilibrium relations for a ternary system by the chromatographic technique. This paper investigates the possibility of extending the method to the determination of equilibrium relations for a quaternary system. Theory A perturbation in the inlet concentration of an adsorbable species propagates through a chromatographic column with a wave velocity that is determined by the equilibrium relation. In a constant-density binary liquid system, continuity considerations show that only a single concentration wave will be observed and the wave velocity will be the same regardless of which of the two components are perturbed. We consider a solute component, designated by subscript i, in a solvent, designated by subscript s, which * Author to whom correspondence is to be addressed. Tel: 65-7722883. Fax: 65-7791936. E-mail:
[email protected]. S0888-5885(96)00427-7 CCC: $14.00
is competitively adsorbed. The wave velocity and hence the retention time for a small concentration pulse are given by
ωi )
L ) τi
v 1- 1+ Ki
(
)
(1)
where
Ki ) (1 - xi)
dq* dq* i s + xi dci dcs
(2)
xi is the mass fraction of the solute in the fluid phase and dq*i /dci, dq*s /dcs are the local slopes of the equilibrium curve. If the densities of the pore and fluid phases remain constant over a small concentration range, which is equivalent to assuming no net mass flux into or out of the porous adsorbent, we have
dq* q dy* i i ) dci c dxi
(3)
dq* dq*s q d(1 - y* i ) i ) ) ) dcs c dc d(1 - xi) i
(4)
Substituting eq 4 in eq 2 gives
Ki )
dq* i dci
(5)
It is clear from eqs 1 and 5 that the measurement of the retention time (or the wave velocity) yields directly the local slope of the equilibrium line, which subject to the constant-density approximation is the same for both solute and solvent. The simplest approach to the correlation of liquid phase equilibrium data is to represent the equilibrium in terms of a concentration dependent distribution coefficient
q* i i ) ki + Aicm i ci
(6)
dq* i i ) ki + Ai(mi + 1)cm i dci
(7)
© 1997 American Chemical Society
408 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997
Linear expressions of eq 6 (mi ) 1) can be used when the curvature of the equilibrium line is not too great (Barker and Thawait, 1984; Ruthven, 1986). ki is the apparent Henry’s law constant, the slope of the equilibrium line at infinite dilution. Thus, ki, Ai, and mi may easily be determined from measurements of retention times over a range of concentrations as can be seen from eqs 1, 5, 6, and 7. The simplest representation of quaternary equilibrium data (c1, c2, c3, cs) is by direct extension of eq 6
q* 1 n12 n13 1 ) k1 + A1cm 1 + B12c2 + B13c3 c1
(8)
q*2 n21 n23 2 ) k2 + A2cm 2 + B21c1 + B23c3 c2
(9)
q*3 n31 n32 3 ) k3 + A3cm 3 + B31c1 + B32c2 c3
(10)
The constants (k1, A1, m1, B12, n12, B13, n13, k2, A2, m2, B21, n21, B23, n23, k3, A3, m3, B31, n31, B32, n32) may, in principle, be evaluated from wave velocity measurements at different compositions. However, a more convenient approach has been suggested by Ching et al. (1990). For a pulse of component 1 in a base solution of component 2 (c1 ) 0, c3 ) 0), eq 8 becomes
dq*1 ) k1 + B12cn2 12 dc1
times can then be made to test the validity of the assumption that the constants derived from the binary data are applicable to the quaternary system. If the equilibrium relations for the quaternary system are rewritten into those for the constituent ternary systems, further tests can be made to check the validity of the assumption that the constants derived from the binary data are applicable to the ternary system. The ternary equilibrium relations are as follows
q* q*1 1 n12 n13 1 1 ) k1 + A1cm ) k1 + A1cm 1 + B12c2 ; 1 + B13c3 c1 c1 (14) q*2 q*2 n21 n23 2 2 ) k2 + A2cm ) k2 + A2cm 2 + B21c1 ; 2 + B23c3 c2 c2 (15) q*3 q*3 n31 n32 3 3 ) k3 + A3cm ) k3 + A3cm 3 + B31c1 ; 3 + B32c2 c3 c3 (16) The behavior of ternary systems has been studied by Glueckauf (1949), who proposed that the wave velocities of the two components are given by
ωi )
(11) ωj )
while for a pulse of component 1 in a base solution of component 3 (c1 ) 0, c2 ) 0), eq 8 becomes
dq* 1 ) k1 + B13cn3 13 dc1
(12)
(
)
v 1 - dq* j 1+ dcj
(
(17)
)
in which
Similar equations can be obtained for component 2 and component 3 for pulses of the components in a base solution of the other two components considered separately. The general equation is as follows
dq* i ) ki + Bijcnj ij dci
v 1 - dq* i 1+ dci
dq*i ∂q* ∂q* i i dcj ) + dci ∂ci ∂cj dci
(18)
dq* ∂q* ∂q* j j j dci ) + dcj ∂cj ∂ci dcj
(19)
(13)
where i ) 2, 3 and j ) 1, 2, 3 (i * j). Measurements carried out over a range of concentrations thus yield k1, B12, n12, B13, n13, k2, B21, n21, B23, n23, k3, B31, n31, B32, and n32 directly. The values of k1, k2, and k3 should of course coincide with the values from the binary system. Measurements with the binary systems (pulses of component 1 in a solution of component 1, pulses of component 2 in a solution of component 2, and pulses of component 3 in a solution of component 3) and measurements with the pseudobinary systems (pulses of component 1 in a solution of component 2, pulses of component 1 in a solution of component 3, pulses of component 2 in a solution of component 1, pulses of component 2 in a solution of component 3, pulses of component 3 in a solution of component 1, and pulses of component 3 in a solution of component 2) thus yield all the parameters. The retention times for pulses of components 1, 2, and 3 in a solution containing all three components (a quaternary system) can then be obtained by simulating the pulse response in a fixed bed using the equilibrium relations derived for a quaternary system. A comparison with the measured retention
The coherence condition requires ωi ) ωj and hence
( ) (
∂q*j dci ∂ci dcj
2
+
)
∂q* ∂q* ∂q* j i dci i )0 ∂cj ∂ci dcj ∂cj
(20)
Equation 20 can be regarded as a quadratic defining dci/dcj. The two roots will give two different values of ωi (or ωj) according to eq 17. These correspond to the velocities for an injected pulse of component i or a pulse of component j, which in a ternary system are not the same. The partial derivatives in eqs 18-20 have the following forms
( )
∂q* i nij i c ) ki + Ai(mi + 1)cm i + Bijcj ∂ci j
( )
(21)
∂q* i c ) Bijnijcicj(nij-1) ∂ci i
(22)
∂q* j nji j c ) kj + Aj(mj + 1)cm j + Bjici ∂cj i
(23)
( )
( )
∂q* j c ) Bjinjicjci(nji-1) ∂ci j
Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 409 Table 1. Details of Column and Adsorbent
(24)
where i and j can be components 1 and 2, 2 and 3, or 1 and 3. The retention times for pulses of both components i and j in a solution containing both components (a ternary system) can then be predicted from eq 17 with dq*i /dci and dq*j /dcj calculated from eqs 18-24. A comparison with the measured retention times can then be made to test the validity of the assumption that the constants derived from the binary data are applicable to the ternary system. Also, the two widely used multicomponent adsorption theories that predict adsorption equilibria for liquid systems, namely the ideal adsorbed solution (IAS) theory (Myers and Prausnitz, 1965; Radke and Prausnitz, 1972) and the potential theory (Grant and Manes, 1966; Greenback and Manes, 1981; Manes, 1980) can be used to predict multicomponent solid loadings from the single-component equilibrium relations (eq 6). These predictions can then be compared with the adsorbed phase concentrations calculated from the equilibrium relations for the quaternary system to check the validity of the equilibrium relations obtained. Experimental Method The adsorption equilibrium in the system glucosesucrose-sorbitol-water (component 1-component 2-component 3-solvent) on 13X zeolite molecular sieves was investigated using the method outlined above. The porous adsorbent in the form of 1/8 in. pellets was purchased from Strem Chemicals, Newburyport, MA. The pellets were crushed and sieved to obtain particles in the size range 105-175 µm. It was characterized by low-temperature nitrogen adsorption, measured using a Quantasorb instrument. Details of the physical properties are summarized in Table 1. D-(+)Glucose monohydrate was obtained from Merck, F.R. Germany, while sucrose was obtained from BDH Ltd., Poole, England. D-Sorbitol was supplied by Sigma Chemical Company. The chromatographic system utilized a Varian HPLC pump to provide a steady flow of liquid through the column, which was a precision bore tube of 316 stainless steel of dimensions given in Table 1, end-capped at both ends with a 2 µm stainless steel frit and compressing fittings of the “Swagelok” type. The column effluent was monitored by a differential refractometer (Waters 410, Millipore, U.S.A.). The sample injection valve was a standard LC valve (Rheodyne 7125) with a 200 µL sample loop. The millivolt signal from the detector was converted to digital form with the aid of an analog-to-digital interface card (Flytech Technology Ltd., Taiwan) interfaced with a microcomputer for data storage and processing. All experiments were carried out under thermostatic conditions at 25 °C. The column voidage was determined by injection of a pulse of blue dextran, a very high molecular weight species that will not penetrate the small pores of the zeolite particles. Retention time measurements for the binary and pseudobinary systems were performed as follows: I. Pulses of glucose, sucrose, and sorbitol were injected into an eluent containing 0-0.35 g/cm3 glucose. II. Pulses of glucose, sucrose, and sorbitol were injected into an eluent containing 0-0.35 g/cm3 sucrose. III. Pulses of glucose, sucrose, and sorbitol were injected into an eluent containing 0-0.35 g/cm3 sorbitol.
column length, cm column diameter, cm porous adsorbent particle size, µm mean pore diameter, Å specific area, cm2/g
20 1.1 13X zeolite molecular sieve 105-175 8.1 2.44 × 106
Table 2. Quaternary System and the Constituent Ternary Systems Composed from the Quaternary System system I II III IV
solutes in eluent glucose/sucrose/ sorbitol glucose/sucrose glucose/sorbitol sucrose/sorbitol
concn of solutes, g/cm3 0.16/0.11/0.06 0.11/0.13 0.10/0.08 0.13/0.08
injected component glucose/sucrose/ sorbitol glucose/sucrose glucose/sorbitol sucrose/sorbitol
The injected pulse in the above three cases was a solution of composition similar to the eluent, but with a slightly higher concentration of glucose, sucrose, or sorbitol. Similarly retention time measurements were performed for the constituent ternaries of the glucosesucrose-sorbitol-water system and also for the quaternary system constituting glucose-sucrose-sorbitolwater. The compositions of the systems are listed in Table 2. The injected sample solution consisted of the eluent with an increased concentration of glucose, sucrose, or sorbitol. The response curves were integrated numerically in order to determine first moments and hence the mean retention times
∫0∞ct dt µ) ∞ ∫0 c dt
(25)
The correction for dead volume was determined from a pulse response measurement with the column removed from the system and the injector connected directly to the detector. Results and Discussion The column voidage, , was determined from retention time measurements with blue dextran, a molecule too large to penetrate the pores of the adsorbent. The distribution coefficient K for blue dextran is thus zero, and eq 1 reduces to
L τ) u
(26)
The retention time measurements were measured for blue dextran over a range of eluent flow rates and plotted against L/u as shown in Figure 1. The slope of the linear plot yields ) 0.53. The retention time measurements with pulses of glucose, sucrose, and sorbitol injected into an eluent containing sorbitol were used to calculate the local */dcG, dqSU * / slopes of the binary equilibrium relations dqG * /dcSO according to eq 1. The calculated dcSU, and dqSO dqi*/dci values were plotted against sorbitol concentration, as shown in Figure 2. The relationship is essentially linear for each solute, and the best fit lines were found by the method of least squares. Hence, according to eq 7 with mSO ) 1 for a linear fit, the intercept and the slope of the plot for sorbitol yield kSO and 2ASO, respectively. Similarly the parameters kG, BGSO, kSU, and BSUSO were determined from the inter-
410 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997
Figure 1. Retention time vs L/u for blue dextran.
Figure 3. Local slopes of equilibrium relations dqi*/dci for sorbitol, glucose, and sucrose vs glucose concentrations.
Figure 2. Local slopes of equilibrium relations dqi*/dci for sorbitol, glucose, and sucrose vs sorbitol concentrations.
Figure 4. Local slopes of equilibrium relations dqi*/dci for sorbitol, glucose, and sucrose vs sucrose concentrations.
cepts and slopes of the plots for glucose and sucrose in Figure 2, respectively, with both nGSO and nSUSO set to 1. Figures 3 and 4 show the dqi*/dci values for the three solutes plotted against glucose and sucrose concentrations, respectively. The data in both figures, showing curvature in the low-concentration range, were fitted according to the equilibrium relations of eqs 6, 7, and 11-13 as follows. 1. Pulses of solute injected into a base solution containing the same solute: Since the distribution coefficients at infinite dilution kG and kSU have been determined previously, eq 7 may be written in the following form
ln
(
)
dq* i - ki ) ln Ai(mi + 1) + mi ln ci dci
(27)
The data for glucose injected into a base solution of glucose (Figure 3) and sucrose injected into a base solution containing sucrose (Figure 4) were plotted according to eq 27 as shown in Figure 5. The slopes of
these linear plots yield mG and mSU directly, while the intercepts give AG and ASU. 2. Pulses of a solute injected into a base solution containing a different solute: The equilibrium relations for these cases are represented by any of eqs 11, 12, or 13. Using the same approach as in 1 above, the data for sorbitol and sucrose injected into a base solution of glucose (Figure 3) were analyzed according to the following equation
(
ln
)
dq* i - ki ) ln BiG + niG ln cG dci
(28)
while the data for sorbitol and glucose injected into a base solution of sucrose (Figure 4), were analyzed according to the equation
ln
(
)
dq* i - ki ) ln BiSU + niSU ln cSU dci
(29)
Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 411 Table 4. Simulated and Experimental Retention Times for the Quaternary System (Table 2) system
a
retention time, s simulated exptl
injected component
I
glucose sucrose sorbitol
1890 1755 1968
1891a 1772a 1997a
I
glucose sucrose sorbitol
1126 1051 1167
1107b 1057b 1171b
Eluent flow rate, 0.5 cm3/min. b Eluent flow rate, 0.8 cm3/min.
Table 5. Theoretical and Experimental Retention Times for the Ternary Systems (Table 2) ternary system
Figure 5. Plots for glucose and sucrose according to eq 27. Table 3. Equilibrium Relations for Glucose, Sucrose, and Sorbitol eluent medium
injected component
sorbitol
sorbitol glucose sucrose
sucrose
sorbitol glucose sucrose
glucose
sorbitol glucose sucrose
equilibrium relation qSO/cSO ) kSO + ASOcSO ) 0.49 + 0.1901cSO qG/cG ) kG + BGSOcSO ) 0.4281 + 0.2636cSO qSU/cSU ) kSU + BSUSOcSO ) 0.3588 + 0.1586cSO nSOSU qSO/cSO ) kSO + BSOSUcSU 0.6893 ) 0.49 + 0.3363cSU nGSU qG/cG ) kG + BGSUcSU 0.6848 ) 0.4281 + 0.2189cSU mSU qSU/cSU ) kSU + ASUcSU 0.9105 ) 0.3588 + 0.116cSU qSO/cSO ) kSO + BSOGcnGSOG ) 0.49 + 0.275c0.724 G G qG/cG ) kG + AGcm G ) 0.4281 + 0.123c0.643 G qSU/cSU ) kSU + BSUGcnGSUG ) 0.3588 + 0.2171c0.7272 G
correlation coefficient 0.9751 0.9724 0.9698 0.9987 0.9980 0.9809 0.9940 0.9958 0.9969
Table 3 summarizes the equilibrium parameters for each solute. Using the equilibrium parameters derived from the binary data, the retention times for glucose, sucrose, and sorbitol in an aqueous quaternary solution containing 0.16 g/cm3 glucose, 0.11 g/cm3 sucrose, and 0.06 g/cm3 sorbitol were determined by simulation of the pulse response in a fixed bed. The partial differential equations in the model equations were solved by the software package PDECOL (Lawrence Livermore Laboratory, U.S.A.) which uses finite element collocation methods to the solution of partial differential equations. These values are compared in Table 4 with the experimentally measured retention times for this solution. The ternary equilibrium relations derived from the binary data were then used to determine the retention times for glucose and sucrose in an aqueous ternary solution containing 0.11 g/cm3 glucose and 0.13 g/cm3 sucrose using eqs 17-24. These values are compared in Table 5 with the experimentally measured retention times for this solution. The same procedure was used for the other constituent ternaries, namely the glucose-
a
retention time,a s theory exptl
injected component
II
glucose sucrose
1285 1193
1282 1167
III
glucose sorbitol
1239 1311
1233 1296
IV
sucrose sorbitol
1175 1330
1169 1295
Eluent flow rate, 0.7 cm3/min.
sorbitol-water system (system III) and the sucrosesorbitol-water system (system IV). In the case of the quaternary system and all the three cases of the ternary system, there is good agreement between the predicted and measured retention times confirming that the equilibrium data for the quaternary system and the ternary system conform to the predictions derived from the binary data. The two major multicomponent adsorption theories that predict adsorption equilibria for liquid systems, namely, the potential theory and the IAS theory, were used to predict multicomponent solid loadings from the pure component adsorption equilibrium relations. The IAS theory has been used for predicting multicomponent liquid adsorption equilibrium (Jossens et al. 1978; Tien, 1986). The method of incorporating potential theory into multicomponent liquid adsorption calculations has been published (Moon and Tien, 1988). Both theories predict multicomponent solid loadings from the singlecomponent adsorption equilibrium data. The accuracy of the predictions for a given system over the entire concentration range can be expressed by the mean deviation defined as
∑S |qi
σi ) [
pre
- qical|S/(qical)S]/N
(30)
where the subscripts pre and cal denote the predicted and calculated values (from the equilibrium relations) respectively, N indicates the data points, and the subscript S denotes the Sth data point. The results of comparing predictions using IAS theory and the calculations using multicomponent equilibrium relations obtained are summarized in Table 7. Good agreement was found between the predicted and calculated solid loadings as is suggested by the average deviations. To predict adsorption equilibria using potential theory, the single-component adsorption isotherm data of glucose, sucrose, and sorbitol were fitted by an equation of the form
V°ai ) (Vi)0 exp(-biEi)
(31)
412 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 Table 6. Properties of Adsorbates and Single-Component Isotherm Parameters of Individual Adsorbates property/parameter
glucose
sucrose
sorbitol
molecular weight bulk solid density (20 °C), g/cm3 solubility in water (15 °C), g/100 g melting point, °C (Vi)0 (eq 31) bi × 102 (eq 31) Mi
180.16 1.54 154 150 0.198 0.35 1.0
342.30 1.5805 67.9 178 0.013 0.1256
182.16 1.489 235 110 0.3544 0.3617 1.2
Table 7. Comparison of Mean Deviations system
species
IAS theory
glucose(1)-sucrose(2)sorbitol(3)
1 2 3 1 2 1 2 1 2
0.126 0.092 0.183 0.095 0.089 0.041 0.085 0.004 0.121
glucose(1)-sucrose(2) glucose(1)-sorbitol(2) sucrose(1)-sorbitol(2) a
potential theorya M-1 M-2
0.202 0.306
0.027 0.172 Figure 7. Coalescence of single-component adsorption isotherm data of glucose and sorbitol with the use of coalescing factor.
M-1: Mi ) 1. M-2: Mi * 1.
Figure 6. Single-component adsorption isotherm data of glucose, sucrose, and sorbitol.
with
Ei )
RT csi ln V°si ci
(32)
The values of the various parameters are summarized in Table 6. The single-component isotherm data are shown in Figure 6 (in the form of V°ai vs (RT)/V°s* i ln(csi/ ci)) and Figure 7 (in the form of V°ai vs (RT)/(MiV°si) ln(csi/ ci)). Comparing these Figures 6 and 7 points to the effect of using Mi in bringing all these data together. It was found that the isotherm data for glucose and sorbitol could be coalesced quite well but the isotherm data for sucrose could not be coalesced. This could be attributed to the large difference in molecular weight of sucrose (and hence the saturated liquid molar volume, V°si) as compared with those of glucose and sorbitol. For the system glucose-sorbitol (components of comparable molecular weights), however, the predictions of the potential theory compare fairly well (Table 7) with the
adsorbed phase concentrations obtained from the equilibrium relations. This indicates that the multicomponent solid loading predictions of the potential theory are reliable only when the components constituting the multicomponent system are of comparable molecular weights. The equilibrium relations in Table 3 indicate that the distribution coefficients for the three components increase with increasing bulk phase concentration but the rate of change is larger for sorbitol than for glucose and sucrose (ASO > AG and ASU). As changes in solution density with concentration were modest, the concentration dependence of the distribution coefficient must arise from either sorbate-sorbate or sorbate-pore wall interactions (Anderson and Brannon, 1981). Although sorbitol and glucose have the same molecular diameter, sorbitol is adsorbed to a greater extent than glucose purely because of differences in sorbate-sorbent interactions. Sucrose is a larger molecule when compared with glucose and sorbitol, and so the lower adsorption capacity of sucrose could be attributed to the steric hindrance which arises when the molecular size approaches the pore size of the sorbent. Also, it can be observed from the values of Bij that the adsorptive capacity of a component is increased to a greater extent in the presence of a less strongly adsorbed component. For example, sorbitol shows an increased interaction in the presence of sucrose as compared to that in the presence of glucose. Conclusions Equilibrium relations for a quaternary system have been determined by an extension of the method suggested for a ternary system. Although chromatographic retention time measurements provide a simple and straightforward means of determining binary adsorption equilibria, this approach cannot be directly extended to multicomponent systems, since in such cases the derivatives dq*i /dci depend on the concentration changes for all components. This problem has been circumvented by assuming that the sorbate-sorbate interactions in the quaternary and the constituent ternaries are the same as in the constituent binaries. The validity of this assumption is then confirmed by comparing the
Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 413
simulated and measured retention times for the quaternary system and the predicted and measured retention times for the constituent ternary systems. The assumption is further confirmed by comparing the adsorbed phase concentrations predicted by the IAS theory and that calculated from the equilibrium relations. If there is agreement, as found in the present study, one may have some confidence in using the predictions from the binary data over a wide range of ternary and quaternary compositions. However, this approach is obviously restricted to cases in which quaternary and ternary interaction effects can be neglected. But, the higher probability of the binary interactions relative to the ternary and quaternary interactions ensures that in most practical systems this approach will be successful. The potential theory could not be used for predicting multicomponent solid loadings because of the theory’s limitation in predicting adsorbed phase concentrations for components of widely different molecular weights. Notation A ) equilibrium parameter, eqs 6, 7 b ) constant, eq 31 B ) equilibrium parameter, eqs 8-10, cm3/g c ) solute concentration in bulk phase, g/cm3 cs ) saturated adsorbate concentration in solution, g/cm3 E ) adsorption potential, defined as (RT)/(MVs) ln(cs/c) k ) equilibrium parameter, eqs 6, 7 K ) distribution coefficient L ) packed length of column, cm m ) equilibrium parameter, eqs 6, 7 M ) coalescing factor n ) equilibrium parameter, eqs 8-10 N ) number of data points, eq 30 q ) solute concentration based on particle volume, g/cm3 R ) gas law constant t ) time, min T ) absolute temperature u ) superficial fluid velocity, cm/min v ) interstitial fluid velocity, cm/min Va ) volume of adsorbed adsorbate per unit mass of adsorbent (V)0 ) constant, eq 31 Vs ) saturated liquid molar volume x ) solute mass fraction in bulk phase y ) solute mass fraction in pore Greek Letters ) voidage of packed bed µ ) first moment, min ω ) velocity of equilibrium front, cm/min Superscripts * ) equilibrium value ° ) single-component adsorption state Subscripts cal ) calculated value, eq 30 G ) glucose i ) glucose, sucrose, or sorbitol j ) glucose, sucrose, or sorbitol pre ) predicted value, eq 30 s ) solvent S ) data point, eq 30 SU ) sucrose
SO ) sorbitol
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Received for review July 19, 1996 Revised manuscript received October 29, 1996 Accepted October 29, 1996X IE9604278
Abstract published in Advance ACS Abstracts, December 15, 1996. X
