Ind. Eng. Chem. Res. 1988,27, 951-956
95 1
MATERIALS AND INTERFACES Thermodynamics of Multiorganic Solute Adsorption from Dilute Aqueous Solution. 1. The Use of an Equation of State Cheng-Sheng Lee and Georges Belfort* Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590
The thermodynamics of nonideal solutions is used for predicting multisolute adsorption using only single-solute adsorption data from dilute aqueous solutions. T h e adsorptions of two pairs of liquid solutes were studied under competition. T h e first pair contained similar solutes (4-methylphenol and 4-(n-propyl)phenol), while the second pair contained very different solutes (1-pentanol and benzene). T h e experimental results agree better with equilibrium predictions based on a two-dimensional adsorbed phase represented by a van der Waals’ equation of state than based on ideal adsorbed solution theory, especially for the weak adsorbate. It appears t h a t the ideal adsorbed solution theory is reliable for multicomponent systems where solute adsorption loading is low. When solute adsorption loading is high, the nonideality of mixing in the adsorbed phase is related t o constants in the two-dimensional equation of state t o allow for solute-solute interactions on the surface. Although activated carbon (AC) adsorption has been used widely for odor and color removal in the water industry, only recently has it been seriously considered for removal of dissolved organics from water supplies (McCreary and Snoeyink, 1977). This recent interest is caused primarily by the growing concern for potential carcinogenic,mutagenic, and teratogenic compounds found in drinking waters (Shuval, 1977). Three basic approaches to the problem of describing multicomponent adsorption are discernible in the literature: empirical, thermodynamic, and a modified form of the Polanyi potential theory. Each approach is considered below. Fritz and Schliinder (1974) proposed a general empirical equation for calculating the adsorption equilibria of organic solutes in aqueous solution. Under certain simplifying assumptions, the equation reduces to some well-known relationships as special cases. Manes and co-workers (Rosene and Manes, 1976, 1977; Manes, 1980; Greenbank and Manes, 1981) have tested the “thick compressed film theory” (or Polanyi adsorption potential theory, as it is often called) for multicomponent adsorption of organics from aqueous solution onto activated carbon. Only after introducing a fitting parameter or “scale factor” did they obtain reasonably good fits between theory and experiment (Greenbank and Manes, 1981). One of the difficulties in using this theory is obtaining an estimate of the adsorbate density. Using the surface excess approach, Minka and Myers (1973) constructed a method to predict competitive adsorption from mixtures in the liquid phase based on single-solute adsorption of the same components in the liquid and vapor phases. Both for modeling purposes and for reducing experimental work, it is desirable to predict the adsorption of mixed organic solutes in dilute aqueous solution employing only experimental data from single-solute adsorption. A significant advance in the ability to predict multicomponent adsorption equilibria occurred with the publication of Radke and Prausnitz’s work (1972). They 0888-5885/88/2627-0951$01.50/0
extended the method of Myers and Prausnitz (1965) from mixed gas to multisolute aqueous phase adsorption. The method, which assumes ideal behavior of the adsorbates on the solid surface, can be expressed in terms of a Raoult’s type law. However, predicted adsorption equilibria are not always found to be in good agreement with experimental data. Both competition for the adsorption sites of the solid surface and interaction between adsorbed molecules occur. To account for this, the ideal adsorbed solution theory can be refined by modifying Raoult’s law with a two-dimensional equation of state to specify the actual state in the adsorbed phase. In this paper we account for nonidealities by incorporating a two-dimensional analogue of the van der Waals’ equation of state for the adsorbed phase into a thermodynamic method previously proposed by Van Ness (1969). The theory is tested and compared with the ideal adsorbed solution (IAS) theory for two pairs of solutes. The first pair contained similar solutes (4-methylphenol and 4-(npropy1)phenol) so as to test the new theory under conditions where the IAS is thought to be valid. The second pair contained very different solutes (1-pentanol and benzene) so as to provide a stringent test for the new theory under difficult conditions. When only single-solute isotherm data are used, the theoretical algorithm is compared with experiments and predictions of the IAS theory. Theory Thermodynamic ThFory. It is useful to define an adsorbed phase fugacity fr according to Innes and Rowley (1942) dpia = RT d In 7;
(constant T )
(1)
and
?r
lim - = 1 rro Zi7r
(constant T )
(2)
where the adsorbed phase mole fraction is defined by 0 1988 American Chemical Society
952 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 Qi zi = = -Qi
5
(3)
QT
Qi
i=l
The adsorbed phase fugacity is two-dimensional and has the same units as the spreading pressure, ir. The spreading pressure of solute, iri, can be evaluated by using the Gibbs adsorption isotherm: C,o Qi iri
- dC:
=
(4)
Ci
(5)
and dpi = dpia
For each phase we have the expressions (constant T ) dpi = RT d In f i dp? = RT d In 7.;
(constant T )
Thus, we may write d In f i = d In f?
7r=---
a-P
Equilibrium between an adsorbate and a liquid phase requires pi = p?
Ki is a function of temperature only for a given adsorbent and substrate. It is a quantity which characterizes the specific interaction between a particular adsorbed species and a particular substrate. Equation of State. Several two-dimensional equations have been proposed for pure components in gas-phase adsorption; one that has received particular attention is the two-dimensional analogue of van der Waals' equation (Ross and Olivier, 1964): RT a
(constant T )
Integration at constant temperature from an equilibrium state of single solute, i, at very low spreading pressure, irTi*, in the adsorbed phase and very low concentration, Ci*, in the bulk solution to the equilibrium state of interest, where i has mole fraction Ziat spreading pressure ir in the adsorbate and concentration Ci in the bulk solution, gives
a2
where CY and P represent the interaction between adsorbed species and the excluded surface area occupied by adsorbate, respectively. For a binary-solute mixture, we assume that the constants a and are related to those of the pure components by a pair of "mixing rules"-i.e., equations expressing CY and (3 as functions of composition. One pair of rules which has proved useful for a binary system is (Van Ness, 1969) (11) P = ZlPl + Z2P2 and where cyl2 is a constant characteristic of the 1-2 interaction. Following Hoory and Prausnitz (1967),we assume that a12 =
(Ly1a2)1/2.
The calculation of the fugacity of a mixed adsorbate at fixed temperature and composition using the two-dimensional analogue of van der Waals' equation was derived by Van Ness (1969): ZlRT P1 2 In fla = In -+ -- -(cY~ZI + a12Z2) (13a) a-/3 a-P aRT
,.
or
Assuming the bulk solution is ideal, then
Eliminating
fla
and f 2 a with eq 9 gives
and
-
(7)
If we let iri* and Ci*approach zero, then fr(iri*) iri* and the relation between fugacities a t equilibrium becomes
ji"(*)
= (iri*/Ci*)Ci
(8)
A t very low spreading pressures, the ideal gas analogue equation becomes valid. Thus, ri*ai* = R T , or alternatively iri*A = Qi*RT or -Qi* =-
~
i
*
A RT where a and A are molar surface area and surface area per gram of adsorbent, respectively. The analogue Henry's constant, K,, is given by
Thus, iri*/(RT)= KiCi* and eq 8 now becomes f r ( n )= KiRTCi
(9)
In C2 = In
2 2
K2(a-P)
+--0 2 a-P
--(a2Z2 2
+ Ql2Z1)
aRT
(141) If values are assigned to the constants al, a2,a12,P1, p2, K1,and K2 from single-solute isotherms, these equations can be solved for Z1, Z2, and a for any values of C1 and C2 in a binary mixture.
Experimental Section Adsorbent. Filtrasorb 400 granular activated carbon (Calgon Corp., Pittsburgh, PA) was ground in a jar mill so as to pass through U.S. sieve Series 200 (0.074 mm) and to be retained by U.S. sieve Series 400 (0.037 mm). The carbon was cleaned with successive washing and supernatant removal. The carbon was then analyzed in a surface area-pore volume analyzer (Model 2100E, Micromeritics Instruments Corp., Norcross, GA), and the data were fit to a BET model. The BET surface area, using N2gas, was 1031 m2/g, the pore volume was 0.95 cm3/g, the equivalent particle diameter was 37-74 pm, and the bulk density was 0.478 g/cm3. The carbon was dried in a vacuum oven (Isotemp 281, Fisher Scientific, Pittsburgh, PA) at 100 "C and at a pressure of P = 25 mmHg for a 5-day period. Any ex-
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 953 traneous adsorbed organics were volatilized by increasing the temperature to 120 "C for 2 day until the weight of the carbon remained constant. The dried carbon was stored in a desiccator until use. Adsorbates. The alkylphenols, 1-pentanol, and benzene were purchased from Aldrich Co. (Milwaukee, WI), graded 99% pure. Stock solutions were made with 0.01 M phosphate buffer and stored in amber glass bottles. In all cases, the water was prepared by passing tap water through an ion exchanger, carbon filters, and a 10000 molecular weight cut-off polysulfone hollow fiber ultrafilter. Adsorption Isotherm Procedure. Several innovations have been introduced into the adsorption isotherm procedures in an attempt to improve reproducibility and accuracy and reduce solute losses. The temperature was kept constant a t 20 f 0.5 "C during this study. The pH was also buffered a t 7.0 f 0.1 with standard 0.01 M phosphate buffer. To minimize extraneous solute loss and maximize solute/sorbent contact, the following steps and precautions were taken: (a) Adsorption was conducted in completely filled and capped 38mL stainless steel centrifuge tubes (Part 301112, Beckman, Fullerton, CA). The initial concentrations of organic solutes were fixed with variable carbon mass from 0 to 285 mg/L. Thus,there was no headspace for possible solute loss to the gas phase, and the solutions were only exposed to stainless steel surfaces for both minimum adsorption loss and to reduce the possibility of introducing extraneous organics. (b) The completely filled tubes were rotated end-overend 360" a t 2 rpm in a constant-temperature bath (20 f 0.5 "C). This allowed the carbon to fall the length of the tube twice per rotation, resulting in good mixing and increased effective external film mass-transfer coefficient. The time to reach pseudoequilibrium (i.e., >95% of the 14th day saturation value, Q") varied from 2 h to several hours depending on the kinetics of the particular solute and rotation rate. The mixing period employed in this study was 24 h. (c) Once pseudoequilibrium was attained, the tubes were placed directly into an ultracentrifuge (Model L8-55, Beckman, Fullerton, CA) and the carbon was spun down a t 20000 rpm for 20 min. (d) The residual solute concentration was measured from aliquots of the supernatant by UV spectroscopy (Model 552, Perkin-Elmer, Norwalk, CT) or TOC (Model 915A, Beckman, Fullerton, CAI. UV spectroscopy was used for measuring the alkylphenol and benzene concentrations. An automatic sipping device run by a positive displacement pump was used to measure the solution solute concentration directly from the centrifuge tube. The maximum adsorbance wavelength was determined for each alkylphenol and benzene, and a corresponding calibration curve of adsorbance versus concentration was prepared. To identify individual solutes from a mixture of alkylphenols, the solutes were extracted with methylene chloride, separated in the gas chromatograph (Signa 4, Perkin-Elmer, Norwalk, CT) and quantified in a mass spectrometer (Model 1020, Finnigan, San Jose, CA). The concentration of 1-pentanol in a binary aqueous mixture with benzene was obtained by subtracting the benzene concentration as measured by UV spectroscopy from the total solute concentration as measured by total organic carbon (TOC).
Rssults and Discussion Characteristic Constants in van der Waals' Equation. The relationship between Q? and C r for the two
zol 16
-, m
;)I
24
,If----.. : I
0
20
40
60
80
IW
140
* I
180
220
I
n-pmtono1 * I
I
260
300
340
~o~~'immo~e/~iter)
Figure 1. Single-solute adsorption isotherm for adsorbates (a, top) alkylphenols and (b, bottom) benzene and I-pentanol. Table I. Characteristic Constants for Adsorbates adsorbate 4-(n-propyl)phenol 4-methylphenol benzene I-pentanol
a,J.m2/mol2
1.05 X lo9 2.00 X log 5.92 X lo8 6.74 X lo8
p , m2/mol 5.44 X lo5 1.03 X IO6 3.75 X lo6 1.56 X IO6
Ki, m 1.07 X 6.94 X IO-' 2.79 X 8.24 X IO*
alkylphenols and the benzene and 1-pentanol are plotted as single-solute isotherms in Figure 1. The single-solute isotherms are fitted well by a Langmuir model (Langmuir, 1916, 1918). The analogue Henry's constants, Kj, are calculated from the initial slopes of Langmuir isotherms and listed in Table I. Since experimental single-solute adsorption isotherm data fitted the Langmuir model well, this expression was used in integrating eq 4 to give the spreading pressure, T , for each single solute a t different values of molar surface area, a. The data sets of T vs a are fitted into a two-dimensional analogue of van der Waals' equation to get the characteristic constants for each solute by curve fitting. The results are listed and plotted in Table I and Figure 2, respectively. Van der Waals' equation (eq 10) clearly fits the data very well. From Table I it appears that 4methylphenol, the smaller of the two molecules, exhibits a larger excluded surface area (0 value) than 4-(npropy1)phenol in the adsorbed phase. A possible reason for this is that the former molecule is more polar than the latter and can thus hydrate more effectively on the surface. This could result in a larger effective excluded area on the sorbent surface. For an ideal solution, the osmotic pressure II = CRT, where C is the solute concentration. In a two-dimensional analogue of osmotic pressure, the spreading pressure ( n ) and molar surface area (a) can be used in terms of an analogue compressibility factor ( a a / R T )as an index to account for nonideal behavior in the adsorbed phase (Van
+
954 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988
1 4 - n mefhylphenol
'0
04
08
16
I 2
20
2 4
0; lO3(mole/g)
-
'
$-
+
0
s \
e 4 -
lo1
\
7 \
\
t
7 A,
\
-.--_
'\
0
3,
'm-
\
e-
-m---
m--------i. i
E 04
,
-
08
I 2
I6
20
Q:.l0~(mole/q I
Figure 3. Analogue compressibility factor as a function of adsorbed (a, top) alkylphenola and (b, bottom) benzene and 1-pentanol.
Ness, 1969; Ross and Olivier, 1964). Thus, if values of ?ra/RT are larger than 1, nonideal behavior is suspected. To test if the IAS theory is suitable for these systems, na/RT is plotted against surface loading (8:) and shown in Figure 3. The assumption of ideal behavior of the adsorbates on the solid surface is only suitable for very low surface loadings for all four single solutes. Hence, for an adequate description of the binary alkylphenol and 1pentanol-benzene adsorption, a theory should account for nonideal behavior throughout most of the surface loadings. Prediction of Adsorption Equilibria. Although 19 experimental mixture points were measured over a 3decade range in concentration and a 2-decade range in loading, only one point per calculated line was shown and compared in Radke and Prausnitz's work (1972). The reason for this is that they could not control the final solute concentrations and thus compare the IAS theory with experiment. We have devised an iterative scheme using material balances to obtain the final equilibrium concentrations for testing these (IAS and our) theories. See Figure 4. In Figures 5 and 6, the experimental binary adsorption data and the predicted results for two different initial concentrations of the alkylphenols and 1-pentanol-benzene in the mixtures are shown. Solid lines and dotted lines are calculated by using the IAS theory and the theory based on the equation of state, respectively. Agreement between predicted and experimental results is good for the latter theory using van der Waals' equation, especially in the higher surface loading region compared to that predicted by the IAS theory. It appears that, as Radke and Prausnitz (1972) have pointed out, the U S theory is most suitable for quantitative prediction of binary adsorption
I
L
I
Figure 4. Iterative scheme for testing theory.
of small organic molecules with similar adsorption characteristics. The large deviations between experimental and predicted results by the IAS theory for the weak adsorbate
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 955
1 0 1 -
02
0
02
04
06
c
02
0
02
04
06
08 1'0 2 3 ~~~(mmo~e/~iteri
08
1
'
,
2
4
3
4
5
5
6
c
6
c 10'(mmoie/liter) Figure 5. Adsorption from aqueous multisolute system (4-methylphenol and C(n-propy1)phenol) at (a, top) low and (b, bottom) higher initial concentrations.
(4methylphenol or 1-pentanol) can be explained by eq 20, Ci(a,T,zi)= Z j C r ( r , T ) ,in Radke and Prausnitz's paper (1972). Since propylphenol or benzene is preferentially adsorbed, its mole fraction in the surface phase is close to unity. Thus, the mole fraction of the second component, methylphenol or 1-pentanol, is very small. Since the loading of methylphenol or 1-pentanol is calculated from C r = Ci/Zi, a very small value of Ziwill give a very large value in C?, well beyond the limits of its isotherm. No matter what functional form is used, the extrapolation of an isotherm over a long distance could introduce a significant error. In Figures 5 and 6, it can be seen that the predictions using the IAS theory fall above or below the experimental observations. These deviations could also provide a measure of the nonideality of the binary adsorption system. The significant differences between the alkylphenol system and the 1-pentanol-benzene system are the lower loadings for the former pair and the large differential loading for the latter pair. Interactions between adsorbed phase species rise with increasing loading and dissimilar solutes.
Conclusion An iterative scheme, combined with material balances, is used to compare the IAS theory with a nonideal theory incorporating van der Waals' equation for two dimensions to predict multisolute adsorption. Using a two-dimensional analogue of van der Waals' equation to specify the interaction of the components in the adsorbed phase, the experimental and predicted data were found to be in good agreement. This agreement is much better than that obtained for the IAS theory espe-
t0,mmoie IMe?,
Figure 6. Adsorption from aqueous multisolute system (1-pentanol and benzene) at (a, top) low and (b, bottom) higher initial concentrations.
cially for the higher surface loadings and for the dissimilar solutes in multisolute mixtures. The need for a two-dimensional analogue of van der Waals' equation of state in the adsorbed phase has to do with nonideal behavior a t reasonable to high loadings. The use of an equation of state in an iterative scheme provides a convenient procedure for predicting multisolute adsorption from dilute solution. A significant advantage of using this theory is that the single-solute adsorption isotherms required for the model are applicable nearly over the whole range of surface loading.
Acknowledgment The authors thank Karen L. Woodfield for conducting the alkylphenols experiments, Gordon L. Altshuler and Roger J. Weigand for stimulating discussions, and Thomas Neubauer for doing the GC/MS analysis. The activated carbon used in this study was supplied by D. G. Chalmers, the Calgon Corp., Pittsburgh. D. A. Dobbs of the U S . Environmental Protection Agency provided funding of this work with EPA Grant CB809686-01-0.
Nomenclature A = surface area per g r a m of adsorbent, A2/g a = molar surface area of adsorbate, A2/mol C = equilibrium concentration of solute, mmol/L fi = fugacity of i as a pure adsorbate, atm f, = fugacity of a component in a mixed adsorbate, atm K = analogue Henry's constant, m m = mass of adsorbent, g Q = amount adsorbed per gram of adsorbent, mmol/g R = gas constant, 8.314 J/(mol-K) T = absolute temperature, K
Ind. Eng. Chem. Res. 1988,27, 956-963
956
V = volume of solution, L Z = adsorbed phase composition, mole fraction Greek Symbols a = constant of two-dimensional van der Waals' equation,
J-m2/mo12 /3 = constant of two-dimensional van der Waals' equation,
m2/mol chemical potential of solute i, J T = spreading pressure, J/m2 II = osmotic pressure, atm
pi =
Superscripts a = adsorbed phase * = value at very low concentration o = value at single solute ' = value at initial state Subscripts i = solute i T = total
Literature Cited Fritz, W.; Schlunder, E. U. Chem. Eng. Sci. 1974, 29, 1279. Greenbank, M.; Manes, M. J . Phys. Chem. 1981,85, 3050. Hoory, S. E.; Prausnitz, J. M. Chem. Eng. Sci. 1967, 22, 1025. Innes, W. B.; Rowley, H. H. J. Phys. Chem. 1942,46, 548. Langmuir, I. J . Am. Chem. SOC.1916, 38, 2267. Langmuir, I. J . Am. Chem. SOC.1918,40, 1361. Manes, M. Activated Carbon Adsorption of Organics from Aqueous Phase;Ann Arbor Science: Ann Arbor, MI, 1980; Vol. 1, pp 43-64. McCreary, J. J.; Snoeyink, V. L. Am. Water Works Assoc. J . 1977, 69(8), 437-444.
Minka, C.; Myers, A. L. AIChE J . 1973, 19, 453. Myers, A. L.; Prausnitz, J. M. AZChE J. 1965, 11, 121. Radke, C. J.; Prausnitz, J. M. AZChE J. 1972, 18, 761. Rosene, M. R.; Manes, M. J . Phys. Chem. 1976, 80, 953. Rosene, M. R.; Manes, M. J . Phys. Chem. 1977,81, 1646. Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience: New York 1964. Shuval, H. I. Water Renovation and Reuse; Academic: New York, 1977; pp 33-72. Van Ness, H. C. Znd. Eng. Chem. Fundam. 1969,8,464.
Received for review June 9, 1986 Revised manuscript received February 8, 1988 Accepted February 19, 1988
Registry No. C, 7440-44-0; 4-HOC6H4(CHz)zCH3, 645-56-7; 4-HOC6HhCH3, 106-44-5;C6H6, 71-43-2; HO(CH.JiCH3, 71-41-0.
PROCESS ENGINEERING AND DESIGN Predictive Controller Design for Single-Input/Single-Output(SISO) Systems Paul R. Maurath,+ Duncan A. Mellichamp, and Dale E. Seborg* Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, Santa Barbara, California 93106
This gaper presents a fundamental analysis of the stability of single-input/single-output(SISO), closed-loop systems with predictive controllers. T h e analysis can be used t o calculate allowable modeling errors for a given system and controller. Design parameter selection guidelines for predictive controllers in SISO systems have been developed by considering performance, robustness, and ease of tuning. T h e performance and robustness of the resulting controllers are demonstrated in four numerical examples and compared to controllers designed using other parameter choices.
In the late 19705, two new control strategies emerged in the control literature which approached the problem of process control somewhat differently than before. Model Algorithmic Control (MAC) (Richalet et al., 1978) and Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980) both (i) utilize a discrete convolution-type model to represent the process and (ii) control the process by optimizing the process output(s) over some finite future time interval. These techniques share several characteristics in addition to those mentioned here and constitute a new category of process control referred to as "predictive control". A number of papers have appeared on these and related predictive control algorithms. A complete review is given in Maurath (1985). Predictive controllers have also been examined from the framework of Internal Model Control (Garcia and Morari, +Presentaddress: Procter & Gamble Company, Cincinnati, OH 45201.
1982). Their paper approaches the IMC controller based on impulse response models from a different perspective than DMC, particularly for nonminimum-phase processes. They also made observations on the effects of controller design parameters in the IMC controller which are closely related to the parameters of the DMC-type controller presented here. The parameter effects noted in this paper are consistent with their earlier results. The goal of this paper is to present a series of guidelines which will lead to reasonable starting values for the design parameters in a predictive controller. The present paper is restricted to unconstrained single-inputlsingle-output systems. Maurath et al. (1985) present a more powerful predictive controller design technique for multiinput/ multioutput processes. A brief introduction to predictive control will be presented followed by a discussion of the effects of the controller design parameters on closed-loop performance and robustness. A stability analysis is presented which con-
0 1988 American Chemical Society
