Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
32
U = superficial velocity Uj = superficial velocity a t incipience V = voltage t
Literature Cited
= voidage
ep = packed bed void fraction c g-~ = permittivity of free mace ( e n = 8.854 X
F/m)
= particle mass density $ = angle between the average field and the normal to the
pp
interparticle center fl = angle between vertical and the normal to the interparticle
contact = effective shear stress based on unfluidized bed height (Teff = fT/2lpd) rW = shear stress a t the wall (or electrode) Teff
Colver, G. M., "Bubble Control in Gas Fluidized Beds with Applied Electric Fields," Paper 76-HT-69. ASME-AIChE Heat Transfer Conference, St. Louis, Mo.. 1976. Dietz, P. W., Ph.D. Thesis, M.I.T., Cambridge, Mass., 1976. Dietz, P. W., Melcher, J. R., AlChE Symposium Series book on air, in publication, 1978. Johnson, T. W., Melcher, J. R., Ind. Eng. Chem. Fundam., 14, 146 (1975). Katz, H., U.S. Patent 3 304 249 (1967). Katz, H., Sears, J. T., Can. J. Chem. Eng., 47, 50 (1969). McLean, K. J., Department of Electrical Engineering, Wollongong University, Australia, private communication, 1975. Rowe, P. N.. Trans. Inst. Chem. Eng., 39, 175 (1961). Zahedi, K., Melcher, J. R., APCA J., 26 (4), 345 (1976).
Receiued f o r reuiew April 28, 1977 Accepted September 12, 1977
Continuous Flow Equilibrium Staged Model for Cycling Zone Adsorption William C. Nelson, David F. Silarski, and Phillip C. Wankat" School of Chemical Engineering, Purdue University, West Lafayette, lndiana 4 7907
A new theoretical model for travelling wave cycling zone adsorption is developed based on equations derived from the continuous flow, equilibrium staged model of chromatography. The model includes zone spreading effects and can use Langmuir, Freundlich, or linear adsorption equilibrium relationships. Inherent in the model is the ability to use temperature, concentration, or pH as cyclic thermodynamic variables and these cyclic variables can be input to the model as either square or ramp waves in the traveling wave mode. The equations are solved numerically on a computer utilizing a modified fourth order Runge-Kutta integration scheme. The theoretical predictions are compared with experimental results for single and multicomponent separations. Good agreement is observed between theory and experiments when accurate equilibrium data are available. When approximate equilibrium data in the form of linear isotherms or superimposed isotherms for multicomponent systems are used, the model gives qualitative but not quantitative agreement with the experimental results. Columns operating as cycliing zone adsorbers have considerably fewer stages than when used in elution chromatography.
Considerable interest and research have been focused on cyclic separation processes which allow for continuous or semicontinuous feed to chromatographic or adsorption systems. One such process, cycling zone adsorption, separates a continuous feed by periodically changing a thermodynamic variable which affects the distribution of solutes between the stationary and mobile phases. The larger throughput which can be achieved with cycling zone adsorption makes it an alternative for preparative chromatography and of potential industrial value. Pigford et al. (1969) and Baker and Pigford (1971) invented cycling zone adsorption as a method for removing all of the solutes in a feed solution. Using temperature as the thermodynamic variable, separations were achieved in either the direct mode (bed is externally heated and cooled) or the traveling wave mode (entering fluid is heated or cooled). Wankat (1973a, 1974) extended cycling zone separations to counter-current distribution using changes in temperature to force a separation. The technique was extended to lowpressure chromatography, utilizing pH as the thermodynamic variable, by Busbice and Wankat (1975) and to high-pressure liquid chromatography (HPLC), using the concentration of an added chemical as the thermodynamic variable, by Nelson and Wankat (1976).Other experimental results are discussed in a recent review by Wankat et al. (1975). 0019-7874/78/1017-0032$01.00/0
Theoretical models for the cycling zone techniques generally followed the progression of experimental work. Baker and Pigford (1971) developed a local equilibrium model based on assumptions of local equilibrium between the solid and fluid phases, no axial dispersion, and no heat of adsorption. Another local equilibrium model, developed independently by Gupta and Sweed (1971), utilized the method of characteristics to predict separations in the direct mode of operation. Wankat (1973a, 1974) and Busbice and Wankat (1975) modified the theory for counter-current distribution and developed an equilibrium staged model based on discrete transfer and equilibrium steps. In this paper a new theoretical model, based on concepts derived from the continuous flow, equilibrium staged model of chromatography, is developed to explain the separations obtained in cycling zone adsorption. The concept of a continuous flow staged equilibrium model is not new; Hung and Lee (1973) and Wankat (1973b) developed similar models for direct mode parametric pumping and the model is related to the Stop-Go model for parametric pumping (Sweed and Wilhelm, 1969). Two reasons exist for building such a model. First, the earlier models have either neglected potentially important effects or been used to explain systems for which they are not inherently suited. The local equilibrium model has neglected axial dispersion terms and the rate of adsorption
0 1978 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 17, No. 1,
1978 33
a... -...I
Faid
Figure 1. Schematic of staged cycling zone system.
t
A
Theory Now consider again the staged system shown schematically in Figure 1. For a continuous flow, equilibrium-stage system a mass balance can be written to calculate the amount of solute on stage i during cycle j . If i + 1,the mass balance is LX,-i,]dt input
- LXl,Jdt= S(dY,,J)+ H(dX,,]) output
accumulation
(1)
Ifi = 1 L(CFeed)dt - LX1,Jdt = S(dY1,J) I TIME
h
T
H(dXI,j)
(2)
where H is the mass hold-up of moving phase on each stage, S is the mass hold-up of stationary phase on each stage, L is the mass flow rate of the moving phase, and CFeed is the mass ratio of solute in the feed. X,,J and YL,]are the mass fractions of solute in the mobile phase and stationary phase, respectively. The carrier fluid is not adsorbed. The solute concentration in the stationary phase is assumed to be a function of the cyclic thermodynamic variable and of the mobile phase solute concentration only. The thermodynamic variable may be temperature, pH, or the concentration of some other chemical species; however, in the following development, temperature will be used as the cyclic variable. This dependence can be written as
I- CyCLE1
n nB ~
-
TIME
Figure 2. Input temperature waves for travelling wave mode: a, square waves; b, step inputs for multicomponent separations.
and desorption, and thus cannot account for zone spreading effects. The counter-current distribution model is based on discrete transfer steps and thus cannot adequately model continuous flow in columns where the number of stages is small. Secondly, a more general model is needed with the flexibility to handle common types of equilibrium isotherms and to handle different thermodynamic variable wave functions. This paper discusses both the formulation of the continuous flow staged model and the comparison of the model simulations with published experimental results. Before the details of the theory are presented, it is important to obtain an understanding of cycling zone adsorption. Consider a column packed with a stationary solid adsorbent that retains solute at low temperatures and releases the solute a t high temperatures. Figure 1 shows a schematic of such a staged system. T h e solution to be treated is fed continuously to the column using equipment which causes its temperature to vary as a square wave starting with a cold temperature for half the cycle and then switching to a hot temperature for the remainder of the cycle as shown in Figure 2a. For multicomponent separations a step input as shown in Figure 2b is used. For the first half cycle shown in Figure Za, mobile phase enters a t a low temperature and the solute is retained; that is, its velocity through the column is considerably slower than that of the mobile phase. As a result, the exiting fluid will initially have a very low solute concentration. Of course, with long cycle times the exiting solute concentration would rise and finally correspond to the feed concentration. For the last half cycle, mobile phase enters a t a high temperature. As the fluid flows through the column the adsorbent now releases the solute to the fluid since the temperature is high. Thus, the exiting fluid will initially have a high solute concentration. Again, with long enough cycle times the exiting solute concentration would fall and eventually correspond to the feed concentration. When repeated continuously, the above procedure outlines a very simplified example of a cyclic separation. Since the inlet fluid temperature is cycled, the process is said to be operating in the traveling wave mode with temperature as the thermodynamic variable forcing the separation. When the entire column is externally heated or cooled, the operation is in the direct or standing wave mode.
yl,J
=
(3)
f(Xl,],TL,J)
where T,,] is the temperature of the ith stage during cycle j . Using the chain rule of calculus, the time dependence of the solute in the stationary phase can be shown to be
dY,,- dy! 1 dt
dT,
aTl,J
at
I
dYL
J
dX,, at
J
(4)
If the equilibrium type is known, such as linear, Freundlich, or Langmuir, then the terms dYL,J/dT,,land dYl,J/dX,,Jcan be evaluated easily. Equation 4 can now be substituted into eq 1. After rearranging, the result for i # 1 is
dt
For the travelling wave mode the temperature varies throughout the column and must be obtained from an energy balance. Before eq 5 can be integrated, an analytical expression for the time dependence of the thermodynamic variable is needed. For temperature as the cyclic variable, an energy balance on stage i can be developed. After rearranging, the needed expression for dT,,jldt can be shown to be
where C,,, C,,, and C ,, are the heat capacities of the mobile phase, stationary phase, and column walls, respectively, and M is the mass per stage of the column walls. Constant heat capacities have been assumed, the column is adiabatic, and solute contributions to the energy balance have been ignored. HC,, MC,,), has special sigThe quantity, HC,,/(SC,, nificance. This term, which shall be denoted as A , is indicative of the relative velocity of the thermal wave compared to that of the mobile phase. For example, if A approaches 1.0 the terms, SC,, and MC,,, must approach zero and the thermal wave moves a t the velocity of the mobile phase; that is, there is no interaction and no heat is absorbed in the column. If A approaches zero, HC,, must approach zero and the thermal wave does not move; that is, all heat is absorbed in the column. This definition of wave velocity, A , is similar to that of Wankat
+
+
34
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
The analytic evaluation of dTi,;/dt can now be found if the desired thermodynamic input wave form is specified. For the square wave, eq 8 can be differentiated, and the needed expression for the time derivative of temperature is
+ ci-2,;e-q27 2! TIME
(1974a). Baker and Pigford (1971) also defined a wave velocity, u . In terms of eq 6 their concept of velocity would be repre-
sented as u = LAIH. The definition of wave velocity, A, allows eq 6 t o be generalized for any thermodynamic variable, not just temperature. The wave velocity can be found either empirically by experimental measurement or theoretically as in the case of temperature. Thus, eq 6 can be rewritten as
(7) where Ti,jcan represent any cyclic variable but, for simplicity in this derivation, it still refers to temperature. Consider again the cyclic variation of temperature shown in Figure 2a. The thermodynamic variable varies as a square wave starting with a cold temperature, T,, for half the cycle and then switching t o a hot temperature, TH, for the remainder of the cycle. For this type of cyclic variation, eq 7 can be integrated analytically in a stagewise manner. The general solution for any half cycle can be given as
=
(h)
At;
(0 I t I cycle time)
and the constants of integration are
C!,j = TI,]1t=0 -
TFeed
(9)
However, other wave forms are also possible. For the step input shown in Figure 2b, eq 8 and 9 are valid but they are used over each portion of the cycle. Figure 3 shows an example of a ramp or sawtooth temperature wave that starts with a cold until temperature, Tc, and increases a t a constant rate, SL, reaching a maximum temperature, TH.Mathematically, this can be written as TFeed
= SLt
TC
(10)
Equation 7 can be integrated analytically using the ramp input wave. The general solution for any complete cycle can be expressed as
T,,] = Tc + SLt
"-"]
(13)
In a similar manner, differentiation of eq 11 yields a n expression for the time derivative of temperature for a ramp wave input.
Figure 3. Ramp (sawtooth) input temperature wave.
T
+...
, Cl,,e-r[(i - 1 ) ~ l - z( i - I)!
I
I
where
- 72)
- ( S ~ / a )+i C r l , l e - T+ C r I - - ~ , , r e - '
where T = a t , 0 I t I 1 cycle time, a = ( L / H ) A ,and the constants of integration are
dt
+
I
= SL (L/H)A -C'l,Je-i
+ Crl-l,Je-T(l-
7)
Reference t o eq 5 shows that for i = 1 the mass balance can now be solved. The first-order differential equation can be integrated with X I , ]as the dependent variable and time t as the independent variable. In the direct mode of operation the temperature is assumed to be fixed during each portion of the cycle and the energy balance is not required. Then eq 5 can be solved for each portion of the cycle. During each portion of the cycle dT1,]ldt = 0 and the energy balance is automatically satisfied. At the end of each portion of the cycle the solute is redistributed between the mobile and stationary phases when the temperature is changed. This redistribution of solute can be obtained from a mass balance on each stage to find a n initial concentration for the next half cycle. The direct mode analysis follows the staged parametric pumping analysis (Wankat, 1973b) except that flow reversal is not employed. Since the focus of this work was on the travelling wave mode of operation, the direct mode development will not be considered further.
Solution Procedure The system of equations presented in the preceding section was solved in a straightforward manner using a stagewise numerical integration technique. In terms of precedence ordering, for any time, t , equations such as eq 8 or 11 could be solved for the thermodynamic variable. With TI,]known, equations for dYl,/dTl,l and dY,,,/dX,,, were evaluated and substituted into eq 5. As eq 5 shows, the solution for each stage in the column is dependent on information from just the previous stage, and for the first stage only the feed concentration and initial conditions are necessary. Thus, a stagewise solution of eq 5 was possible. The outlet solute concentration from stage 1 allowed calculation of the outlet concentration from stage 2 and so on until the last stage was encountered. Lavie and Reilly (1972) showed theoretically that for linear isotherms the cycling zone system will eventually reach a limiting repeat state, where each cycle, j , is an exact repeat of the cycle before it. Repeating states have also been obtained experimentally in all column experiments. Thus the assumption was made that limiting states would be obtained and the equations were solved for this repeating state. For each stage, starting with stage 1,a numerical integration was performed until the repeating state was obtained. A mass balance on the stage was used as the criteria for convergence. After the limiting state was found for stage 1, these values were then
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
35
g5r---l 90
COLUMN VOLUMES
Figure 5. Comparison of theory and experiment for outlet pH waves in removal of fructose from water. Data from Busbice and Wankat (1975): 3.9 column void volumes/half cycle, T = 25 "C, pH 8.5; (0.05 M morpholine buffer) or p H 5.0 (0.05 M acetic acid buffer), 4 stages; A = 0.528, lag = 1.0 column volumes; -, computer simulation outlet pH; 0-0, inlet pH; A-A, experimental data for outlet pH.
J i 1
4
b
Ib
I;
o;
T I M E (M1N.i
b
o;
35'
Figure 4. Comparison of theory and experiment for separation of acetic acid from water on activated carbon in a thermal traveling wave system. Cycling zone and equilibrium data from Baker and Pigford (1971); a, outlet temperature wave, A = 0.757; b, outlet acetic acid concentration: TC = 7.45 "C. T H = 55.7 "C, cycle = 33.3 min; feed concn = 0.0610 N, 7 stages, lag time = 1min; -, computer simulation; 0-0, experimental data.
used in a new series of numerical integrations for stage 2 and so on. Note that the repeating state is time dependent during a cycle, but the entire startup solution does not have to be obtained. The numerical integrations were carried out on a computer using a modified fourth order Runge-Kutta scheme (Lapidus, 1962).Using this scheme, all stages, except the first, converged or reached a limiting state in three cycles. Depending on the tolerance set in the mass balance convergence criteria, the first stage required between four and seven cycles to converge, the extra cycles being attributable to the initial conditions used to start the program. Computer program listings are given by Nelson (1975) and Silarski (1976).
Comparison of Theoretical Predictions with Published Experimental Results The above theoretical model was used to simulate the experimental separations obtained by Baker and Pigford (1971), Busbice and Wankat (19751, Dore and Wankat (1976), and Nelson and Wankat (1976). The results of these simulations are discussed in the following paragraphs. Baker and Pigford (1971) separated acetic acid from water using activated carbon in both thermal traveling wave and direct mode experimental systems. The equilibrium relationship was of the Freundlich type and could be represented by the equation Yi,i = a ( X L ,) jb
(15)
where a = f(T,,j)and b = f(TL,]). The temperature of the inlet feed stream was varied as a square wave from a hot temperature of 55.7 "C to a cold temperature of 7.45 "C. Using eq 6 and data from Baker (1969),a theoretical wave velocity, A , of 0.757 was calculated. Figure 4a shows a comparison of the experi-
mental temperature wave leaving the column (Baker and Pigford, 1971) and the computer model prediction. The two curves are quite similar and, due to the relatively high wave velocity, the exit waves are distinct and close to the original inlet square waves. Data are shown for only the cold to hot transition since this was the only data reported by Baker and Pigford (1971). Figure 4b shows the comparison of the experimental solute concentration wave exiting the column and its computer prediction. Baker and Pigford did not report the number of theoretical stages in their column; thus, different values for the number of stages had to be tried. The results for seven stages were found to be closest to the experimental curves represented in Figures 4a and 4b. The results of the theoretical model agree very well with the experimental curves; however, the theoretical curves were plotted with a slight lag of 1min. This lack of synchronization between the experimental and theoretical curves occurred in the other comparisons as well and can be attributed to the dead volume between the end of the column and the point of solute concentration measurement. Complete input data for the simulation are given by Nelson (1975). Busbice and Wankat (1975) studied the separation of glucose and fructose from water using a dihydroxyborylphenyl succinamyl derivative of aminoethyl cellulose (DBAE cellulose) as an adsorbent. The travelling wave mode system used pH as the thermodynamic variable with the values pH 8.5 and pH 5.0 as the upper and lower bounds of the inlet square wave. Since their experimental system was buffered and complex interactions including neutralization reactions and ion exchange occurred within the system, the [H+] or [OH-] could not be used directly as the cyclic variable without including these complex interactions in the theoretical analysis. Instead, a semiempirical approach was used and p H was cycled in the calculations using a value of 0.528 for the wave velocity A calculated from experimental results without the sugars present. This value of A was then used for the simulations of the separations and now includes the effect of the buffers and ion exchange. Figure 5 shows a comparison between the experimental outlet pH wave (Busbice and Wankat, 1975) and the theoretical prediction for the same wave. Note that the outlet waves are similar in shape and more rounded than the inlet wave due to the low value of wave velocity. Four stages were used since this gave the best fit with outlet concentrations. Complete input data for the simulation are given by Nelson (1975). The equilibrium isotherms employed were of the Langmuir type and were written in the form
36
Ind. Eng. Chern. Fundarn., Vol. 17, No. 1, 1978 3.5, -
,
,
,
35
g 25
x\ ,
I
,
,
I
COLUMN VOLUMES TIME
-
0 25
1
0,251
0
(MIN)
Figure 7. Comparison of theory and experiment for removal of glycyl-L-phenyl-alamine from water. Experimental data is from Nelson and Wankat (1976):low dichloroacetic acid concentration = 0.0015%, high acid concentration 0.03%, cycle = 10 midhalf cycle, feed concentration 40 mg/L, lag = 3 min, 8 stages, A = 0.9. Linear distribution coefficient for simulation: K = 0.0741 - 1.873 (DCAA %): -, computer simulation; 0-0, experimental data.
k
;
l
;
a
b
b
;
’
COLUMN VOLUMES
Figure 6. Comparison of theory and experiment for outlet fructose concentration for separation of fructose from water on DBAE cellulose using p H as the cyclic variable. Cycling zone and isotherm data are from Busbice and Wankat (1975): inlet pH’s 5.0 and 8 3 , 4 stages; A = 0.528, lag = 1.0 column volumes; a, feed concentration = 1 mg/mL, cycle 2.2 column volumeshalf cycle; b, feed concentration = 4 mgimL, cycle = 3.9 column volumesihalf cycle (corresponds to Figure 5 ) ;-, computer simulation; 0-0, experimental data.
y. . = L.J
U‘Xi,j
l+b’Xi,j
where a’ = f(pH) and b’ = constant. Figures 6a and b show comparisons of the theoretical model with experimental data for the separation of fructose from water. The agreement between theory and experiment was not as good as that obtained with the acetic acid-water system of Baker and Pigford (1971).This lack of agreement occurred because the Langmuir isotherms could not completely represent all the interactions occurring in the column, and the empirical fit of the pH wave was not as good as the theoretical fit of the temperature wave. The agreement was better for shorter cycle times and lower fructose concentrations, indicating that the complex interactions discussed above were present since short cycle times and low fructose concentrations would tend to minimize the interactions not handled by the theoretical model. Nelson and Wankat (1976) separated dipeptides in a preparative high-pressure liquid chromatographic system using the concentration of an added chemical as the thermodynamic variable which forced the separation. Specifically, glycyl-Lphenylalanine and glycyl-L-tyrosine were removed from water using different levels of dichloroacetic acid as the cyclic variable in a column packed with a nonpolar C18 bonded packing. At higher concentrations of the acid (-0.03% V/V) the dipeptides were held up in the column and a t low acid concentrations (-0.0015% V/V) the dipeptides were eluted from the column. Since adsorption isotherms were not measured an approximate linear relationship for the equilibrium isotherms of the form
where K is a linear function of dichloroacetic acid concentration was used. The linear isotherm data were obtained by standard methods from pulse chromatography experiments (Silarski, 1976), and the best linear fit of K as a function of dichloroacetic acid concentration was found by a least-squares linear regression. The resulting isotherms are only approximate since much higher dipeptide concentrations were used in the cycling zone experiments than in the chromatography experiments. The dichloroacetic acid wave velocity of A = 0.9 was also obtained from chromatographic experiments. Figure 7 shows the comparison of the theoretical prediction with experimental data for the separation of glycyl-L-phenylalanine using square input waves of dichloroacetic acid. Although 125 equilibrium stages were reported for the column during pulse experiments, it is known that the cycling zone technique greatly overloads the packing compared to an analytical separation. This is clearly evident in Figure 7 where the theoretical curve is based on only eight equilibrium stages. Considering the approximate nature of the equilibrium isotherms, the agreement with the experimental data is credible, but could certainly be improved by using nonlinear isotherms. Complete input data for the simulation are given by Silarski (1976). In the cases studied up to this point, the separations have involved the removal of only one component. Recent research by Wankat (1975), Dore (1975),and Dore and Wankat (1976) has developed a multicomponent cycling zone technique. If the assumption is made that no interactions exist between the different solutes, then the continuous flow, equilibrium-stage model can readily be adapted to predicting multicomponent separations. For a two-component separation, the computer model is run twice, one run for each component. The behavior of the thermodynamic variable is kept the same in both simulations, and the individual solute concentration waves are superimposed to predict the separation. For multicomponent, separations, the cyclic variable is input into the column in a series of steps, one plateau for each component as shown in Figure 2b, or as a continuous change in level similar to the ramp wave in Figure 3. The reason for such steps is discussed in detail elsewhere (Wankat, 1975; Wankat e t al., 1975). Briefly, however, the step sizes must be chosen so that some components will move faster than the wave velocity and some slower. Each component concentrates a t that step where its velocity first becomes faster than the velocity of the cyclic variable. As mentioned previously, Dore and Wankat (1976) studied multicomponent separations; their experimental system was
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978 37
001
00
I
/ n----h-n---n--
IO
20 COLUMN
30 40 VOLUME
50
60
Figure 8. Comparison of theory and experiment for separation of glucose-fructose-water system on DBAE cellulose using pH steps. Cycling zone data is from Dore (1975): pH1 8.5, pHz 7.35, pH3 5.0,2.0 column volumes/each portion of cycle, T = 25 "C; A = 0.528, feed concentration = 0.25 mg each sugar/mL, 4 stages, lag = 0.5 column volumes; -, computer simulations; 0 ,fructose experimental data; 0,glucose experimental data.
the same as that studied by Busbice and Wankat (1975) except that they used three pH levels to separate glucose and fructose from water. Figure 8 shows the experimental outlet concentration waves (Dore, 1975) for one such separation and the theoretical prediction for the same waves using the continuous flow, equilibrium stage theory. Langmuir isotherm data developed by Dore and Wankat (1976) were used. The separation between glucose and fructose is generally poor due to interaction effects and a low capacity adsorbent. The agreement between theory and experiment is quite good for glucose. However, the predicted fructose values predict too sharp a peak maximum and not enough fructose removal a t the high pH values. Part of this lack of agreement was probably due to the assumed superposition of isotherms used in the simulation. Support for this hypothesis was shown in Figure 6 where much better agreement between theory and experiment was obtained when only fructose was present. The ability of the model to predict the double peaks shows that the major effects are included. The complete input data for the simulation are given by Silarski (1976).
Discussion and Conclusions The results in Figures 4 through 8 have shown the flexibility and accuracy of the continuous flow equilibrium stage model. The model can handle various cyclic variables, including temperature, pH, and the concentration of an added chemical, and has the ability to handle different equilibrium isotherms, including Freundlich, Langmuir, and linear types. The accuracy of the model is in direct proportion to the accuracy of the equilibrium data available. Where the interactions can be modeled successfully, as in the acetic acid-water system of Baker and Pigford (1971), the theoretical curves are in very good agreement with the experiment results. However, if the equilibrium data are approximate in nature, as in the case of the dipeptide-water system of Nelson and Wankat (1976), the theoretical curves are only in qualitative agreement with experimental results. One of the more promising uses of the numerical model is in the area of multicomponent separations. As the results in Figure 8 illustrated, the model can predict the separations involving more than one solute. Multicomponent separations using the technique of cycling zone adsorption are difficult to accomplish experimentally; however, the advantages of such continuous separations are numerous. T o aid in solving this problem, the continuous flow model can be used to screen the more important variables, such as cyclic wave type, cyclic
variable levels, or timing of the thermodynamic variable waves, In this way, an optimal set of operating conditions can be defined which can then be tried out in the laboratory. As the number of components increases, the restrictions that have to be met for a multicomponent separation become more stringent. The equilibrium stage model can be used to evaluate different sets of conditions that would be impractical to run experimentally. T o predict the separation the number of stages in the column or the H E T P value is needed. In the simulation results reported the number of stages was determined by a trial and error matching of theoretical and experimental results. Once N has been determined the model can be used to predict separations for a variety of different operating conditions or different designs. This procedure does provide a good fit to the experimental data but would be quite awkward for design. For design a prediction of H E T P from simple experiments would be very desirable. Several different methods for predicting the H E T P can be tried. The number of stages can easily be determined for a pulse chromatography experiment. Unfortunately, the experimental results of Dore and Wankat (1976) and Nelson and Wankat (1976) show that the number of stages for a cycling zone separation are significantly less than the number obtained for the same column operating as a chromatograph. This decrease in number of stages occurs because of the increased sample size, increased concentrations which make the isotherms nonlinear, and increased dispersion because of nonisothermal operation. Thus pulse chromatography results are not directly comparable. Another approach to predict the H E T P is to compare theoretical solutions for breakthrough curves for different theoretical models. This was done by Broughton et al. (1970) for a discrete transfer, counter-current distribution type model to approximately predict HETP in terms of the mass transfer coefficient. Since the closed form solutions for the models are available only for the limiting cases of isothermal operation, no dispersion and linear equilibrium isotherms, the predicted H E T P is valid only for these limiting conditions. Another approach is to obtain isothermal breakthrough curves and then fit these curves with the theoretical models using the appropriate nonlinear isotherm. The cycling zone model for the start-up conditions can simulate breakthrough curves if A = 0. Alternatively, eq 5 can be solved for isothermal operation which is similar to the direct mode simulation. This approach was not implemented here since our solution procedure is for limit cycle operation and does not easily simplify to breakthrough results. This procedure would be similar to the procedure of fitting cycling zone data that was employed. Separate computer simulations are required for each breakthrough curve. The procedure for fitting breakthrough curves to find the number of stages in a column can be greatly simplified if the isotherm can be approximated as linear, eq 17. For isothermal operation with linear isotherms eq 5 simplifies to a form analogous to eq 7. Thus the solution obtained for eq 7 can be used for isothermal breakthrough if the appropriate changes in variables are made. Equation 8 is the solution for a step input for a series of noninteracting first-order systems. These systems have been extensively studied and the solutions with N as a parameter are available (e.g., Douglas, 1972; Coughanowr and Koppel, 1965). The advantage of this procedure is that the breakthrough curves do not have to be generated on the computer for each run. The disadvantage is that since a linear isotherm must be assumed and since the additional dispersion caused by temperature gradients in a cycling zone system are ignored, the predicted values of N will probably be too high and the technique will not be applicable for systems with highly nonlinear isotherms.
38 Ind. Eng. Chem.Fundam., Vol. 17, No. 1, 1978
These approaches will give different N and Hh'TP values when applied to different species or thermal breakthrough curves. The N which gives the best overall fit has to be employed. Direct simulation of cycling zone runs will provide the best N while with the other methods some compromise is necessary. This problem is caused by the basic shortcoming of the staged model in that it is not a realistic description of what is actually occurring in the column. This model can also be used for staged systems such as extraction. For staged systems a stage efficiency could be incorporated into the model. In addition, conversion of this model to the direct mode is straightforward for both continuous contact and staged equipment. Multizone systems can also be modeled by reshaping the temperature wave before each zone.
Nomenclature A = HC,,/(SC,, HC,, MC,,) = thermal wave velocity a,b = Freundlich isotherm parameters, eq 15 a',b' = Langmuir isotherm parameters, eq 16 CLj, = constant of integration given by eq 9 C'. . = constant of integration given by eq 12 CFeed = solute mass ratio in feed, g of solute/g of diluent C = heat capacity, cal/g "C = holdup of moving phase on each stage, g i = stagenumber K = X / Y , linear isotherm, eq 17 L = flow rate of moving phase, g/s M = mass of the column walls per stage, g N = total number of stages S = holdup of stationary phase on each stage, g SL = slope of ramp input, in eq 10, "C/s t = time, s T , j = temperature of stage i during cycle j, "C u = thermal wave velocity = L A / H in Baker and Pigford (1971). X i , j = mass fraction solute in moving phase of stage i during cycle j Yj,, = mass fraction solute in stationary phase of stage i during cycle j
+
LJ
H"
+
Greek Letters = LA/H 7 = (L/H)At N
Subscripts C = cold H = hot i = variable stage number j = cyclenumber m = mobile phase s = stationaryphase w = wall Literature Cited Baker, B., Ph.D. Thesis, University of California, Berkeley, Calif., 1969. Baker, B., Pigford, R. L.. Ind. Eng. Chem. Fundam., IO, 282 (1971). Broughton, D. B., Neuzil, R. W., Pharis, J. M., Brearley, C. S., Chem. Eng. frog., 66 (9),70 (1970). Busbice, M. E., M.S. Thesis, Purdue University, 1974. Busbice, M. E., Wankat, P. C., J. Chromatogr., 114, 369 (1975). Coughanowr, D. R., Koppel, L. B., "Process Systems Analysis and Control", pp 74-79, McGraw-Hill, New York, N.Y., 1965. Dore, J. C.,M.S. Thesis, Purdue University, 1975. Dore, J. C.,Wankat, P. C., Chem. Eng. Sci., 31, 921 (1976). Douglas, J. M., "Process Dynamics and Control, Vol. 1. Analysis of Dynamic Systems", pp 187-190,Prentice-Hall, Englewood Cliffs, N.J., 1972. Gupta, R., Sweed, N. H., Ind. Eng. Chem. Fundam., 10,280 (1971). Hung, Y. C., Lee, V. J., Paper 17c presented at 68th National Meeting of AIChE, Houston, Texas, Mar 1, 197 1 Lapidus, L., "Digital Computation for Chemical Engineers", pp 86-92, McGraw-Hill, New York. N.Y., 1962. Lavie, R., Rielly, M. J., Chem. Eng. Sci., 27, 1835 (1972). Nelson, W. C., M.S. Thesis, Purdue University, 1975. Nelson, W. C., Wankat, P. C., J. Chromatogr., 121, 205 (1976). Pigford, R. L., Baker, B., Blum, D. E., Ind. Eng. Chem. Fundam., 8, 848
(1969). Silarski, D. F., M.S. Thesis, Purdue University, 1976. Sweed, N. H., Wilhelm, R. ti.,Ind. Eng. Chem. Fundam., 8,221,(1969). Wankat, P. C., Sep. Sci., 8,473 (1973a). Wankat, P. C., lnd. Eng. Chem. Fundam., 18,372 (1973b). Wankat, P. C.,J. Chromatogr., 88,211 (1974). Wankat, P. C.,Ind. Eng. Chem. Fundam., 14, 96 (1975). Wankat, P. C.,Dore, J. C., Nelson, W. C., Sep. f u r i f . Methods, 4, 215
(1975).
Receiued ,for reuieu: M a y 6, 1977 Accepted August 19,1977 Research was p a r t i a l l y supported by N a t i o n a l Science F o u n d a t i o n G r a n t No. GK43282.
