Multicomponent Cycling Zone Separations Phillip C. Wankat School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
The local equilibrium theory and an equilibrium staged theory are used to show that multicomponent mixtures can be separated into their individual components by a traveling wave cycling zone technique. To force this multicomponent separation the inlet temperature is varied as a series of steps with one step for each component. The local equilibrium theory predicts infinite separations under certain circumstances while the equilibrium staged theory predicts finite separations. Restrictions which must be satisfied to achieve good separations are delineated. Since the multicomponent cycling zone technique concentrates the products and has a high throughput, it has the potential of becoming a major technique for preparative chromatography.
In this paper the cycling zone adsorption technique invented by Pigford et al. (1969) is extended to separation of multicomponent mixtures. This extends cycling zone adsorption into the realm of preparative chromatography. Multicomponent mixtures are separated in the traveling wave mode of operation by utilizing several changes in the cyclic thermodynamic variable. Two theoretical approaches are used to predict the separations which will occur and to determine the restrictions which must be satisfied to achieve these separations. Interest and research in cyclic separation techniques has been high ever since the introduction of parametric pumping by Wilhelm et al. (1966). Since these cyclic techniques have been reviewed by Wankat (1974b), only a few cycling zone papers will be discussed here. Baker and Pigford (1971) utilized a local equilibrium theory to predict the separations that can be obtained in both the direct or standing wave mode (bed is externally heated and cooled) and in the traveling wave mode (entering fluid is heated or cooled). The very interesting theoretical prediction that infinite (practically speaking, very large separations) could be achieved in the traveling wave mode by adjusting the thermal wave velocity resulted from this theory. This prediction was not checked experimentally although other experimental results were in good agreement with the theory. Wankat (1973, 1974a) developed an equilibrium staged theory for cycling zone extraction in both the direct (1973) and traveling wave modes (1974a). Large but finite separations were predicted for the traveling wave mode when the thermal wave velocity was optimized. Agreement between theory and experiment was fair. In the work up to now all solutes would be removed together from the feed. In this paper both the local equilibrium model of Baker and Pigford (1971) and the equilibrium staged model of Wankat (1974a) are used to study the separation of multicomponent mixtures (two or more solutes) in traveling wave cycling zone systems. Both models predict that solutes can be recovered individually if the inlet temperature wave varies as a series of steps (see Figure 1)instead of as a simple square wave. Technique for Multicomponent Cycling Zone Separation The basic idea used to obtain multicomponent separations is that the cyclic variable is input into the column in a series of steps. The step sizes are chosen so t h a t some components will move faster than the wave velocit'y of the cyclic variable and other components will move slower. 96
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
The next step causes one additional component to move faster than the cyclic variable. The net result is a multicomponent separation with each component concentrating at that step where its wave velocity first becomes faster than the velocity of the cyclic variable. For instance, consider adsorption with temperature as the cyclic variable with two components (A and B in a nonadsorbed carrier) to be separated. Assume component A is less strongly adsorbed than B a t all temperatures. As temperature increases both components are adsorbed less strongly and both concentration waves move through the column a t a faster speed. At some low temperature, Tc, both components will move slower than the thermal wave. Since thermal wave velocity is relatively insensitive to the temperature, the concentration wave velocities will eventually become greater than thermal wave velocity if temperature is continually increased. A temperature T I is chosen so that A moves faster than thermal wave velocity but B still moves slower. Component A will now tend to concentrate a t temperature TI since the thermal wave overtakes all A input a t Tc and is over taken by A input a t T I . Another temperature T2 is chosen so that both components move faster than the thermal wave velocity. Component B will now concentrate near step a t T2. Component A does not concentrate here, but moves past this thermal wave to temperature T I . The inlet temperature profile for this two-component separation is shown in Figure 1. The inlet temperature profile is repeated cyclically. If the column is long enough and the times for each temperature are set properly, very large separations can occur. If more than two components are present the feed can still be separated if additional temperature plateaus are added. The separation can also be obtained if instead of temperature plateaus a continuous change in temperature (either linear or nonlinear) is utilized. Now each component will tend to concentrate a t that temperature where it moves a t the same velocity as the thermal wave. This argument assumes that the thermal wave does not move so fast that there is no feasible temperature a t which the concentration waves will move faster than the thermal wave. This may require some technique to slow down the thermal wave as discussed by Baker and Pigford (1971). Other thermodynamic variables may be substituted instead of temperature as the cyclic variable. For instance, an additional polar or nonpolar solvent which changes the distribution coefficients of the components can be added to the feed a t differing concentrations. The concentration of this added solvent now replaces temperature as the cyclic variable. By changing the solvent employed the sol-
vent wave velocity can be varied. In a very rapid cycling system pressure might be utilized as the cyclic variable.
Local Equilibrium Theory Baker and Pigford (1971) developed a local equilibrium theory for both direct and traveling wave modes of operation. This theory can easily be modified to apply to multicomponent separations with an inlet temperature profile as shown in Figure 1. In addition to the assumptions that solid and fluid phases are locally in equilibrium and that axial dispersion and heat of adsorption effects may be neglected made by Baker and Pigford (1971), we will assume that all of the isotherms are linear and that adsorption of each component is independent of the other components present. Nonlinear adsorption isotherms can be handled by the method of Baker and Pigford (1971). Interactions between the various components arise from competition for the adsorption sites and would require a somewhat more complicated analysis than that given here. These interactions might aid or hinder the multicomponent separation. These assumptions will be reasonable if the solutes are in small concentrations in a nonadsorbed carrier and if separations are relatively small. With the assumptions made above the energy balance is independent of the mass balances and is
TIME
Figure 1 . Feed temperature profile for two solutes.
TIME
Figure 2. Characteristic curves for two solutes. B is more strongly adsorbed.
where
mine for linear isotherms where it is given as This nomenclature is the same as that of Baker and Pigford (1971). For an adiabatic column with feed temperatures shown in Figure 1, the solution to eq 1 is a wave with velocity Uthermal and a shape similar to that shown in Figure 1. Thus the model predicts that thermal waves are translated through the column without changing shape. Following Baker and Pigford (1971) the mass balance for a given component is [l
+=
€ + - -1 - --
f f (1
-
CY
CY
=
az
-
4PS
(%)I
at aC +
1-CY ff
Since eq 3 is exactly the same as eq 7 in Baker and Pigford (1971) and has the same boundary conditions, it will have the same solution. The solution was obtained by first making the change of variables
= f - Z’Lrthermal and then utilizing the method of characteristics. This solution says that concentration is constant along characteristic lines given as ‘T
dz d7
(4)
where z’
~r,,Ilc
=
,+ 1
1-
E
+- 1-CY (1 E
(5) - E)Ps(%)
as long as the temperature is constant. When the temperature changes (i.e., a t the intersection of a concentration and a temperature characteristic), the concentration will change. This change in concentration is easiest to deter-
. -C=
CrCi-l-i-Uthermal-‘
Ci-1
Lrci-i
(6) - ‘thermal-’
When the thermal wave velocity lies between the two concentration wave velocities, the right-hand side of eq 6 will be negative. In this case the concentration waves do not cross the thermal wave but instead follow along the thermal wave once they have intersected it. This is shown in Figure 5c of Baker and Pigford (1971) and in Figure 2. This “trapping” of the concentration wave along the thermal wave causes infinite separations (in actual practice large, but not infinite). The characteristics can also be drawn in a z-t plane where characteristic is d z l d t = L’c for concentration waves and d z / d t = VThern,al for temperature waves. Outlet concentrations can now be determined by following characteristic lines and determining the appropriate concentration changes. If the thermal wave velocity and the temperatures are chosen so that (“A2)
LrB2) ‘A1)
>
‘thermal
>
( u B l > CrAC> u B c )
(71
where U A is~ the velocity of component A a t temperature 7’2, then the model will predict infinite separations if the times for each portion of the cycle satisfy certain inequalities. Characteristic lines for a case with infinite separations are shown in Figure 2 . Figure 2 is drawn so that inequality (7) is satisfied and so that all of component A will exit a t the boundary between TC and T I and all of component B will exit a t the boundary between T I and Tz. Thus this simplified model predicts that all of component A will exit a t a point in time and all of B will exit a t a different point (infinite concentrations). During the rest of the cycle the fluid contains no A or B. These infinite separations will not occur in practice because of nonlinear isotherms and dispersion Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
97
effects. Despite these limitations large separations of multicomponent mixtures can be expected. The restrictions on the times for each portion of the cycle can be determined by following the concentration characteristics in Figure 2 and determining the distance a characteristic will travel before intercepting a thermal wave boundary. In order to obtain an infinite separation the following restrictions must be satisfied. Restriction 1. All of component A input during the portion of the cycle when T = Tc must intercept the Tc-Ti boundary before exiting from the column. This is represented by the inequality 0
5
t,
5
L
("., 1
-
-
Restriction 2. All of component B input during the portion of the cycle when T = T Z must intercept the Ti-Tz boundary before exiting from the column, or
Restriction 3. All of component A input when T = Tz should cross both the Tz and the T I regions and intercept the Tv-Tl boundary before exiting from the column. This condition can be stated as a restriction on t i .
Restriction 4. All of component B input when T = TC should cross both the Tc and the Tiregions and intercept the T1-r~boundary before exiting from the column. This condition can be stated as a restriction on ti.
(mixed with nonadsorbed carrier) of some finite concentration will exit from the column. If restriction ( 2 ) is not satisfied there will be a time period when pure B of some finite concentration will exit from the column. Neither of these problems is serious if the purpose of the separation is to separate A and B. However, if restrictions (3) and (4) are not satisfied it is possible to get intermixing of components A and B when the fluid of temperature T1 exits from the column. These arguments can be clearly seen from Figure 2. In acdual practice dispersion, mass transfer effects, and nonlinear isotherms will prevent infinite separations from occurring. From eq 2 and 5 it is obvious that the thermal and concentration wave velocities must be less than u for the traveling wave mode. If U t h e r m a l is large and is close to u , then restrictions ( 2 ) and (3) become difficult to satisfy, and it may not be possible to find temperatures for which inequality (7) is satisfied. If the restrictions and inequality can be satisfied operation with very short time periods is still undesirable since it is difficult to obtain fully developed temperature waves (this point is discussed later). Baker and Pigford (1971) found that for liquid systems U t h e r n l a l was too large to optimize the separation, and they discussed methods for decreasing U t h e , m a l . If CTtheIn,al is made too low then it may be difficult to satisfy restriction (1)and (4) and the thermal wave is likely to be too diffuse due to dispersion. A too low thermal wave velocity can easily occur in adsorption of gases. The arguments presented here can easily be extended to three, four, or more components. For each additional component another temperature plateau must be added in FiguFe 1 and two additional restrictions are required to prevent intermixing of components. For a system with three adsorbed components in a carrier there are a total of six restrictions. These restrictions will not be presented here, but they can easily be determined with the aid of a diagram similar to Figure 2 . As the number of components is increased the restrictions become more and more stringent.
The stronger of the two restrictions ( 3 and 4) on tl must Equilibrium Staged Model be satisfied. The local equilibrium model discussed previously is Several interesting conclusions can be drawn from these based on several simplifying assumptions. Limitations of four restrictions and from the definitions of the thermal the local equilibrium model when applied to parametric wave velocity, eq 2 , and the concentration wave velocity, pumping are well known (for instance, see Wankat, eq 5. First, in the standing wave or direct mode of opera1974b). To obtain a more realistic picture of separations tion, U t h e l m a l = m , restriction ( 2 ) cannot be satisfied. which might be achieved experimentally, a model which Thus, the usual direct mode method of heating or cooling includes dispersion effects must be used. Besides the local the entire bed cannot be used for multicomponent separaequilibrium model the only published theory for cycling t ions. zone systems is the staged model for application of cycling Second, if the equality signs are used to calculate t L zone techniques to counter-current distribution (Wankat, and t 2 for insertion in restrictions (3) and (4) then all cal1973, 1974a). This model is an exact model for the staged culated times are directly proportional to L . If the equalcounter-current distribution system, and is an approxiity is used to calculate t z then restriction (3) will be satismate model for cycling zone adsorption in columns. This . is the expected order fied if and only if Vi2 > U H ~This model is related to the STOP-GO algorithm developed for for the component wave velocities, but is not required to parametric pumping (Sweed and Wilhelm, 1969). satisfy inequality (7) and is not required to satisfy restricThe staged model with discrete transfer and equilibrition (3) if a value of t z less than the right-hand side of reum steps assumes that the column can be divided into N striction (2) is used. If the equality is used to calculate t~ equilibrium stages. Each equilibrium stage contains a stathen restriction (4) will be satisfied if and only if U A > ~ tionary phase (adsorbent or stationary solvent) and a moCBC.If tC is calculated as less than the right-hand side of restriction (1) then it is no longer necessary that U A ~ bile phase. At each transfer step the mobile phase from each stage is transferred to the next stage. Mobile phases be greater than UBc. The practical significance of this from different stages do not mix during the transfer step. argument is that the order of selectivity of the adsorbent After each transfer step, the operation is stopped and all for the components can change with temperature without stages are allowed to reach equilibrium. Once equilibrium limiting the separation. This point is illustrated later by has been attained a new transfer step is done. In the travexample. eling wave mode of operation the temperature is varied by Third, if all the restrictions are not satisfied a good but changing the temperature of the material fed to the first not infinite separation is predicted. Thus, if restriction (1) stage. This model is an exact model for counter-current is not satisfied there will be a time period when pure A 98
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
distribution type equipment (for example, see King, 1971). It is an approximate model for continuous flow continuous contact equipment (columns) which becomes more accurate as the number of stages and the number of transfer steps in each portion of the cycle increases. This model is presented in detail elsewhere (Wankat, 1973, 1974a) for removal of a single solute from a carrier. T o apply this model to multicomponent separations only the feed temperature format needs to be varied. With a feed temperature varying as shown in Figure 1 multicomponent mixtures can be separated. A brief development of the model is given below for a one-zone system. The nomenclature follows Wankat (1974a). We assume that densities, heat capacities, volumes of both phases in each stage, and the weight of each stage are constant. These assumptions imply that the solutes are held in a nonadsorbed carrier. In the analysis used here distribution coefficients are assumed to be functions of temperature only, but concentration dependence can be included. With these assumptions the energy balance is independent of the mass balances. Assuming that the stages are well insulated, the energy balances can easily be written as a recursion equation. For a general stage i after transfer S the energy balance can be written as
0 00
4
8
TRANSFER
122 STEP
16
20
NUMBER
24
28 1
(TIME)
Figure 3. Results of staged theory calculation for separation of two solutes. 40 stages, T, = 0, 7'1 = 100, T2 = 400, S, = 13, SI = 12, S 2 = 2, A = 0.6, K4 = 1.0 0.022', K c = 0.3333 + O.O0333T, vM/vs = 1.
+
where
3.2
Equation 8 can easily be solved by incrementing S and i. With a cyclic temperature input as shown in Figure 1 the outlet temperature will eventually reach a repeating state where the temperature repeats itself. Characteristic temperature profiles for square wave inputs were presented earlier (Wankat, 1974a). The mass balances can be developed as recursion relations if we first define fl,.y as the fraction of a given solute that is in the moving phase in stage i after transfer step S.
x
28
n
;
*A
24-
\
'
u 20-
16-
t-
2 I20
a 08-
(10) 04
Defining M l , s as the mass of the given solute in stage i after transfer step S , the mass balance is Mi,s
= fi-i,S-iMi-i,S.i
+
(1 - fi,s-i)'fi,s-i
(11)
Equations 10 and 11are written for each solute. The mass balances are solved by using the temperature distribution calculated from eq 8 and 9 to calculate the f,,. for each stage in each part of the cycle and then using these values in eq 11 to calculate the mass of solute in each stage. Again a repeating state will be attained. Methods for solving the repeating state equations without doing the start-up calculations are given by Wankat (1973, 1974a). All simulations shown here were obtained from digital computer solutions of eq 8-11. The staged model introduces dispersion and is in this respect more realistic than the local equilibrium model. The actual dispersion present in a physical system can be modeled by adjusting the number of stages. Care should be taken that this is done for cycling zone measurements since the number of stages will be less than the number obtained from pulse measurements (Busbice, 1974). The countercurrent distribution model becomes more accurate as the number of stages increases. The predicted separations for two different systems are shown in Figures 3 and 4. These systems do not correspond to any real chemical system, but were chosen for
-
Figure 4. Results of staged theory calculation for separation of two solutes. 40 stages, T , = 0, T I = 100, T P = 250, Sc = 13, SI = 5, S z = 12, A = 0.6, K A = 1.0 + O.O2T, K H = 0.25 + O.OlT, VM/V. = 1.
convenience. All the values required to generate the curves are given in the figure captions. The time for each portion of the cycle were selected so that restrictions (1) to (4) are satisfied. For the staged system these restriction are written by relating N to L, A to Crthermal, and f to U L . This gives the staged equivalent of the continuous system with the velocities normalized so that they have a maximum value of unity. In Figure 3 the two components are well separated from each other, but the separation of component B from the carrier is small. In Figure 4 both components are well separated from the carrier, but there is considerable overlap between the outlet peaks. Neither of these separations is close to being infinite even though the restrictions set by the local equilibrium model are satisfied. Thus, there are obviously other factors involved which are not modeled by the local equilibrium model. These factors are the same factors which cause zone Ind. Eng. Chem., Fundam., Voi. 14, No. 2 , 1975
99
8ot
i
70-
i -
A
-
3 0-
TRANSFER
STEP
NUMBER
(TIME)
Figure 3. Results of staged theory calculation for separation of three solutes. 160 stages, T , = 0, T I = 100, TZ = 250, 7'3 = 400. S , = 40, SI = 32. S z = 32, SB = 24, A = 0.6,K \ = 1.0 + 0.02T, K,c = 0.25 + O.OlT, K , = 0.3333 0.003337: V\I/V. = 1.
+
broadening in elution chromatography and are discussed in detail by Giddings (1965). A computer study of the outlet profiles for binary separations and a study of Giddings' (1965) discussion of zone broadening shows t h a t three additional restrictions can be identified. Satisfaction of these restrictions will produce good but not infinite separations. These additional restrictions are in more qualitative form than those given previously, although they could be put in quantitative form by modeling a specific experimental system. Restriction 5, All portions of the cycle must last for a long enough period of time so that the thermal waves are reasonably well developed. The larger C r t h e l m a l (or A ) the shorter the time periods can be. This restriction is necessary to prevent dissipation of the thermal wave before the column outlet. Since temperature drives the separation, dissipation of the thermal wave will reduce the separation. The following restrictions are caused by zone spreading. Restriction 6. The sum of t( + t 2 must be large enough to prevent the trailing edge of the slower moving component, B or C, from overlapping with the leading edge of the faster moving component. A. Restriction 7 . Time tl must be large enough to prevent the trailing edge of the faster moving component, A, from overlapping with the leading edge of the slower moving component, B or C. In Figure 3, t 2 is quite small and restriction 5 is not satisfied. The result is that a large separation of component C does not occur. In Figure 4 tl is relatively small and restriction 7 is not satisfied. The result is overlap between the trailing edge of .4 and the leading edge of B. Any method' which allows these restrictions to be satisfied will improve both the maximum purification and resolution of the components. Maximum purification from the carrier and resolution of the two components is greatly increased by increasing N (or L ) and a t the same time making the appropriate increases in t( , tl, and t 2 in accordance with restrictions (1) to (4).This allows considerably more time for each portion of the cycle so that restrictions (5) to (7) can be more closely satisfied. Maximum purification can also be increased with small change in resolution by merely doubling tc, tl, and t z without changing N In this case restrictions (1) to (4)are no longer satisfied. but more separation occurs. These increases in maximum purification from the carrier agree with earlier results obtained for one component (Wankat. 1974a). 100
Ind. Eng. Chem., Fundarn., Vol 14, No. 2, 1975
Purification and resolution can also be increased by optimizing the temperatures and the thermal wave velocity so that all times are of sufficient duration and still satisfy restrictions (1) to (4). The temperatures and thermal wave velocities shown in Figures 3 and 4 have not been optimized. Variations in the adsorbent or solvent properties could also be used to change the dependence of the distribution coefficients on temperature and to optimize the separations. With two components seven restrictions must all be satisfied simultaneously in order to obtain good separations. This makes the separation fairly difficult. When more components are t o be separated the number of restrictions increases by three for each additional component. For a ternary separation there will be six restrictions resulting from the local equilibrium theory, one restriction equivalent to restriction ( 5 ) , and three restrictions to prevent overlap of peaks due to zone spreading phenomena. This totals ten restrictions which should be simultaneously met to obtain a good ternary separation. The characteristic curves for a ternary system will not be shown here, but they can easily be developed by following the principles used to construct Figure 2 . For a ternary system four temperature levels are used in the feed and components will concentrate a t the boundaries between Tc and TI, 2'1 and 2'2, and T2 and T B The equilibrium staged model can easily be used to predict ternary separations. A ternary separation of the components separated as binary pairs in Figures 3 and 4 is shown in Figure 5 . Note that a t 0" component B is more strongly adsorbed than component C while a t 100. 250, and 400" component C is more strongly adsorbed. Thus, the order of selectivity can change without affecting the separation. The cycle times used for this separation do not satisfy the six restrictions derived from the local equilibrium theory. The separation achieved with these restrictions satisfied did not result in a good purification of component C since the cycle times were too short. In Figure 5 the zone spreading restrictions were satisfied and times were chosen to increase in the same way as the values predicted by the local equilibrium theory. This increased the separation, but the separation could undoubtedly be increased further by optimization. One major reason that separations are limited is dispersion of the thermal wave. For the separation shown in Figure 5 a maximum inlet temperature of 400" and a minimum inlet temperature of 0" were used. The maximum outlet temperature was 319" and the minimum outlet temperature was 27". Thus. the stages near the outlet undergo a much smaller change in temperature than those near the inlet and as a result the separation is decreased. If a higher thermal wave velocity were used there would be less dispersion and a larger temperature difference on the stages.
Discussion and Conclusions The theories presented in this paper show that a continuous multicomponent feed can be separated by a cycling zone technique. As the number of components increases the restrictions become more stringent, but theoretically any number of components can be separated as long as their distribution coefficients are different from each other and are temperature dependent. Separation would be enhanced by using chromatographic grade adsorbents with low HETP and high efficiencies. The theoretical arguments employed here are not restricted to adsorption with temperature as the cyclic variable. They are also applicable to other chromatographic systems, extraction, absorption etc. Other thermodynamic variables such as pressure or concentration of a n added solvent can be used instead of temperature.
This technique requires that the thermal wave travel faster than the solutes when the adsorbent is cold and slower when the adsorbent is hot. For most liquid systems this would require slowing down the thermal wave while in gas systems the thermal wave would probably have to be speeded up. Slowing down the thermal wave was discussed by Baker and Pigford (1971) and could be done by either mixing an inert solid of high heat capacity with the adsorbent or by developing an adsorbent with an inert inner core of high heat capacity material. When other thermodynamic variables are used, the velocity of the cyclic variable wave may naturally fall within the desired range. Thus Busbice (1974) found that the velocity of pH waves was in between the concentration wave velocities for fructose and glucose (multicomponent separations utilizing the technique reported here were not tried by Busbice (1974)). Research is continuing on the experimental verification of the theories presented here. The assumptions used for both models are not valid if the adsorbed components are present in large amounts or if there is no nonadsorbed carrier. In these cases flow rate changes can become significant and the isotherms would no longer be linear. Even though the models are not valid the physical reasoning behind the models is still valid. Thus, multicomponent separations would be expected for concentrated solutions although the separations may be less sharp than those shown here. Without a nonadsorbed carrier there will always be overlap between different components. In this paper it has been assumed that all distribution coefficients decrease when the temperature is increased. If some of the distribution coefficients increase while some decrease (which can happen with extraction or when the cyclic variable is concentration of some added solvent) the multicomponent separation is even easier. In this case Wankat (1974a) showed that two solutes can be separated from each other using the usual square wave technique. It was mentioned earlier that the usual direct mode or standing wave technique cannot be used for multicomponent separations since all of the restrictions cannot be satisfied. If some technique of heating the column so that the temperature varies along the length of the column is used then multicomponent feeds could be separated. This has been done by Zhukhovitskii (1960), who used a moving heater. An alternative would be to have several heaters and coolers along the length of the column so that a traveling wave of specified velocity could be simulated. These direct mode techniques have the advantage of simple control of the thermal wave velocity, and the disadvantages of complicated equipment and inefficient use of energy. In practice the advantage may outweigh the disadvantages since the thermal wave velocity has not yet been experimentally controlled in a cycling zone system. This cycling zone technique is related to many other separation techniques such as pressure swing adsorption. parametric pumping, temperature programmed chromatography, and gradient elution chromatography. The differences and similarities between cycling zone techniques and the latter two techniques are of considerable interest. All three techniques can use a variety of adsorbents, solvents, and coated solids as the stationary phase. A variety of different physical arrangements such as columns, countercurrent distribution apparatus, or thin layers could be employed. In cycling zone techniques any thermodynamic variable can be employed to force the separation. In chromatography the same thermodynamic variables are employed to speed up or improve the separation. The differences between these techniques are that preparative chromatography utilizes intermittent feed pulses while in cycling zone the feed is continuous, and in chromatography
additional thermodynamic gradients (such as temperature programming) only speed u p or improve the separation while in cycling zone the thermodynamic changes force and produce the separation. Cycling zone techniques also have the advantage that products are concentrated instead of diluted as in chromatography. From these advantages it appears that cycling zone techniques have the potential of becoming one of the major preparative chromatographic techniques.
Nomenclature The nomenclature used for the local equilibrium theory follows that'of Baker and Pigford (1971). The nomenclature for the staged equilibrium theory follows that of Wankat (1974a). The essential parts of these nomenclatures are repeated below.
Local Equilibrium Theory T = temperature, "C zo = lengthof bed, cm C = fluid concentration, mol/l. q = solid-phase concentration, mol/kg of dry solid t = time, min u = wave velocity, cm/min u = interstitial fluid velocity, cm/min z = axial distance. cm Greek Letters = interparticle void fraction = intraparticle void fraction ps = structural density of solid, kg/cm3 T = t-z/CTrhermal,time, min
CY
Subscripts C = cold A,B = componentsA,B conc = concentration related i, i + 1 = conditions before and after a concentration characteristic crosses a temperature boundary S = solid thermal = thermal 1,2,3 = refer to conditions at temperatures TI, Tz, T3 Equilibrium Staged T h e o n A = thermal wave velocity defined by eq 9 CP%,, Cps, = thermal conductivities of mobile phase, solid phase and tube or column wall, cal/g "C K = distribution coefficient = conc solute in mobile phase/conc solute in stationary phase M = mass of solute in stage, g T = temperature, "C VV, V. = volume of mobile and stationary phases per stage, ml WI = weight of each stage or portion of column wall assigned to each stage, g f = fraction of solute in mobile phase, defined by eq 10 oh,, p, = densities of mobile and solid phases, g/ml Subscripts A, B, C = components A, B. C i = number of stage S = transfer step number S C , SI, ' 3 2 , '33 = number of transfer steps a t temperatures, TC, TI, Tz,T3
Literature Cited Baker. B . . Pigford. R L.. Ind. Eng. Chem.. Fundam.. 10, 283 (1971) Busbice, M., M . S Thesis. P u r d u e University, 1974.
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
101
Giddings, J. C., "Dynamics of Chromatography, Part I, Principles and Theory," pp 26-35, Marcel Dekker, New York, N.Y., 1965. Pigford. R. L.. Baker, B., Blum. D. E., Ind. €ng. Chem., Fundam., 8, 144 (1969). Sweed, N. H., Wilhelm, R. H.. lnd. Eng. Chem., Fundam., 8.221 (1969). Wankat, P. C., Separat. Sci., 8, 473 (1973). Wankat, P. C., J . Chromatog.. 88. 211 (1974a). Wankat, P. C.. Separat. Sci., 9, 85 (1974b). Wilhelm, R. H., Rice, A . W., Bendelius, A. R., Ind. Eng. Chem., Fundam., 5, 141 (1966).
Zhukhovitskii, A . A , , in "Gas Chromatography-1960,'' W. Scott, Ed., Butterworths. London, 1960.
pp 293-300, R. P.
Received for review J a n u a r y 31, 1974 A c c e p t e d N o v e m b e r 8, 1974 &search was p a r t i a l l y s u p p o r t e d 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. GK-32681.
The Independent Reactions in Calculations of Complex Chemical Equilibria Pehr H. Bjornbom The ROyal lnsfitute of Technology. Department of Chemical Technology, S- 700 44 Stockholm, Sweden
From the atom matrix infinite many sets of linearly independent reactions can be constructed, a priori, which can describe all t h e composition changes d u e to t h e complex reaction. However, these sets cannot automatically be used for equilibrium calculations. T h e independent reactions for the latter purposes might be fewer than those from t h e atom matrix and t h e y are defined by t h e following requirements: ( 1 ) t h e reactions are linearly independent; ( 2 ) t h e y can describe all the available composition changes of t h e system; (3) all t h e extents of reaction can vary independently. T h e number of reactions, defined in t h i s way, must be found from f u r t h e r experimental information than is stored in the atom matrix. This n u m b e r decreases when passivation of molecular processes occurs; for example, a group in t h e molecules being inert. In such cases m a n y complex equilibrium calculation methods, proposed in t h e literature, will fail, especially free energy minimization with element material balances constraints.
Introduction The concept of independent (or distinct) chemical reactions has been introduced in the discussion of complex chemical equilibria (Jouguet, 1921; Aris and Mah, 1963). The equilibrium composition of a chemically reacting system can be obtained by analytical or computational minimization of the Gibbs free energy (1)
G = G(T,P,nj)
where n, (i = I, 2, . . . N ) refers to the number of moles of species i in the system. Zeleznik and Gordon (1968) have reviewed the field of complex equilibria. Let the following T reactions be the independent reactions
C b i j B i= 0 i
( j = 1,2,.
.. , T )
(2)
where B, refers to the compound i and b,, is the stoichiometric coefficient of B , in reaction number j. By definition these reactions are linearly independent. According to reactions 2, the composition changes in the system must satisfy the following equations
[,
where refers to the extent of reaction for the j t h reaction. In matrix notation eq 3 becomes An = BAE
where B refers to the column vector sition). An refers 5r1.v)~ and 5E to 102
(4 )
the matrix, the j t h column of which is ( b ~ , ,bz,, . . . b , ~ , () T~ denotes transpoto the column vector ( A n i , An2, . . . , (551,5 5 2 , . . ., 5 5 ~ )B~ is . called the
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
stoichiometric matrix of the set of reactions 2. The equations 3 define the constraints under which (1) is to be minimized. According to Brinkley (1946), a set of T = N-R reactions can be constructed from the formula matrix (see below), where R refers to the rank of this matrix. This would indicate that there are N-R degrees of freedom in the An, values, if the 55' values are chosen independently. As Zeleznik and Gordon (1968) have pointed out, Brinkley's approach is equivalent to minimizing ( l ) ,using the material balance over the elements in the system as the constraints. A different view is held by Aris and Mah (1963), who implied that the number of independent chemical reactions is an empirical entity; they proposed a method for experimental determination of this number. The purpose of the present paper is to clarify this contradiction in the literature. The Stoichiometric Behavior of a Complex Chemical Reaction Restrictions Due to the Conservation of Atoms. The assumption that a complex reaction stoichiometrically can be described by the reactions 2 must be shown by showing that there are extents of reaction which satisfy eq 3 for all composition changes. This means that it is equivalent to say that (3) can be satisfied for all composition changes and that reactions 2 describe the composition changes. Assume that there are M species of atoms in the system denoted AI, Az, . . . , A.w. The stoichiometric coefficients of a reaction must satisfy the material balances over the atoms partaking in this reaction. Thus if the reaction is xbiBi = 0 i
(5)
