Ind. Eng. Chem. Res. 1988,27, 1066-1073
1066
Composition Profiles in Multicomponent Perturbation Gas Chromatography Marc J. DeBarro,?Sharon B. McGregor, and Charles J. Glover* Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122
Recent work has extended perturbation gas chromatography theory to multicomponent systems and has demonstrated the procedure for determining multicomponent equilibrium isotherms from response peak retention times. For accurate use of the technique for equilibrium and rate measurement and for processes involving membrane and zeolite separations, an understanding of the response curves is helpful. This paper presents a tractable model for calculating multicomponent perturbation gas chromatography elution curves which incorporates both phase equilibrium and interphase mass-transfer rates. Example calculated curves agree well with experimental data and provide insight into peak shapes and their development. Eigenpulse injections can be used to simplify responses experimentally, enabling accurate retention times (for measuring equilibrium) and peak dispersion rates (for evaluating mass-transfer effects) to be obtained.
Gas perturbation chromatography is an effective means of obtaining vapor-liquid and vapor-solid interphase transport properties. The response of a column which is initially at steady state to a composition perturbation, introduced upstream, depends upon the phase equilibrium and upon rate properties such as diffusion within the two phases and axial diffusion and flow dispersion. In principle, by measuring the response composition elution curves which result from a known input (injection) curve, one can determine these equilibrium and rate properties. This basic concept has been recognized for a number of years and exploited in many chromatographic measurements. Early work established these basic principles and demonstrated them for single-component sorption (binary) systems. The rate of propagation of a composition change (peak) through the column is related to equilibrium, and the rate of broadening of a peak is a measure of rate effects. The propagation rate has commonly been reported in terms of a retention volume (corrected to standard conditions of temperature and pressure) per unit mass of the stationary phase. In terms of equilibrium isotherms, this retention volume represents a composition partial derivative of the isotherm and at infinite dilution is directly related to the Henry’s law constant for the system, provided the sample injected into the column is small enough. Analyses with more complicated inputs such as large step changes have also been made for a single sorbing species. If an isotherm is nonlinear to the changes in composition within the column, then the analysis is considerably more complicated in detail although the basic concepts remain intact. In multicomponent systems where there is more than one sorbing species at finite concentration, the analysis is complicated considerably by the isotherm interactions which can occur and by the coupling of the transport equations. Deans et al. (1970) and Helfferich and Klein (1970) laid some of the groundwork in analyzing multicomponent systems and in understanding the interferences which can occur. These efforts provided some general formulations which serve as a starting point for later analyses but did not present end-use equations for multicomponent systems. More recently, Valentin and Guiochon (1976) provided an analysis for ternary systems and Glover and Lau (1983) developed an analysis for general multicomponent systems which yielded end-use equations * Corresponding author.
Present address: Rockwell Intemational, Downey, CA 90241.
for obtaining multicomponent vapor-liquid equilibria. This latter analysis relies upon the assumption of small perturbations so that the isotherms and the transport equations can be linearized about the steady-state condition existing before injecting the perturbation. For this situation, the various composition peaks which evolve travel at velocities which are determined by all the isotherm composition partial derivatives with respect to composition evaluated at the steady-state composition. A sufficient number of experiments at a variety of operating conditions for the column can allow calculation of the various sorption isotherms (Ruff et al., 1986a,b). In all of these studies, the propagation rate of peaks has been used to evaluate equilibrium. The actual peak curves, or peak shapes, have not been used or related to equilibrium. While in binary systems with only one sorbing component the response peak is fairly uninteresting, in multicomponent systems, the responses can be very intriguing indeed. Injecting a component upstream of a column can produce responses downstream which are measured as ”positive” peaks, “negative” peaks, and interesting combinations of positive and negative (Ruff et al., 1986b). This type of behavior is primarily dependent upon the notion of eigenpulses (or eigencompositions)as discussed by Deans et al. (1970), also referred to as coherence pulses (Helfferich and Klein, 1970). In the fully developed response peaks for a given perturbation, the individual species responses in each of the several peaks are related to each other in very specific ways according to the equilibrium isotherm. The collection of these specific concentrations which travel together coherently we will refer to as eigenpulses. Deans et al. (1970) showed that these eigenpulses which exist when true local equilibrium holds also exist for the asymptotic conditions of response of a column, even in the presence of significant masstransfer effects. That is to say that, in the presence of these rate effects, if the peak elution times are long enough (i.e., if the column is sufficiently long or the velocity sufficiently low), then the response observed will be individual peaks, each traveling at its own asymptotic or characteristic velocity, in accordance with equilibrium, and each consisting of eigenpulses or perturbations, also in accordance with equilibrium. These responses must be understood for multicomponent analysis techniques and applications and can provide a very useful consistency check on the equilibrium calculations which use only peak retention time values and not actual compositions. Such an understanding also has im-
OSSS-5SS5/SS/2627-1066~01.~0/0 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1067 plications on understanding and designing a variety of separations processes, such as those using membranes and zeolites. It is the objective of this paper to present results of such elution curves calculated for multicomponent systems. In the model used for these calculations, dispersive rate effects are accounted for by a lumped resistance using a two-phase film theory. Additionally, numerical dispersion mimics axial diffusion and flow dispersion. Results are presented for column responses of (a) a ternary system with one of the two sorbing components at finite concentration, (b) a ternary system with both sorbing species at finite concentration, and (c) a quaternary system. Comparisons of some calculated and experimental column responses are given. Finally, the concept of making eigenpulse injections to the column is considered. Such perturbations have the potential of experimentally producing separate response peaks where otherwise multiple peaks overlay and interfere with the equilibrium analysis. The model used for the numerical study is similar to that of Glover and Lau (1983) but includes mass-transfer effects. The numerical solution of the model generates elution curves for individual components and a combined curve. These elution curves can then be used to examine peak behavior in various systems and the implications on VLE measurement and on more general applications of multicomponent chromatography.
Theory Perturbation gas chromatography involves flowing a gas of constant composition (flowing phase) over a stationary phase until a steady state is achieved. Then an infinitesimal perturbation is introduced upstream and allowed to propagate down the length of the column. The perturbation is small enough that physical parameters of the column may be assumed to remain at their steady-state values. The perturbation in the flowing phase causes a corresponding perturbation in the stationary phase. The perturbation separates into a number of composition perturbations, each moving at its own velocity, which are detected as peaks as they emerge from the column. The number and composition of the peaks depend upon the composition of the flowing phase and composition of the injection and upon equilibrium and mass transfer between the phases. The system of equations used in this work to describe the process begin with those of Glover and Lau (1983) but also incorporate interphase mass-transfer resistance. The flow is modeled as one-dimensional, with the concentration of species i in a phase representing the average concentration over the cross-sectional area. Plug flow is assumed with negligible axial diffusion. The column temperature, pressure, and total flowing phase concentration are constant, and the nonvolatile component of the stationary phase is uniformly distributed. Flowing phase species concentrations are expressed in terms of mole fractions. The stationary-phase compositions are expressed on the basis of moles per mass of inert stationary phase; this dry-basis definition simplifies both the manipulation of transport equations and the results. The flowing phase consists of n components which may, in general, be sorbed by the stationary phase. Transport Equations. With these assumptions and definitions, the transport equations for the flowing phase are
a
a
at
az
- (yica) = -- (yicaV) - ri
i = 1,2,...,n
and for the stationary phase are
axi
m, dt = ri
i = 1,2,...,n
Following the procedure of Glover and Lau (1983), the n flowing phase equations are summed and substituted back into eq 1. The resulting n-1 transport equations for the flowing phase are ca
aYi
- = -caV at
aYi n - ri + y i C r j az j=l
i = 1,2,...,n-1
(3)
If the nth component is an inert and does not sorb into the stationary phase, rn = 0, and only n-1 independent equations are necessary to describe the stationary phase. Modeling interphase mass-transfer resistance by using a two-phase film representation gives the n-1 transport rates as
where hi is the overall mass-transfer coefficient based on vapor-phase mole fractions and y? is the vapor-phase mole fraction which would be at equilibrium with the stationary-phase average concentration. This approach, as opposed to including radial diffusion terms, simplifies the mathematics of the problem and does not affect the equilibrium aspects of the peak behavior which we are concerned with in this paper. Linearization. This system of equations can be linearized about the steady state, taking advantage of the experimental condition of having small perturbations from the steady-state conditions. Accordingly, eq 3 becomes aAy. n-1 a AYi = -c*a*V* 2 c*a* - Ari + yi*CArj at az j=1 i = 1,2,...,n-1 (5) and eq 2 becomes
aax,
m, -- Ari i = 1,2,...,n-1 at The last term in eq 5 represents the effect of perturbations in the carrier gas flow rate and in the gas-phase cross-sectional area which accompany the composition perturbations as the result of the interphase transport of the sorbing species. These flow rate and area perturbations affect the gas-phase composition of species i in direct proportion to its steady-state composition in the carrier gas. This effect on the composition of species i has been termed the “sorption” effect. If that species is at infinite dilution, then the local variations in carrier gas velocity are not responsible for any influx of that species and therefore the sorption effect is absent. For species at finite concentration, however, it can be a significant effect. If the net interphase transport is zero (Le., Cy;:Arj = 0) then the sorption effect would again be zero, even though species i may be present in the carrier gas at a finite concentration. If the flowing phase velocity were slow enough, equilibrium would be approached for a perturbation and the stationary-phase compositions could be directly related to flowing-phase concentrations through an appropriate sorption isotherm, gi, where X i = gi(T,P,y,e,y,e,...,yn-le) (7) If T and P are constant, then for sufficiently small perturbations, AXi = (ag,/ayle);p,y,, Ayle + (agi/aY2e)tTga,,Ayze + ... + (agi/ayn-le);p,y: Ayn-le i = 1,2,...,n (8)
1068 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988
This result relates the Ay; values which result from linearizing eq 4 to the A X i values in eq 6. Matrix Equations. Written in matrix form, eq 5, for the flowing phase, is dAY = -c*a*V* c*a* at ~
-- AR az
+ Y*AR
J = C D E - - (v * I
0 A z o 0
(9)
and, for the stationary phase, eq 6 is
m, d A X -- AR
(10)
AR = H ( A y - A p )
(11)
at
partially, in order to reduce the number of numerical calculations required to solve the (2n - 1)X (2n - 1) matrix at every time step. Let
then if J is made triangular, eq 16 becomes decoupled, leading to an easier solution of the next 2n-1 equations. Let
t=(2)
where (from eq 4) In these equations, A y and AX are ( n- 1)X 1vectors, and
... y* = Yn-1
Yn-1
.
.
**.
Yn-1
A p = B*-'AX
o
) Za ("> AX
(Y* - I)H*
(18)
(12)
where /3* is an ( n - 1) X ( n - 1) matrix of the sorption derivatives for the first n-1 components. If eq 9 and 10 are written together using eq 11 and 12, then m,I
T = L-lJL
where ( = L-'(S). The system is solved in the { domain and transformed back into the t domain. The Crank-Nicholson approximation is used to approximate the time derivative, where
( a g i / a pI ) *T?,Yke
Equation 8 gives A y e in terms of A X
(;*a*'
and where T is the triangular matrix. Then J = LTL-'. Substituting this expression for J and eq 17 into eq 16 and multiplying by L-' yields
Letting
p> =
)
and 0 = 112. This gives the following finite difference approximation:
+ ( c * a * ~ * ~ 0) 0 o az ( AX 0 )-
(I - Y.IH*B*-')( -H*@*-1
ti)
(13)
or
Numerical Procedure The partial differential equation can be solved numerically for Ay and AX by using a finite difference technique. For time step p , the spatial derivative is approximated with a backwards difference expression:
where the subscript j represents the grid point location. Substituting the above expression into eq 14 and multiplying by A-l gives
Equation 16 represents a system of coupled differential equations. It is desirable to decouple this system, at least
Because the matrix equation is made triangular, matrix inversion is not required at every step; thus only n-1 inversions are required to obtain the triangular matrix. The Crank-Nicholson approximation is an implicit method that averages the time derivative over two time steps. Because of this, larger time steps may be used without introducing instabilities into the solution. In order to eliminate instabilities which may be associated with the convection term, a backwards difference formulation was used to approximate the spatial derivative (Basco, 1984). Unfortunately, the backwards difference approximation is only of first-order accuracy. Thus, an artificial diffusion (numerical dispersion) is introduced into the solution. This diffusion is proportional to the mesh size; that is, the finer the mesh, the less numerical dispersion. According to Von Neumann stability criterion, the system is stable for all Courant numbers (V*At/Az). However, at larger time steps, oscillations may occur in the solution (Roache, 1982).
Results and Discussion The perturbation gas chromatography experiment consists of observing the responses of a chromatographic column to an upstream composition perturbation. Prior to perturbation, the column is assumed to have reached a steady-state condition with the carrier gas, which, in a multicomponent situation, consists of a number of components. The stationary phase is in equilibrium with this carrier gas and contains the carrier gas species in amounts
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1069 Table I. Peak Elution Times for the System at 355 K steady-state compositions perturbations (X104) CBH6 0.0000 0.0000 0.0000 0.2000 0.2000 0.2000
C6H12
C6H14
0.2436 0.2436 0.2436 0.1646 0.1646 0.1646
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
(+) Indicates a positive peak.
CBH6 0.0000 1.0000 0.0000 0.8000 -0.2000 -0.2000
C6H1'2 1.0000 -0.2436 -0.2436 -0.1646 0.8354 -0.1646
numerical time, min
C&14 0.0000 0.0000 1,0000 0.0000 0.0000 1.0000
(+Ia
(+)
3.00 2.90 2.05 3.00 3.02 2.10
exptl time, min (+)
(-)b
3.07 3.08 1.98 3.09 3.08 2.16
3.70 3.00 3.75 3.85 3.10
(+)
(-)
3.76 3.14 3.85 3.93 3.14
(-) Indicates a negative peak.
according to their sorption. In this work, one of the carrier gas components is assumed to be nonsorbing. The perturbation can be accomplished simply by injecting a desired component into the flow stream upstream of the column. This injection, being done at constant pressure, displaces an equal amount of carrier gas. Consequently, the perturbation that actually enters the column consists of an increase in the amount of the material injected, superimposed upon negative perturbations (relative to the steady-state composition) of all of the carrier gas components, each in proportion to its steady-state concentration. In a multicomponent system, the simple process of injecting a single component actually results in a multicomponent perturbation. The question that we have addressed in this work is, "how do these multiple perturbations distribute themselves among the n-1 response peaks which are observed in an n-component system?" Our previous experimental work shows that this distribution can occur in a nonobvious way. In our previous work, we have proposed a model and demonstrated the procedure for ascertaining the various species adsorption isotherms in a multicomponent system (Glover and Lau, 1983; Ruff et al., 1986a,b). This procedure uses the propagation velocities or retention times or retention volumes for the various peaks that occur in a multicomponent gas chromatography experiment. It should be emphasized that it is the velocities of the peaks which are used in this procedure and that the actual response peak shapes or the individual species concentration curves are not necessary for the calculations. Experimentally, in finite concentration systems, upon injecting a given species into the column, one may observe positive response peaks, negative response peaks, or both, depending upon the systems studied and upon the sorption isotherms. One such example is reported in Ruff et al. (1986b). In this work we take the isotherms thus determined and use them in the model outlined above to calculate detailed species composition curves or column responses as functions of position and time. From these species compositions, we can calculate peak propagation velocities and total composition perturbations (the sum of the individual species) for comparisons with the experimentally measured times and responses. Elution Curves. In this work we report data and make numerical calculations for two steady-state compositions, (1) finite in cyclohexane and (2) finite in both cyclohexane and benzene. Calculations and data for the finite benzene concentration system are not reported, in that they are similar to those of cyclohexane. For both of these situations, column responses were calculated for injections of cyclohexane or benzene or for n-hexane, a third sorbing species which is at infinite dilution. To obtain the elution curves, the finite difference routine was implemented using the initial perturbation sizes in Table I. Physically injecting a species upstream of the column displaces a like amount of carrier gas which is at the steady-state composition. If the carrier gas contains some of the species
I
Time
Figure 1. Experimental chromatograms for system 1 when perturbed by injecting n-hexane (n-H), benzene (B), cyclohexane (C), or air (A).
C
1
Time
Figure 2. Experimental chromatograms for system 2 when perturbed by injecting n-hexane (n-H), benzene (B),cyclohexane (C), or air (A).
which is injected, then the injection of that species is reduced in accordance with this displacement and composition. Numerically, this process is modeled using the net injection sizes in Table I. The individual response curves are Ayi divided by the initial total (not net) perturbation of the injected component. The total resize (1 X sponse curve is the s u m of the individual response curves. In this work, the individual response curves are not scaled for differences in detector sensitivity to the individual species, as would be required for more quantitative comparisons. The two systems at 1 atm and 355 K have the following steady-state compositions: system 1, y c A = 0.0, Y C ~ H= , ~0.2436, Y C ~ H= ~0.0; ~ system 2, YC,& = 0.2, Y C ~ =, 0.1646, yc6Hl4 = 0.0.
~
--
1070 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 040
0 35
I
I
ia
0301
W
0 25
0 25
v,
z
E?
020
0 20
W v)
OI
z
I
0 15 0 10
(r
3
k
0 05
W
a
0 00
0 10 -005 -015
I 1 00
-0 15 10
30
20
40
50
I 00
60
10
20
Figure 3. Superposition of peak elution curves for system 1. Curve a, n-hexane injection; curve b, benzene injection; curve c, cyclohexane injection.
040 035
-0
to
-Oo5 015
040
la
I 00
40
50
60
Figure 5. Species response curves for a positive benzene injection in system 1. Curve a, benzene; curve b, cyclohexane. Curves a and b sum to give curve b in Figure 3.
7
t
30
TIME (MIN)
TIME (MIN)
7
-005 -010
I 10
20
30
40
50
60
TIME (MINI
Figure 4. Superposition of peak elution curves for system 2. C w e a, n-hexane injection; curve b, benzene injection; curve c, cyclohexane injection.
Experimental responses for these systems are shown in Figures 1 and 2. For system 1 (Figure 1) when cyclohexane is injected, a single response is observed since this is actually only a binary system. This result is easily understood, and the perturbation represents simply the variation of the cyclohexane concentration about its steady-state value. On the other hand, when either nhexane or benzene is injected, two response peaks are observed; for n-hexane one of these is positive and the other negative, whereas for benzene injection both response peaks are positive. Of the two peaks which elute in these instances, one elutes at the same velocity and therefore at the same time as does the response peak for the cyclohexane perturbation. For system 2 (Figure 2), each injection, including cyclohexane, produces two response peaks. Figures 3 and 4 show numerically calculated response curves for the situations of Figures 1and 2. The calculated and experimental curves agree quite well, a strong indication that the model contains the essential features of the
I
-015
00
10
20
30
40
50
6C
TIME (MINI
Figure 6. Species response curves for a positive n-hexane injection in system 1. Curve a, n-hexane; curve b, cyclohexane.
physical situation. Qualitative similarities are striking, and differences are in quantitative detail only, likely arising from differences in the detector sensitivity to the various species. The agreement in the calculated peak elution times with the experimental values (Table I) is within experimental error. A rational understanding of the nature of the response curves in a multicomponent environment can best be obtained by examining individual species curves in the context of eigenpulses. For system 1, Figures 5 and 6 give individual species curves for injections of benzene and n-hexane, respectively. Cyclohexane is the only species at finite concentration; therefore, it possesses the only non-zero off-diagonal components in the equilibrium matrix ( B c ~ H- c~ ~ Hand ~ ~ +6H11-C6H ), has the only non-zero sorption ejfect term ~ C 6 H 1 2 and ~is; the i $ only ~ ) ,sorbing species which suffers a perturbation in its steady-state composition simply by injecting another species. Figure 5 reveals a single benzene peak accompanied by both a positive peak and a negative peak of cyclohexane. The benzene pulse carries along a negative cyclohexane
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1071 0 40 0 35
0 30
u
025
0 25
020
0 20
015
0 15
010
0 10
005
0 05
000
0 00
-0 05
-0 05
?
2
m W
E 2
g
LT
3 k
a W
a
I
-0 10
-O -0 l15 o 00
- 0 15 00
10
20
30
40
50
60
10
20
peak in the amount dictated by the concept of eigenpulses. Then, to satisfy overall cyclohexane conservation, a positive cyclohexane peak is induced and travels at the cyclohexane peak velocity. The two peaks do not exactly cancel in area because some cyclohexane is displaced by the benzene injection, resulting in a net negative cyclohexane perturbation. Summing these benzene and cyclohexane curves gives curve b in Figure 3. Figure 6 illustrates species responses for an injection of n-hexane. In this case, the n-hexane carries along a positive cyclohexane peak (again in accordance with eigenpulses), inducing a larger negative cyclohexane peak (to satisfy the cyclohexane balance) at the cyclohexane elution time. The induced negative cyclohexane peak again comes out at the elution time for pure cyclohexane. Summing these two species curves gives curve a in Figure 3. With two sorbing species at finite concentration, as there are in system 2, the situations become more complicated. Now only BC H , ~ - c ~ Hand ~ Bc~H -~ c ~ and H~~ Y L ~ Hare~ ~identically zero. Tke result is that the peaks induced to satisfy mass balances will now carry with them an appropriately sized peak of the other component. Figure 7 illustrates the response to a cyclohexane perturbation. A positive cyclohexane peak carries along a positive benzene peak in accordance with eigenpulses. The mass-balance-induced negative benzene peak now carries a positive cyclohexane peak (in accordance with eigenpulses) which also affects the amount of cyclohexane traveling in the first peak through the mass balance. Figure 8 illustrates the result of a benzene injection. Again, a benzene perturbation must carry along with it an eigenpulse-satisfying cyclohexane perturbation, and vice versa. However, the positive-induced cyclohexane peak now carries a positive benzene peak. Summing the individual responses in Figures 7 and 8 gives curves c and b in Figure 4. Calculations were made to examine the behavior of the column if all species were present. System 2 was subjected to a perturbation in n-hexane, producing peaks traveling at three velocities. Now, with coupling between benzene and cyclohexane, induced peaks of each of these species also must carry along perturbations in the other. Figure 4 (curve a) gives a summed curve of the eluting peaks. Note that the last peak (centered at 3.8 min) is hardly noticeable. This is made clear by examining the species curves given in Figure 9. The n-hexane peak carried along
40
5 0
50
TIME (MIN)
TIME (MINI
Figure 7. Species response curves for a positive cyclohexane injection in system 2. Curve a, cyclohexane;curve b, benzene.
30
Figure 8. Species response curves for a positive benzene injection in system 2. Curve a, benzene; curve b, cyclohexane. 0 40 0 35 0 30
025
?
8
020
2 2
Q
015
+
2U
0 10
+ a
005
a
000
3
U
-0 05 -0 10
-0 15
1
00
10
20
30
40
50
60
TIME (MINI
Figure 9. Species response curves for a positive n-hexane injection in system 2. Curve a, n-hexane; curve b, benzene; curve c, cyclohexane.
positive benzene and cyclohexane peaks of approximately equal magnitudes. The induced negative cyclohexane peak (centered at 3.1 min) carries along a negative benzene peak, resulting in a strong second peak. On the other hand, a perturbation in benzene eluting at 3.75 min will carry along a cyclohexane peak of opposite sign, resulting in a very weak third peak. Eigenpulse Injections. The experimental situation may arise in which two peak’s retention times are close enough to be difficult to measure accurately due to peak overlap. In these instances, it would be desirable to be able t o ndectivelv eliminate or at least, att,enuate the indnced
peaks in the column so that a more accurate characterization of the perturbation could be achieved. The following discussion considers this possibility. Consider a ternary system which is finite in one or both of the species. Injecting one of the species into the system will result in two peaks eluting from the column at the characteristic elution times of each species. Depending upon the coupling in the equations, each peak may contain one or both of the species. Based on the theory of Glover
1072 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988
W
040
I
0 30
I
035 0 25
benzene and numerical error. A peak of this sort, if achieved experimentally, would be quite acceptable for calculating a retention time.
(0
z
B
0 20
W (0
a
z 0
0 15
2
0 10
3
0 05
E
k
Lu a
0 00
.n,F. “
I
I
00
10
20
30
40
50
60
TIME (MINI
Figure 10. Peak elution curve for system 1subject to a benzene and cyclohexane eigenpulse injection. 0 40 0 35
0 30
Acknowledgment
CZ5
Financial support for this work, provided by National Science Foundation Grants CPE-8111272 and CBT8615703 and by the American Association of University Women is gratefully acknowledged.
(0
z
g 2 z
020 0’5
0
i
01°
U
3
Conclusions The perturbation gas chromatography technique has been extended in recent years to multicomponent systems. In these systems, column behavior, in response to upstream perturbations and in terms of measured elution concentration curves, can be quite complex. Although the details of these response curves are not necessary for calculating isotherms, in principle, they can be very useful, or even essential, in practice. This is especially true when multiple response peaks overlap and present complications in measuring the retention times. The model presented in this work provides the means of calculating or predicting column responses. The calculations for specific systems agree quite well with experimental responses, indicating that the important features of the physical problem are incorporated in the model. An important feature of the model is that it shows that, if an appropriate perturbation is made to the column, then interfering response peaks can be reduced or eliminated, so that accurate retention times, which are necessary for calculating sorption isotherms, can be measured accurately.
005
t;
0 15
L TIME (MIN)
Figure 11. Species response curves for system 1 subject to a benzene and cyclohexane eigenpulse injection. Curve a, benzene; curve b, cyclohexane.
and Lau, and following the work of Huang (1984), the eigenpulse for system 1for a peak which contains benzene must obey Figures 10 and 11 provide calculated curves for this situation. In this instance, a “negative” quantity of cyclohexane needs to be injected into the column to cancel out the positive-induced cyclohexane peak. This is accomplished by injecting an inert along with the benzene into the column such that the appropriate amount of cyclohexane is displaced. This displaced cyclohexane would appear as a negative peak on the chromatogram. This negative amount of cyclohexane will travel along a t the characteristic velocity of the cyclohexane along with the positively induced peak. If the input composition is correct, then these two peaks will cancel out, eliminating the induced peak completely. For system 1, the cyclohexane peak was never fully eliminated although it is greatly attenuated. This is probably due to a combination of both the sensitivity of the cyclohexane to perturbations in
Nomenclature A = total interfacial area a = flowing phase cross-sectional area c = total flowing phase molar concentration, a constant g, = sorption isotherm for component i H, = mass-transfer coefficient of component i, H , = h , A / L H = ( n - 1) X (n - 1) diagonal matrix of H,’s h, = overall vapor side mass-transfer coefficient for species i I = (n - 1) X ( n - 1) identity matrix L = column length L = ( n - 1) X ( n - 1) submatrix of A m, = amount of inert stationary phase per column length n = total number of flowing-phase components P = column pressure AR = (n - 1) X 1 vector of sorption terms r, = net rate of transfer of component i from the flowing phase to the stationary phase, i = 1,...,n T = column temperature T = transformation matrix t = time At = time step in finite difference algorithm V = flowing phase molar average velocity X , = sorption of component i (moles sorbedlm,), i = 1,...,n AX, = deviation from steady-state of X, AX = ( n - 1) X 1 column vector of the AX, Y* = ( n - 1) X ( n - 1) matrix of stationary-phase mole fractions y , = mole fraction of component i in the flowing phase, i =
1,...,n y I e = mole fraction of component
i in the flowing phase in equilibrium with the stationary phase y * = (n - 1) x 1 column vector of mole fractions Ay, = deviation from steady state of y , A y = ( n - 1) X 1 column vector of Ay, z = distance in flow direction Az = spatial step size in finite difference algorithm
Ind. Eng. Chem. Res. 1988,27, 1073-1084 Greek Symbols
/3 = (n - 1) x (n - 1)matrix of sorption derivatives { = transformed vector = 2(n - 1) X 1 vector of A y and AX
Superscript * = steady-state value
Literature Cited Basco, D. R. “Introduction to Computational Fluid Dynamics”. Ocean Engineering Program, Texas A&M University, College Station, TX, 1984. Deans, H. A.; Horn, F. J. M.; Klauser, G. AIChE J. 1970,16, 426.
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Received for review July 21, 1987 Accepted February 8, 1988
An Equation of State for Electrolyte Solutions. 1. Aqueous Systems Containing Strong Electrolytes Gang Jin and Marc D. Donohue* Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218
A new equation of state for mixtures containing electrolytes has been derived using perturbation theory for the two fundamental interactions between various species in solution: short-range interactions between particles and long-range Coulombic interactions (charge-charge interactions) between ions. In this equation, short-range interactions between molecules are calculated by using the Perturbed-Anisotropic-Chain theory (PACT) of Vimalchand and Donohue. A perturbation expansion based on Henderson’s restricted primitive model is used for charge-charge interactions between ions. The solvation effects caused by charge-molecule interactions near ions are taken into account through the dielectric constant. Additional new expressions, a third-order perturbation expansion for charge-dipole interactions and a first-order perturbation expansion for charge-induced-dipole interactions, are derived here for the interactions of ions with molecules in the bulk of the solution. Preliminary calculations involving mean ionic activity coefficients for 50 strong electrolytes in water, specific volumes of several binary electrolyte solutions, and K factors for argon and methane in aqueous solutions of NaCl show the usefulness of this new equation of state. In these calculations, only one adjustable parameter C, is used over a range of molarities from infinite dilution to 6 M. Average absolute errors are less than 6% for activity coefficients for most binary systems, 2% for specific volumes of two binary systems, and 5% for K factors for the ternary systems.
I. Introduction Electrolyte solutions, especially aqueous electrolyte solutions, are common in the environment, living organisms, and industrial processes. Their thermodynamic properties play an important role in separation processes in chemical and petrochemical refining, purification processes in environmental engineering, and fermentation processes in biochemical engineering. Because the applications are numerous and important, the interest in prediction of thermodynamic properties of electrolyte solutions has grown over the past decade. A number of important models have been developed for electrolyte solutions in recent years. For example: Meissner and Tester (1972) proposed an empirical model to calculate mean ionic activity coefficients; the model of Pitzer (1973, 1979) is widely used (Edwards et al., 1975, 1978); and the NRTL model has been applied to ionic solutions by introducing the Debye-Huckel term for long-range Coulombic interactions (charge-charge interactions) into the theory (Cruz and Renon, 1978; Renon, 1985; Chen and Evans, 1986). The utility of these empirical or semiempirical models is well established. Several theoretical models also have been developed for electrolyte solutions. One recent advance in statistical
* Author to whom correspondence
should be addressed.
mechanical treatment of ionic systems is the application of perturbation theory. Henderson and Blum (1980,1981) and Henderson (1983) derived a perturbation expansion for charge-charge interactions based on a restricted primitive model. Later, they extended their work to a mixture of dipolar hard spheres and charged hard spheres and proposed a nonprimitive model (Henderson et al., 1986). Though this nonprimitive model (in which the dielectric constant is taken as unity) is based on a more fundamental picture of an electrolyte solution, it cannot be used for actual calculations of electrolyte behavior because not enough is known yet about the details of the physical structure of the fluid. In primitive models, interactions between ions and molecules very near ions (solvation effects) are taken into account through the dielectric constant. However, even in a primitive model, interactions between ions and bulk molecules must be considered explicitly; it is not adequate to assume that all effects of ion-molecule interactions over all ranges are accounted for by the dielectric constant. There are many other primitive models for electrolyte solutions which include ion-molecule interactions (Edwards et al., 1975,1978; Cruz and Renon, 1978; Chen and Evans, 1986). A very good review of both theoretical and empirical models for electrolyte solutions has been given by Renon (1986). The purpose of this work is to present a new model for electrolyte solutions by adding contributions for mole-
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