Ind. Eng. Chem. Res. 1995,34, 2625-2633
2625
Effects of Nonlinearity in the Theory of Multicomponent Chromatography Anatolyi Kalinitchevt Institute
of
Physical Chemistry, Russian Academy of Science, Lenin Avenue 31, GSP 117915 Moscow, Russia
A number of properties connected with nonlinearity of multicomponent sorption systems are theoretically investigated. Two types of such systems are considered: system I, a Langmuir p-component sorption system defined as the system with constant separation factors of the components and uniform flow velocity; system 11, a system of n-component with the so-called “sorption effect” and linear sorption isotherms of the substances. By use of the ideal equilibrium model for this system, the relation is deduced that describes the dependence of the flow velocity on the concentrations of the components. With this relation, it is established that the principles of calculations of both shock and diffuse concentration waves in system I1 are formally the same as in system I (Langmuir p-component system, p = n - 1). The theoretical descriptions of concentration profiles across the stationary waves have been obtained for systems I and I1 including nonideality factors. It has been proved that the dependence between concentrations of components in a constant-pattern profile is linear (and coincide with composition path grid) in concentration space when the values of the nonideality factors for each of the component are the same. In this case, the analytical solutions have been obtained that describe the concentration profiles of the components.
Introduction Taking into account that this issue is published in honor of Prof. F. Helfferich and besides the topic of this paper concerns multicomponent theory, the author would like to stress the oustanding contribution of Prof. Helfferich to the advance of nonlinear theory of multicomponent chromatography. His publications in this field, written in the 1970s and 1980s, enhance the theory to significant extent. Prof. F. Helfferich introduced into the theory of multicomponent chromatography a number of fundamental conceptions, which play a crucial role in the description of the dynamics of multicomponent systems: the interference of the species, the description of the behavior of the systems in terms of “concentration waves”, and, finally, the concept of “coherence”of the multicomponent chromatographic systems (Helfferich, 1967; Helfferich, and Klein 1970). Prof. F. Helfferich put in operation the so-called “htransformation” (in other words, transformation to Rieman invariants), which is suitable for the equilibrium model of chromatography with constant separation factors. This transformation orthogonalizes composition path grids in composition space. Accordingly,the simple and clear interpretation of the behavior of the nonlinear multicomponent chromatographic systems has been obtained without solutions of differential equations of mass balance. The problem is solved on the basis of the solution of algebraic equations through the so-called “h-functions”, The main achievements of nonlinear theory of multicomponent chromatography were observed and generalized by Prof. F. Helfferich in his state-of-the-art report (Helfferich, 1986a) with conclusions concerning recent applications in multicomponent systems. Successful solutions of the problems of both “interference” of species and concentration waves become possible with description of the multicomponent system behavior in terms of concentration waves including the concept of “coherence”. The concept of coherence +
E-mail:
[email protected].
(Helfferich, 1967, 1968, 1984; Helfferich and Klein, 1970) has been developed t o facilitate the qualitative understanding and quantitative calculations of multicomponent chromatography, particularly under arbitrary initial and feed conditions. His works in the 1980s dealt with conceptual understanding and generalization of the fundamental coherence concept (Helfferich, 1984, 1986a,b, 1989). This key and general concept is a fundamental concept, which is not restricted t o the systems with constant separation factors. It is applicable to a wide variety of multicomponent nonlinear systems including, for example, multiphase flow in porous media (Helfferich, 19811, continuous countercurrent mass transfer processes (Hwang and Helfferich, 19891, chromatographic reactors, and nonsteady state distillatioon (Nandakumar and Andres, 1981). For two-component systems and with equilibrium theory, a rigorous proof of attainment of coherence state, based on the method of characteristics, has been given by Helfferich (1986a,b),although the most rigorous proof for development toward coherence state in systems with quasi-linear partial differential equations was presented previously in mathematical works (Rozhdestvensky, 1960; Liu, 1977). The phenomenological theoretical approach t o multicomponent fured bed column performance is used in this paper. This approach includes the solution of the appropriate differential nonideal material balance equations under specified simple initial and feed conditions: so-called “frontal and frontal-displacement sorption dynamics” (Rachinsky, 1964). In frontal-displacement analyses, a column presaturated with a mixture of arbitrary composition {cj, ..., c,} receives as influent a mixture of another composition {ci, ..., ci}. The purpose of the present paper is t o demonstrate a number of properties of multicomponents chromatographic systems, which are connected with their nonlinearity and with interference effects of sorbable species. These properties are the consequences of the analytical solutions followed below for the system of differential mass balance equations, which describe the
0 1995 American Chemical Society 0888-5885/95/2634-2625$09.00/0
2626 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
behavior of nonlinear multicomponent chromatographic systems, including nonideality factors. These solutions are presented in the second part of the paper. They yield the constant-pattern profiles in multicomponent concentration waves, which correspond to shock waves in ideal equilibrium theory. The behavior of a solute molecule of any component in a dispersed phase of a chromatographic column is affected by the presence of molecules of other species through their competition for the limited number of available sorption sites in the sorbent. Thus the various species will “interfere” with one another in the column especially at high concentrations of the solute. The term “interference” was relevantly used in the book by Helfferich and Klein (1970) for the mutual influence of the sorbable components in dispersed phase of the column. It is interesting to note that the same term “interference”is used in thermodynamics of irreversible processes for the description of mutual influence of coexisting (co-working) cross effects (Glensdorf and Prigogin, 1973). In addition, there is another direct aspect of interference that was also discussed and solved partly (Helfferich and Klein, 1970; Helfferich 1984, 1986a): collision (or crossover) of concentration waves of various kinds in the column during the chromatographic process. The inclusion of the interference into the multicomponent theory leads to simultaneous partial differential equations of mass balance. Any perturbation in initial and feed conditions of the multicomponent system generates the complex response, which is described adequately in terms of multicomponent concentration waves (Helfferich and Klein, 1970; Helfferich, 1967, 1984; Kuznetsov, 1967; Rhee et al., 1970). The style of propagation, broadening, and interference of the multicomponent waves depends on the types of sorption multicomponent isotherms (static factors) and on various nonideality factors (disturbances) describing the column efficiency (deviation from local phase equilibrium, nonideal flow pattern, etc.). The nonideality factors smooth the effect of the nonlinearity factors in multicomponent systems (Helfferich and Klein, 1970). For example, at favorable stability conditions, instead of shock waves, asymptotic steady state of propagation of the concentration waves with finite widths can be formed, which will be considered here. However this smoothness cannot change principally the nonlinear character of the static interference factors influence (Helfferich and Klein, 1970).
Concepts and Basic Equations in Chromatographic Theory The theoretical approach in this paper deals with the general problem of percolation of a multicomponent mobile phase through a dispersed phase with transfer of sorbable species between two phases. The paper discusses the effect of nonlinear multicomponent equilibria in the propagation dynamics in k e d bed sorption processes. The multicomponent nonlinear model presented here assumes differential formulation including (1)axial dispersion, (2) temperature-independent equilibrium with constancy of separation factors, and (3) a constant mass transfer coefficient for each component based on overall resistance across the sorbent phase. The premises of the model are the systems are isothermal, the flow is unidimensional, the fractional
pore volume of the medium is uniform and constant, and the mobile phase may be liquid or gas. The nonlinear system of mass balance partial differential equations, describing the movement of sorbable components in a sorbent, including dispersion effects (longitudinal dispersion) and a nonuniform flow velocity, has the form (Rachinsky, 1964; Helfferich and Klein, 1970)
(1) Mass exchange kinetics in the sorbent is described here by the phenomenological kinetic equations on the basis of the models of mass transfer control with linear driving force (Glueckauf, 1955, Helfferich, 1962):
(a) for intraparticle diffusion control
or (b) for film diffusion control
G m =fm- 1
m = 1 , 2 , . . . , p (2b)
where f m ( C ) are the competitive multicomponent isotherm equations and Gm(a)are the reverse competitive isotherm equations. In the condition of negligible nonideality factors, the theory of chromatography is described on the basis of ideal equilibrium theory, which has been extensively worked out in many works, especially for Langmuir multicomponent competitive sorption isotherms (Helfferich, 1967; Klein et al., 1967; Tondeur and Klein, 1967). The bibliography of the works is presented and discussed in detail in a number of publications (see, for example, the book by Helfferich and Klein (19701, the state-of-the-air report (Helfferich,1986a1,and the paper by Lin et al. (1990)). Recent application of the theory of ideal elution chromatography to the competitive Langmuir two-component system has been presented recently (Golshan-Shirazi and Guiochon, 1989). As mentioned earlier, the main factors that influence the behavior of concentration waves in the equilibrium theory are connected with the competitive multicomponent sorption isotherms fm(C). In this case, the interference of the species causes the nonlinear effects in the propagation of concentration waves, which cannot be principally obtained by the superposition of the solutions of the single-component mass balance equations (Hewerich, 1986a). An important mathematical tool for constructing solutions of the multicomponent system is the characteristic equation, which can be derived from the mass balance differential equations. The solutions of this equation (eigenvalues &, (k = 1, 2, ..., p ) of the corresponding concentration matrix) play an essential role in the description of the multicomponent system behavior in terms of concentration waves. The eigenvalue & determines the eigenvelocity of the corresponding multicomponent wave, which will be denoted here by the term “k-wave”. In the composition space (c1, ...,
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2627
cp}, the characteristic 12-direction connects concentrations of the components across the multicomponent 12-wave. The characteristic directions arise in the process of the solution of the multicomponent mass transfer problem. By the use of basic principles of thermodynamics and through the stability properties of multicomponent thermodynamic systems, it has been proved (Kvaalen et al., 1985)that, for physically plauible systems, these directions (and corresponding eigenvalues ;Idare always real, distinct, and in number equal to the variance (number p , here) of Gibbs phase rule: ;I1 < A2 < ... ;Ip. Various shock or rarefaction multicomponent 12-waves (lz = 1,2, . . . , p may travel through the column. In this case, the equations of the ideal model represent mathematically a first-order quasi-linear and totally hyperbolic system. The problem, which is described by the system of hyperbolic equations with initial and feed compositions, corresponding to the frontal or frontal-displacement types of chromatography, is named “the Rieman problem” in the more refined language of mathematics (Lax, 1957) or “the splitting of the initial shock in Russian mathematical literature (Rozhdestvenskyand Janenko, 1968). The solution of the Rieman problem is essentially simplified when the functions of special type (invariants) are found. Functions of such types, Rieman invariants (Helfferich, 1967; Helfferich and Klein, 1970; Kuznetsov, 1967)and general Rieman vector invariants (Rhee et al., 1970; Rhee, 19811,have been found for the problem of nonlinear multicomponent chromatography. On the basis of the ideal equilibrium model, it has been shown (Kuznetsov, 1967; Helfferich and Klein, 1970; Rhee et al., 1970; Rhee, 1981) that Lax’s stability conditions (Lax, 1957) are fulfilled for propagation of the shock waves in the competitive Langmuir multicomponent sorption systems. Accordingly, the steady state of concentration waves will be asymptotically attained in the nonideal nonequilibrium model of the chromatographic process for definite relations between initial and feed compositions of a mixture in a column. The system of partial differential equations (1)and (2a) or (1) and (2b) will be considered below for two particular types of nonlinear multicomponent chromagraphic systems: System I: A multicomponent Langmuir sorption system (constant separation factors of the competitive components and uniform flow velocity U):
U uniform; f,(C) = L,C,N,
V=1
m = 1 , 2, . . . , p (3)
m
For the parameters of the reverse competitive Langmuir multicomponent isotherm equations G,(a), the relations
U = 1N; I,
= l/L,;
g , = -L,/b,,
CC, = C, = m
constant
m = 1,2, ..., n ( 5 )
The second relation in (5) assumes negligible pressure drop and validity of the ideal gas laws in gas chromatography. The sorption effect was first reported by Bosanquet (Littlewood, 1962) and has been studied in various contexts (Zhuhovitsky et al., 1965;Peterson and Helfferich, 1965; Valentin and Guiochon, 1975). The sorption effect is essential for gas chromatography and is usually neglected in liquid chromatography, though in the last case the second condition in (5) describes the condition of incompressibility with conservation of summary concentration COof all n components. In liquid chromatography the effect is usually negligible, because the net volume change of the combined mobile and stationary phases upon sorption-desorption tends to be quite small.
Results and Discussions For the ideal model of the so-called “equilibrium theory” (at D m = 0; 1/pm = 0 or l/cm= 0), the system of equations (1) and ( 5 ) is transformed t o a quasilinear system of differential equations
where (7) is obtained by summarizing (6) with indexes m = 1, ..., n. The purpose of the first part of the paper is to show that the principles of calculations of both shock and rarefaction (diffuse)concentration chromatographicwaves are formally the same as in system I (competitive Langmiurp-component system, p = n - 1). The fundamental differential equation of the Rieman problem (Rozhdestvensky and Janenko, 1968; Rhee et al., 1970; Rhee, 1981) is used here for the systems (6) and (7)
+
P
u = 1 + &?,a,
n
u nonuniform; ~,cc) =y,~;
m = 1,2, ...,p
(4) are valid, System 11: A system of n-components with linear sorption isotherms of the species including of the socalled “sorption effect”:
where A is the characteristic velocity. The species are numbered here in the sequence of decreasing affinity for stationary phase: y1 > y2 > ... > y m > ... > yn. In view of condition (5), only ( n - 1)functions C , can be varied independently, therefore one function (here Cn) is excluded from (8). After rearrangement of (8) and with differentiation with respect t o R, the relations P
e, - e,,
= J,(R - R,); R = ynCo + &,;
Ro =
m
m = 1, 2, ... , p (9) can be deduced (Kalinitchev, 1985a). Constants e,o, J,, and Ro are constants of integration, which gives relations (9) from the transformed equations (8).
2628 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
The system of relations (9) describes “straight lines” passing through the fixed point { e m 0 1 in ( n - 1)dimension concentration space. Such topology of composition path grids is presented by Helfferich and Klein (1970) and Helfferich (1981, 1986a, 1989). Figure l a shows one of the examples of such topology for twocomponent Langmuir system. After substitution of relations (9) in (8) the parameter p = u / A can be written as
where 1 < j ,m < p. Constant H,derived from (9) and (lo), is not dependent on number m:
This constant can be expressed through the parameter
J, by use of (9)-(11): P
+
H = 1 y n + C y m J m m = 1 , 2 , ...,p (12) m
/
Summation of relations (10) over indexes m = 1, 2, ..., p gives the first relation in (13). Then, expressing the parameter Jmthrough the factor p from (lo), the characteristic equation for eigenvalues pk = U / A k of the system of differential equations (6) can be obtained (second relation in (13)) D
D
D
m
m
m
Jmk
= f?,/((l
-k Y m
- pk)co)
k
= 1 , 2 , . . . , p (13)
After substitution of the relation (9) into (8) and with integration of the obtained differential equation, the relation for the nonuniform flow velocity u is derived:
u = Q/(H - R/Co)= Q/p; d(l+ y m ) = Qm4H - R/Co) (14) where Q is constant of integration, which is dependent on initial conditions. The derived relation (14) presents the dependence of the flow velocity u on the concentrations o f p components of the mixture (Kalinitchev, 1985a). Taking into account the relation (141, the system of differential equations (6) coincides with the system of mass balance equations describing Langmuir system I
-+ at aE,
a(ue,/(l+ y m ) )
i h
=0
m = 1,2, . . . , p (15)
where the dependences Uem/(l+ Y m ) = Q m e m / ( H - R E O ) play in (15) the role of the multicomponent Langmuir reverse isotherms (3). The characteristic equation (13) coincides formally with the corresponding equation of multicomponent Langmuir system I, where the separation factors YmlYi of the components m and i are the same as the separation factors amiused in theory by Helfferich and Klein (1970). The invariants pk/y1 have the same features as the invariants l/hk, where hk-functions
Figure 1. Composition routes and profiles in frontal-displacement pattern for two-component competitive Langmuir system. (a) Simplex P1PzP3 with composition path grid for parameters: LI= 9, LZ = 2, bl = 2, bz = 1 in the Langmuir equations (3); W, watershed point; -, “slow” paths, which correspond to 1-waves; - - - ,‘‘fast’ paths, which correspond to %-waves. Composition route PzMS corresponds to the frontal-displacement pattern presented in (b). (b) Concentration rarefaction 1-wave (PzM, - -1 and selfsharpening 2-wave (MS, -) calculated at the moment 2’ = Ct = 60 on the base of ideal equilibrium model. Velocities u11= 0.222, U I Z = 2, and w z = 3; plateau zones CZI= 1,C11 = 0 (Pz), CIZ= 2.286, CZZ= 0.428 (M), c13 = 1, c23 = 0 (SI. (c) Dependence N~k(Nlk) calculated from numerical distributions Ci(x)a t the moment T = {t = 60 and analytical solution, eq 23 (-); nonequilibrium film diffusion model (at 51 = 5 2 in eqs 2b). ( * * a )
according to the procedures presented in multicomponent theory by Helfferich and Klein (1970), taking into account the correspondence of the separation factors am1 = ym/yl ( m = 1, 2, ..., n). The composition path grid for system I1 has the same features as in system I (example of path grid for twocomponent system is presented in Figure la). Arrows in Figure l a show the characteristic K-directions, in which the eigenvelocity dk (and corresponding hk function) encreases. The invariant hk and vector invariant {Jkk} are changed but invariants h, ( mf 12) and vector invariants { J m k } (m f k) are conserved across the characteristic k-direction in the composition path grid. The corresponding change of the hk-invariant determines the type of multicomponent K-wave (shock or rarefaction) in the chromatogram. A large number of examples of the composition path grids for multicomponent Langmuir
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2629 systems are presented in the book by Helfferich and Klein (1970). The identical examples of composition path grids on simplex PI, ..., P, attributed t o system I1 in composition space {CI, ..., C,) can be considered. The concentration distributions across k-waves together with the velocities of shock or rarefaction k-waves (for example, Figure lb) are described by use of the invariants (hk-functions)according to the relations presented in the book by Helfferich and Klein (1970). These conclusions establish the first part of the sought result. It is known that when the definite stability conditions are fulfilled the steady state of the moving concentration waves is attained asymptotically. In this state, the sharpening effect of sorption equilibrium and the dispersive effects of disturbances balance one another, so that the width of the k-wave and the advancing concentration profiles remain unchanged from then on (Rachinsky, 1964;Helfferich and Klein, 1970). This, the so-called “constant pattern” of a self-sharpening wave, is a stable state. Then the stationary multicomponent k-wave with constant-pattern profiles travels through the column. The criteria of attainment of the steady state have been analyzed (Kuznetsov, 1967; Helfferich and Klein, 1970; Rhee et al., 1970) for multicomponent Langmuir sorption systems on the basis of the general theory of quasilinear hyperbolic equations. The relations, which describe the velocities of the shock and rarefaction (diffuse) concentration waves through Rieman invariants (hk-functions),are obtained in the book by Helfferich and Klein (1970). The criteria of establishing of shock k-waves (expressed through hk-function) were derived from these relations. Stationary waves with finite widths and constant-pattern profiles will be formed in the column instead of shock waves, when the nonideality factors are included into consideration. Let us consider the multicomponent concentration stationary waves. At the steady state, the mass balance equations are transformed into usual differential equations by the Zeldovitch-Todes approach. Such equations are presented in the book by Rachinsky (1964) for the multicomponent frontal (or frontal-displacement) sorption dynamics at the steady state. With transformation to a moving coordinate system (z = x - Wkt) and using conditions for concentrations { Cmk} on plateau zones, the material balance (1) for system I (with uniform flow velocity, U)yields
(20a)
for all 1 c m , j c p . With equal dispersion factors of the components (D, = Dj D andp, = pj = p ) or (D, = Dj D and 5, = f) it has been proved (Kalinitchev, 1994) that for the stationary k-wave the linear relations (21) where
are the solutions of differential (19) and (20a) or (19) and (20b). The reduced (dimensionless) concentration distributions
across the stationary k-wave are not dependent on a number (m) of any component due to the relations (21). From (21) and (22), the relations for the constantpattern profile across the stationary k-wave can be easily deduced where U m = b,C,; amk= LmCmkNk;Umk = b,C,k; vk = 1 xcu,k and Cmk and Cm(k+l) are the values of concentrations of the mth component on plateau zones on the left and on the right, respectively, of the k-wave. The relations (17) for the stationary k-wave velocity (Wk) correspond to Renkin relations for shock waves and have been usually obtained in fields of gas dynamics or fluid mechanics from the integral form of mass balance (Rozhdestvensky and Janenko, 1968). Kinetic equations (2a) and (2b) are written for the moving coordinate system ( 2 ) as follows:
+
Nlk = N 2 k = ... = NmkE Nk; 0 d Nk d 1 (23) The linear relations (21) describe the dependence between concentrations of components across the constant-pattern profile of the k-wave when the dispersion factors for each of the components are the same. These linear dependences coincide with the straight composition paths obtained for the ideal model in composition space (in the ideal model the parameters &m represent the Rieman vector invariant (Rhee et al., 1970). A large number of examples of these composition path grids are presented in the book by Helfferich and Klein (1970)
2630 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
(for example here, see Figure la). The linear composition path grid can be orthogonalized by the h-transformation. The relations of the same type can be deduced for system 11:
c, - cmk= Bk,(R - Rk)
(24)
where Bkm
= ( C m ( k + l ) - Cmk)/(Rk+l - R k )
n
Rk = Cam,; amk = y,ck
m, k = 1, 2, ...,p
m
when the values of the dispersion factors are the same for any component (Kalinitchev, 1985a). After substitution of relations (21)into the equations (16) and (Ma) or (16) and (18b) the analytical solutions can be derived by integration over z (Kalinitchev, 1985b,c, 1986). These solutions describe the constantpattern profiles across the concentration k-wave for various combinations of dispersion factors: system I
Ia.
D, = D and l / p m = 0 for all m; HETP
Ib.
3
2D/U
D, = D and 1/5, = 0 for all m; HETP
IC.
simplex P1P2P3 in Figure l a , the values of concentration plateau zones {Cmk} were previously calculated theoretically (Kalinitchev, 1985~).The theoretical values {Cmk} coincide with the computerized ones. Then, the stationary 2-wave, which includes concentration profiles of both displacer (CZ)and displaced component (C1) (Figure lb), was considered. Figure ICshows that, in contrast to theoretical linear dependence (23) (solid line), the numerical concentration dependence NZk(Nlk) across the stationary k-wave is not linear (dotted line). It is obvious, therefore, that the steady state of the two-component k-wave (k = 2) (Figure lb) was not yet achieved in the numerical calculations (Tsabek, 1978) to the analyzed moment T = 60 (all numerical calculations were made at T 6 60). The conclusion was made (Kalinitchev, 1985c, 1986) that the analytical solution (21)-(23) can be used as the reliable criterion for check up of the numerical scemes and of the results of computer-aided simulations of multicomponent nonlinear chromatographic systems. The widely used empirical parameter (Rachinsky, 1964) effective width ( b k ) can be used for the description concentration profiles. This parameter of the multicomponent k-wave has been expressed from the solutions (25) and (26)
2D/U
D, = 0 and p, = p for all m; HETP
f
2(1 - w f l ) w k / P
m = 1, 2, . . . , p
where 6 is an arbitrarily-selected small dimensionless concentration value (usually in practice 6 = 0.1). With relations (24) for system 11, the simultaneous material balance equations can be integrated over z to give the constant-pattern profile (Kalinitchev, 1985a,c): system 11:
vk h ( N k ) - vk+1h(1 - Nk) =
IIa.
2(vk - Vk+l)(l- Wfl)Z/HETP-k Kk Kk
Id.
D,
= D and l/pm = 0 (or 1/5, = 0) for all m; ln(Nk) - ln(1 - Nk) =
(Rk - Rk+l)W&(DCo) (28)
(25)
= vk+l - vk
D, = 0 and 5 = 5, for all m; HETP
E
2 ( 1 - wk/U)2u/
to concentration routes, which follow the “slow” paths (in the simplex P1P2P3) with connection of the sides P1W and P2P3 or (b) to concentration routes that follow the “fast” paths with connection of the sides PlP2 and WP3 (as an example, see Figure la). The qualitative examples of concentration frontaldisplacement profiles in a two-component system are presented in Figure 2. Figure 2 shows additional broadening of the width ( h z k ) of a two-component k-wave for the displacer profile (Cl = 1- Nk) (solid lines) in comparison with the width of the wave ( & I ) obtained in a one-component system (dashed lines). It can be deduced from the solution (27) that the static effects of the interference of species (described by constants bl and b2 in two-component Langmuir isotherm equations (3)) lead to the additional broadening of both the displacer’s and the displaced component’s profiles in comparison with the one-component wave profile (see profile of the displacer in Figure 2). The additional broadening of the profiles is the larger the higher is the curvature (b2)of competitive sorption isotherms fi(C)(Kalinitchev,1985~). The solution, the describing constant-pattern profile for a one-component Langmuir system, is obtained by Glueckauf (1955), who noted that, in the case of a strongly curved sorption isotherm (bl>> 11, the concentration profile C(z)is sharp in the field of small frontal concentrations and is diffuse in the field of large frontal concentrations for intraparticle diffusion kinetics ((2a)) and vise versa for film diffusion kinetics ((2b))(dashed lines in Figure 2, parts a and b, respectively). For the strongly curved Langmuir isotherm of the displacer 1 (large values bl >> 1 in (311, (25) and (26) can be transformed into the solutions, which describe constant-pattern profiles in the two-component frontaldisplacement pattern (solid lines in Figure 2)
z
0
z
Figure 2. Frontal-displacement constant pattern profiles N k ( Z ) for strongly curved (bl >> 1) Langmuir isotherms, eqs 33 and 34 (- - -, one-component frontal pattern; -, two-component frontal, displacement pattern): (a) intraparticle diffusion kinetics (at & = /?z in eqs 2a; (b) film diffusion kinetics (at 51 = 5 2 in eqs 2b).
+
1 h ( N k ) - (Vk+lNk)h(1- Nk) = 2(1 - wk/U)z/HETP for intraparticle kinetics (33) -1
+ (Vk+l/vk)h(Nk) - h(1- Nk) = 2(1 - wk/U)z/HETP
for film diffusion kinetics (34)
By use of (33) and (341, one can deduce (Kalinitchev, 1985~1,that in two-component frontal-displacement constant pattern the additional broadening of the displacer’s profile (1 - Nk) falls into the field of small concentrations for intraparticle diffusion kinetics (solid line in Figure 2a) and vise versa for film diffusion kinetics (solid line in Figure 2b).
Conclusions The approach developed by Helfferich and Klein (1970)permits the calculation of concentrations plateau zone (Cmk} between (k - 11th and kth waves over the entire chromatogram for Langmuir multicomponent systems. These calculations can be derived by use of the ideal model of chromatography presented in the book by Helfferich and Klein (1970). The power of these calculation predictions lies in the fact that the values {Cmk} do not depend on the values of nonideality factors (Helfferich and Klein, 1970). The criteria expressed through the corresponding hkfunctions (Helfferich and Klein, 1970) can be used to determine whether or not constant-pattern profiles will be formed in the chromatogram for given initial and feed compositions. The solutions (21)-(34) presented here describe the concentration multicomponent stationary k-waves with a finite width (‘‘shock layer with finite thickness” in terminology by Golshan-Shirazi and Guioshon (1989)) instead of shock k-waves in ideal equilibrium theory. According to (27) or (31) and (32), the width of the stationary multicomponent k-wave is proportional to the combination of the nonideality factor (HETP factor) and static interference factors, presented in multicomponent Langmuir equilibrium equations (3) by the isotherms parameters (bi).
2632 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 The approach of Helfferich and Klein (1970) is related to the ideal model and suffers from the unrealistic assumption of an infinitely efficient column. Nevertheless, these results can serve as a basis for the solutions presented in this paper. Although the profiles obtained in ideal theory are unrealistic, because real columns have a finite efficiency (usually expressed through the HETP parameter), the ideal model presents an excellent first approximation, especially for concentration distributions in diffuse (rarefaction)k-waves. As far as final patterns are concerned, disturbances can be ignored in rarefaction waves, but must be included in any realistic treatment of self-sharpening waves (Helfferich and Klein, 1970). The simple analytical solutions (21)-(34) derived give the possibility to include the influence of nonideality factors into the estimations of the constant-pattern profiles in multicomponent chromatograms. The approach used here includes various nonideality (dispersion) factors, describing the column efficiency (deviation from local phase equilibrium, nonideal flow pattern). The parameter characterizing the column efficiency, height equivalent to a theoretical plate (HETP), is connected by simple relations with the nonideality factors used here. More accurate results could be obtained, when needed, by proceeding to numerical calculations. It has been shown that, in this case, the analytical solutions derived can serve for testing numerical schemes of computer simulations of multicomponent nonlinear chromatographic systems. For this purpose, the corresponding linear concentration dependencies (211, (231, and (24) in multicomponent composition space can be used easily, as was demonstrated. On the other hand, the results derived from the analytical solutions (21)-(34) have the major advantage of demonstrating some general dependencies of influences of interference effects in multicomponent nonlinear systems.
Nomenclature dimensionless sorbate concentration of mth component calculated in relation to the unit volume of the mobile phase of the column (taking into account the porosity of the sorbent) C, = dimensionless concentration of the mth component in the mobile phase c,k and a m k = values of concentrations c, and a, on concentration plateau zone between the multicomponent (k - 1)th and kth waves U = interstitial linear flow velocity of the mobile phase (system I) u = nonuniform linear flow velocity of the mobile phase (system 11) u k = value of velocity u in kth plateau zone HETP = height equivalent to a theoretical plate L,, b, = dimensionless coefficients in the competitive multicomponent Langmuir isotherm equation (eq 3) I, g, = dimensionless coefficients in the competitive multicomponent reverse Langmuir isotherm equation (eq a, =
3) wk = velocity of the multicomponent stationary k-wave for
system I (eq 17) = velocity of the multicomponent stationary k-wave for system I1 (eq 29) z = x - Wkt, moving coordinate system x = distance variable in adsorbent bed t = time wk
Nk = normalized dimensionless concentration constantpattern profiles of any component in the stationary k-wave h z k = z2 - z1 effective width of the stationary k-wave z2 = Z ( N k = c ) coordinate, corresponding to the “small” concentrations (Nk = c) in the k-wave z1 = z(Nk=l-c) coordinate, corresponding to the “large” concentrations (Nk = 1 - E ) in the k-wave { J m k } = Rieman vector invariants C, = sum of components with indexes m, where m = 1 , 2 ,
...
ln(N) = denotes natural logarithm (to the base e = 2.72...; loge N) Greek Symbols ym = Henry coefficients of the sorption isotherms for system I1 & = kth solution of the characteristic equation (velocity eigenvalue of the concentration matrix) ,uk = Rieman invariants Pm (or 5,) = mass transfer kinetic coefficients for mth component
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Received for review November 21, 1994 Revised manuscript received March 17, 1995 Accepted March 24, 1995@ IE940689E
Abstract published in Advance A C S Abstracts, July 1, 1995. @
