Ind. Eng. Chem. Res. 1996,34, 2782-2788
2782
Paradigms and Paradoxes in Modeling Adsorption and Chromatographic Separations Daniel Tondeur Laboratoire des Sciences du G n i e Chimique, CNRS-ENSIC, Nancy, France
Modern approaches to modeling adsorption and chromatographic operations rely, implicitly or explicitly, on a restricted number of “paradigms”, such as the equilibrium-stage approach and the theoretical plate, the kinematic wave, the constant pattern, the Langmuir isotherm, and a few others. Their role in organizing knowledge and experience in adsorption research is considerable, and the set of currently used paradigms expresses the state of mind, the “dominant ideology” in the field at a given instant. The present communication proposes a nonexhaustive inventory of these paradigms, analyzes their origin and their scientific supports, and discusses their role in the progress of the field, their conservative function, and their contribution to strengthening the concerned community or to opening it to others. Illustrations are given that paradigms are sometimes paradoxes, and finally, a discussion of the missing paradigms is proposed, i.e., paradigms which have not yet been established or recognized, but which may be the keys to future progress.
Introduction Generic concepts or models play a central role in the development, the structuring, and the recognition of a scientific field. By a deviation of the original meaning, such generic concepts have been called “paradigms” (Greek: nqadaypa = example, model). As Thomas Kuhn (1970) has deeply analyzed, the scientific revolutions are usually associated t o the emergence of new general paradigms; in more steady situations, the building of theoretical frameworks, the professional organization of a field of scientific activity, and the dominant scientific ideology are usually aggregated around a few prevailing paradigms. Important properties of a paradigm are therefore the acceptance by a human community, and the development of a teaching corpus. Figure 1 illustrates schematically a part of the analysis of T. Kuhn: scientific breakthroughs, represented by sharp changes, are preceded by perturbations and a struggle for imposing new concepts or destroying old ones, and are followed by perturbations for elimination of spurious effects. Between two scientific breakthroughs are periods of consolidation and gradual progress, sometimes perturbed by eddies that do not develop into new breakthroughs. A paradigm, at its highest level, may be a fulltheoretical framework: T. Kuhn quotes for example quantum theory as a whole as a paradigm of physics. It should be understood nevertheless that the qualitative aspects (basic axioms, representations, and hypotheses) are essential in the notion of paradigm, rather than the mathematical or experimental “hardware”. The latter are in fact determined and structured by the former. Concerning chemical engineering, it is commonly understood that its very emergence is tied to the concept of unit operation, which may thus be considered as the paradigmic foundation. A later milestone is certainly the concept of unified transport phenomena, as approached by Bird, Stewart, and Lightfoot (1961). In recent times, it is worth noting that prospective thinking about the future of chemical engineering is organized around the notion of paradigm (Wei, 1985; Villermaux, 19931, and finding new paradigms is set as a goal for basic research. It is clear that the paradigms of unit operation and of transport phenomena define (at least historically) the structure of the whole field of chemical
Y 0
T
scientific breakthrough
‘cold fusion“
0
3 5
8Y Y
s4
appearance of new concepts (perturbation, struggle)
consolidation organisation (dominant ideology)
T
TIME
Figure. 1. Schematic illustration of the time evolution of scientific knowledge.
engineering, each at a different level of detail, but with a similar level of generality. The question then arises whether there are other paradigms t o be invented at this level of generality. On the other hand, it is clear that these paradigms do not suffice for working out concrete problems and therefore others are necessary, and have evolved, that are more or less specific of a subfield. The purpose of the present paper is to discuss the prevailing paradigms in the field of adsorption engineering, restricted here essentially t o the unit operation aspect. In other words, we do not emphasize here the physicochemical aspects or the materials aspects, although they are briefly mentioned. A list of concepts and models which in the author’s mind may be considered as paradigms, and have played, still play, or should play an important role in structuring adsorption research, is as follows: discrete equilibriumstage approach; continuum or homogeneous medium approach; local equilibrium model; kinematic wave approach; flux-concentration analogies; constant-pattern and shocks; Langmuir paradigm; adsorbed solution model; coherence. I have chosen to discuss here some aspects which in my opinion offer interesting features for present and future thoughts.
0888-5885/95I2634-2782~09.QQlO 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2783 I Counter-flow
4-y MIX.
EQUIL.
SEPAR.
EQUIL.
w
I1 One moving phase
MIX.
SEPAR.
I11 CSTR
Figure 2. Models for equilibrium stages.
The Discrete Equilibrium Stage: The Historical Role of the First Paradigm The importance of this paradigm is both historical and conceptual. It comprises what is probably the first attempt t o give a mathematical representation of the chromatographic process (Martin and Synge, 1941). In doing so, it contributed to modify profoundly the field of chromatography in its methodology, in its status as a unit operation, in the population of scientists that feel concerned, and in its industrial developments. From a purely empirical activity, chromatography evolved into a field which could be approached conceptually, and with some rather general concepts no longer specific t o the analytical chemist. Therefore, engineers and other scientists became concerned, with obvious consequences. There is thus a qualitative jump associated to this development, much more important than the mathematical model of Martin and Synge itself, which was soon to be overrun, and which is no longer in use today. It is this type of qualitative change, affecting the field as a whole, which characterizes the role of a paradigm of high level. Note that the Martin and Synge approach met soon the competition of the continuum and wave approach discussed below and which I consider contributed to a similar extent t o this qualitative breakthrough. At a lower level of impact, the equilibrium-stage approach did pave the way for a broad line of modeling approaches, which still lead today t o useful subparad i p s and operational tools. It is of some interest to discuss the conceptual and practical implications of this model. Figure 2 illustrates three of its classical implementations . The first two involve a discretization of both phases into fractions that are contacted successively. The second implementation, adapted for fixedbed operation, can be viewed as a special case of the first. The third (continuous stirred tank reactor model) involves discretization of the fixed phase only, with continuous flow of the other phase. The following remarks can be made on this modeling: (a) These models are “unrealistic” with respect to the “continuous” character of the chromatographic process (macroscopically,the bed is continuous, and the flow is also continuous). (b) However, the models are physically sound, in the sense that one can build a system, with discrete stages that works exactly as described (for example with Craig contactors, or CSTRs).
(c) An interesting aspect of the discrete approach is that it brings chromatography closer to other operations, physically staged, such as distillation. Hence there have been some attempts to develop adsorption processes “resembling” distillation, such as parametric pumping (Grevillot, 1976, 1986). Clearly, the field of chemical reactor engineering is also solicited. (d) The epistemologic question is then whether the discrete process legitimately represents the continuous one, without spurious effects. This question is not fully understood; in most cases, the answer seems positive but in some situations discrete stage models exhibit oscillations which express a physical behavior (not a numerical instability) not encountered with continuous models (Schweiger and Le Van, 1993; Bailly, 1994). Curiously, the reciprocal question also arises: continuous models (infinite number of stages) used in chromatography are not guaranteed to represent consistently, even as limiting Gases, countercurrent staged operations, unless special care is taken of boundary conditions (Hwang and Helfferich, 1989). (e) Stages in these discrete models need not be equilibrium stages. Mass-transfer limitations may be incorporated easily, in detail, or globally through a stage efficiency. Therefore, there is a broad family of nonequilibrium models, leading to analytical or compact solutions in the case of linear equilibria (e.g., Villermaux, 1987). (f) The discrete model furnishes not only a picture of the process, but also an algorithm for the numerical solution. Such algorithms are convenient and widely used because they are extremely easy to program, and they meet no convergence problems. On the other hand, they are likely to be less efficient than elaborate numerical methods used by specialists. However, keep in mind that the discrete-stage models, numerically speaking, are just a simple way to discretize a continuous process. (g) The staged approach is convenient for characterizing architectures more complex than the one-dimensional cascades. In summary, the equilibrium-stage paradigm, aside the deep qualitative change induced in the field, has brought about quite a few important effects, in terms of giving an intuitive picture of the process, of showing analogies with other operations, of developing more general and more advanced models, of developing simple computational procedures, and of paving the way to systemic approaches for more complex architectures.
The Continuous Medium Approach: The Paradoxes of Continuity and Discontinuity The basic notion discussed in this section is that of continuum, as opposed to the discrete approach discussed above. This discussion will in fact also introduce the next paradigm, that of kinematic wave. The main contributors to this approach were probably De Vault (1943) and Glueckauf (1949). The obvious conceptual basis of this approach is to consider the adsorbent bed as a continuum with respect to the spatial coordinates: the axial or radial coordinates disregard the fact that, along them, an observer or a molecule meets successively intergranular porous space, solid phase, and intragranular porous space. In expressing the conservation laws, the concentration in the phases will be considered as continuous and differentiable, and of course, this is a priori as questionable as the discretization of the previous section.
2784 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
There is an interesting paradox in this situation: at the scale of the particle and the pores, where the space is really discontinuous, we homogenize it (we make it continuous), whereas at the scale of the bed, where the process is really continuous, we discretize it. A deeper look a t this paradox brings us to the question of how discretization and homogenization should be done, in relation to the structure of the process at the respective scales. Let us just mention at this point that the continuum coordinate is legitimate inasmuch as the homogenization is done “properly”, i.e., the concentrations are properly averaged over the respective phases (note: a conventional arithmetic spatial average is sufficient for mass conservation purposes; for kinetic modeling, such may not be the case). For example, eq 1 is a legitimate form of material balance over a slice of bed of finite thickness Az and a finite time interval At. Besides defining the average concentrations q and c in the adsorbed phase and the fluid phase, respectively, the only other requirement is to accept a first-order Taylor type expansion for the average concentration profile (but this is a matter of accuracy, not of legitimacy). t+At
S,z + h t [
“ “1
( l - ~ ) % + ~ ~ + u ~ d z d t (1) = O
However, another qualitative jump is required to go from eq 1 to the conventional partial differential conservation equation, eq 2,
(1 - E ) % +
“ a c u -= 0 at az
E-+
and it is not just a matter of letting the thickness of the slice and the time interval, Az and At, become infinitesimal. Going from eq 1 t o eq 2 requires in addition the assumption of ergodicity. Ergodicity is a statistical concept which we shall not comment on further here, except to say that it implies the equality of the time averages and the space averages expressed in the integrals of eq 1. At the scale of description corresponding to eqs 1and 2, the statistically large number of particles involved, and their supposedly Markovian behavior, guarantee ergodicity. This discussion is therefore pertinent only when one gets to the microporous scale (smaller population with possibly long-range interactions), or when one considers applying the continuum approach to other, chromatographic-like phenomena, such as particle flow, multiphase flow in porous media, or car traffic (see below). In summary the continuum approach, which is often taken for granted, does raise some legitimacy questions that may be worth investigating when a finer understanding is sought, or in areas such as materials design. But its main role, in my opinion, is to have generated a series of more specific and powerful paradigms which are discussed below, especially the equilibrium model and the kinematic wave approach. An important effect has certainly been t o open the field of chromatography to physicists, mechanical engineers, and mathematicians, through the formulation in a mathematical framework familiar t o these scientific areas.
The Local Equilibrium Paradigm: The Paradoxes of Quasi-Static Processes The idea underlying the local equilibrium approach is to consider that at any abscissa along a chromato-
4
I
constant state
/
abscissa along bed
’
discontinuity or “weak solution“
I
*
abscissa along bed Figure 3. Behavior of shock profiles in a fixed-bed. (a) The overlapping profile that would result from a nondispersed local equilibrium model, and the discontinuous “weak” solution. (b) Propagation of a constant pattern shock.
graphic column, viewed as a continuum, and at any time, the adsorbent phase and the flowing phase are at thermodynamic equilibrium (De Vault, 1943;Glueckauf, 1949). This is in a sense the continuous version of the equilibrium stage approach. As simple as this may seem a priori, some conceptual and theoretical difficulties had to be surmounted to establish this approach in chemical engineering, and it is still not generally accepted outside this field. The first difficulty lies in the very notion of thermodynamic equilibrium in a dynamic system: how is it conceivable that in a system where flow in and out takes place, and mass transfer occurs, equilibrium may prevail everywhere? Thermodynamics furnishes a suitable conceptual tool to answer that apparent paradox: that of reversible or quasi-static process (Kvaalen et al., 19851, i.e., an infinite sequence of events that are all infinitely close to equilibrium, but displace this equilibrium. Again this is nothing else than the limit of the equilibrium stage approach. But even calling on this notion, I still find it difficult to convince some colleagues that finite zones of “quasi-equilibrium” (concentration plateaus or constant states) may exist in a chromatographic column (Figure 31, or a fortiori in other dynamic exchange processes such as heat storage, electrophoresis, etc. The second difficulty lies in the paradox of shock waves that arise within this theoretical framework. De Vault (1943) has shown that if the local equilibrium, or let us rather say the continuum quasi-static, approach is applied to the adsorption under a concave (Langmuir type) isotherm, “overlapping“ or “breaking” fronts are generated which are physially meaningless (other point of discussion!). The way out is what the mathematicians call “weak solutions”,i.e., solutions involving finite jumps or ”discontinuitiesin the dependent variables (the concentrations) and constant states (Figure 3a). S o the continuum quasi-static approach leads t o the paradox of having t o artificially introduce discontinuities! A common assumption in sorption operations, well supported by practice and theory, is that of “constant
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2785 pattern” or “steady shock”. It states that a more realistic picture than the discontinuity just mentioned is a sharp but continuous front, which conserves its shape a t it travels along the bed (Figure 3b). This behavior is known in aerodynamics (propagation of pressure waves) as a shock, and is encountered in many other nonlinear dynamic systems (Wallis, 1969; Whitham, 1974). An interesting remark is that the constantpattern adsorption front implies a linear relation between the adsorbed phase concentration q and the fluid phase concentration c. Since it also requires that the equilibrium isotherm be concave, this means that local equilibrium cannot prevail along the constant-pattern front. Thus, two of the most common modeling approximations of chromatographic theory, local equilibrium and constant-pattern behavior, are in a sense incompatible. A thermodynamic statement narrowly related but not equivalent to the above is that there exists no reversible path between two constant states, when the conditions for a shock prevail (adsorption with a concave isotherm). In other words, entropy must be created in such a front, even if it is infinitely sharp (see Frey, 1987,1990). The paradox of the local equilibrium approach is thus that it cannot be used without introducing dissipative (entropy creating) zones. The mathematical difficulties related to the above paradoxes are largely resolved in the framework of the theory of quasi-linear, hyperbolic partial differential equations (Courant and Hilbert (19621, for example), and the quasi-static paradigm has permitted very powerful theories to be developed (Helfferich and Klein, 1970; Rhee et al., 1970, 1971). Nevertheless the conceptual difficulty somehow remains and prevents this paradigm from spreading into other areas, or even from being admitted by all the community of chromatographic processes. This is an example where a powerful paradigm remains restricted to a specialized community and does not contribute to its opening toward others.
The Kinematic Wave Paradigm: Wave Versus Corpuscules and Flux Versus Concentration We have introduced in the previous sections the pioneering work of De Vault, expanded by Glueckauf, in terms of the continuum and quasi-static approach. A paradigmic outcome of this approach is the notion of concentration front or wave, already introduced above, but which deserves further comments. The term “kinematic wave” is apparently due to Whitham (19741, who was concerned not with chromatography, but with car traffic and with hydraulic waves. The fact that Whitham addressed these two domains is of epistemologic interest: we look at the behavior, on the one hand, of discrete individuals (cars) and, on the other hand, of a continuous medium. Nevertheless, both cases are described by the same wave equation, eq 3,
~at +
w
~
=
o
(3)
where w is the wave velocity, or velocity of propagation of a specified value of the dependent variable q (representing the concentration of cars, or the thickness of the water layer):
w = (&/at), (4) The analogy between these propagation phenomena relies on considerations similar to that of light pro-
pagation: the propagation of a population of discrete corpuscules or cars can be described as a wave phenomenon. Waves in hydraulics, sedimentation, multiphase flow, and mass and heat transfer arise from conservation equations that may be written in the form of eq 3. A differential conservation equation is generally of the form %/at
+ div J = 0
(5)
where div J is the divergence of the flux J. Equation 5 is transformed into eq 3 if the flow is unidimensional (div = a/&) and if one lets: dJfdq = w
(6)
Equation 6 implies that there exists a differentiable relation between flux and concentration of the conserved species. Such a relation may be established experimentally for processes such as car traffic, sedimentation, bubble flow, multiphase flow in porous media (Wallis (1969)and Helfferich (1981)for example). In adsorption and chromatography, the starting conservation equation is eq 2. If we assume a constant flow velocity u and local equilibrium, and put:
J = uc
(convective flux)
and q = q(c)
(isotherm equation)
the derivative of J appearing in eq 6 is essentially the inverse of the isotherm slope:
wac=
= u dciaq
Thus, under the quasi-static approach, the flux versus concentration curve is essentially the isotherm, and the conservation equation may be written as a kinematic wave equation, bringing thus adsorption and chromatography into that conceptual framework. The importance of the paradigm of kinematic wave lies precisely in this fact of bringing together, in one unique conceptual framework, with corresponding mathematical tools, all the different physical phenomena mentioned before, and obviously quite a few others amenable to this formalism. This paradigm thus contributes t o open the field of chromatography to others, and also to teach the wave phenomena as a unifying field (Tondeur, 1987).
Multicomponent Langmuir: Paradigms of Competition The Langmuir model for heterogeneous kinetics and adsorption equilibrium is probably the most widely used model in these fields. I should like t o insist on three aspects of Langmuir’s approach, which in my opinion have an important impact on our present view of adsorption processes and, more generally, of “competition dynamics”: the kinetic approach t o equilibrium, the special form of Langmuir’s isotherm, and the implications beyond adsorption. Let us recall the basic mechanistic aspects of this approach: adsorption equilibrium is viewed as the equality of the desorption rate and the adsorption rate, locally and for each component. Langmuir’s isotherm is obtained when one assumes that the desorption rate of species i is proportional to the fractional coverage 8i = qJqm of the surface by that species, and that the
2786 Ind. Eng. Chem. Res., Vol. 34, No. 8,1995
adsorption rate is proportional t o the fluid phase concentration ci and to the uncovered fractional area 1 - C8j. The equality of these rates yields (7) where K,is equal t o the ratio of the adsorption rate constant to the desorption rate constant of species i. Equation 7 has the property of being “invertible”to give the more usual multicomponent Langmuir isotherm:
Langmuir’s isotherm is rightly criticized for oversimplifying the physico-chemical phenomena, especially for assuming that the maximal capacity qmis the same for all components, for neglecting “lateral” interactions between adsorbed molecules, and for neglecting surface heterogeneity. But from the point of view developed here, the essential is elsewhere. First, the kinetic approach to equilibrium provides the kernel for a large family of sorption models, which can be generated by modifylng the elementary rate laws. The energy and thermal aspects can be introduced into the kinetic parameters, accounting for the energies of interactions between molecules and between molecules and surface. Surface heterogeneities can be accounted for, just as in the thermodynamic and statistical approaches, by introducing appropriate distribution functions of the properties, and integrating over the surface area. Thus, Langmuir’s model may not be satisfactory, but it does pave the way for practically unlimited sophistication, while keeping a physical meaning of the parameters. Second, this approach leads to interesting connections between equilibrium and internal diffusion kinetics, a question that appears relatively unexplored (but I must confess my ignorance of the literature of heterogeneous catalysis). An interesting hint in this direction is given in recent papers by Chen and Yang (1992, 1993), building on older ideas of Riekert (1971). Chen develops rate equations for multicomponent surface diffusion based on jump probabilities. This allows him to relate binary diffusion coefficients a t high coverage to single component coefficients at low coverage, and to the coverage, in a set of multicomponent Fickian rate laws:
where [Dl is a nondiagonal matrix of the coefficients mentioned above, Vq is the adsorbed concentration gradient, and J is the diffusion-flux vector. Curiously, Chen did not exploit the equilibrium isotherm which results by equating their rates of adsorption and desorption. Using a notation consistent with the above, this isotherm is: (10) where A, are interaction parameters between species i and j . The analogy is obvious with eq 7, which is obtained as a special case by letting A, = 1 for all j , meaning no interaction. We thus have a Consistent formulation of a generalized Langmuir-like adsorption isotherm, and of a generalized Fickian surface diffusion. Further connections may be found between the Langmuir form and other equilibrium or transport processes, if one rewrites eq 8 or eq 10 in the form:
K, e. e. 4 .= - = L/--S Kj ci cj
(11)
This is formally analogous to the definition of relative volatility in distillation, of relative mobility in electrophoresis, and of relative permeability in multiphase flow through porous media. In the two latter transport processes, 8 has then the meaning of a transport number or fractional flux. It has been shown (Tondeur and Bailly, 1987) that Langmuir adsorption shares mathematical and dynamic properties with StefanMaxwell diffusion; I conjecture that this in turn implies that the matrix of Stefan-Maxwell coefficients is the inverse of the Fickian matrix [Dl introduced in eq 9, for the case where A, = 1 (but I have not been able to demonstrate this yet). All these features indicate that the “Langmuirian” relations relie on some deeper and more general substrate, going beyond adsorption. In my opinion they express “the laws of fair competition for a limited resource”: 1. A constant hierarchy exists between species (here, an order of decreasing adsorptivity), expressing their competitiveness for the access to the resource (here, the adsorption sites). 2. Fair, but competitive distribution takes place: the access of a species t o the resource is proportional to the population of that species, and t o the availability of the resource. 3. No monopoly, no trust: all species have access to all the available resource; no species or group of species can monopolize the resource. Beyond this amusing generalization, but in a less obvious fashion, multicomponent Langmuir systems have peculiar mathematical properties, which have attracted the attention of mathematicians (Canon and James, 1993): uncoupling and linearization under a particular so-called H-transformation (Helfferich and Klein, 19701, special spectrum of eigenvalues, orthogonality of eigenvectors under the H-transform, simple topology of state-space, etc. All this entails that the dynamic behavior of Langmuir adsorption systems is particularly well understood, as illustrated by the book of Helfferich and Klein. The Langmuir paradigm is interesting and important because of the essential role it still plays in modeling adsorption processes, because of the fascinating mathematical properties attached to it, because it brings closer together different sorts of processes, but also because all the implications and analogies have probably not been elucidated yet. This concerns especially the diffusional mechanisms. The Adsorbed Solution Paradigm The ”adsorbed solution” approach (whether ideal or not) has in my opinion the status of a paradigm, above and beyond the specific tool it provides, because of its conceptual bases and the qualitative way of thinking it induces. The central point which contributed decisively to develop it is that adsorption equilibrium may be described with concepts and methods transposed from liquid-vapor equilibrium. From there, not only the ideal adsorbed solution approach, but also many nonideal two-phase equilibrium approaches, results immediately (conceptually speaking that is). The ideal adsorbed solution theory is also a good example of a paradigm whose legitimacy question seems
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2787
Absuissu
Figure 4. Schematic movement of a coherent distribution (the waves progress and spread out, but the set of compositions is conserved).
well posed: it is understood that the thermodynamics of continuous, three-dimensional, two-phase equilibria cannot apply without limitations to three-phase systems with one phase being bidimensional. However, it is clear where the basic assumption lies, where the qualitative jump is, and hence also where further progress may be made.
Coherence Multicomponent systems are the general case in practical situations. Although some “preparadigmic” approaches exist for the dynamics of multicomponent systems (Helfferich and Klein, 1970; Tondeur and Bailly, 1987), a deep qualitative understanding, going beyond particular models (like Langmuir) is still pending. There are probably some general “logistic”rules of behavior of competitive dynamic systems to be discovered that are almost model independent and give kinetic as well as equilibrium information. The coherence concept, as introduced, used and advocated by Helfferich (1970,1989), is a key t o such understanding. Borrowing from Helfferich (1989) his approach to coherence in the context of chromatographic waves: “Since the wave velocities with respect to all components must be equal a t any point in a coherent wave, a complete set of values of dependent variables coexisting a t such a point in space and time will travel jointly, in the same direction and at the same speed, and so remain in each other’s company [see Figure 41. A possible definition is the conservation of such sets. This definition includes equilibrium and steady-state, conditions in which no value moves with time, so no existing set is ever broken up”. More generally, he considers coherence as a state, or rather a dynamic regime, toward which a dynamic system will naturally tend when relaxing after a finite-time perturbation. In terms of dynamics, one might thus define coherence as a property characterizing a ”stable, finite attractor” in the state space of multicomponent distributed systems (the stability may be normal or asymptotic; the finiteness means that the attractor is not a node, or an infinite curve, or a fractal). However, coherence not only refers to the end state of relaxation after a perturbation, but also t o the way the relaxation occurs, the way in which an uncoherent perturbation breaks up into coherent modes. In my opinion, coherence is one of the most deep reaching and powerful concepts in process dynamics, at least as far as multicomponent systems are concerned. However, it is also an example of a concept that does not emerge into wide recognition, and thus perhaps has not yet reached the status of paradigm. The reasons are probably the lack of a relatively simple definition, the ignorance by the population who could make the best use of it, namely the specialists of process control
and dynamics, and probably also an unprecise view of the extension of the concept. Is there some thermodynamic “principle” hidden behind coherence, at least in cases like adsorption where thermodynamic equilibrium is influent? I personally believe it. This principle may be one of least dissipation, or minimization of entropy production, generalizing Prigogine’s theorem on minimum dissipation of steady states of open systems (Prigogine, 1947). Some thoughts in this direction can be found for example in Tondeur (1991).
Closure: Missing Paradigms
I have tried t o illustrate the qualitative role played by paradigms in structuring and in establishing the scientific bases of adsorption engineering. Paradigms arise sometimes a posteriori as the concrete expression of a conceptual advance, sometimes a priori as the tool for advances to come, and sometimes both together. They are often a t the heart of a conflict of concepts, which they resolve or bypass. In any case, they are keys t o scientific progress. Their relation to other forms of progress, in instrumentation, in materials design, or in industrial processes, for example, has not been discussed here, but I believe that they also play an essential role in these domains, perhaps not in the gradual process of empirical improvement of existing tools and methods, but in preparing the breakthroughs. The question that comes to mind is of course what are the paradigms on which the future evolutions (revolutions) will rely. Obviously, if we knew for sure the answers to this question, a good part of the job would already be done. The follovrJing examples are thus merely some personal thoughts, and by no means authoritative or exhaustive statements. Progress is certainly expected to continue in the design of adsorbents and catalysts. Interestingly, although the appearance of synthetic zeolites can be considered as a major technical revolution in separation and catalysis, no paradigm seems associated to this breakthrough, probably because zeolites were not “invented” but copied from nature. Aside from their chemical composition and structure, we know that the overall properties of these materials are strongly determined by their transport properties, which couple thermodynamic , diffusional, and geometric properties. Theories and tools, and ultimately paradigms are expected which would allow us to understand, describe, and monitor this triple complex coupling. Current interest in the binary couplings of these types seem to point in that direction. Cyclic steady state (that is, the periodic repetition of a transient process) is the normal way of operating industrial adsorption and chromatographic processes. The cyclic steady state is predictable essentially through the transient processes preceding it (in spite of attempts to predict it more directly (Croft and Le Van (1994) for example). Rules, tools, and more profoundly paradigms are needed in order to characterize, predict, and analyze the possible cyclic steady states, the optimal strategies to reach them, and their dynamics with respect to perturbations. None of the paradigms discussed in this text really imply a revolution: they have just added and complemented each other. The adoption of any one of them does not imply the rejection of any other, and probably does not contradict any paradigm in other fields. This is equivalent to saying that there has been no revolu-
2788 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
tionary breakthrough in modeling adsorption; progress has instead been gradual. The two ideas that come closest t o a revolutionary character are, in my opinion, the equilibrium model and the coherence concept. Both require, in some way, a qualitative break with the usual way engineers analyze processes. Both meet difficulties in gaining widespread acceptance and understanding. Both are found “counterintuitive” by some, and go without saying for (a few) others. If paradigms are the keys to scientific revolution, in their phase of spreading and increasing acceptance, they are also the ingredients of scientific conservatism once they are established: they set frames to what one is “allowed” to think. This unavoidable dialectic is nevertheless generating progress. Quoting T. Kuhn (1970):“The net result of a sequence of such revolutionary selections, separated by periods of normal research, is the wonderfully adapted set of instruments we call modern scientific knowledge”.
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