An Invited Paper 167

Unifying Concepts in Non-linear Unsteady Processes Part I: Solitary Travelling Waves Concepts Unificateurs des Processus Transitoires Non-liniaires Partie I. Ondes Migratoires Solitaires

DANIEL TONDEUR

Laboratoire des Sciences du GPnie Chimique, CNRS-ENSKT, 1 rue Grandville, 54042 Nancy Ckdex (France)

Abstract

Non-linear unsteady processes, as different as car traffic, adsorption, sedimentation, packed-bed heat transfer and hydraulic waves, can be analysed in a unified fashion by using the concept of the travelling wave. The fundamental aspects of the ‘solitary’ waves are discussed, as well as their distinctive features, such as overall and local propagation velocities, shape modifications, appearance of shocks, and ‘coherent’ asymptotic shape. These fea- tures are governed by the velocity distribution along the wave, itself determined by the relationship between the flux and the concentration variables of the process. Qualitative rules result from the analysis of ‘operating lines’ and ‘equilibrium lines’ in the flux versus concentration diagram, in a way somewhat similar to the McCabe-Thiele analysis for steady counter-current operations.

RCsumk

Des processus transitoires et non-linkaires t&s diffkrents, tels que la circulation de vkhicules, l’adsorption, la stdimentation, le transfert de chaleur en lit fixe, les vagues hydrauliques, peuvent Etre analysies d’une man&e unifike ?I l’aide du concept d’ondes migratoires. On discute les aspects fondamentaux des ondes migratoires ‘solitaires’, et on met en Cvidence leurs traits distinctifs principaux: vitesse de propagation moyenne et locale, modification de forme de l’onde, formation de chocs, forme asymptotique ‘cohkente’. Ces traits sont r&is par la distribution des vitesses le long de l’onde, elle-mEme d&erminCe par la relation entre la variable de flux et la variable de concentration du pro&d& L’ttude des ‘courbes optratoires’ et des ‘courbes d’lquilibre’ dans le diagramme flux-concentration (comparable au diagramme de McCabe-Thiele) fournit un ensemble de rkgles qualitatives de comportement d’une onde.

Synapse Le pr2sent texte propose une approche un$Ce d’un

certain nombre de processus du domaine de l’ingt+nit+ie d&pendant du temps et d’une variable d’espace, tels yue l’ad.yorption et Ze transfert de chaleur en lit jixe, la stdimentation et I’kcoulement diphasique, les ondes hy- drauliques, . . , . L’exemple de la circulation des vk- hi&es sur une route, qui est rkgie par des rggles analogues, est utilisk pour illustrer le caractlre g&Cral de cette approche, dans laquelle le concept d’onde mi- gratoire (ou onde cinkmatique, ou encore onde de conti- nuiti) jouo un r61e central [l-7]. .On s’intkressera d’abord duns cette premiPre partie aux ondes ‘soli- taires’, qui sont engendrtes par une perturbation monotone dans un systPme ri une seule variable dkpen- dante (7es systPmes multiconstituants, dans lesquels une perturbation monotone engendre un train d’ondes, seront discutPs dans la 22 partie). Des exemples deper-

turbation en crtineau (deux perturbations successives de sens oppose!) seront aussi Pvoqakes britkement.

On montre comment la conservation de la mati$re (ou de I’enthalpie, ou du nombre de tkhicules) s’exprime par une ‘Cquation d’onde’ (eqn. (4)) oli apparait la vitesse de propagation w, d’une valeur locale c de ia concentration. L’onde considP&e traduit done le d&placement le long de la variable d’espace z d’une distribution, d’un projil de concentration engendrkpar uneperturbation (Figs. 3-5). La uitesse locale w, est 6gale ci la d&i&e dJ/dc duflux de mati&e (de chaleur, de vthicules) par rapport ri la concentration (eqns. (5), (6)). La vitesse moyenne w de l’onde est &gale au rapport AJ}Ac des variations de Jet de c le long de I’ona’e (eqn. (7)).

Les propriMs qualitatives et quantitatives de I’onde (vitesses de propagation, modiJication de forme, forma- tion de chocs, forme asymptotique3nale) sont done d& terminkes par la relation entrejlux et concentration.

On examine un certain nombre d’exemples dans

02552701/87,%3.50 Chem. Eng. Process., 21 (1987) 167-178 0 Elsevier Sequoia/Printed in The Netherlands

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lesquels cette relation a une _forme connue, indepen- dame des conditions initiales et aux limites du prob- leme. Ehe peut alors etre d&rite par une courbe et une relation algibrique. C’est le cas notamment de la sedi- mentation et de la circulation de vehicules, qui ont un comportement anaiogue, caracteristique de la ‘migra- tion’ genie (Fig. 1); c’est le cas aussi de l’adsorption, et plus generalement de kc migration accompagnee d’echanges rapides avec un milieu immobile de capacite limittie. Pour l’adsorption. si l’on admet que [es trans- fert.s dtffiisionnels sont rapides, on montre que l’isotherme d’adsorption (Fig. 2) joue ie role de la courbeflux-concentration; 1eju.x ici est leproduit de la concentration en phase mobile par le debit de cette phase (eqn. (13)).

Le signe de la pente locale dJ/dc determine ie signe de w,., et done le sens de propagation des ondes. Ainsi en sedimentation et circulation automobile (Fig. I), dans les zones a faible concentration, fe pente positive engendre des ondes qui se propagent dans le meme sens que Ies particules ou les otihicules: c’est Ie cas de I’onde qui &pare la suspension initiale B du liquide clair A dans l’exemple de la Fig. 10. Dans les zones a.forte concentration, la pente est negative, et les ondes dam cette rigion se deplacent ‘en sens inverse des particules: c’est le cas de i’onde ‘de choc’ qui se produit lors dun carambolage de voitures (onde separant les particules arrtWe.s D des particules en mouvement C dam la Fig. 10). Dans ie cas de l’adsorption, la cow-be flux-concentration est monotone croissante, et les ondes se propagent dans le sens des molecules, c’est-a-dire dans le sens de l’ecoulement de la phase mobile.

La courbure de la relation J(c), c’est-a-dire la d&iv&e seconde d2J/dc2, d&ermine comment la vitesse locale IV, varie avec la concentration. Ainsi lorsque d2Jldc2 est posittf (d2q /dc2 negatty en adsorption: isotherme de Langmuir), w, = dJ/dc augmente avec la concentration, et par consequent les faibies concentra- tions se propagent moins vite yue les concentrations &levees. II en resulte qu’une onde de desorption, resul- rant d’une diminution de la concentration d’entrke, est ‘dispersive’, c’est-a-dire s’etale de plus en plus a mesure qu’elle se propage (Fig. 3). On montre l’analo- gie de ce phenomhne avec une ‘onde de drainage’ en Bcouiement iaminaire (Fig. 4). A l’inverse, une onde de sorption (augmentation de la concentration d’entree) se ‘comprime’ et donne naissance a un ‘choc’. Un choc chromatographique n ‘est autre qu’une onde a t&s fort gradient de concentration, qui tendh conserver saforme en se propageant (Fig, 5) de la m&ne faGon qu’un choc acoustique.

Dans certains cas (sedimentation, Fig. 1; stockage de chaleur latente, Fig. 7; Pcouiement diphasique en milieu poreux, Fig. 8) la courbeflux-concentration prt- sente un ou plusieurs points d’inflexion. et par conse- quent la distribution des vitesses locales n’est pas unimodale. L’onde correspondante peut ators avoir a la fois des parties dispersives et des chocs (Fig. 6).

Dans le diagrammeJlux-concentration on distingue la ‘courbe d’equiiibre’ qui exprime la relation locale entrepux et concentration, et qui est independante des

conditions aux limites et initiales, et la ‘courbe optra- toire’ qui est ie lieu des valeurs duJEux et de la concen- tration qui coexistent r~ellement dans une onde, et qui depend des conditions aux limites. La courbe operatoire d’une onde, dam sa forme asymptotique (atteinte au bout d’un temps sufisamment long apres la perturba- tion qui Pa crPPe), est l’enveloppe convexe de la courbe d’equilibre (Figs. 7et 8) construite du c&P ori la ‘condi- tion d’entropie’ est satisfaite, c’est-a-dire, telle que le transfert se fait dans [e sens du potentiel thermody- namique decroissant.

On discute enfn la nature des regimes transitoires examines, qui peuvent posseder une ‘structure’, lice a la distribution des vitesses. Cette distribution peut &re untforme et constante (cas du choc stabiiise), non- untforme, mais monotone et constante (onde dispersive simple), non-uniforme, semi-monotone et constante (onde composee coherente), non-uniforme et non- monotone (onde instable qui se r&out en plusieurs ondes), non-untforme et non-constante (onde non- cohhtrente). Le regime coherent est dejmi comme i’aboutissement de i’kvoiution du processus migratoire, au bout d’un temps sufisamment long. Cette notion g&u?ralise les notions d;ttat d’equilibre et d’etat sta- tionnaire. On conjecture que le regime coherent obPit a une loi de minimum analogue au principe du chemin optique minimum de Fermat, ou au theoreme du mini- mum de production d’entropie de Prigogine.

Introduction

The purpose of the present paper is to emphasize some analogies between one-dimensional non- stationary processes encountered in various areas of engineering, such as heat transfer in packed beds, ad- sorption, sedimentation, two-phase flow, hydraulics, compressible gas flow, traffic flow, etc. In many limit- ing cases, and with some approximations, all these systems are amenable to a common analysis, based on the mathematical form of the conservation equation (first-order quasi-linear partial differential equation), on classical tools of chemical engineering (equi- librium and operating diagrams), and some physical intuition. It is the author’s belief that a synthetic view of these processes may be of some use for teaching and research purposes, and also of help to the engi- neer to understand the basic concepts underlying the design tools he is likely to use.

The first part of this contribution is restricted to ‘one-component’ systems, in other words, systems where only one entity such as a chemical species, heat or momentum is being transported, and therefore a single conservation (or continuity) equation is writ- ten. In such systems, a single monotonic perturbation will generate a single ‘solitary’ wave. The second part will be devoted to multicomponent systems which re- quire the introduction of additional tools and con- cepts, and where multiple waves are generated.

The developments presented here may be found in independent and specific forms in the literature of the engineering areas mentioned above. Let us cite, how-

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ever, some references which bear a certain character of generality: Aris and Amundson [ 11, Helfferich and Klein [2], Prigogine and Hermann [3], Rodrigues and Tondeur [4], Whitham [5, 61, and especially the re- markable book by Wallis [7]. The purely mathematical aspects have been the object of a large number of studies in the mathematics literature [8-141.

From particle motion to wave motion (or from statistical physics to thermodynamics through traffic flow)

As in the propagation of light or electromagnetic energy, the duality wave/particle plays an important role in picturing processes where individuals move collectively. Typical examples are solid particle flow in a fluid in a gravity field (sedimentation, centrifuga- tion), drops or bubbles in multiphase flow (such as in liquid-liquid extraction columns), flow of cars on a road, etc.

Let us use the last example as an illustration. Con- sider a stretch of a one-way road of length AZ and write the conservation of cars on this stretch over a time interval At:

(cr + Lb - c,) AZ = (J, - J; + aZ) At (1)

where c, is the number of cars per unit length in the stretch at time t and J, is the flux of cars into the stretch considered (this assumes no cars going in or out through side-roads). If the traffic is dense enough, that is, if the sample observed is large enough, c and J can be considered as continuous and differentiable func- tions of time t and abscissa z. Equation (1) may then be rewritten as

1 gdt+ s divJdz=O

* -_ (2)

where the divergence operator reduces here to ajaz. For small enough time and distance intervals, the differential form of this equation may be used:

(3)

If we postulate a continuous differentiable relation J(c) between the flux J and the concentration c, or between the average car velocity u and the concentra- tion of cars, as is usually done in studies on traffic flow, eqn. (3) may be expressed as the classical form of a travelling or kinematic continuity wave equation [6]:

g+w+o with the velocity w, given by

dJ w, = -

dc

(4)

This velocity is that of a given value of c, as can be seen

by rearranging eqn. (4) and applying the chain rule for derivatives:

dJ (acpt), a2 z=W'.=-(acjaz>;= at ( 0

(6)

w,. is thus a local wave velocity, as opposed to the actual velocity of the cars. Equation (4) may also be consid- ered as a special form of the Hamilton-Jacobi equa- tion (ref. 8, for example), familiar in mechanics.

An ‘integral’, or ‘averaged’, form of this equality is obtained by averaging w, over a concentration inter- val, or alternatively by rearranging the finite difference form, eqn. (1):

s w dc ic = c2 - Cl

(7)

Equations (4), (5) and (7) are the forms used clas- sically in sedimentation [7], in chromatography [ 1, 21, and in hydraulic wave flow [5]. Prigogine and Her- mann [3] and Whitham [6], among others, have used this formulation in traffic flow. It is easily understood that this approach is approximate for small numbers of cars, and becomes rigorous as statistically larger numbers are considered, as is the case when the indi- viduals considered are molecules, or possibly solid par- ticles in a suspension. It appears that the passage from particle motion to the wave description meets some of the problems of statistical physics. Although it is not our purpose to investigate this, let us consider a simple illustration of this relationship.

The flux J used in the previous formulation might be expressed as the product of an average car velocity u and the concentration c, u being considered a func- tion of c:

J = UC

Then. from eqn. (S), the wave velocity is

(8)

du wc=u+cZ

and the problem is expressed in terms of average car velocity rather than flux, a possibly more convenient quantity to investigate. However, eqn. (8) is me&ring- ful only if the system is ergo&c with respect to the car velocities, that is, if the time-averaged velocity is equal to the space-averaged velocity.

Although ergodicity is a very important concept in statistical physics [ 151, the author does not know whether this is of great concern to traffic engineers, let alone the average car driver. Concerning chemical en- gineering, there may be some pertinence to these ques- tions in hydraulic problems such as the collective movement of bubbles or drops, or rivulets in two- phase systems (liquid-liquid extraction, bubbly flow, boiling problems, behaviour of distillation plates, fluid distribution in contactors), where a relatively small number of individuals is concerned.

The choice of car traffic as an example should not be considered as acceptance by the author that the

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collective behaviour of human beings can be ac- counted for by the laws of statistical physics. Readers interested in this aspect should definitely read the book by Prigogine and Hermann [3].

In the following, we shall be concerned essentially by families of situations which can be expressed by wave equations of the form of eqn. (4).

Relations between flux and concentration

In the preceding section, we saw that the local wave velocity w,. associated with a value of concentra- tion c was simply related to the derivative of the flux (or the average individual velocity) with respect to concentration. Similarly, the average wave velocity r~ over some concentration interval was related to the finite change ratio AJlAc. It is thus important to in- vestigate the basic physical interactions which govern the flux versus concentration relationship. Several families of interaction may be distinguished.

In the first, the dependence of the flux on concen- tration is due to a hindrance effect of the individuals among themselves: the more individuals there are per unit volume, the greater the hindrance. We therefore call it the ‘mutual hindrance’ case. A functional rela- tion may then sometimes be postulated between flux and concentration. The second family is one in which the individuals interact with some immobile medium, such as stationary phase in chromatography, or park- ing space along a highway. The flux then depends on how the individuals distribute between moving and immobile. Another family implies a dependence of the flux on the gradient of concentration: this is the case of diffusional phenomena.

Mutual hindrance

In the example of car traffic and of hindered set- tling (sedimentation), flux versus concentration rela- tions have been established empirically, and the author does not know whether any detailed physical or statistical approach accounts for the observed be- haviour. The form of the relationship is presented on Fig. 1 (for one type of particles, or cars, that is, a relatively homogeneous population). This curve ac- counts for different properties. Near zero concentra- tion, individuals behave as if they were alone (non-hindered traffic), their velocity u is independent of concentration, and the flux is therefore a linear function of concentration. The system is then defin- itely ergodic, and eqn. (8) applies. On the contrary, at very high concentrations, the hindrance is such that a ‘traffic jam’ exists, and the flux tends toward zero, or toward some minimum ‘plug-flow’ value. In between, the flux thus goes through a maximum, and the con- centration distribution is therefore di-valued: there are two values of c which correspond to any value of J between J,,,,, and J,,,,,. For concentrations above that of J,,,, the slope ddldc is negative, and thus negative wave velocities are possible.

5 b

s 1 I/\-

I ConancrUm

c

Fig. I. The schematic flux versus concentration curve for sedimen- tation and car traffic.

Such behaviour is also known to occur in liquid- liquid extraction columns, where the same flux of dis- persed phase may be obtained under two distinct regimes: ‘dense bed’ where the drops of dispersed phase barely touch each other, and ‘loose bed’ where these drops are almost independent [7]. Both regimes are stable and, as a matter of fact, may occur simulta- neously at steady state in two regions of the same column [ 161. The passage from one regime to another is easily achieved by introducing some resistance to propagation; for example, a toll or a bridge on a high- way will produce and maintain a high concentration zone, bounded upstream and downstream by low concentration zones which may correspond to the same flux of cars.

I propose to label this collective behaviour ‘mutual steric hindrance’, because the hindrance comes from the competition of individuals for an open space, and to distinguish it from ‘configurational steric hin- drance’, stemming from the interaction of the individ- uals with a geometrically restricted space (micropores of a zeolite, openings of a sieve). The latter behaviour may be included in the second family of interactions.

Viscous hindrance (fluid flow)

Viscous hindrance may be considered as a special case of mutual hindrance, in the sense that the migra- tion of the individuals (molecules or particles) is gov- erned essentially by the direct momentum transfer between them and possibly also with a stationary medium. The flow of solid particles may also, under certain circumstances, be included in this family.

Interaction with the medium (adsorption)

Let us consider another version of the car traffic problem, assuming now that not all cars move, but some are stopped at parking spaces distributed along the road. In addition, assume that the traffic is in the low concentration regime, with u constant, and that eqn. (8) holds. Let q be the concentration of cars stopped per unit length of road. The material balance (eqn. (3)) must then be rewritten to account for these cars:

(10)

171

If q is a continuous differentiable function of c, the derivative 8q/at in the equation may be replaced by (dq/dc)(&/at) and the equation again takes the kine- matic wave form (eqn. (4)) with the wave velocity given this time by

u

“” = 1 + dq/dc (11)

It can be verified that this expression is nothing but the derivative of the flux UC with respect to the total concentration c + q, in consistency with eqn. (5).

In addition to the average velocity of the moving cars u and the wave velocity w, a third so-called ‘ap- parent velocity’ u may be defined, also called ‘species velocity’ [2]. It is defined as the velocity averaged over all cars (moving and not), and is thus equal to

c u u=np=p c+9 1+9/c

(12)

With this definition, the flux is such that

J = UC = u(c + q) (13)

Notice the analogy of form between eqns. (I 1) and (12).

The propagation will thus be governed by the dis- tribution q(c) of the individuals between moving and non-moving, in other words, by the occupancy of parking spaces, and not by the crowding of the road as in the first no-parking version. For cars, it may be anticipated that q will be a monotonically increasing function of c with an asymptotic tendency toward a maximum (which corresponds to saturation of the parking spaces). This is also true for situations where thermodynamic equilibrium exists between the fixed and mobile ‘phases’, as in adsorption. The q(c) rela- tion is then simply the ‘sorption’ isotherm (sorption may be understood as physical or chemical adsorp- tion, or absorption, or ion exchange, or any reversible mechanism of capture). The equivalent of Fig. 1 in that case is obtained simply when the adsorbed spe- cies is very diluted in an ‘inert’, that is a non-adsorbed solvent or carrier gas. Then the velocity u is practi- cally that of the inert and can be considered as con- stant. The flux of solute is then simply proportional to c, and the equivalent of Fig. 1 is essentially the curve c versus c + q, that is, a form of the adsorption isotherm, as illustrated in Fig. 2. The adsorption

9 q+c

Fig. 2. Representations of the adsorption isotherm

isotherm no longer represents the pertinent flux versus concentration relation when u is a variable. An exam- ple of such a situation is outlined briefly in the next section.

We include in this family the case of immiscible displacement in porous media. The propagation of the wetting or non-wetting phase in the porous struc- ture is governed by interfacial forces, in many respects similarly to adsorption. A detailed example is given in the next section (see Fig. 8).

In all such situations, the relation between flux and concentration is monotonic, there is no extremum, nor multivalued situation. All wave velocities are positive.

D#krential dependence (diffusion)

In the case of common diffusion, the relation be- tween flux and concentration is Fick’s law:

J= -Da” aZ (14)

thus, a differential equation, and no longer an alge- braic type of relation. Substitution of eqn. (14) into the continuity equation (3) leads to the diffusion equation

ac a% z=D&Y (15)

which is not, to the author’s knowledge, trans- formable into a kinematic wave equation. Diffusion thus does not seem to fit into the wave approach, although some attempts have been made to use wave- like solutions.

It is interesting, however, to observe how diffusion is affected by simultaneous adsorption. Consider the example of longitudinal diffusion of a species in a pore of a solid catalyst, accompanied by adsorption on the walls. The governing equations are Fick’s law (eqn. (14)), the material balance accounting for ad- sorbed molecules (as in eqn. ( IO)), and the adsorption isotherm q(c). Combining these, and eliminating q as in the treatment of eqn. (lo), one obtains

ac D a% t= 1 + dq/dc dz2 (16)

In this diffusion equation, the ordinary pore diffusion coefficient is modified by adsorption in the same way as the wave velocity in eqn. (11). The diffusion is slowed down, the steeper the slope of the isotherm, dq/dc. We shall discuss diffusional situations again in the second part of this paper, dealing with multicom- ponent systems.

The movement of a sand dune under the effect of a horizontal wind may be included in this family (see Fig. 12) : the flux of sand entrained at any level on the rear of the dune is a decreasing function of the slope of the dune, thus of the derivative of its local height, considered as a concentration. Of course, the mechan- ical cohesion between the sand grains also plays a determining role.

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Dispersive and compressive waves-shocks

Dispersive desorption waves

Considering the case of adsorption, and the ex- pression of the local wave velocity, eqn. (11) it can easily be seen that, if the isotherm is of Langmuir type (Fig. 2) the derivative dy/dc decreases when q and c increase (in other words d2q/dc2 is negative). The wave velocity w, corresponding to low concentrations is therefore smaller than that corresponding to high concentrations. Consider the consequence of this property for a desorption wave; such a wave is gener- ated, for example, when, pure non-adsorbed helium flows into a bed of active carbon previously equili- brated with a gas containing, say, hydrocarbons. Fig- ure 3 shows successive positions of such a desorption wave in the bed. Since the high concentrations move faster than the lower ones, the desorption wave will tend to spread more and more as it moves down the bed: this behaviour may be qualified ‘dispersive’, or ‘diffusive’, or ‘spreading’. Another representation of such a wave is in the distance-time plane: to each concentration is associated a line (a so-called charac- teristic) which represents its trajectory in this plane. In this example, the desorption wave is represented by a ‘fan’ of straight characteristics which all converge to a point, taken here as origin of both time and distance. The wave is thus initially a perfectly sharp jump, and is called a ‘centred wave’ [ 11.

The dispersiveness of the desorption wave just considered was seen to result from the curvature of the adsorption isotherm, thus of the term dq/dc in eqn. ( 11). Dispersiveness is also obtained with dq/dc constant (linear isotherm) but a variable flow rate affecting U. This occurs, for example, by the desorp- tion of a species present in a large amount using a given flow rate of inert carrier gas. Ahead of the des- orption wave, the flow rate is that of the inert aug- mented by the flow rate of the desorbed species. Behind the desorption wave, the flow rate is simply that of inert gas. The flow rate, and thus the velocity U, varies continuously along the desorption wave in such a way that the lowest concentrations of desorbed species correspond to the lowest velocities. Qualita- tively, this so-called ‘desorption effect’ disperses the profile in a way similar to the curvature of a Lang- muir-type isotherm (d’q/dc’ < 0), and the two effects enhance each other. An opposite curvature generates a ‘compressive’ effect (discussed below) which coun- teracts this effect. A detailed description of the effect of variable flow rate requires solving simultaneously the conservation equation for the adsorbed species and the inert carrier [ 17-191. This leads to a modified expression of the wave velocity:

uoF(c) w = l + (c, ~ c) dq/dc

(17)

where u. is the velocity at the column inlet, ct the total molar concentration of the gas and F(c) a known function of c.

I \Y 4 Time

Fig. 3. Successive concentration distributions along an adsorbent bed and the corresponding fan of characteristics in the distance- time plane, illustrating the dispersive propagation of a desorption wave (the adsorption isotherm has the curvature of Fig. 2).

Dispersive waves in &id mechanics

Let us consider the example of laminar liquid flow on a flat inclined plate. The steady-state fully devel- oped velocity profile is given by the familiar parabolic law (Fig. 4)

u = ; (S2 ~ 2)

with

A= g Ap sin CL

(19) P

6 is the film thickness, x the distance from the surface normal to the plate, and CI the inclination angle of the plate. For each value of x, there is thus an associated velocity u(x) and, with the usual hypothesis, u(6) = 0.

Consider now the effect of a sudden decrease of flow rate Q at the entrance of the inclined section. Between the zones downstream where the initial con- ditions prevail (with a flow rate Q,), and the zones upstream where a new steady profile establishes, corresponding to the new flow rate Q2 < Q,, a tran- sient ‘drainage wave’ builds up, in which the film thickness 6 varies, and the local velocity also (Fig. 4). This wave is dispersive because the velocities of the upper layers of the film are larger than those of the lower layers.

An explicit expression for the velocity profile of the drainage wave may be obtained under the approxima- tion of quasi-steady flow, in which dynamic effects, such as acceleration, are neglected (see, for example, ref. 7) and the local velocity of any point of the sur- face of the wave depends only on the local film thick- ness. The continuity equation in the wave, written for a unit width of film, is (see eqns. (3)-(6))

a6 aJ a6 3t+i=S1+w,g=0 z

(20)

173

Steady flow profile

Steady flow prorile

Fig. 4. The dispersive drainage wave. obtained in laminar flow hy a decrease in flow rate.

where dJ

wg =cis (21)

is the velocity of propagation of a given value of film thickness. Here, the film thickness plays the role of a ‘concentration’. The flux corresponding to a given 6 is obtained by integrating the velocity profile (eqn. (18)):

and the local wave velocity is then, from eqn. (21), u’g = Aa? (23) Notice that the velocity profile of the wave is parabolic, but its curvature is opposite to that of the steady velocity profile.

Dispersive waves are also generated in compress- ible fluid flow, and are usually called expansion or rarefaction or dilatation waves [20].

Compressir:e waues und shocks

We designate by ‘compressive’ the behaviour op- posite to dispersive. Considering the adsorption ex- ample, the same arguments used for the case of desorption of a solute hold, that is, with a Langmuir- type isotherm, high concentrations tend to move faster than small concentrations. If a progressive con- centration increase is imposed at the sorbent bed inlet, a ‘catching up’ of the small concentrations by the high ones will occur, and the concentration distribution in the bed, instead of spreading, will tend to ‘compress’ and become sharper and sharper, as illustrated in Fig. 5. If this tendency, due to the isotherm curvature,

were not in physical reality counteracted by diffu- sional phenomena, the wave would tend to ‘overlap’ or ‘break’ (broken profile in Fig. 5), as do the sea waves on the beach. From a mathematical point of view, the solution of the differential conservation equations becomes multivalued. In one-dimensional space this is not acceptable, and mathematicians con- struct one-valued (but no longer smooth) solutions by introducing a discontinuity, a jump. Much attention has been given to the conditions and construction of such solutions (refs. 8-12, for example), and we shall introduce some of the fundamental results in the fol- lowing paragraphs.

In the physical problem, owing to the action of diffusion, mass transfer resistances and hydrody- namic dispersion, which all have a spreading effect, the adsorption wave tends to take a smooth constant shape, a so-called ‘constant pattern’, which is actually a moving steady state. It can be shown (ref. 21, for example) that along this steady concentration distri- bution, a linear relation exists between flux and con- centration, or, in the present example, between fixed phase and mobile phase concentrations: this property is the equivalent of the well-known ‘straight operating

Fig. 5. Successive concentration distributions along an adsorbent bed resulting from a progressive increase of inlet concentration. Illustrates the compression of the adsorption wave and the final constant pattern.

174

line’ property of steady-state counter-current opera- tions.

The analogue in compressible fluid flow is of course the familiar shock wave, in which a sharp change in pressure is propagated at velocities which can be much larger than the velocity of sound [20]. The hydraulic analogue is obtained in the example of the laminar film by an increase in flow rate Q. The same phenomenon underlies the behaviour of ‘tsunamis’ or ‘race waves’ in the sea, and of flooding waves in rivers, under the effect of heavy rains in the collecting basin, or the rupture of a dam; the forma- tion of the sharp edge of a moving sand dune (Fig. 12) and of ‘hydraulic jumps’ in the flow of granular mate- rial [22] probably obey similar laws. Shock waves in all these physical situations are governed by the gen- eral conservation property expressed by eqns. (1) and (7) which can be rewritten as

= -L\J shock AC

(24)

where the A designate the change across the shock. Such relations are known as the Rankine-Hugoniot equation in compressible fluid flow (c is then the den- sity p, and J is the product pu of density times veloc- ity). In the case of adsorption, this condition becomes

u w

shock = 1 + Aq/Ac (25)

Composite waves-the convex envelope rule

By ‘composite’ waves, we mean waves which have both compressive and dispersive parts. This will occur in adsorption when the isotherm has a point of inflec- tion, that is, a point where the second derivative d2q/ dc2 changes sign. The distribution of velocities is thus no longer monotonic. Let us consider three somewhat different physical situations.

The first occurs in packed-bed heat transfer when a phase change is involved. For example, some mate- rials have recently been developed for heat storage purposes with encapsulated fusible chemicals, such as hydrated calcium chloride [23] or paraffin [24]. When a heat carrier fluid is passed through a bed packed with such material, a temperature wave is created, which can have surprising shapes. Figure 6 is an ex- perimental example obtained on beds of carbon parti- cles impregnated with paraffin, and percolated by hot air. The temperature curve measured at the bed outlet (or at any abscissa) shows a composite wave, involv- ing two zones with a strong temperature gradient (compressive shocks), separated by a zone with a weak gradient (dispersive part). This behaviour can be explained if one considers the shape of the so- called ‘enthalpy curve’, that is, the curve representing the specific enthalpy h, of the immobile storage mate- rial versus the specific enthalpy h, of the mobile fluid phase (Fig. 7). The latter can usually be considered as linear with temperature (the specific heat C,r is con- stant). This curve is the equivalent of the adsorptiun

Fig. 6. Heat storage in a bed packed with paraffin-impregnated carbon, percolated by hot air. The temperature measured at the bed outlet shows shocks and the dispersive wave (from ref. 24).

isotherm, and is thus the appropriate flux-concentra- tion relation, the heat flux at given flow rate being proportional to the temperature of the fluid. The heat transfer problem is governed by a continuity equation similar to eqn. (10) with c replaced by T (C,, being assumed constant), and q by h,. The local thermal wave velocity is thus, analogous to eqn. ( 11)

w’ = 1 + (l/C,r)(dh,/dT) The velocity profile generated by this equation and a given initial temperature distribution will not be monotonic, and there will be compressive and disper- sive zones. These zones have to be pieced together properly to construct an admissible one-valued solu- tion.

Since dispersive zones obey eqn. (26) and shocks obey the counterpart of eqn. (25), the connecting con-

t / 0 1 .

Tempeaure ,T

M M 40 50 6r Fig. 7. The enthalpy-temperature curve of paraffin-impregnated carbon. The broken curve is the operating line of the wave of Fig. 7.

175

dition between these elements is that at their ‘intersec- tion’ we have

Ah, _ dhs Ahf - dh, (27)

In other words, the slope of the enthalpy curve at such a point should be equal to the slope of the ‘jump line’ or ‘operating line’ of the shock. (This condition is actually valid only in the final so-called ‘coherent’ regime of the response, a notion which we introduce in the following section.) Geometrically, this condi- tion amounts to taking the convex envelope of the enthalpy curve; in Fig. 7, the ‘operating line’ which describes the profile of Fig. 6 coincides with a string stretched on one side of the enthalpy curve. The side is that of the fluid enthalpy axis, if the heat is trans- ferred from fluid to fixed phase; the enthalpy of the fluid at any point in the real system should always then be larger than what it would be if the fluid were in local equilibrium with the solid. If the fluid cools the solid, the convex envelope is taken on the side of the solid enthalpy axis. These conditions are equiva- lent to the so-called ‘entropy condition’ introduced by mathematicians and physicists [ 1, 8-121 which basi- cally expresses that a shock generates entropy.

The second example deals with immiscible dis- placement in a porous medium. The fluxconcentration curves between two values of concentration have in that case the general shape of Fig. 8 and display hys- teresis (the wetting and the ‘drying’ follow different paths). Consider, for example, a drying operation, such as filter cake dewatering by air flow [25]. The operating line of this operation is obtained by taking the convex envelope of the fluxconcentration curve on the descending side (broken line in Fig. 8). The straight part of this line corresponds to a shock be- tween complete wetting (S = 1) and the intermediate value S,,,; the curved part corresponds to a dispersive front between S,,, and some residual value S, (liquid remaining in ‘pendular’ state, which is no longer re- moved by the hydrodynamic process). The corre- sponding concentration profiles at various times in the bed are represented in Fig. 9. This approach is the so-called Buckley-Leverett theory [ 261, which thus falls into the general wave approach presented here.

The third example is sedimentation (Fig. IO). Con- sider a batch sedimentation process in a closed vessel, starting frotn a well-stirred uniform suspension of solids corresponding to a concentration cg as studied by Kynch [27]. This process will give rise to a clear solution A at the top of the vessel, and a compact sediment D at the bottom of the vessel. Between these states A, B, D, different travelling waves arise, as illus- trated in Fig. 10, which also shows the relation be- tween the various forms of representation. We can observe a shock moving downward, between the orig- inal suspension B and clear liquid A, a shock moving upward between the compact sediment D and a layer of C of varying concentration (described by a disper- sive wave), and an upward moving shock between the latter zone C and the original suspension B. This pat- tern of behaviour may be deduced from the flux-

lo

09.

pr b 07. 4 $ as. I

I

+/

i

I

od

a3 1 +

@;. _,:’ 7

I

a2

Cl1 f

I s,

,’ 0

/? . a4 as a6 a7 us 03

Level of raturatlon s

Fig. 8. Flux versus saturation curves showing hysteresis for immis- cible displacement in porous media (wetting phase : water; non- wetting phase : air; from ref. 25).

concentration curve by drawing the proper operating lines connecting the supposedly known states A, B and D, using the convex envelope rule.

Note: the difference between this type of behaviour and adsorption, say, lies not only in the non- monotonic fluxconcentration curve, but also in the type of boundary conditions. The equivalent of a finite chromatographic pulse, for example, would be the sedimentation of a finite slug of suspension in a very deep cylinder (the boundary condition of a com- pact sediment is then removed).

i--W water

S t

I

m -Q8 water

QS

air

Fig. 9. Progression of the desaturation wave during dewatering of a filter cake by air (from ref. 25).

176

- T,me to cn Fig. 10. Representations of a batch sedimentation process: operat- ing lines in the flux versus concentration diagram (top right); characteristics in the distance--time diagram (bottom left); concen- tration profile at a time r0 (bottom right).

Qualifying transient regimes-the notion of coherence

Waves are by essence transient phenomena, in the common meaning of this term, that is, unsteady regimes. We feel there is a definite need for qualifying more precisely transient phenomena, and distinguish various degrees of ‘transientness’. To illustrate this, consider first the case of shocks. We have mentioned that a shock may exhibit a constant pattern, that is, a concentration distribution moving while conserving its shape. This is obviously very close to the common notion of a steady state, and is just a matter of refer- ence frame. We might include under the term ‘steady regime’ processes whose basic features are constant with respect to some Galilean reference frame (in movement at constant velocity and without dilatation or contraction of coordinates, with respect to some ‘fixed’ reference). This definition might include peri- odic steady regimes.

A dispersive wave is not a steady regime by this definition, however it does have a distinct feature which warrants qualification. For example, the con- centration distributions of Fig. 3 are related to one another by a linear relation between the abscissa of a given concentration and the corresponding time: (AZ/At), = ~1, = constant. We shall therefore call this behaviour the ‘linearly dispersing regime’. In the spe- cial case of centred dispersive waves, as in this Figure, the linearity is a simple proportionality ((z/t), = w, = constant), hence the name ‘proportionate pat- tern’. Linearly dispersing waves are also called recti- linear [28] or simple waves [ 1, 14, 291. Of course, the velocity distribution in such waves is constant and monotonic.

A more general point of view is needed to analyse examples involving composite waves, such as that of heat storage (Fig. 6). Both steady and linearly dis- persing regimes are present. In the Figure shown, however, the distinctive property is that the velocity

distribution is constant and semi-monotonic (the term ‘semi’ allows for parts with uniform velocity, which are the shocks); the fast moving temperatures are al- ways downstream of the slower moving temperatures, and no catching-up or interference process occurs. This property is equivalent to the convex envelope rule. We shall characterize this general behaviour by the term ‘coherence’.

The important notion of coherence has been intro- duced by Helfferich and Klein [2] in connection with multicomponent chromatographic systems, where it plays an essential role, and we shall come back to it in the second part of this paper. The coherent regime is that toward which a dynamic system tends asymptot- ically after a single perturbation (if left enough time). It is not necessarily a steady state. In the cases of one-dimensional waves considered here, the coherent regime is characterized by a constant and semi- monotonic velocity distribution. We may conjecture that the coherent regime possesses some dynamic properties analogous to those of equilibrium states, steady states, and some categories of dissipative struc- tures [30], namely structural stability with respect to small perturbations, and the minimum of some ‘action integral’ probably related to entropy produc- tion. These views are in accordance with modern ideas in thermodynamics tending to generalize Fermat’s theorem of minimum optical path: the notion of ‘co- herent evolution’ introduced by Schmid [28] is closely related to the coherence of Helfferich and Klein.

The coherent regime being the final ‘stable’ regime attained by a system, it may be preceded by various non-coherent events, depending on the nature of the perturbation. A single step perturbation will lead quasi-instantaneously to a coherent regime in most cases, whereas a ramp perturbation will give rise to more or less long non-coherent transients, depending on the direction and the rate of change. A sequence of two steps, each of which would separately give a co- herent regime, gives rise to non-coherent interferences between the two regimes. The simplest such example is probably the movement of a finite chromatographic pulse [ 2, 29, 311 illustrated by Fig. 11. With a convex adsorption isotherm (d2q/dc2 > 0, see Fig. 2), a shock is formed at the front end of the pulse, and a disper- sive (coherent) wave at the tail end. The velocity profile of the complete pulse is not monotonic: the shock is slower than some concentrations of the dis- persive wave, and therefore the latter will start inter- fering with the shock. The amplitude of the shock then starts decreasing, and so does its velocity. The velocity distribution of the chromatographic signal is then no longer a constant: this regime is not coherent.

fl I

I

i ‘0 ___ ___ c

Fig. 11. Schematic progression of a finite chromatographic pulse.

Fig. 12. A sand dune showing the steep front and gentle rear slope: Mount Pilat, in the south-west of France (by courtesy of Editions Cornbier, Maqon. France).

The breaking of sea waves on a slightly sloping beach, and the formation and migration of a sand dune (Fig. 12) may be interpreted in somewhat similar terms. More complex processes occur, for example, when the flow in the heat storage experiment of Fig. 6 is re- versed, turning the heat storage step into a reversed flow ‘de-storage’ step. The final coherent pattern of this step has its operating line on the solid enthalpy side of the curve of Fig. 7 (by the convex envelope rule and the entropy condition already discussed) and is very different from the storage pattern [24]. But before this pattern is attained, complex non-coherent transients occur, which involve such processes as re- compressing dispersive parts, dispersing and/or split- ting shocks [32].

Conclusions and prospects

The approach used throughout this paper is char- acterized by putting together the classical chemical engineering tools of equilibrium lines and operating lines, and the mathematical tools of conservative first- order partial differential equations having wave-like solutions. We can see that this approach furnishes a rather general conceptual framework for picturing, understanding and teaching a wide family of different, but related, engineering processes. We would not dare recommend this approach as an accurate design tool, although it may sometimes be useful even in this re- spect. We emphasize that the basic assumption under- lying this whole treatment is that of quasi-static processes, in which, for example, diffusional or capil- lary resistances, and inertial effects are neglected, and in which the hydrodynamic pattern is statistically steady. In many situations the presence of resistance will merely round off the results of the quasi-static model, without changing its basic qualitative pattern.

Another basic feature is that we have considered essentially single-shot non-periodic phenomena. In the language of hydraulics and mechanics, we have con- sidered ‘solitons’, that is, waves produced by a single

177

perturbation. To what extent can this approach be connected to the classical treatment of periodic phe- nomena, as encountered, for example, in electromag- netism? Can concepts such as oscillation, resonance, and wave reflection be transposed? We leave these questions open, and risk only a few remarks and con- jectures.

Firstly, the systems considered here are all strongly non-linear (owing to the non-linear nature of the flux- concentration curves). The classical theories of peri- odic phenomena are essentially linear, and therefore not transposable. Secondly, whereas the notions of resistance and capacitance are common in chemical engineering, the notions of inductance and of inertia have, to the author’s knowledge, no obvious chemical engineering counterpart, and it seems therefore out of the question to build a chemical circuitry analogous to RLC electric circuits or to mechanical oscillators. Thirdly, however, feedback effects exist and are well known in chemical engineering, and are able to gener- ate oscillations, resonance and reflection. To cite but one example of wave reflection, consider the enriching section of a distillation column with reflux, submitted to a step perturbation in the feed condition. This per- turbation generates a composition/enthalpy wave that moves up toward the top of the column. This wave is somewhat damped out by the condenser, and an at- tenuated wave is reflected back into the downward flowing liquid, which may possibly interact with an- other incident wave. See, for example, ref. 33 for a recent investigation of such phenomena.

Let us acknowledge the fact that a large body of mathematical literature exists dealing with the conti- nuity wave equation (4) [8-141. The results are not usually expressed in terms accessible to engineers, and have therefore been ‘rediscovered’ and reformulated case by case. Multicomponent systems, which will be considered in the second part of this study have un- dergone a similar process, and require the introduc- tion of still different concepts, with an even wider character of generality.

Nomenclature

coefficient in velocity profile, defined by eqn. (19), m-‘sP’ concentration of a mobile species (moles, mass or number of individuals, n, per unit volume or distance) concentration at given value of time t diffusion coefficient, in eqns. ( 14)- ( 16), mz s-’ gravity constant, in eqn. (lY), m s - ’ enthalpy of fluid phase and solid phase, re- spectively, in eqns. (26) and (27), kJ m - 3 flux, in moles, mass or number, n m - 2 s ~ ’ flux at given position z concentration of immobile species (moles, mass or number per unit volume or distance) fractional saturation of porous medium by wetting phase (Fig. 9)

178

t T 24

V

W z

:

CL P

time, s absolute temperature, K velocity of fluid, or of moving individual, ms-’ apparent, or averaged, velocity, in eqns. (12) and (13) ms-’ wave velocity, m s - ’ abscissa, m

inclination angle of plate (Fig. 5) film thickness, m viscosity, kg m - ’ s - 1 density, kg m - 3

Subscripts

c, 6, T refer to a given value of concentration, film thickness, or temperature, respectively.

References

1

2

3

4

5

6

7

8

9 10 11

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