Ind. Eng. Chem. Res. 1995,34, 2849-2864
2849
Wave Propagation in Mass-Transfer Processes: From Chromatography to Distillation YngLong Hwang Union Carbide Corporation, P.O. Box 8361, South Charleston, West Virginia 25303
This article reviews the advancement of the dynamic theories based on wave propagation phenomena for two major classes of separation processes: fixed-bed sorption processes such as chromatography and countercurrent mass-transfer processes such as distillation. Aiming at providing a common perspective, this article begins with a brief review of the h d a m e n t a l physics of wave propagation in these processes with notations common to both classes. Then, for singlevariance systems (with one independent component), it reviews the well-developed theories for f x e d beds and, as the main focus, the recent efforts to adapt these theories to countercurrent processes, in particular, distillation. The last part reviews the equilibrium theories of multicomponent chromatography with emphasis on the coherent wave theory originated by Helfferich and briefly covers the most recent attempts to apply this theory to multicomponent countercurrent processes. The review is summarized with prospects of applying the wave propagation theory to distillation control and design.
Introduction Chemical processes involving mass transfer from one phase to another are very common from laboratories to plants in chemical and related industries. Mass transfer between phases is indeed the most important mechanism utilized in separation processes. Many masstransfer processes are conducted in a fashion in which the bulk phases at large flow in a single dimension. These include two major classes. One class consists of a mobile phase flowing over a stationary phase as in chromatography and its variants such as fured-bed adsorption, fmed-bed ion exchange, pressure swing adsorption, etc. The other comprises two mobile phases flowing countercurrently as in moving-bed sorption, liquid-liquid extraction, gas absorption, stripping, and distillation. As a prototype, such a process in either class can be considered as a column in which diffusional mass transfer occurs between two phases with a relative convective transport in the axial direction. Although the two classes bear close physical resemblance, they were historically advanced in separate fields between which little communication was established. This article attempts to provide a common perspective of the two important classes of separation processes by reviewing the theories based on a dynamic phenomenon common to both classes: wave propagation, namely, travel of composition variations in the column. The physical analogy between chromatography and distillation was recognized to some extent in the early development of the former. Until recently, however, the similarity between the two classes of mass-transfer processes was scarcely exploited to advance the technology of countercurrent processes. Chromatography and its variants are dynamic operations by nature. In contrast, countercurrent processes in continuous operation have typically been analyzed and designed on the basis of steady states. There have been a few studies of the dynamic behavior of such processes mainly for the purpose of control, but most of these employed linearized and lumped models to fit into the linear control theory. In practice, many countercurrent processes display strong nonlinear and distributed characteristics, which were disregarded by the linearized and lumped approach. A better understanding of such
characteristics is crucial to better design and operation of such processes. Dynamic behavior of chemical processes has recently attracted more and more attention owing to the strong demands of safer, cleaner, and more efficient operation of chemical plants. Such demands tend to make chemical processes more difficult to control using conventional strategies based on linearized models. A common effort today is inventory reduction because chemical inventory is usually a threat to both safety and environment. This typically makes process units more sensitive to disturbances and interactions among one another, and thus more difficult to operate. Furthermore, such an effort may change processes from continuous operation to more flexible batch or campaign operation (a “campaign” operation uses the same equipment to produce several chemicals with each made in a continuous mode for a few weeks or months). Batch processes and the transitions within a campaign operation usually cover wide ranges of operating conditions and therefore may exhibit strong nonlinear behavior. For separation processes, severe nonlinearity may also result from high-purity specification for the demands of not only product quality but also waste minimization. In addition, the effort to conserve resources often leads t o highly integrated processes (e.g., reactive distillation), of which the dynamic behavior is usually critical to the success of such an effort. For the two classes of mass-transfer processes of interest here, a better understanding and description of their dynamic behavior may be acquired by exploiting the concept of wave propagation. With emphasis on physical insight instead of mathematical manipulation, this article will review the well-established wave theories for chromatography (fured-bed sorption) and, as the main focus, the recent efforts to incorporate these theories into countercurrent processes, in particular, distillation. The review will be presented in three sections. The first section will briefly review the fundamental physics of wave propagation in masstransfer processes, and the other two will review the advancement of the dynamic theories for single-variance (with one independent component)and multicomponent systems, respectively. With no intention to exhaust the literature on the subject, this review will focus on
a888-588519512634-2849~ag.aaia 0 1995 American Chemical Society
2850 Ind. Eng. Chem. Res., Vol. 34, No. 8,1995 (c) x ory
Saturated Zone Saturation Front
BUlk- Flow Front
I
Equilibrium Convection
Dissipation
Figure 1. Concentration wave and mechanisms of material transport: (a) saturation front and bulk-flow front in a sorption process, (b) trajectories of both moving fronts, and (c) a concentration wave representing material transport in the axial direction.
studies with emphases on general concepts instead of particular applications.
Wave Propagation Phenomena This section offers a brief review of the fimdamentals of wave propagation in both fixed beds and countercurrent processes. As described at the outset, these two classes of processes can be represented by a prototype column containing two phases with a relative convective transport in the axial direction. One can view a process of this type as transporting certain key materials, which can be transferred between phases, from one end of the column to the other (Hwang, 1991); actually, the purpose of such a process is to hold back such transport. The material transport can be described by a wave, which is defined as the travel of a variation of composition (or other properties) along the column. For demonstrating wave propagation, this section will use single-solute sorption (adsorption or absorption) as an example to discuss how a key component (the solute) is transported through the column. Material Transport Mechanisms. Consider for convenience an x phase (fluid) flowing in the axial direction denoted by z and a y phase (solid or fluid) either stationary or flowing countercurrently (in the -2 direction). First, let us focus on a simple fixed bed with a stationary y phase and an x phase flowing a t a bulkflow velocity uo in an ideal plug-flow fashion. In a typical sorption process, the solute carried into the column by the x phase is partially transferred (sorbed) to the undersaturated y phase. Assuming that such mass transfer between phases is extremely fast so that a “local equilibrium”is established (Wilson, 1940), the y phase will be saturated layer by layer as soon as a layer is in contact with the x-phase feed. Before getting into a deeper analysis, which will be discussed later, one may for now imagine that such layer-by-layer saturation will result in a sharp front of the saturated zone, as shown by Figure la. As Figure l b illustrates, the saturation front, called a “boundary” by DeVault (1943) and a “wave” by Klotz (1946), will travel a t a linear velocity U A behind the front of the x-phase bulk flow because the sorbed portion of the solute has become immobilized. In other words, the solute is convectively transported in the axial direction at the velocity UA. This mechanism of material transport, referred to as the “equilibrium convection mechanism” (Hwang, 1991), results entirely from the intrinsic equilibrium properties of the two phases. For a countercurrent process, the
rate of such an equilibrium convective transport is a net result of the x-phase transport of solute and the y-phase conveyance of “vacancy” for the solute. Note that the wave velocity may become zero or even negative if one keeps raising the y-phase flow rate in the -z direction. Although many countercurrent processes are stagewise instead of continuous in space, the wave propagation behavior is much the same in a stagewise process with many stages. In reality, mass transfer between phases needs some time to attain equilibrium and the flow of a phase usually deviates from the ideal plug-flow condition owing to the difhsion and hydrodynamic dispersion in the axial direction. For demonstrating the general concept instead of giving a detailed analysis, this article will use “axial dispersion” to refer to the overall effect of axial molecular diffusion and axial contributionsfrom all hydrodynamic effects (including mixing in stages of a stagewise process). Axial dispersion pushes some solute molecules ahead of the sharp saturation front while moving others backward. Moreover, a finite mass-transfer rate allows some solute molecules to escape from being sorbed and move with the bulk flow. Both mechanisms make the wave spread, symmetrically or asymmetrically, as illustrated by the concentration (x or y) profile in Figure IC.Such “dissipation mechanisms’’ are subject to equipment and operating conditions such as phase contacting, solid particle size, flow pattern, flow rates, and so on. Typically, these mechanisms make only a minor contribution to the net rate of solute transport in the axial direction, except in the vicinity of the column ends. At a column end, the dissipation can stretch only into the column and therefore leads to a net contribution to the solute transport. This “column-end effect” plays a critical role in countercurrent processes (Hwang and Helfferich, 1988), as will be reviewed later. Wave Velocity. The travel of a wave can be described quantitatively on the basis of material balance. As qualitatively discussed above, the rate of the axial material transport depends mainly on the equilibrium convection mechanism. Accordingly,let us fist consider an ideal column with the assumptions of local equilibrium and no axial dispersion; the effects of the dissipation mechanisms will be discussed later. With the assumptions of no axial dispersion as well as uniform flow rates and phase holdups, the material balance around a thin layer of the simple prototype column leads to a partial differential equation as follows (see Nomenclature for symbols):
This equation applies to a fixed bed with a zero bulkflow velocity uoof the y phase and to a countercurrent process with a positive uo. With local equilibrium, the y-phase concentration y can be related to the x-phase concentration x via the phase equilibrium relation y = fix). Substituting such a relation in eq 1 results in a first-order hyperbolic partial differential equation:
where dy/& is the slope of the equilibrium curve. Kvaalen and Tondeur (1988)proved that dy/& is always positive for a thermodynamically stable equilibrium relation y = fix). The basic wave propagation behavior
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2861 entered the column as a step change, it maintains its sharpness during its travel. Such a step, known as a “shock wave”, is mathematically a “weak solution” (Lax, 1954) of the hyperbolic partial differential equation 2. The shock wave travels at a velocity which can be derived from an integral material balance across the wave, and expressed in parallel with eq 3 as (uo= 0 for a fixed bed):
I 1
Figure 2. Wave sharpening and spreading resulting from a nonlinear equilibrium: (a) a nonlinear equilibrium curve, (b) a self-sharpening wave and a constant pattern, and (c) a nonsharpening wave and a proportionate pattern.
can be deduced from this equation without actually solving the equation [methods for solving first-order partial differential equations can be found in several textbooks (Courant and Hilbert, 1962; Jeffrey and Taniuti, 1964; Aris and Amundson, 1973; Whitham, 1974; Rhee et al., 1986,1989)l. One can define a wave velocity u by tracking the movement of a specific value of x (or y> within the wave and relating u t o the phase equilibrium via eq 2: a2
(at),, =
u o - u”r(dy/dx)
+
1 r(dy/dx)
(3)
This expression reveals that the wave velocity in general varies with concentration. Wave Sharpening and Spreading. For a system with a linear equilibrium relationship within the concentration range of interest, eq 3 indicates that the local wave velocity is independent of concentration. Therefore, a wave maintains its shape under the ideal equilibrium and plugflow conditions. However, many practical systems cover concentration ranges within which the phase equilibria are nonlinear. For illustrating the effect of a nonlinear equilibrium on the shape of a wave, let us focus on a common type of equilibrium curve as exemplified by Figure 2a. Such a type of equilibrium, referred to as “favorable” in the fields of adsorption and ion exchange, includes Langmuir adsorption, ion exchange with nearly constant separation factors, distillation of nearly ideal mixtures, etc. Let us start the discussion with a fixed bed (u”= 0). First, consider a wave with concentration decreasing monotonically from a higher level A in its upstream portion to a lower level B in its downstream portion (a sorption process). Because the slope dyldx of the equilibrium curve increases from A to B, as revealed by Figure 2a, the local wave velocity is higher in the upstream portion than in the downstream portion of the wave according to eq 3. As a result, the wave becomes sharper and sharper during its travel (DeVault, 19431, as Figure 2b illustrates. Such a wave was referred to as “self-sharpening” by Glueckauf and Coates (1947). Without the dissipation effects, the wave eventually becomes a discontinuous step (DeVault, 1943), as indicated by the dashed line in Figure 2b; if the wave
where A denotes the difference between the two sides of the shock wave. In reality, the sharpening effect of the nonlinear equilibrium is counteracted by the dissipation effects, and therefore the wave eventually attains a constant shape known as a “constant pattern”, as illustrated by Figure 2b. The constant-pattern wave travels at the shock wave velocity UA. In contrast, for a wave with concentration increasing from B in its upstream portion to A in its downstream portion (a desorption process), the local wave velocity is higher in the downstream portion of the wave. Consequently, as Figure 2c demonstrates, the wave spreads during its travel and was referred t o as “nonsharpening“ by Glueckauf and Coates (1947). Eventually, the dissipation effects become unimportant compared with the equilibrium spreading effect, and therefore the extent of the spreading becomes proportional to the time elapsed. Such a phenomenon has been known as a “proportionate pattern” (illustrated by a dashed line in Figure 2c). The fundamental sharpening and spreading characteristics of nonlinear waves are applicable to more complicated situations. For example, one may view a pulse as consisting of two waves with opposite concentration changes. With a nonlinear equilibrium, a pulse becomes asymmetric with a self-sharpening and a nonsharpening flank. The spreading of the nonsharpening flank plus the dissipation effect keeps reducing the peak height of the pulse. For a system with a more complicated equilibrium relationship, such as a sigmoid equilibrium curve, one may also analyze the sharpening behavior of a wave part by part (Glueckauf, 1947). In a countercurrent process (u” > 01, a wave may travel in either the +z or -z direction or even stand still (Hwang, 1987). Furthermore, the local wave velocities within a wave may head toward different directions. Nevertheless, eq 3 reveals that the local wave velocity u decreases monotonically as the slope dyldx of the equilibrium curve increases, as in a fxed bed. Accordingly, the sharpening and spreading behavior of such a wave is qualitatively the same as one in a fixed bed. Wave Dissipation. As mentioned above, there are basically two kinds of wave dissipation effects: nonequilibrium and axial dispersion. The effect of nonequilibrium can be quantitatively described by imposing on the material balance equation 1a model for the rate of mass transfer between phases. For example, one can write the y-phase material balance with a linear drivingforce model for the mass-transfer rate:
Then, the wave propagation dynamics resulting from a specific set of initial and entry conditions can be established by solving eqs 1 and 5 with a specific
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equilibrium expression fix). In some cases, one may need a more complicated model for the mass-transfer rate, for example, the two-film diffusion model, the penetration model, the particle-diffusion model, etc. In a simple model, the effect of axial dispersion in each phase can be expressed using Fick‘s law of diffusion with an effective dispersion coefficient for the overall effect of dispersion and diffusion. Superposition of these effects of both phases on the convective transports results in a revised material balance equation: & -+ at
& uo-
az
-
To account for both dissipation mechanisms, one can use the above simple models and obtain the column dynamics by simultaneously solving eq 6 and the following one:
9 - uo at
E
a2
- D,--$ = K(f(x) - y) az
(7)
To include the dissipation effects, one may also employ a stagewise model, which is certainly a natural choice for a stagewise process. Although such a model typically consists of equilibrium stages, it can actually include not only the dispersion (back mixing) effect but also the nonequilibrium effect, as demonstrated by Colburn (19411,who related the “number of transfer units” (a representation of mass-transfer coefficient) to the “number of theoretical plates” (number of ideal equilibrium stages). Wave Propagation Dynamics. The above discussion shows that the concept of wave propagation provides a clear picture of the dynamic behavior of a masstransfer process. Viewing such a process as transporting material from one column end to the other, one can express the rate of the axial material transport in terms of the travel of a wave contributed by two types of mechanisms. The equilibrium convection mechanism, stemming from the intrinsic property of the two phases, dictates the velocity of the wave and its sharpening or spreading tendency. This mechanism is the main source of nonlinear behavior. On the other hand, the dissipation mechanisms, dependent on the equipment and operating conditions, either counteract the equilibrium sharpening tendency or augment the spreading tendency. Usually, these mechanisms are nearly linear and often approximated by simple models as shown by eqs 5-7. For many applications in which wave dissipation is unimportant, one can quickly sketch the wave propagation dynamics using only the wave velocity equations 3 and 4, without the trouble of solving differential equations. The dissipation effects, however, may be crucial in a few applications, for example, calculating pulse widths and resolutions in analytical chromatography, predicting breakthrough curves of fixed beds, and dealing with column-end effects in countercurrent processes. The above equations clearly indicate a difference between the wave propagation behavior in a fixed bed and in a countercurrent process. In the latter, a wave driven by one stream has to travel against the opposite flow of the other stream, and therefore it may travel in either direction or stand still. Such an obvious difference, however, is not the most critical one. The most critical differences stem from the boundary conditions rather than the differential equations. In a fixed bed,
the wave propagation results from disturbances (composition changes) entering from only one column end. The downstream part of the column that the wave has not yet reached has no effect on the wave dynamics. In contrast, disturbances can enter a countercurrent process from both ends. Furthermore, disturbances can be induced at a column end when a wave, after traveling through the column, arrives a t that end; this leads to wave “reflection” (Jaswon and Smith, 1954; Hwang, 1987)and complicates the column dynamics. In addition, there are crucial differences regarding continuous countercurrent processes (excluding batch distillation, etc.). As a wave approaches a column end, it travels out of a k e d bed unhindered, but it is slowed down by the column-end effect of a continuous countercurrent process and eventually arrested to establish a steadystate profile. In a fured bed, a nonsharpening wave keeps spreading on its entire travel through the bed so that its sharpening behavior is very different from that of a self-sharpening wave. In a continuous countercurrent process, which is typically much shorter than a fixed bed relative to the practical width of a wave, a nonsharpening wave is contained in the column and eventually attains a steady-state shape, which is usually not drastically different from that of a self-sharpening wave. Moreover, a wave in a fixed bed usually travels on a background of uniform composition. In a continuous countercurrent process, a disturbance wave has to travel on a nonuniform steady-state background (Hwang and Helfferich, 1988).
Single-VarianceSystems With the general concept of wave propagation in mass-transfer processes outlined in the preceding section, this section will review existing theories for singlevariance systems such as single-solute adsorption and absorption, binary ion exchange, and binary distillation. Although some applications, e.g., chromatography, are by nature multicomponent, all linear systems will be treated as single-variance ones because such a system is an exact additive superposition of its single-variance subsystems. As discussed earlier, the wave dynamics of a system with a linear equilibrium is straightforward if there is no dissipation. Thus, most investigations of such systems spent major effort on the dissipation effects, which can be more clearly examined when the equilibrium has no effect on the wave shape. On the other hand, many studies of nonlinear waves focused on the effects of nonlinear phase equilibrium. For chromatography and fixed-bed sorption, most theories for single-variance systems were established before 1960, as appeared in several classical reviews (Vermeulen, 1958;Helfferich, 1962;Vermeulen et al., 1984; Helfferich and Carr, 1993). In a recent enlightening review, Tondeur (1987) related the nonlinear wave phenomena in fixed beds to those in various physical systems and provided a unified conceptual framework. In addition, Marquardt (1990)reviewed the studies of the nonlinear wave behavior in chemical processes, covering both separation and reaction processes. Chromatography with Linear Equilibrium. The simplest model of chromatography may be derived from the premises of a linear equilibrium relationship, the local equilibrium condition, and the absence of axial dispersion. Such a model was initiated by Wilson (1940) for analyzing chromatography. This simple model has been used to calculate the retention time (or retention volume) in analytical chromatography, in which solutes
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2853 are in low concentrations and therefore can usually be described by linear equilibrium relations. To account for the dissipation effects, Martin and Synge (1941) formulated a plate model by borrowing the concept of HETP (height equivalent t o a theoretical plate) from distillation. Mayer and Tompkins (1947) also applied a similar model to fxed-bed ion exchange. The plate model, originally under limited operating conditions, was later generalized mathematically by Glueckauf (1955a)and Said (1956). Glueckauf(l955c) then related the HETP of ion-exchange columns to the mass-transfer rate. Based on Gaussian pulses in elution chromatography, Klinkenberg and Sjenitzer (1956) related the HETP to both mass-transfer rate and axial dispersion. Extending their work, van Deemter et al. (1956) formulated a well-known equation for estimating the HETP of chromatography columns. These plate models were reviewed in detail by Villermaux (1981). As mentioned earlier, the nonequilibrium effect can be taken into account either in stagewise models like the above ones or in differential models like eqs 1 and 5. The latter were actually developed earlier. Before mathematical models were established for the dynamics of chromatography, researchers in another field had investigated the dynamics of heat-transfer beds, which are mathematically analogous to fxed-bed sorption columns with linear equilibrium relations. Modeling a heat-transfer bed with equations similar to eqs 1and 5 (with uo = 0 and A x ) = Kx,viewing x and y as thermal energy per unit mass or volume), Anzelius (19261, Schumann (1929), and Furnas (1930) independently obtained an analytical solution for the transient temperature profiles resulting from a step change of temperature at the bed entrance. Such a solution, later referred to as a “Jfunction” (Hiester and Vermeulen, 1952), was applied to ion-exchange beds by Beaton and Furnas (1941) and to adsorbers by Klotz (1946) and by Hougen and Marshall (1947). Later, Goldstein (1953) and Klinkenberg (1954) elaborated the J function with its mathematical properties and asymptotic approximations. A plot of the J function (Furnas, 1930) gave a sigmoid wave shape similar to that in Figure ICand demonstrated the character of a linear wave. A linear wave spreads during its travel because of the dissipation effects and lack of the equilibrium sharpening effect, but it spreads to a lesser extent than a proportionate pattern because there is no equilibrium spreading effect (the spreading is approximately proportional to the square root of the time elapsed). In addition to these efforts, researchers also developed fixed-bed theories with more complicated models for the mass-transfer rate. The majority of these rate models were based on either solute diffusion or chemical reactions. Among the diffusion-type rate models, Wicke (1939) and Glueckauf (1955b) dealt with intraparticle diffusion while Rosen (1952, 1954) and Kasten et al. (1952) accounted for diffusion in both phases. Reaction-type rate models typically lead t o nonlinear waves, and thus will be reviewed later. The simplest way to incorporate axial dispersion into a differential model of a fixed bed is a linear form exemplified by eq 6 (with uo = 0 and D, = 0). With a linear phase equilibrium, Lapidus and Amundson (1952) analytically solved such an equation in two cases for an infinitely long column. In one case, they assumed a local equilibrium to examine the dispersion effect alone; in the other, they employed a linear rate model of the form of eq 7 t o show the combined dissipation effect of
both dispersion and nonequilibrium. From their work, Vermeulen (1958) showed that, asymptotically (after a wave has traveled for a long time in a long column), axial dispersion and nonequilibrium make similar contributions t o the dissipation of a wave. The local equilibrium case was later extended to a column of finite length by Bastian and Lapidus (1956)and by Hashimoto et al. (19641, who analyzed the extent of dispersion relative t o the convective transport in terms of the Peclet number. More recently, Vink (1977) dealt with the nonequilibrium case with an intraparticle-diffusion rate model using the method of moments. Rasmuson (1981)provided a mathematical treatment for fixed beds involving axial dispersion and nonequilibrium effects from both intraparticle and fluid-phase diffusion. In addition, there are a few theories based on eq 6 but involving a nonlinear phase equilibrium; these will be reviewed later. Linear Waves in Countercurrent Processes. Before discussing nonlinear waves, let us examine linear waves in countercurrent processes. Under the assumptions of local equilibrium and no axial dispersion, a linear wave travels at a constant velocity and maintains its shape during its travel according to eq 3. The wave velocity may be positive, negative, or even zero. A wave with zero velocity would stay a t the location where it was created. This implies that the resulting steady state would depend on its history of establishment. Such an unrealistic artifact arises because the simple model without dissipation fails to account for the effects of boundary conditions, which affect the column behavior via asymmetric dissipation near the column ends (Hwang and Helfferich, 1988). This underlines that one cannot completely neglect the dissipation effects in a countercurrent process. As mentioned earlier, the dissipation effects can be more easily examined for systems with a linear phase equilibrium. Since the convective transport rates in a countercurrent process are typically high relative to both the mass-transfer rate between phases and the contribution of axial dispersion, the nonequilibrium effect usually dictates the wave dissipation. Accordingly, the following review will focus on this effect. To study a countercurrent process operated continuously, one needs to examine not only the main wave, which ultimately becomes a steady-state profile, but also the disturbance waves that travel on the background of a nonuniform steady-state profile. For linear systems, fortunately, the nonuniform background can be handled easily by expressing disturbance waves in terms of perturbation variables that are deviations from the steady-state background. The most common disturbances investigated are steps and impulses as well as sinusoidal functions for frequency responses. Among pioneer studies of linear dynamics of countercurrent processes based on differential models, Cohen (1940) investigated a packed column for isotope fractionation using a model of the form of eqs 1 and 5 (with f i x ) = I&). He obtained a solution in the Laplace-transform domain and series approximations in the time domain for specific cases. Later, Lees (1968) applied a similar approach with frequency response to a gas absorber. Blalock and Clements (1975) extended such an analysis to a model including both nonequilibrium and axial dispersion effects similar to eqs 6 and 7. To obtain solutions in the time domain, a few works relied on various numerical techniques (Buell et al., 1972; Bradley and Andre, 1972; Tan and Spinner, 1984a,b). Of
2864 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
more interest are several studies that established analytical solutions of eqs 1 and 5 and interpreted the dynamic behavior with the concept of wave propagation. Working on a packed rectifying column, Bowman and Briant (1947) initiated the wave concept for countercurrent processes and derived a lengthy integral solution. Jaswon and Smith (1954) constructed a solution of a complicated series form, which was intuitively proposed. Using examples of gas absorption, liquid extraction, continuous and batch distillation, and heat exchange, they also provided a general picture of wave propagation with reflection at the column ends. Chernyshev (1966) presented an interesting approach of superposition of two waves traveling in opposite directions, but he oversimplified the boundary conditions. Modeling a double-pipe heat exchanger with equations equivalent to eqs 1 and 5 , Friedly (1972) obtained a series solution similar to the one given by Jaswon and Smith; he also derived the beginning part of an integral solution (with a minor error) using the method of characteristics along with the Riemann representation (Courant and Hilbert, 1962, Chapter V). These earlier studies established the fundamentals of wave propagation phenomena in countercurrent processes. However, none revealed any mathematical analogy to the solutions of fxed beds, in particular, the J function. Applying the Riemann representation as employed by Friedly to eqs 1 and 5 along with a special Laplace transformation, a recent study (Hwang, 1987) yielded a complete integral solution. For the dynamic response to a step disturbance, this solution is reduced to the J function as the y phase becomes stationary. This directly proved the analogy in mathematics between fixed beds and countercurrent processes and thus laid a foundation for incorporating the well-developed wave theories for the former into the latter. More importantly, the solution also clearly demonstrated the critical difference between the two types of mass-transfer processes: the complex dynamics of a countercurrent process resulting from its two-end boundary conditions. By introducing “wavelets”as partial contributions from disturbances and reflections at column ends, the study clarified the wave reflection picture given by Jaswon and Smith (1954) and the wave superposition scenario proposed by Chernyshev (19661, as will be briefly reviewed below. Figure 3 illustrates the propagation of linear waves in time-distance (t-z) space. Consider a disturbance introduced a t the x-phase entrance at a time represented by point 0. For comparison, Figure 3 parts a and b shows the wave propagation phenomena in a fixed bed and in a countercurrent process, respectively. In a fxed bed, the leading edge of the disturbance wave travels with the bulk-flow front at the velocity uo toward the far end, which is of no importance. The wave affects only the t-z space behind the bulk-flow front. In that domain, the behavior a t any time-distance point such as C results from the propagation of the initial condition at A and the entry condition (disturbance) at B. This applies to the entire t-z domain behind the front, as indicated by the propagation of the conditions at points A and D t o point E. In contrast, in a countercurrent process, the leading edge of the x-phase disturbance wave soon reaches the other column end (as indicated by line OM), at which point a y-phase disturbance is induced and carried back into the column by the y-phase flow at the velocity uo (with the leading edge represented by line MN). The primary wave generated by
(a) Fixed Bed
(b) Countercurrent
Process
X X
D
Y
O
‘ V O
Uo-
Y
IG
r
M
L
Figure 3. Linear wave dynamics in time-distance space: (a) wave propagation in a fxed bed and (b) wave propagation, reflection, and superposition in a countercurrent process.
the x-disturbance was called “wavelet”1(Hwang, 1987); the secondary wave resulting from the reflection of wavelet 1was called wavelet 2. In principle, the wave reflection will keep going forever as if the two column ends were two mirrors face to face. Such a phenomenon divides the entire time-distance domain into “subdomains” as shown and numbered in Figure 3b. Subdomain 0 (triangle OLM) is of little interest since the column has not yet experienced the disturbance. In subdomain 1(triangle OMN), a wave consists solely of wavelet 1and can be determined much as it is in a fured bed except that the initial condition is conveyed by the y-phase flow, as illustrated by points A, B, and C in Figure 3b. The dynamics is getting more and more complicated afterwards. For example, the behavior at a point E in subdomain 2 results from a superposition of two wavelets: wavelet 1 resulting from the xdisturbance value at time D and wavelet 2 induced by the reflection a t point G of another wavelet 1resulting from the x-disturbance value at an earlier time F. The behavior a t a later time will involve more wavelets and more reflections. Fortunately, the study showed that the contributions beyond wavelet 2 are normally negligible. It also pointed out that the reflection is a result of the dissipation effect and that there will be no reflection under local equilibrium. Nonlinear Waves in Chromatography. Now, let us look into nonlinear waves in fxed beds, of which an in-depth review was recently given by Helfferich and Carr (1993). For single-solute sorption, DeVault (1943) and Weiss (1943)independently pioneered basically the same nonlinear wave theory under the premises of local equilibrium and no axial dispersion. They demonstrated the wave sharpening and spreading tendency stemming from nonlinear phase equilibrium. In addition, they and Walter (1945a) showed that a nonsharpening wave asymptotically attains a proportionate pattern, for which the dissipation effects are relatively unimportant. DeVault also discussed the formation of a shock wave from a self-sharpening wave. Later, Sillen (1950)investigated both self-sharpening and nonsharpening waves based on the idea that, under the local equilibrium condition, a wave resulting from a single step change of feed concentration can be expressed by a function of a single time-distance combined variable z l t (a “Riemann problem” in mathematics). Glueckauf
Ind. Eng. Chem. Res., Vol. 34, No. 8,1995 2855 (1947) extended this "equilibrium theory" t o systems with more complicated phase equilibria such as one with a sigmoid isotherm. The equilibrium theory provides a handy tool for predicting the velocity and the sharpening and spreading tendency of a wave. However, many practical applications need to take into account wave dissipation, which is especially important for situations involving self-sharpening waves . The nonequilibrium dissipation effect was investigated around the time when the equilibrium theory was established. Considering adsorption as an irreversible reaction, Bohart and Adams (1920) and Wicke (1939) showed that a wave in such an adsorber attained a constant pattern. Glueckauf and Coates (1947) examined the final patterns of self-sharpening and nonsharpening waves with a simple rate model as in eq 5 along with a Freundlich or Langmuir equilibrium relation. For an ion-exchange column, Thomas (1944,1948) replaced the simple rate model in eq 5 with a nonlinear expression viewing ion exchange as a second-order reversible reaction and obtained an analytical solution which is a nonlinear combination of the J function. Although the reaction-type rate model might be somewhat unrealistic for sorption processes, this solution mathematically demonstrated that a nonlinear wave asymptotically attains either a constant pattern or a proportionate pattern. Employing the same rate model for ion exchange, Walter (1945b) postulated a constant pattern and derived an expression that turned out to be a special case of the Thomas solution. The Thomas solution was extended to other initial and entry conditions for various applications by Hiester and Vermeulen (19481,Baddour et al. (1954), and Goldstein and Murray (1959). On the basis of this solution, Hiester and Vermeulen (1952) also developed a numerical solution for a more realistic rate model in terms of a mass-transfer coefficient and a nonlinear phase equilibrium expressed by a separation factor. With such a rate model, Michaels (1952)derived an analytical solution for the constant-pattern case using a pseudo-steady-state approach. Later, Vermeulen and co-workers (Hiester et al., 1956; Hall et al., 1966) dealt with ion-exchange columns with masstransfer rates determined by both intraparticle and fluid-phase diffusion. As for the effect of axial dispersion, many researchers employed models of the form of eq 6 (with uo = 0 and D, = 0). A few studies of the dissipation of nonlinear waves owing to axial dispersion alone were based on such an equation along with the assumption of local equilibrium. Lightfoot (1957) presented a simple formula for situations where constant patterns were attained. Employing an equilibrium isotherm represented by a second-degree polynomial, Houghton (1963) investigated the band shapes in chromatography. Rhee et al. (1971b) provided an extensive mathematical analysis of the formation and width of a constant pattern in an infinitely long column. Yamaoka and Nakagawa (1975) used the method of moments and the perturbation theory t o analyze the peaks in chromatography. Kalinichev et al. (1978) derived approximate analytical solutions that may be used to calculate the time needed for establishing a constant pattern. In addition, several studies covered both nonequilibrium and axial dispersion effects on nonlinear waves in fixed beds. The majority of them were based on eqs 6 and 7 (with uo = 0 and D, = 0). Acrivos (1960) gave approximations of the fixed-bed behavior for limiting cases of nonlinear phase equilibrium. Cooney and
Lightfoot (1965) mathematically proved the existence of asymptotic solutions representing the final pattern of a nonlinear wave with either one or both dissipation effects. Rhee and Amundson (1972a) extended their mathematical analysis of constant patterns in infinite columns (Rhee et al., 1971b) to the situation with both nonequilibrium and dispersion effects. In addition to the dissipation effects reviewed above, there are other nonidealities which may have significant effects on the dynamics of some fixed beds. One of those is the variation of the bulk-flow velocity uo, which has been assumed constant in eqs 1-7. In gas chromatography with high solute concentrations, the bulk-flow velocity may vary considerably with the extent of sorption and therefore may affect the sharpening behavior of waves, as discussed by Bosanquet and Morgan (1957), Golay (1964), Peterson and Helfferich (19651, Tsabek (1981a,b), Ruthven (1984), and LeVan et al. (1988). Another nonideality is the variation of temperature owing to a substantial heat of sorption or an external heat supply or removal. This may significantly affect the fmed-bed dynamics, typically because of a major effect on the phase equilibrium and minor effects on the mass-transfer rate and the axial dispersion. In such a case, one needs to include an energy balance equation in the mathematical model, which therefore includes both the concentration and the temperature as state variables and should be mathematically manipulated in a way similar to that for a two-component system. Detailed discussions of nonisothermal fixed beds were given by Leavitt (19621, Amundson et al. (1965), Pan and Basmadjian (1967, 19711, Rhee et al. (1970b), Ruthven et al. (1975), Ozil and Bonnetain (1978), Jacob and Tondeur (1981, 19831, Helfferich (1982), and Yoshida and Ruthven (1983). Binary Distillation. One may anticipate that the behavior of a nonlinear wave in a countercurrent process would be much the same as that in a fixed bed. This is true only to some extent. Rhee and Amundson (1973) adapted the theories for fixed beds to countercurrent moving beds by assuming an infinitely long column. Such an adaptation described well the velocity, the sharpening tendency, and the shape of a wave. However, because it ignores the column-end effect as an equilibrium model does, it also leads t o the same mathematical artifact that a steady state would depend on the history of its establishment. This underlines again the critical role of the column ends in a countercurrent process. Since distillation is the most widely practiced countercurrent process, the review here will focus on binary distillation columns, especially on those producing highpurity products. By nature, a high-purity column exhibits severely nonlinear behavior as reflected by its sharp composition and temperature profiles (Rose et al., 1956; Moczek et al., 1965; Mohr, 1965; Luyben, 1971; Fuentes and Luyben, 1983; Kapoor et al., 1986; Skogestad and Morari, 1987), because only a small portion of the column is utilized for the major separation while other parts are used only for refining the products. Observing the existence of such a sharp temperature profile, Luyben (1972) proposed a profile-position control strategy for high-purity distillation columns. Such a strategy was later reinvented and applied t o plant columns by Boyd (1975) and Silberberger (1977). The last work was fortified and applied to extractive distillation by Gilles and Retzbach (1980, 1983) with a moving-front model, which essentially implies a shock
2866 Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995
(leaving and entering streams are nearly in equilibrium). This leads t o an operating line FD in Figure 4a with both F and D on (actually, close to) the equilibrium curve. Comparing the local wave velocities at F and D according to eq 8, one verifies that the wave is selfsharpening. Such a wave soon becomes a constantpattern wave traveling at a shock wave velocity that can be derived from an integral material balance: A(Vy - Lx) = A(Wx Uy)
+
VI
i
Figure 4. Constant-pattern wave in binary distillation column: (a) an equilibrium curve and an operating line of steady state and (b) a constant-pattern wave with its location represented by a stagnation point.
wave. Further enhancing their approach, Marquardt (l985,1988a,b)utilized the concept of constant-pattern waves in his model with wave position and wave-shape parameters as state variables. He also took into account the shape change of a wave near a column end by allowing the wave-shape parameters to vary with the wave position. Incorporating the critical impact of the column ends in the nonlinear wave behavior, Hwang and Helfferich (1988)provided a cause-and-effectanalysis of the dynamics of single-variance countercurrent processes in general and binary distillation in particular (Hwang, 19911, as will be reviewed below. Much as a fured-bed adsorber is t o hold solutes from reaching the far end (breakthrough), a distillation column is to keep the light components from getting to the bottom or, more commonly, t o keep the heavy components from reaching the top. In some cases, one column may be used to achieve both goals. For binary distillation, one can use a wave to describe the upward transport of the heavy component by the vapor phase or the coupled downward transport of the light component by the liquid phase. Such a wave is typically nonlinear because of the nonlinear vapor-liquid equilibrium. With the y phase representing the vapor, eq 3 can be rewritten as follows using notations specific to distillation (see Nomenclature):
Here, for convenience, the wave velocity u is defined in the vapor (y-phase) flow direction with s representing the distance from the bottom (-2 direction) in terms of either length (for packed columns) or number of trays (for tray columns). Note that this equation assumes uniform molar flow rates and holdups within a wave. Normally, the numerator in eq 8 dictates the dependence of the wave velocity on the slope dyldx of the equilibrium curve since the vapor holdup U is usually negligible [U(dy/dx)
