Experimental Study of Wave Propagation Dynamics of

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Experimental Study of Wave Propagation Dynamics of Multicomponent Distillation Columns Jack Ting† and Friedrich G. Helfferich Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802

Yng-Long Hwang,* Glenn K. Graham, and George E. Keller II‡ Union Carbide Corporation, P.O. Box 8361, South Charleston, West Virginia 25303

Distillation columns with sharp separations exhibit severely nonlinear behavior, which has been known to cause difficulties in column control and design. Such a column is characterized by sharp composition and temperature variations in the column. Previously, the binary distillation case was thoroughly analyzed using a nonlinear wave theory and such an analysis was experimentally validated. For multicomponent distillation, the complicated nonlinear dynamics of the movement and interference of multiple sharp composition variations can be elucidated with a coherent-wave theory developed earlier for general countercurrent separation processes. With a ternary alcohol mixture, the present study has experimentally verified the theory by demonstrating the existence and propagation of constant-pattern coherent waves in a 50-tray stripping column in response to a step disturbance of feed composition, feed flow rate, or reboiler heat supply. The study has also tested the theory’s predictions of composition profile, wave velocities, and asymmetric dynamics. Introduction Distillation columns with sharp separations have become more and more important in the chemical process industries, owing to the growing demands of high product purity, high recovery of products and unconverted reactants, and stringent waste control. Such columns, including high-purity columns which have attracted some attention, are characterized by composition and temperature profiles with relatively sharp variations dividing pinched zones. By nature, these sharp-separation columns exhibit severely nonlinear behavior, which has been known to cause difficulties in column control and design. Although the nonlinear behavior of such columns has been recognized for several decades (Rose et al., 1956; Moczek et al., 1965; Mohr, 1965), there have been relatively few studies aiming at providing fundamental insight into such interesting behavior. For application to control of binary or pseudo-binary high-purity columns, there were several studies exploiting the characters of the sharp temperature variations to formulate improved control strategies such as profileposition control (Luyben, 1972; Boyd, 1975; Silberberger, 1977), model-based control with a moving-front model (Gilles and Retzbach, 1980), and that with a traveling wave model (Marquardt, 1988). Recently, in a series of fundamental studies, the nonlinear behavior of sharpseparation columns has been elucidated by a nonlinear wave theory that depicts the movement of the sharp composition/temperature variations as “constant-pat* To whom correspondence should be addressed. † Current address: Union Chemical Laboratories, Industrial Technology Research Institute, 321 Kuang-Fu Road, Section 2, Hsinchu, Taiwan. ‡ Current address: Director, Business and Industrial Development Corporation, Charleston, West Virginia 25301.

tern waves” (or “shock waves”) whose evolvement and propagation are governed by the nonlinear phase equilibrium. Adapted from the theory for fixed-bed sorption, the wave theory was developed earlier for countercurrent separation processes in general (Hwang, 1987; Hwang and Helfferich, 1988, 1989b, 1990) and more recently for binary distillation in particular (Hwang, 1991, 1995) by including the effects of a side feed as well as reflux and reboil. Lately, an experimental study (Hwang et al., 1996) demonstrated the existence and propagation of constant-pattern waves and the resulting asymmetric dynamics in binary distillation columns by tracking the movement of sharp temperature profiles. The results validated the nonlinear wave theory in both qualitative description of wave behavior and quantitative prediction of wave velocity. This theory was recently applied to the control of high-purity columns with encouraging results (Han and Park, 1993; Balasubramhanya and Doyle, 1997); so was a similar traveling wave model (Marquardt and Amrhein, 1994). In multicomponent distillation, the evolvement and propagation of nonlinear waves become much more complicated because a column in principle contains multiple waves, which divide intermediate zones with unknown compositions. So far, the nonlinear wave theory for multicomponent distillation has been established only for a simple section with no side feed or withdrawal. In such a simple section, the theory for distillation is mathematically similar, or identical for some cases, to countercurrent moving-bed ion exchange or adsorption. For adsorption with Langmuir isotherms, Rhee et al. (1971) applied their “equilibrium theory” for fixed beds (Rhee et al., 1970) to moving beds under the assumption of uniform initial composition profiles. This work provided an extensive mathematical analysis of the basic wave behavior resulting from the countercurrent flow pattern, but it oversimplified the process in

10.1021/ie990038o CCC: $18.00 © 1999 American Chemical Society Published on Web 09/03/1999

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two critical aspects. First, a countercurrent process in reality has nonequilibrium boundary conditions at one or both ends (the entering stream is not in equilibrium with the leaving stream) so as to supply a driving force for mass transfer; failure to account for these nonequilibrium conditions will leave the steady-state location of the key separation wave either undetermined or trivial (Hwang and Helfferich, 1988). Second, for such a process, most dynamic phenomena of practical interest are initiated from steady states with nonuniform profiles. Taking these critical aspects into account, Hwang and Helfferich (1989b, 1990) adapted the “coherentwave theory” for fixed beds (Helfferich, 1967, 1968; Helfferich and Klein, 1970) to countercurrent separation processes in general using moving-bed ion exchange as an example for demonstration. They also pointed out that the theory is readily applicable to distillation since ion-exchange equilibrium is mathematically similar to vapor-liquid equilibriumsthe former with constant separation factors is formally identical to the latter with constant relative volatilities. In the present form, the theory provides fundamental insight into the nonlinear dynamic and steady-state behavior of multicomponent distillation and is directly applicable to simple columns such as stripping columns and batch distillation columns. For application to fractionation columns, the effects of side feeds (and withdrawals if any) as well as of reflux and reboil need to be taken into account in the future. To verify the theory, the present study has aimed at experimentally demonstrating the existence and propagation of constant-pattern coherent waves in a simple multicomponent distillation column. Although there were some previous experimental studies on distillation dynamics, none of them provided adequate data for illustrating multicomponent wave behavior. Among very few studies on multilocation dynamics of a distillation column, Howard (1970) recorded the composition histories at 26 points in a 14-tray ternary distillation column during a start-up under total reflux. However, he focused on the responses at individual locations instead of the holistic dynamics of the entire column. To demonstrate the fundamental behavior of coherent waves, we assembled a stripping column with 50 trays for the present work using analogous equipment to that in a previous study on binary distillation (Hwang et al., 1996). With a ternary alcohol mixture, we tracked the movement of the sharp temperature variations in the column during a transition from one steady state to another in response to a step change of feed composition, feed (liquid) flow rate, or reboiler heat supply (vapor flow rate). In addition to confirming the existence of constant-pattern coherent waves, this study has also tested the theory’s predictions of composition profile, wave velocities, and asymmetric dynamics. Theory This section provides a brief review of the coherentwave theory for multicomponent countercurrent separation processes (Hwang and Helfferich, 1989b, 1990), in the language of distillation. As mentioned in the Introduction, sharp-separation columns are of major interest here, and only simple columns or column sections with no side feeds or withdrawals will be considered. Since a sharp-separation column typically has a large number of stages, whether it is equipped with trays or packings is of little difference to the wave propagation behavior;

also, axial dispersion (mixing) is typically unimportant. Although the coherent-wave concept is, in theory, applicable to any type of vapor-liquid equilibrium, mixtures with azeotropes are beyond the scope of the present study because of their mathematical complexity. For simplicity, this article will only discuss the simple case with uniform molar flow rates of both liquid and vapor throughout the column section (the “constant molar overflow” assumption; this also requires uniform molar holdups to maintain consistent material balances). This assumption provides a satisfactory approximation to many distillation columns in practice. For such a case, the column behavior can be determined without considering the energy balance. The theory can be extended to columns with nonuniform flow rates by taking the energy balance into account as in previous studies on binary distillation (Hwang, 1995; Hwang et al., 1996); however, this is beyond the scope of this article. Waves. Consider either a tray column or a packed column with a mixture of n components (n - 1 independent), which are indexed in the sequence of decreasing volatility (i.e., from light to heavy). Let s denote the distance (in terms of number of trays or length) from the column base; accordingly, the trays in a tray column are counted from bottom up with the reboiler, if any, numbered as stage 0. Because lighter components are enriched and heavier components are depleted from bottom to top, there are some composition variations in the column, which are accompanied by temperature variations due to vapor-liquid equilibrium. A “wave” here is defined as a monotonic variation of composition or temperature, as in a fixed-bed sorption process. In response to a change from an entering stream, a wave may travel either upward or downward. The propagation of a wave can be observed by tracking the movement of each specific value of a concentration (e.g., liquid mole fraction xi) within the wave. Such a “wave velocity” can be derived from the dynamic material balance. With negligible axial dispersion, the wave velocity can be expressed as follows (see Nomenclature for symbols):

vxi ≡

(∂s∂t)

) xi

V(∂yi/∂xi)/ - L W + U(∂yi/∂xi)/

i ) 1, 2, ..., n

(1)

where (∂yi/∂xi)/ represents a derivative along a “characteristic line” (for details, see Hwang and Helfferich, 1989b; Hwang, 1987) and is dictated by the phase equilibrium. Accordingly, the wave velocity is primarily governed by vapor-liquid equilibrium. The wave tends to spread if vxi increases with s and tends to sharpen if vxi decreases with s; the former case has been known as a “nonsharpening” wave and the latter as a “selfsharpening” wave (a severely nonideal mixture may lead to a “composite” wave with different sharpening properties in different parts of the wave; however, that is beyond the scope of this article). Under ideal equilibrium conditions, a self-sharpening wave eventually becomes a discontinuity traveling at a “shock wave velocity” which can be derived from a material balance across the discontinuity:

v∆xi ≡

(∂s∂t)

∆xi

)

V(∆yi/∆xi) - L W + U(∆yi/∆xi)

i ) 1, 2, ..., n (2)

where ∆xi ≡ x′′i - x′i with ′ and ′′ denoting the lower and upper sides of the wave, respectively. In reality, with

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dissipation from finite mass-transfer rates and axial dispersion, such a wave attains a “constant pattern” (fixed shape) when the sharpening tendency from phase equilibrium is balanced by the dissipation effects. Such a constant-pattern wave travels at the same shock wave velocity because the dissipation affects mainly the wave shape. Coherent Waves. In theory, the wave velocities in eqs 1 and 2 can be calculated by solving a set of dynamic material balances, which are partial differential equations, along with a set of mass-transfer equations relating the compositions in the two phases. This, however, requires a tremendous mathematical effort. The coherent-wave theory is intended to avoid such a laborious task by supplying a practical simplification based on the concept of “coherence” (Helfferich, 1967, 1968; Helfferich and Klein, 1970). When a column is disturbed by a step change of an entering-stream condition, a physically stable column will soon resolve to an asymptotically stable dynamic state, which is called “coherence”. A wave in a multicomponent system can be viewed as a collection of single-component waves, one for each component. When a coherent state is attained, all component waves travel jointly, i.e., have the same wave velocity:

vx1 ) vy1 ) vx2 ) vy2 ) ‚‚‚ ) vxn ) vyn

(3)

v∆x1 ) v∆y1 ) v∆x2 ) v∆y2 ) ‚‚‚ v∆xn ) v∆yn

(4)

Substituting eqs 1 and 2 in the above requirements, respectively, leads to the following differential and integral “coherence conditions”:

(∂yi/∂xi)/ ) dyi/dxi ) λ ∆yi/∆xi ) Λ

for all independent i (5) for all independent i

(6)

Equations 5 or 6 will become much easier to solve if the relations between {xi} and {yi} are independent of time and space. For a sharp-separation column, of which a considerable portion is typically occupied by pinched zones, the equilibrium assumption offers a good approximation. Under this assumption, combining a set of vapor-liquid equilibrium relations with eq 5 results in an eigenvalue problem:

det(Φ - λI) ) 0

where Φ ≡ {∂yi/∂xj}(n-1)×(n-1) (7)

Given a set of thermodynamically stable phase equilibrium relations for a system of n - 1 independent components, eq 7 results in n - 1 distinct real positive eigenvalues {λk} of the Jacobian matrix Φ at all composition points except for some possible degenerate cases at certain singular points (Kvaalen et al., 1985; Kvaalen and Tondeur, 1988). Accordingly, there are a sequence of n - 1 “coherent waves”, each associated with an eigenvalue. The eigenvalue λk governs the local velocity vk of coherent wave k:

vk )

Vλk - L W + Uλk

k ) 1, 2, ..., n - 1

(8)

Some or all of the coherent waves can be self-sharpening and eventually become constant-pattern waves, which at the coherent state are constrained by eq 6. This integral coherence condition has a set of n - 1 distinct

real positive solutions {Λk} compatible with the “entropy condition” (Lax, 1957). Each of these physically plausible solutions {Λk} determines the shock wave velocity of a constant-pattern coherent wave:

v∆k )

VΛk - L W + UΛk

k ) 1, 2, ..., n - 1

(9)

With the components indexed in the sequence of decreasing volatility, both {λk} and {Λk} are in decreasing sequences. Since the vapor holdup U is typically much smaller than the liquid holdup W, eqs 8 and 9 easily indicate that both {vk} and {v∆k} (positive for upward and negative for downward) are also in decreasing sequences. Accordingly, coherent wave 1 travels at the top of the sequence and wave n - 1 travels at the bottom. Coherent Waves and Separation. In a distillation column section with a nonazeotropic mixture of n components, each of the n - 1 coherent waves corresponds to a separation, or “cut”, between a pair of adjacent components. For instance, coherent wave k represents a k|k + 1 cut, across which from bottom up components 1 to k become enriched and components k + 1 to n become depleted. Since a distillation column typically serves to achieve one key separation, the key cut and the corresponding coherent wave is of major interest in practice. For a binary distillation column with a nonazeotropic mixture, it has been shown that the sole wave is normally self-sharpening (Hwang, 1991), which is the only type that can provide a sharp separation. For a multicomponent sharp-separation column, it is taken as a premise here that all coherent waves are self-sharpening and thus eventually become constant-pattern waves. This will be demonstrated with our experimental results, but a rigorous theoretical discussion is beyond the scope of this article. Column-End Effect and Standing Waves. Consider a sequence of n - 1 coherent waves at a transition state traveling at their respective “natural velocities” (from eq 9) governed by vapor-liquid equilibrium. Eventually, all upward traveling waves will reach the top column end and all downward traveling waves will reach the bottom end. Near a column end, the intrinsic tendency for these waves to leave the column will be counterbalanced by the nonequilibrium condition between the leaving and entering streams. Consequently, these waves will be slowed by the column-end effect and eventually bunched near the top and bottom column ends, respectively, at the new steady state. There is an important exception. If there exist one wave with a zero shock wave velocity, such a “balanced wave” (Hwang and Helfferich, 1988, 1989b) will eventually stand in the middle portion of the column section. The steadystate location and shape of such a “standing wave” is determined by the nonequilibrium column-end effect. In practice, it is desirable to make the wave for the key separation balanced in order to maximize the separation by adjusting the flow rates L and V. Balanced Wave and Asymmetric Dynamics. Although a balanced wave is most desirable for the key separation, it is also most sensitive to disturbances and therefore most difficult to maintain. The departure from such a steady state is always faster and usually much faster than the return to it; such “asymmetric dynamics” can be clearly explained by the nonlinear wave theory (Hwang and Helfferich, 1989a, 1990; Hwang, 1991). A

Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3591

disturbance to a balanced steady state will lead to an unbalanced wave with either a positive or negative velocity, which therefore travels upward or downward until being arrested by the column-end effect. Such a transition is dictated by the resulting wave velocity. If the condition is restored, the unbalanced wave standing near a column end will become balanced with a zero natural velocity. The resulting balanced wave has no intrinsic tendency to travel, but it will be pushed inward by the nonequilibrium column-end effect. As the wave moves away from the column end, it travels more and more slowly because the column-end effect decreases rapidly with the distance from the end. Therefore, such a returning transition is usually much slower than the departure from a balanced steady state. Composition Paths and Composition Routes. An important step in understanding the behavior of a multicomponent distillation column is to obtain its composition profile. The composition profile of a sharpseparation column contains multiple coherent waves dividing intermediate pinched zones with compositions that need to be determined. These compositions serve not only as prerequisites for predicting the coherentwave velocities but also as anchors for sketching the composition profile. For computing the composition profile of a multicomponent column, Helfferich (1967, 1968) devised a general and handy tool. For a mixture of n components, the n - 1 distinct eigenvalues of the Jacobian matrix Φ are associated with n - 1 eigenvectors. These eigenvectors define n - 1 directions at each point in the composition space, represented by either {xi} or {yi}. As a result, the eigenvectors at all composition points construct n - 1 families of “composition paths”, along which composition variations are coherent waves. For instance, coherent wave k follows a composition path of family k. The set of composition paths depend only on vaporliquid equilibrium and is independent of column configuration and operating conditions. Therefore, a composition path grid can be established for a specified mixture and then utilized for various column designs and operations. Given a set of boundary conditions (typically the conditions of the two entering streams), the path grid serves as a road map to establish the “composition routes”, which represent the composition variations in the composition space, of the coherent waves. Such a procedure also determines the compositions of the intermediate pinched zones. With such information, one can quickly sketch the composition profile of a distillation column with little computation effort. This provides the compositions needed for calculating the velocities of constant-pattern waves and should be very helpful to the design of a distillation column. The application of this tool will be demonstrated later with our experimental results. Ideal Mixture and h-Transformation. In general, the composition paths can be nonlinear. For mathematical simplicity, such a general case will not be discussed in this article. Since we selected a nearly ideal mixture for the present experimental study, this article will discuss only ideal mixtures with constant relative volatilities, which will be denoted by {R1i} with reference to the lightest component 1. Such an ideal mixture has linear composition paths; this has been proved for a mathematically identical ion-exchange system (Helfferich, 1967, 1968; Helfferich and Klein, 1970). It has also been demonstrated that such a linear grid can be

orthogonalized (decoupled) by applying the “h-transformation”, which transforms the set of n concentrations {xi} or {yi} to a set of n - 1 independent coordinates {hi}. Such a linear composition path grid and its orthogonal h-grid will be illustrated later with our experimental system. A brief summary of the h-transformation is given in Appendix A. With the aid of the h-transformation, both eigenvalue sets {λk} and {Λk} can be obtained analytically. Because the h-coordinates are orthogonal, only hk varies across coherent wave k while all other h-coordinates remain constant. For coherent wave k, the respective eigenvalues λk and Λk can be related to {hi} as follows (Helfferich and Klein, 1970):

[ ]/[ ] [∏ ]/[ ∏ ] n

λk )

n-1

R1i ∏ i)1

hi ∏ i)1

hk

n

Λk )

n-1

R1i

h′kh′′k

i)1

i*k

hi ) xλ′kλ′′k

(10)

(11)

Since only one h-coordinate varies across each coherent wave, one can immediately obtain the h-coordinates for all intermediate pinched zones from given {hti } and {hbi } at the top and bottom ends, respectively:

{

zone 1

t t t , htk, hk+1 , ‚‚‚, hn-1 } {ht1, ‚‚‚, hk-1

zone k

{hb1,

l ‚‚‚,

b hk-1 ,

t t htk, hk+1 , ‚‚‚, hn-1 }

b t t , hbk, hk+1 , ‚‚‚, hn-1 } zone k + 1 {hb1, ‚‚‚, hk-1

(12)

l zone n

{hb1,

‚‚‚,

b hk-1 ,

b b hbk, hk+1 , ‚‚‚, hn-1 }

The zones are indexed in the way that coherent wave k is the composition variation between zones k and k + 1 while zones 1 and n are the top and bottom ends, respectively. Note that eq 12 is derived from the equilibrium premise. In a multicomponent column section, at least one end is not pinched (the entering stream is not in equilibrium with the leaving stream) as discussed above. For nonequilibrium column ends, {hti } and {hbi } should in theory be calculated from the leaving stream compositions {yti } and {xbi }, respectively, which are consistent with the compositions inside the column but are normally not known a priori. Fortunately, as demonstrated by Hwang and Helfferich (1989b), {hti } and {hbi } can be calculated from the entering-stream compositions {xti } and {ybi }, respectively, and still give the same results for the key coherent wave, which is typically away from the column ends (this maneuver affects only those uninteresting waves bunched near the column ends). With this approach, one can easily predict the natural velocity of the key constant-pattern coherent wave using eqs 12, 11, and 9. This will be demonstrated later with our experimental system. Experimental Section In this study of multicomponent distillation, we used the same experimental approach as in our previous work on binary distillation (Hwang et al., 1996) with

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Figure 1. Laboratory stripping column for studying coherentwave propagation in multicomponent distillation.

analogous apparatus, materials, instrumentation, computer software, and techniques for disturbance introduction and response measurement. For conciseness, these will only be briefly reviewed in this section. To demonstrate the fundamental behavior of coherent waves, we conducted our experiments with a nearly ideal ternary mixture on a laboratory glass column with 50 trays in a stripping column configuration. Specifically, our objectives were as follows: (1) To demonstrate the existence and propagation of constant-pattern coherent waves in a simple multicomponent distillation column in order to confirm the theory qualitatively; (2) To compare the column temperature profile (as an alternative for composition profile) and the wave velocities with those predicted so as to verify the theory quantitatively or recommend refinements if needed; (3) To observe the asymmetric dynamics between the departure from and the return to a balanced steady state as predicted. We conducted our experimental work at Union Carbide Corporation’s Technical Center in South Charleston, West Virginia. Apparatus. Figure 1 presents a schematic diagram of our laboratory stripping column. The column itself consisted of several vacuum-jacketed glass column segments with Oldershaw sieve trays of 50 mm in diameter and 25 mm in tray spacing (adaptation of LG5631 from Lab Glass, Inc.). For monitoring the temperature profile of the column, we modified these Oldershaw segments by installing thermocouple wells on every other tray to measure the vapor temperatures. At the top, a glass total condenser was employed, and a glass liquid-dividing head (LG-6171 from Lab Glass, Inc.) was inserted below it to withdraw all condensate

as the distillate product (with no reflux). For reboil, we designed a low-holdup thermosiphon reboiler consisting of a vacuum-jacketed glass column base and a set of four stainless steel boiling tubes parallel to each other, which were heated with an electric heating tape and were wellinsulated. The liquid level in the column base was regulated by a multilevel pressure balancer. Two feed cans were used for making a step change of the feed composition. The feed was transported by a highprecision metering pump (Digifeeder DF-165-C from IVEK Corp.) to the column through an in-line preheater, which was heated with an electric heating tape. For data acquisition and device control, we coded a program in a data-acquisition software package (LabVEW 2.2 from National Instruments Corp.) and ran it on a microcomputer (Macintosh Quadra 950 from Apple Computer, Inc.). The temperatures were measured with type-T thermocouples. The weights of feeds and products were recorded by balances for computing their flow rates. The base pressure was monitored with a pressure sensor (PX425-015AV from Omega Engineering, Inc.) to estimate the pressure drop across the column. The heat supplies to the thermosiphon reboiler and the feed preheater were controlled via ac-voltage power controllers (model 18DZ from Payne Engineering) with PI feedback control loops coded in the computer program. In addition, a gas chromatograph (model 5890 II from Hewlett-Packard Co.) with a capillary column (type DB-1 from J&W Scientific) was used for mixture composition analysis. Materials. We selected a nearly ideal ternary mixture in this initial study of multicomponent systems to avoid the complexities from thermodynamic nonideality. Along with concerns of safety and other practical aspects, we chose a mixture of methanol, 1-propanol, and 1-pentanol for the following reasons: (1) Physical properties of these alcohols and their mixtures are well-studied and documented (Daubert and Danner, 1991; Gmehling et al., 1977; Hill and Van Winkle, 1952). (2) The vapor-liquid equilibrium of this mixture is nearly ideal with nearly constant relative volatilities. (3) These alcohols are relatively safe to human health and the environment. (4) These alcohols have normal boiling points in a proper range for boiling and condensing at ambient pressure and for measuring with regular thermocouples. The physical properties of these alcohols are given in Appendix B. In this study, we used alcohol materials with purity above 99% (from Aldrich Chemical Co.). Pressure. For simplicity, we conducted all our experiments at ambient pressure, which is usually 730760 mmHg at our location. We monitored the base pressure at all times to calculate the pressure drop across the column; the head pressure was recorded from the base pressure sensor before starting an experiment. Reboiler Characteristics. The effective reboiler heat supply QB (net boilup energy) was determined with 1-propanol in our previous study (Hwang et al., 1996). This variable was correlated to the heat-transfer driving force, that is, the difference between the heating-tape temperature TQ and the reboiler temperature T0. Within our operating range, this correlation turned out to be linear. A PI feedback controller was coded in the computer control program to regulate the temperature difference TQ - T0 in order to maintain a desired QB. The reboiler holdup (excluding the column base) was

Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3593 Table 1. Operating Data of Dynamic Experiments on Ternary Stripping Column run index

1A

1B

2A

2B

3A

3B

4A

4B

5A

5B

6A

6B

ref 1a

ref 2b

disturbance {xi,F} {xi,F} {xi,F} {xi,F} F F F F QB QB QB QB step changec +0.0953 -0.0953 +0.0741 -0.0741 +7.34% -6.78% -12.7% +15.3% +8.59% -8.76% +16.3% -16.8% variables changed {xi,F}, Fw {xi,F}, Fw {xi,F} {xi,F} Fw Fw Fw Fw TQ TQ TQ TQ key wave initial locationd new locationd

wave 1 wave 1 wave 2 wave 2 wave2 wave 2 wave 2 wave 2 wave 1 wave 1 wave 1 wave 1 wave 1 wave 2 balance top balance top balance bottom bottom top balance top bottom top balance balance top balance top balance bottom balance top bottom top balance top bottom

initial steady state feed x1,F composition x2,F (mole frac.) x3,F

0.4172 0.2976 0.2852

0.3463 0.3053 0.3484

0.4042 0.3013 0.2945

0.3529 0.4049 0.4024 0.4013 0.4005 0.3880 0.3986 0.3911 0.3945 0.4000 0.4000 0.2992 0.2986 0.2983 0.2972 0.2972 0.2991 0.3006 0.2978 0.2982 0.3000 0.3000 0.3479 0.2965 0.2993 0.3015 0.3023 0.3129 0.3008 0.3111 0.3073 0.3000 0.3000

F (mol/h)e Fw (g/h)

33.71 1901

32.82 1974

24.94 1422

23.92 1434

24.75 1412

26.43 1512

26.28 1506

22.95 1316

32.71 1897

32.80 1881

32.88 1902

32.96 1900

33.23

24.84

QB (kJ/h)f TQ (°C) T0 (°C)

895 192.0 115.1

896 194.0 117.0

1064 227.0 139.0

1068 227.1 138.9

1057 227.0 139.5

1063 227.0 139.1

1063 227.0 139.1

1061 227.0 139.2

881 192.0 116.0

968 198.0 116.3

805 186.0 115.0

953 198.0 117.3

890

1061

D (mol/h)e Dw (g/h)

16.40 622.7

14.76 579.3

18.04 834.2

16.51 782.2

18.14 835.8

19.04 878.6

19.16 885.0

16.91 786.2

16.12 617.3

15.10 658.0

15.76 597.6

16.84 675.2

16.29

18.09

distillate x1,D composition x2,D (mole frac.) x3,D

0.8164 0.1562 0.0274

0.7790 0.1853 0.0357

0.5364 0.4207 0.0429

0.5088 0.5433 0.5418 0.5400 0.5360 0.8061 0.6359 0.8169 0.7441 0.4354 0.4133 0.4141 0.4155 0.4132 0.1647 0.3170 0.1566 0.2248 0.0558 0.0434 0.0441 0.0445 0.0508 0.0292 0.0471 0.0265 0.0311

bottoms x1,B composition x2,B (mole frac.) x3,B

0.0000 0.4329 0.5671

0.0000 0.3917 0.6083

0.0000 0.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0021 0.0000 0.0000 0.0000 0.0000 0.0000 0.0014 0.0052 0.0000 0.4190 0.4099 0.4276 0.3827 0.4155 0.0000 1.0000 1.0000 0.9986 0.9948 1.0000 0.5810 0.5901 0.5703 0.6173 0.5845 1.0000 N/Ag

new steady state feed x1,F composition x2,F (mole frac.) x3,F

N/A

N/A

0.3463 0.3053 0.3484

0.4172 0.2976 0.2852

0.3529 0.2992 0.3479

0.4042 0.4024 0.4049 0.4005 0.4013 0.3880 0.3986 0.3945 0.3911 0.3013 0.2983 0.2986 0.2972 0.2972 0.2991 0.3006 0.2982 0.2978 0.2945 0.2993 0.2965 0.3023 0.3015 0.3129 0.3008 0.3073 0.3111

F (mol/h)e Fw (g/h)

32.63 1963

33.71 1901

23.80 1427

24.94 1422

26.57 1520

24.75 1412

23.12 1326

26.76 1533

32.78 1901

32.80 1881

32.96 1900

32.86 1901

QB (kJ/h)f TQ (°C) T0 (°C)

895 194.0 117.1

895 192.0 115.1

1066 227.1 139.0

1064 227.0 139.0

1058 227.0 139.4

1057 227.0 139.5

1061 227.0 139.2

1063 227.0 139.1

957 198.0 117.0

890 192.1 115.5

950 197.8 117.3

803 186.0 115.1

D (mol/h)e Dw (g/h)

14.61 575.0

16.40 622.7

16.45 780.9

18.04 834.2

19.21 884.8

18.14 835.8

16.86 786.2

19.10 878.5

16.92 663.9

16.22 618.9

16.84 675.2

15.15 575.4

distillate x1,D composition x2,D (mole frac.) x3,D

0.7751 0.1894 0.0355

0.8164 0.1562 0.0274

0.5084 0.4329 0.0587

0.5364 0.5438 0.5433 0.5332 0.5459 0.7749 0.8100 0.7441 0.8147 0.4207 0.4124 0.4133 0.4140 0.4105 0.1940 0.1622 0.2248 0.1587 0.0429 0.0438 0.0434 0.0528 0.0436 0.0311 0.0278 0.0311 0.0266

bottoms x1,B composition x2,B (mole frac.) x3,B

0.0000 0.3879 0.6121

0.0000 0.4329 0.5671

0.0000 0.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0006 0.0000 0.0000 0.0008 0.3913 0.3772 0.3827 0.4330 1.0000 0.9994 1.0000 1.0000 0.9992 0.6087 0.6228 0.6173 0.5670

733 38

737 36

743 33

0.0999 368

N/Mi N/Mi

0.0734 339

pressure and holdup P (head, mmHg) pres. drop (mmHg) W (mol/tray)h col. holdup (g)

742 34

741 34

742 35

N/M 0.0744 N/M 343

N/M N/M

744 35

741 34

743 37

745 36

N/M 0.0749 0.0987 0.0990 N/M 345 364 365

736 38

736 37

740 37

742 34

N/M 0.1048 N/M 383

a Reference steady state 1: each condition other than feed composition is the average among the initial conditions of runs 1A and 5A and the new conditions of runs 1B and 5B. b Reference steady state 2: each condition other than feed composition is the average among the initial conditions of runs 2A and 3A and the new conditions of runs 2B and 3B. c Step changes of composition {xi,F} are expressed in terms of the norm (size in mole fraction) of the change vector with “+” and “-” signs arbitrarily assigned to A and B runs, respectively, to denote the opposite directions of the change vectors; step changes of feed rate F in runs 3A-4B with focus on wave 2 are expressed as percentages of the feed rate of ref 2; step changes of reboiler heat supply QB in runs 5A-6B with focus on wave 1 are expressed as percentages of the heat supply of ref 1. d Standing wave locations: “balance” denotes a balanced wave in the middle portion of the column while “bottom” and “top” denote unbalanced waves near bottom and top column ends, respectively. e Molar flow rates F and D are calculated from measured mass flow rates Fw and Dw, respectively. f Effective reboiler heat supply QB is calculated from the recorded reboiler temperature difference TQ - T0. g “N/A” denotes that the new steady state was not attained when the experiment was terminated in runs 4B, 5B, and 6B. h Molar liquid tray holdup W is assumed to be uniform throughout the column and is calculated from measured column mass holdup. i “N/M” denotes that the column holdup was not measured in the respective runs.

also measured in that study. It concluded that there was no significant dependence of the holdup on the heat supply, and the average reboiler holdup within our operating range was 50 g. Column Holdup. In addition to the compositions and flow rates, tray holdups are also required for calculating the wave velocity. The molar vapor holdup can be estimated from the ideal gas law. For the liquid holdup, we assumed the average tray holdup remains the same throughout an experiment and determined it by measuring the entire column holdup at the end of an experiment. Upon finishing a dynamic experiment, we

collected all liquid in the column and the reboiler and then calculated the column holdup by subtracting the reboiler holdup. Input Disturbances. With a ternary mixture, there are two coherent waves, but the wave other than the key separation wave stays near a column end and has little impact on the column dynamics, as discussed earlier. Each of our dynamic experiments was intended to demonstrate the propagation of a chosen key wave in response to a step change of either feed composition {xi,F}, or feed flow rate F (i.e., the liquid rate L entering the top of the stripping column), or reboiler heat supply

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QB (which determines the vapor rate V entering the bottom of the column). For all runs, we chose a reference feed composition of {x°i,F} ) {0.4, 0.3, 0.3} to obtain easily measurable temperature profiles and product flow rates. For conservation of material, we recycled feed by recombining the products collected from a previous run and adjusting the composition with the aid of a gas chromatograph. Although such a practice introduced some errors to the feed composition, these errors were acceptably small. In all experiments, we preheated the feed to 78 °C, which is about 5 °C below the bubble point of the reference feed; this was to obtain a nearly saturated liquid feed and in the meantime avoid the risk of partial vaporization. In each dynamic experiment, we first established an initial steady state with an appropriate combination of feed rate and reboiler heat supply that led to not only a proper tray loading but also a desired location of the standing key wave. To introduce a step change of feed composition, we prepared two cans of feeds with different compositions and switched from one can to the other. To introduce a step change of feed rate, we changed the mass feed rate Fw by altering the feed pump speed. The ideality of these sharp step changes was slightly softened by the fact that the feedback control of the feed preheater took 2-5 min to compensate for the accompanied change of the feed temperature. In addition, the feed composition change was inevitably delayed by 1-2 min, which is the time required to transport the liquid from the feed can to the feed port of the column. We created a near-step change of the reboiler heat supply by changing the set point of TQ - T0, which typically overshot the new set point in 2 min and settled at that temperature in about 5 min. Data Acquisition. We observed the propagation of the key wave during a transition from one steady state to another by tracking the movement of the temperature variation. The data-acquisition program scanned the thermocouples on every other tray and the reboiler in cycles of 3-5 s. When a steady state was approached, we monitored the most sensitive tray temperature (the one at the sharpest part of the temperature variation) to examine the slow change in such a situation. At the beginning and the end of a transition, we ensured the attainment of a steady state by keeping such a temperature on a flat trend for at least 20 min. The computer program also recorded the weights of feed, distillate, and bottoms in each cycle and calculated the flow rates based on weight changes in corresponding time intervals. In addition, we collected distillate and bottoms samples at both initial and new steady states and analyzed their compositions with a gas chromatograph. Dynamic Experiments. We conducted six pairs of dynamic experiments. The operating conditions and the measured flow rates and compositions are listed in Table 1. In each pair of experiments, the “B” run was intended to be the reverse of the “A” run. As mentioned earlier, there are two coherent waves in a ternary system and one of them is the key separation wave. We designed our 12 experiments with half of them (run pairs 1, 5, and 6) focusing on wave 1 and the other half (run pairs 2-4) focusing on wave 2. Since a balanced wave is of major practical interest, we designed each group of six runs around a set of reference conditions that gives a balanced key wave. Both reference sets employed the reference feed composition {x°i,F} ) {0.4, 0.3, 0.3} mentioned earlier. For each reference, we chose

a feed rate F° within the range for a proper tray loading and then determined the appropriate reboiler heat supply Q°B to establish a balanced standing key wave at steady state. Once the pair of F° and Q°B was obtained from one run, the reference can be used for all other runs with the same key wave. In practice, however, these conditions needed to be adjusted slightly from one run to another, owing to slight changes of the environment (e.g., the ambient pressure and temperature) and the feed composition (due to feed recycling). For the purpose of presentation, the F° and Q°B of ref 1 for wave 1 are averaged among the balanced steady states (either initial or new) of runs 1A, 1B, 5A, and 5B, while those of ref 2 for wave 2 are averaged among the balanced steady states of runs 2A-3B. The two reference sets are also given in Table 1 (ref 1 and ref 2). In addition to the above operating conditions, the distillate rate D° and the bottoms composition {x°i,B} of each reference are also calculated with the same averaging procedure. For predicting the column behavior, the distillate rate will be used to estimate the vapor flow rate through the column while the bottoms composition will be used to calculate the composition of the vapor entering the bottom end. In each run, we first established an initial steady state with a set of proper conditions around one of the references to have the chosen key wave stand at a designed location (balanced in the middle portion or unbalanced near a column end) with the other wave (unbalanced) standing near a column end (wave 1 at top or wave 2 at bottom). Then, we introduced a step change to make the key wave travel. During such a transition, the other wave stayed near the column end because the step change was not large enough to change the sign of its natural velocity. Eventually, the column attained a new steady state (except that runs 4B, 5B, and 6B were terminated slightly earlier, owing to experimental difficulties). The new operating conditions (feed composition, feed flow rate, and reboiler heat supply) are listed in Table 1 under the new steady state. In addition, the distillate rate and the compositions of distillate and bottoms at the new steady state (or the end of the run) were also measured (for reference only, not used for predicting the column dynamics). We observed the propagation of the key wave by tracking the movement of its associated temperature variation. In each of runs 1A-2B, we introduced a step change of the feed composition to examine the travel of wave 1 in runs 1A and 1B and the travel of wave 2 in runs 2A and 2B. In each case, the magnitude of the composition change is less than 10 mol % in terms of the norm of the composition change vector. To maintain a constant molar feed rate F, we also adjusted the mass feed rate Fw in accordance with the change of the feed molecular weight in runs 1A and 1B; such an adjustment was not carried out for 2A and 2B. In addition, as the key wave traveled, the reboiler temperature T0 varies slightly due to the slight composition change in the reboiler. Our control program automatically adjusted the heatingtape temperature TQ accordingly to maintain a constant reboiler heat supply QB, as indicated in Table 1. In each of runs 3A-4B, we introduced a step change of the feed rate to observe the travel of wave 2, while in each of runs 5A-6B, we introduced a step change of the reboiler heat supply to examine the travel of wave 1. In Table 1, these step changes are expressed in terms of a percentage of the respective reference value. We at-

Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3595

Figure 2. Composition paths and composition routes for coherent waves in a stripping column with an ideal ternary mixture: (a) in liquid composition space; (b) in h-space.

tempted to carry out these runs with the reference feed composition. However, owing to our feed recycling practice, the feed composition varied slightly. Moreover, for these runs except runs 5A and 5B, we anticipated to consume more than one can of feed. Therefore, after consuming some feed in one can to establish the initial steady state, we switched to the other full can at the same time we introduced the step change of either feed rate or reboiler heat supply. Such a procedure avoided the undesirable disturbance during the transition period from switching feed can or adding feed, but it introduced a minor step change of feed composition, as indicated in Table 1. Results and Discussion This section will start with an outline of how the coherent-wave theory can be applied to our experimental system. Following such an outline, our experimental results will be presented and compared with the predicted results. Coherent-Wave Analysis. The first step of a coherent-wave analysis is to construct the composition path grid for the specified mixture. As discussed in Appendix B, the vapor-liquid equilibrium of the methanol/1propanol/1-pentanol ternary mixture was considered to be ideal with constant relative volatilities of {R1i} ) {1, 3.0, 13.4}. For such an ideal mixture, Figure 2a,b presents the composition path grid in the liquid composition space and in the h-space, respectively. Note that the border with an absent intermediate component (x2 ) 0) consists of two paths in different families divided by a “watershed” point W with x1 ) (R12 - 1)/(R13 - 1); the watershed point serves as an additional apex in the h-grid for orthogonalization (for details, see Helfferich and Klein, 1970). The second step is to establish the composition routes of the coherent waves on the composition path grid from the boundary conditions: first for the initial steady state and then for the resulting coherent state after the step disturbance. With the operating data given in Table 1, the composition routes of the coherent waves in each initial steady state (including the references) can be

established starting from the entering-stream compositions: {hti } from {xti } and {hbi } from {ybi }, as indicated by eq 12. For the stripping column, {xti } is the feed composition {xi,F} while {ybi } is in equilibrium with the bottoms composition {xi,B} so that {hbi } can be calculated from {xi,B} directly. In our experiments, we were intended to investigate the departure from the reference steady state 1 in runs 1A and 5A and that from the reference steady state 2 in runs 2A and 3A. For the reference steady state 1, the column-end compositions are represented by points F° and B° in Figure 2; for ref 2, F° is the same but B° is at the apex x3 ) 1. With these two end points, the composition routes of the two coherent waves can be easily constructed by requiring coherent wave 1 to follow a path of family 1 and coherent wave 2 to follow a path of family 2. This coherence requirement quickly determines the unknown composition of the intermediate pinched zone between the two coherent waves, as marked by point M° in Figure 2. The entire composition route F°M°B° in the h space (Figure 2b) is actually a graphical representation of eq 12. Along the composition route, the eigenvalues {λk} can be calculated with eq 10 and then the local wave velocities {vk} can be obtained with eq 8 (actually, only those at the three points F°, M°, and B° are needed). The local wave velocities confirmed that both waves in each reference are indeed self-sharpening and thus become constant-pattern waves. Now, let us consider the dynamic response of the column to a step disturbance. For runs 5A and 3A with a step change of reboiler heat supply QB (or vapor rate V) and of feed rate F (or liquid rate L), respectively, the composition route F°M°B° stays the same. Each flow rate change takes practically negligible time to reach the standing wave of the key separation in the column (demonstrated for binary distillation earlier, see Hwang et al., 1996) and thereby makes the key wave start to travel according to eq 9. For runs 1A and 2A with a step change of feed composition {xi,F}, the new feed composition is indicated by point F in Figure 2. This disturbance FF° quickly resolves into two coherent disturbance waves, which soon merge into the standing waves F°M°

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Figure 3. Constant-pattern coherent wave 1 departing from balanced steady state in response to a step change of feed composition (run 1A).

Figure 4. Constant-pattern coherent wave 1 returning to balanced steady state in response to a step change of feed composition (run 1B).

and M°B°, respectively (Hwang and Helfferich, 1989b). Such wave interference leads to a new set of coherent waves as represented by FM and MB in Figure 2 (point B is on the same path as route M°B°). For run 1A with wave 1 for the key separation, composition B may be slightly different from B°, but this is of little consequence to the behavior of the key wave. For run 2A with wave 2 for the key separation, composition B stays essentially the same as B° until wave 2 (originally balanced) reaches a column end. In either case, the coherence condition requires composition B to be on the same path as route M°B°. As a result, the composition of the intermediate pinched zone M can be readily determined, regardless of the exact composition B. With the new composition routes FM (wave 1 in run 1A) and

MB° (wave 2 in run 2A) constructed, one can verify the self-sharpening tendency of the two resulting coherent waves in the same way as that for the initial standing waves F°M° and M°B°, and then calculate their natural velocities with eqs 11 and 9. To quantify the entire transition time completely, one also needs to estimate the time required for the disturbance waves to merge into the initial standing waves. This can be estimated using a similar composition route analysis with the step disturbance FF° resolving into two routes along two paths of different families. This usually gives only an order-of-magnitude estimate since the initial standing waves on which the disturbance waves travel are not ideal shock waves (discontinuities) as under the equilibrium condition. Fortunately, such an estimate is

Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3597

Figure 5. Constant-pattern coherent wave 2 departing from balanced steady state in response to a step change of feed composition (run 2A).

Figure 6. Constant-pattern coherent wave 2 returning to balanced steady state in response to a step change of feed composition (run 2B).

typically sufficient in practice because this is normally a relatively short transient in comparison with the travel of the resulting waves. The coherent-wave behavior in the other runs can be analyzed by using the same approach. Runs 1B, 2B, 3B, and 5B were the reverse of their respective A runs, and the resulting key coherent wave should become balanced with a (nearly) zero natural velocity and should be pushed inward solely by the nonequilibrium columnend effect. For such a case, of most interest is the qualitative behavior of asymmetric dynamics rather than the exact wave velocity, whose prediction would require a much more laborious computation involving the nonequilibrium wave dissipation effect. In runs 4A, 4B, 6A, and 6B, the initial and new operating conditions

embrace a reference set of conditions. Although these runs were not initiated from a reference balanced steady state, the resulting coherent waves should behave similarly to those in the other A runs, and thus can be analyzed in the same way. Experimental Results. Figures 3-14 present the recorded snapshots of temperature profiles at various times in our six pairs of dynamic experiments. Each graph demonstrates the travel of the key coherent wave during the transition from one steady state to another initiated by a step change of either feed composition, feed flow rate, or reboiler heat supply. In Figure 3 for run 1A, an initial steady state with a balanced wave 1 (designed based on ref 1) was established. Then, a feed composition disturbance was introduced to the top of

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Figure 7. Constant-pattern coherent wave 2 departing from balanced steady state in response to a step change of feed flow rate (run 3A).

Figure 8. Constant-pattern coherent wave 2 returning to balanced steady state in response to a step change of feed flow rate (run 3B).

the column; this started to change the tray temperatures near the top end. It took about 10-15 min for the disturbance to reach and merge into the standing waves. The resulting key coherent wave 1 represented a composition variation between two new pinched zones on its two sides, as indicated by the slightly higher temperatures of those zones. It traveled upward in a fixed shape and at a practically constant velocity in the first 70 min of its travel (20-90 min of elapsed time); after that it was slowed by the column-end effect. Figure 4 for run 1B illustrates the reverse transition shown by Figure 3. As Figure 5 shows, run 2A was started from a steady state with a balanced wave 2 (designed based on ref 2) and disturbed by the same feed composition change as that in run 1A. Part of the disturbance crossed over wave 1 near the top end and took about 10

min to reach and merge into the key wave 2 in the middle portion of the column. The resulting coherent wave 2 traveled in a way similar to wave 1 in Figure 3, but it eventually was bunched with wave 1 near the top end. Figure 6 for run 2B presents the reverse transition shown by Figure 5. Figures 7-10 for runs 3A-4B demonstrate the travel of wave 2 from or around the reference steady state 2 in response to a feed rate disturbance. The liquid flow rate change affected the wave velocity essentially immediately. The resulting wave 2 had a natural velocity different from that of the initial standing wave, but it represented the same composition variation as indicated by the unchanged temperatures of the pinched zones on its two sides. Figures 11-14 for runs 5A-6B illustrate the travel of wave 1 from or around the reference steady state 1 in

Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3599

Figure 9. Constant-pattern coherent wave 2 traveling from bottom to top in response to a decrease of feed flow rate (run 4A).

Figure 10. Constant-pattern coherent wave 2 traveling from top to bottom in response to an increase of feed flow rate (run 4B).

response to a vapor flow rate disturbance generated by a step change of the reboiler heat supply. The behavior was much the same as shown by Figures 7-10 for that from a liquid rate disturbance. Constant-Pattern Coherent Waves. The temperature profiles in Figures 3-14 clearly demonstrate the existence of two constant-pattern coherent waves with an intermediate pinched zone in our ternary stripping column. These figures illustrate that the temperature profile at a steady state (e.g., initial) consisted of two coherent standing waves, of which one could be balanced in the middle portion of the column to maximize the key separation and the other stayed near a column end. In response to a step disturbance, the figures show that the resulting wave of the key separation traveled in a fixed shape, namely, constant pattern. As illustrated by Figures 3, 5, 7, 9-11, 13, and 14, an unbalanced key

wave (with a nonzero natural velocity) traveled at a practically constant velocity in the middle portion of the column, where the nonequilibrium column-end effect is insignificant. In contrast, as shown by Figures 4, 6, 8, and 12, when a balanced wave was established near a column end by restoring a balanced condition, it traveled inward but slowed as it moved away from the column end. These results confirm the qualitative prediction of the theory. Composition/Temperature Profile. As discussed earlier, the composition profiles for the initial standing waves and for the resulting coherent waves in each run can be quickly sketched by establishing the corresponding composition routes on the composition path grid or by using eq 12 for an ideal mixture. The {hi} obtained for the column ends and the intermediate pinched zone (denoted by a superscript “m”) are listed in Table 2. The

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Figure 11. Constant-pattern coherent wave 1 departing from balanced steady state in response to a step change of reboiler heat supply (run 5A).

Figure 12. Constant-pattern coherent wave 1 returning to balanced steady state in response to a step change of reboiler heat supply (run 5B).

compositions of both liquid and vapor phases of this zone can be calculated from {hm i } with the inverse h-transformation as presented in Appendix A; the liquid-phase composition {xm i } is given in Table 2. However, this composition was not measured for verifying the prediction, owing to the difficulty of accurate sampling from trays of a small column. Alternatively, supporting evidence can be supplied by comparing the predicted and measured values of the intermediate pinched zone temperature Tm. For measuring Tm, we identify the pinched zone as a range of trays (shown in Table 2) on which the temperatures at the initial steady state were approximately the same and then take an average of the temperatures on these trays. To predict Tm, we employ Raoult’s law with the component vapor pres-

sures calculated using the Antoine equation (Appendix B) and with a pressure Pm, estimated by assuming a uniform pressure drop across the column. As listed in Table 2, the predicted Tm agrees very well with the measured one in all runs with discrepancies in most cases less than 1 °C, which is about the same as the measurement error of the thermocouples. Wave Velocity. With the composition routes of the resulting coherent waves constructed, the velocity of the key wave k can be easily predicted by first calculating its eigenvalue Λk using eq 11. Table 3 presents the Λk along with the required h-coordinates for the resulting key wave in each run as well as for each reference standing wave; runs 1B, 2B, 3B, and 5B are excluded since these runs resulted in balanced waves which were

Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3601

Figure 13. Constant-pattern coherent wave 1 traveling from bottom to top in response to an increase of reboiler heat supply (run 6A).

Figure 14. Constant-pattern coherent wave 1 traveling from top to bottom in response to a decrease of reboiler heat supply (run 6B).

driven by the column-end effect. The next step is to calculate the shock wave velocity using eq 9. For the molar holdups, we employ for all cases an approximate average liquid holdup W ) 0.1 mol/tray (see Table 1) and a rough estimate of vapor holdup U ) 0.001 mol/ tray, which has little impact on the computation anyway. For the molar flow rates, we assume uniform flow rates L and V throughout the column, as mentioned earlier. Although we measured the flow rates F and D, these tend to be significantly lower than the average L and V in the column due to heat loss, as experienced previously from binary distillation experiments (Hwang et al., 1996). As a reasonable approximation, we first estimate L° and V° for each reference balanced standing wave by requiring its natural velocity to be zero; this requirement results in V°Λk ) L°. Assuming L° and V° are the flow rates at a certain location of the column,

the material balance from that location to the top end gives L° - V° ) F° - D°. These two equations lead to L° and V° for each reference standing wave, as listed in Table 3. Then, we estimate L and V for each run based on the step changes of feed rate and reboiler heat supply by assuming the fractional deviation of L from L° is the same as that of the feed rate, and the fractional deviation of V from V° is the same as that of the reboiler heat supply. With these estimated molar flow rates, the predicted velocities of the resulting key waves are listed in Table 3. From the recorded temperature profiles shown in Figures 3-14, the velocity of the resulting key wave in each run is measured over a time period (shown in Table 3) within which the wave traveled at an approximately constant velocity (before being slowed by the columnend effect). These measured wave velocities are also

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Table 2. Composition Routes of Coherent Waves and Intermediate Pinched Zone Composition and Temperature run index key wave

1A

1B

2A

2B

3A

3B

4A

4B

5A

5B

6A

6B

ref 1a

ref 2b

wave 1 wave 1 wave 2 wave 2 wave 2 wave 2 wave 2 wave 2 wave 1 wave 1 wave 1 wave 1 wave 1 wave 2

initial location travel direction int. pinch (trays)b Pm (mmHg)c

balance top balance top balance bottom bottom top balance top bottom top balance balance up down up down down up up down up down up down up up 6-14 763

6-14 766

34-42 34-42 34-42 34-42 34-42 34-42 751 750 749 750 752 749

6-14 773

6-14 774

6-14 766

6-14 766

6-14 769

34-42 750

initial standing waves {hti } from {xi,F}

ht1 ht2

2.133 10.136

2.016 9.453

2.109 10.036

2.036 9.452

2.114 10.012

2.111 9.981

2.111 9.956

2.110 9.947

2.089 9.832

2.103 9.966

2.096 9.851

2.100 9.893

2.105 9.975

2.105 9.975

{hbi } from {xi,B}

hb1 hb2

1.000 7.502

1.000 7.074

1.000 3.000

1.000 3.000

1.000 3.000

1.000 3.015

1.000 3.054

1.000 3.000

1.000 7.358

1.000 7.263

1.008 7.465

1.000 6.980

1.000 7.321

1.000 3.000

b hm 1 ) h1 t hm ) h 2 2

1.000 10.136

1.000 9.453

1.000 10.036

1.000 9.452

1.000 10.012

1.000 9.981

1.000 9.956

1.000 9.947

1.000 9.832

1.000 9.966

1.008 9.851

1.000 9.893

1.000 9.975

1.000 9.975

0.0000 0.0000 0.0029 0.0000 0.6570 0.6698 0.6561 0.6628 0.3430 0.3302 0.3410 0.3372

0.0000 0.6706 0.3294

0.0000 0.6706 0.3294

{hm i }

{xm i } from {hm i }

xm 1 xm 2 xm 3

Tm predicted (°C) Tm measured (°C) Tm error (°C)

0.0000 0.0000 0.6861 0.6205 0.3139 0.3795 104.8 104.6 0.2

106.8 105.9 0.9

0.0000 0.0000 0.6766 0.6204 0.3234 0.3796 104.7 105.1 -0.4

106.3 N/Ed

0.0000 0.0000 0.0000 0.0000 0.6742 0.6713 0.6688 0.6680 0.3258 0.3287 0.3312 0.3320 104.6 105.3 -0.7

104.8 105.3 -0.5

104.9 105.3 -0.4

104.8 N/E

106.0 105.4 0.6

105.7 105.1 0.6

105.5 N/E

105.6 104.8 0.8

105.5 105.3 0.3

104.8 105.2 -0.4

resulting coherent waves {hti } from {xi,F}

ht1 ht2

2.016 2.133 9.453 10.136

2.036 2.109 9.452 10.036

2.111 2.114 9.981 10.012

2.110 9.947

2.111 9.956

2.089 9.832

2.103 9.966

2.100 9.893

2.096 9.851

2.030 9.430

2.030 9.430

{hbi } from {xi,B}

hb1 hb2

1.000 7.502

1.000 3.000

1.000 3.000

1.000 3.015

1.000 3.054

1.000 3.000

1.000 7.358

1.000 7.263

1.008 7.465

1.000 6.980

1.000 7.321

1.000 3.000

1.000 1.000 9.981 10.012

1.000 9.947

1.000 9.956

1.000 9.832

1.000 9.966

1.008 9.893

1.000 9.851

1.000 9.430

1.000 9.430

0.0000 0.0000 0.0000 0.0029 0.6570 0.6698 0.6628 0.6561 0.3430 0.3302 0.3372 0.3410

0.0000 0.6183 0.3817

0.0000 0.6183 0.3817

107.0 105.8 1.3

106.3 106.1 0.2

{hm i }

b hm 1 ) h1 t hm 2 ) h2

{xm i } from {hm i }

xm 1 xm 2 m x3

Tm predicted (°C) Tm measured (°C) Tm error (°C)

1.000 7.074

1.000 3.000

1.000 1.000 9.453 10.136

1.000 1.000 9.452 10.036

0.0000 0.0000 0.6205 0.6861 0.3795 0.3139

0.0000 0.0000 0.6204 0.6766 0.3796 0.3234

106.7 105.6 1.1

104.9 104.6 0.3

106.3 106.1 0.2

104.6 N/E

0.0000 0.0000 0.0000 0.0000 0.6713 0.6742 0.6680 0.6688 0.3287 0.3258 0.3320 0.3312 104.7 105.1 -0.4

104.7 105.3 -0.6

104.9 105.3 -0.4

104.8 N/E

106.0 105.1 0.9

105.7 105.1 0.6

105.6 N/E

105.5 104.7 0.8

a References 1 and 2: initial conditions are listed in Table 1; the resulting coherent waves are the response to a designed step change of feed composition {xi,F} from {0.40, 0.30, 0.30} to {0.35, 0.30, 0.35}; the measured Pm and Tm are the average values from the runs designed accordingly. b Intermediate pinched zone is identified with a range of trays on which the temperatures at the initial steady state are practically the same. c Pressure at the intermediate pinched zone Pm is estimated assuming a uniform pressure drop across the column. d “N/E” indicates that the intermediate pinched zone did not exist because the two coherent waves overlapped with each other near a column end.

presented in Table 3 along with the prediction errors relative to the measured velocity. Note that such an error up to (100% may be considered acceptable for a dynamic phenomenon with high sensitivity in nature and initiated by a merely 5-10% change of an input variable. These errors indicate that the simple eq 9 tends to overpredict the wave velocity. As indicated by the results from a previous study on binary distillation (Hwang et al., 1996), the major source of the prediction error likely comes from the assumption of uniform molar flow rates (which implies the omission of the column heat loss) and uniform molar holdups. A laboratory glass column, especially with many thermocouple wells, tend to have a relatively large heat loss compared with a well-insulated plant column. For practical application, it is recommended to begin with the simple eq 9 and test its prediction against observed data. If its prediction is unsatisfactory, then the wave velocity equation can be extended to the cases with nonuniform molar flow rates by incorporating the energy balance, as previously demonstrated for binary distillation (Hwang, 1995; Hwang et al., 1996). Asymmetric Dynamics. Figures 3-8, 11, and 12 for run pairs 1-3 and 5 clearly demonstrate the asymmetric dynamics that the departure from a balanced steady state (A runs) is always faster than the return

to it (B runs). Note that the exact new steady state for the B runs (the balanced one) might not have been attained before we terminated the experiment because the movement of the temperature profile was extremely slow near the end of each run. These results verified the qualitative nature of this interesting dynamic behavior as explained by the nonlinear wave theory (Hwang and Helfferich, 1989a, 1990). However, the dynamic asymmetry in these results appears to be not as severe as that from previous computer simulations. One reason is that we chose a relatively moderate system so that we could complete the slow return transition in a reasonable time period. In addition, nonidealities such as nonideal tray behavior and heat loss might have alleviated the dynamic asymmetry because they tend to loosen the pinches around the column ends. Conclusions A coherent-wave theory was established previously (Hwang and Helfferich, 1989b, 1990) to provide better physical insight into and a concise mathematical model of the nonlinear behavior of multicomponent countercurrent separation processes. To test this theory in its application to distillation columns with sharp separa-

Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3603 Table 3. Velocity of Constant-Pattern Coherent Wave for Key Separation run index

ref 1

ref 2

1A

2A

3A

4A

4B

5A

6A

6B

wave 1 balance

wave 2 balance

standing

standing

wave 1 balance {xi,F} up

wave 2 balance {xi,F} up

wave 2 balance L+ down

wave 2 bottom Lup

wave 2 top L+ down

wave 1 balance V+ up

wave 1 bottom V+ up

wave 1 top Vdown

h′k ) hbk h′′k ) htk h2 ) ht2 for wave 1 h1 ) hb1 for wave 2 Λk

1.000 2.105 9.975

3.000 9.975

1.000 2.016 9.453

3.000 9.452

3.000 9.981

3.054 9.947

3.000 9.956

1.000 2.089 9.832

1.008 2.100 9.893

1.000 2.096 9.851

1.914

1.000 1.343

2.109

1.000 1.418

1.000 1.343

1.000 1.323

1.000 1.346

1.957

1.919

1.947

reference flow rates F° (measured, mol/h) D°

33.23 16.29

24.84 18.09

reference flow rates L° (adjusted, mol/h)a V°

35.48 18.53

26.43 19.67 0.00% 0.00% 35.48 18.53

0.00% 0.00% 26.43 19.67

+6.96% 0.00% 28.26 19.67

-6.92% 0.00% 24.60 19.67

+7.70% 0.00% 28.46 19.67

0.00% +7.55% 35.48 19.93

0.00% +6.70% 35.48 19.78

0.00% -9.79% 35.48 16.72

35.4

14.4

-18.3

14.1

-19.6

34.6

24.3

-28.7

7.4 20-90

5.6 10-90

-12.3 0-20

12.7 20-90

-12.7 20-90

13.8 0-40

29.4 10-40

-9.6 20-120

+378%

+157%

+49%

+11%

+54%

+151%

-17%

+199%

key wave initial location disturbance travel direction predicted wave velocity wave k

deviationb (L - L°)/L° from ref. (V - V°)/V° flow rates (mol/h) L V velocity (tray/h)c

v∆k

0

0

measured wave velocity velocity (tray/h)d v∆k elapsed time (min) prediction error (%)

Reference flow rates L° and V° are adjusted from measured F° and D° to satisfy both v∆k ) 0 for each reference balanced standing wave and the overall material balance. b Fractional deviation of liquid rate L from L° is assumed to be the same as that of the feed rate; fractional deviation of vapor rate V from V° is assumed to be the same as that of the reboiler heat supply. c Predicted velocity of the key wave is calculated with an average liquid holdup W ) 0.1 mol/tray and an estimated vapor holdup U ) 0.001 mol/tray. d Measured velocity of the key wave is the average over a period as given in elapsed time within which the wave traveled at an approximately constant velocity. a

tions, we have conducted dynamic experiments on a laboratory stripping column with 50 sieve trays. For investigating the fundamentals of wave propagation dynamics, we used a nearly ideal ternary mixture of methanol, 1-propanol, and 1-pentanol in the present study. Using the same experimental approach as in a previous study on binary distillation (Hwang et al., 1996), we tracked the movement of sharp temperature variations in response to deliberately introduced step changes of feed composition, feed flow rate, and reboiler heat supply. Our results can be summarized as follows: (1) This study has provided an experimental observation of the dynamic behavior of a multicomponent distillation column from a holistic point of view. (2) The study has demonstrated the existence and propagation of two constant-pattern coherent waves with an intermediate pinched zone in a ternary stripping column and thereby confirmed the qualitative aspect of the theory. (3) The study has verified that the theory satisfactorily predicts the composition of the intermediate pinched zone in a ternary column and thereby facilitates a quick sketch of the composition profile. (4) The study has tested the theory in predicting the velocity of the coherent wave for the key separation with discussions of possible error sources and recommendations for refinements if needed. (5) The study has verified the asymmetric dynamics that the departure from a balanced steady state is always faster than the return to it, as elucidated by the theory. These results have fortified the coherent-wave theory for applications to design and control of multicomponent distillation columns with sharp separations.

Acknowledgment We gratefully acknowledge the financial supports of the National Science Foundation under Grant CTS9020749 and of Union Carbide Corporation. We are also indebted to Mr. M. Y. Nehme and Ms. L. A. Patterson of Union Carbide for their help in our experimental work. Nomenclature B ) molar flow rate of bottoms (mol/h); as a subscript, bottoms or reboiler D ) molar flow rate of distillate (mol/h); as a subscript, distillate Dw ) mass (weight) flow rate of distillate (g/h) F ) molar flow rate of feed (mol/h); as a subscript, feed Fw ) mass (weight) flow rate of feed (g/h) hi ) i-th coordinate resulting from h-transformation I ) identity matrix of order (n - 1) × (n - 1) i, j ) index of component k ) index of coherent wave or component L ) molar flow rate of liquid stream (mol/h) m ) number of components absent from a zone in a column n ) total number of components P ) pressure (mmHg, 1 mmHg ) 133.32 Pa) p/i ) vapor pressure of component i (mmHg) QB ) effective heat supply to reboiler (kJ/h) s ) upward distance from bottom of column (trays or m) T ) temperature (°C) TQ ) reboiler heating-tape temperature (°C) T0 ) reboiler temperature (°C) t ) time (h) U ) molar vapor holdup (mol/tray or mol/m) V ) molar flow rate of vapor stream (mol/h)

3604

Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999

Table A1. Physical Properties of Methanol, 1-Propanol, 1-Pentanol, and Their Relative Volatilities Compound: (index) name

(1) methanol

weighta

molecular normal boiling point (°C)a vapor pressure (Antoine equation: p/i in mmHg, T in °C)b: log p/i ) A - B/(T + C) A B C

(2) 1-propanol

(3) 1-pentanol

32.042 64.7

60.096 97.2

88.150 137.8

8.08097 1582.27 239.726

7.74416 1437.69 198.463

7.39824 1435.57 179.798

R12 methanol/1-propanol

R13 methanol/1-pentanol

3.0

13.4

relative volatility: temperature range: 80-140 °C a

b

Data are gathered from DIPPR databank (Daubert and Danner, 1991). Parameter values are obtained from DECHEMA databank (Gmehling et al., 1977).

vxi ) local wave velocity for specific value of xi, defined in upward direction (tray/h or m/h) v∆ ) shock wave velocity, defined in upward direction (tray/h or m/h) W ) molar liquid holdup (mol/tray or mol/m) xi ) mole fraction of component i in liquid yi ) mole fraction of component i in vapor R1i ) relative volatility of component 1 to component i ∆ ) prefix for difference between two sides of shock wave Λk ) eigenvalue for coherent shock wave k λk ) eigenvalue for coherent wave k Φ ) Jacobian matrix of order (n - 1) × (n - 1) for vaporliquid equilibrium relations of an n-component mixture

tantly, there are m trivial roots:

{

h1 ) 1 hj-1 ) R1j hj ) R1j hn-1 ) R1n

Appendix A: h-Transformation For an ideal mixture of n components with constant relative volatilities {R1i}, the h-transformation (or “hyperplane transformation”) orthogonalizes the linear composition paths in the composition space of either liquid or vapor phase by converting the n concentrations {xi} or {yi} to n - 1 independent coordinates {hi} (for details see Helfferich and Klein, 1970). If all components are present (i.e., xi > 0 and yi > 0 for all i), one can obtain n - 1 roots {hi} from either of the following equations: n

xi

∑ i)1h - R

n

) 0 or

1i

yi

∑ i)11/h - 1/R

)0

xi )

(A2)

In the case with absent components, say, xj ) yj ) 0 for m components, eqs A1 are adjusted as follows (with n - m terms in summation) to give n - m - 1 nontrivial roots:

∑ i*j h - R

1i

n

) 0 or

yi

∑ i*j 1/h - 1/R

[∏ n-1

yi )

]/[ ]/[∏

]

n

n

(1/hj - 1/R1i)

j)1

(A5)

]

(1/R1j - 1/R1i)

j*i

(A6)

Appendix B: Physical Properties The physical properties of the three alcohols used in our experiments are given in Table A1. The vapor-liquid equilibrium of the ternary mixture of these alcohols is nearly ideal. For mathematical simplicity in predicting the coherent-wave behavior, the vapor-liquid equilibrium is considered to be ideal with constant relative volatilities. According to Raoult’s law for ideal vaporliquid equilibrium, the relative volatility R1i is the vapor pressure ratio of component 1 to component i at a given temperature: R1i ) p/1/p/i . With the vapor pressures computed from the Antoine equation in our experimental temperature range 80-140 °C, R12 actually decreases with temperature from 3.6 to 2.6 and R13 decreases from 18.2 to 10.1. For our analysis, we choose the average R12 and R13 over that temperature range, as listed in Table A1. Literature Cited

R1i < hi < R1,i+1

xi

[

n-1

(hj - R1i) ∏(R1j - R1i) ∏ j)1 j*i

(A1)

1i

Each of these roots falls between two adjacent values of R1i

n

(A4)

The inverse h-transformation can be given as follows:

Superscripts b ) bottom end of column section m ) intermediate pinched zone between two coherent waves in ternary system t ) top end of column section ° ) reference balanced steady state ′ ) lower side of wave ′′ ) upper side of wave

if x1 ) y1 ) 0 if xj ) yj ) 0 and hj nontrivial if xj ) yj ) 0 and hj-1 nontrivial if xn ) yn ) 0

)0

(A3)

1i

All nontrivial roots satisfy inequality (A2). Concomi-

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Received for review January 19, 1999 Revised manuscript received June 18, 1999 Accepted June 23, 1999 IE990038O