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Batch distillation column low-order models for quality program control
Chemical Engineering Science 55 (2000) 3187}3194
Batch distillation column low-order models for quality program control B. H. L. Betlem* Department of Chemical Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands Received 3 March 1998; accepted 19 November 1999
Abstract For batch distillation, the dynamic composition behaviour can be described by the dominant time constant and the bottom exhaustion. Its magnitude is determined by the change of the composition distribution and is maximal when the in#ection point of the molar fraction pro"le is located in the middle of the column. Then, the tray interactions are minimal. The distribution change during the batch run strongly depends on the applied control strategy. Under constant quality control, the dominant time constant proved to be nearly constant whereas under constant re#ux control the dominant time constant can vary by more than a factor of four. To calculate the dominant time constant from static design calculations, two di!erent methods are discussed: the retention time and the `change of inventorya time method. These methods are experimentally veri"ed. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Batch distillation; Dominant time constant; Distillation control; Distillation modelling
1. Introduction 1.1. Dynamic behaviour A signi"cant proportion of the industrial production by volume and a much larger production by value is made in batch plants. Batch production is #exible and particularly suitable for low volume products with a high added value. The interest on batch distillation in industry as well as in academia is growing fast. However, analysis studies about process dynamics still lag behind in comparison with the continuous counterpart. Research in the area of continuous distillation dynamics has a rather long history. Relatively recent overviews describing loworder models are Sa gfors and Waller (1995), and Skogestad and Morari (1987). Usually, batch distillation studies restrict themselves to optimal design or optimal operation, using quasi-steady-state models or short-cut models (Kim, 1999). However, this may lead to solutions that are not accurate enough. This article focuses on the composition dynamics of batch distillation. Principles about inter-tray dynamics and the dominant time constant developed for continuous distillations are adjusted
* Tel.: 0031-53-4893043; fax: 0031-53-4893849. E-mail address: [email protected] (B. H. L. Betlem)
for batch operations. This leads to a few interesting notions. In addition, the consequences for control are discussed. Roughly speaking, the composition dynamics of continuous distillation columns due to manipulation of the re#ux fraction or the vapour rate can be divided into two time (or frequency) regimes. Both regimes can best be illustrated by the linear discrete tray composition equation. If the pressure is assumed to remain constant, then the tray relation reads (Rademaker, Rijnsdorp & Maarleveld, 1975, pp. 35, 127, 283): !dx
G>
ddx G !R dx # (1#R )dx #q G G V dt G G\
"(x !x ) G> G
d¸ d< G> ! G\ , ¸M
(1)
in which
MM dy < G G . q " , and R " (2) V G ¸M dx ¸ G G In the equations dx indicates a perturbation around the mean value x , q is the tray mixing time, and & is the V stripping factor incorporating the tray e$ciency. The relation between the composition of the sequential trays is described by the stripping factor. The di!erence between the terms (dx #R dx ) and (dx #R dx ) G G G G> G G\
0009-2509/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 5 8 4 - 9
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B. H. L. Betlem / Chemical Engineering Science 55 (2000) 3187}3194
Nomenclature M ¸ n RH < x y R q
liquid mass mol liquid #ow mol/s number of trays, dimensionless re#ux fraction (L/V), dimensionless vapour #ow mol/s liquid molar fraction, dimensionless vapour molar fraction, dimensionless stripping factor, dimensionless time constant s
can be considered to be a measure for the composition interaction between adjoining trays. The short-term response of the tray composition dx G depends on d¸ and d< , which behaviour is determined G G by the relatively fast liquid and vapour dynamics. This can be described by three #ow dynamic parameters: the tray hydraulic time constant q , the sensitivity of the * liquid over#ow to the vapour tray load, and to some smaller extent, the column pressure time constant (Betlem, Rijnsdorp & Azink, 1998a). Since the compositions change slowly and the terms (dx #R dx ) and G G G (dx #R dx ) tend to cancel each other, all tray G> G G\ compositions react equally for a considerable long time and Eq. (1) can be reduced to
ddx d¸ d< G "(x G> ! G\ . q !x ) V dt G> G ¸M
(3)
The initial response is linear and correlates closely with the local gradient (x !x ). However, the "nal reG> G sponse is nonlinear. When the average stripping factor of the column section is used, the equation for a rectifying Eq. (3) can be simpli"ed to !q
V
d ln(1!x ) d¸ d< G "(1!RM ) G> ! G\ . dt ¸M
(4)
The logarithmic transformation corrects for nonlinear behaviour (see Skogestad & Morari, 1988, pp. 1855). According to Eq. (4), all tray compositions react according to the same time constant. This pattern is disordered at the top, bottom, and feed tray. When the average stripping factor remains in tact along the column, this short-term time constant will become apparent. This condition is only ful"lled, when the #ow variations are restricted to internal #ows (d¸"d<) and #ow dynamics can be neglected. However, then the slope of the operating line remains nearly constant and, consequently, the tray compositions will hardly change. Therefore, in practice, this time constant becomes poorly detectable or identi"able (Skogestad & Morari, 1988, p. 1860; Sa gfors & Waller, 1995, p. 2047). The medium-term response results from the relatively slow column settling time for a new composition distri-
q* qV
hydraulic tray time constant *MG /*¸G tray mixing time M M /¸M
Indices B D F i 0 n#1
bottom distillate feed tray number bottom accumulator
bution pertaining to the new stationary #ows. This can be described by the left-hand side of Eq. (1) !dx
G\
ddx G !R dx "0. # (1#R )dx #q G G V dt G G>
(5)
The settling time is determined by all composition interactions in the column. When the #ow dynamic e!ects are neglected, the overall behaviour becomes "rst order (Rademaker et al., 1975, p. 140), which can be described by the dominant time constant. For batch distillation, three di!erences compared to continuous operation in dynamic behaviour apply. E In addition to the short-term and the medium-term dynamic regimes mentioned, the bottom exhaustion causes a continuously transient behaviour by gradually shifting the operating point. This long-term behaviour can be accelerated or slowed down by the production rate, which is the result of the vapour and re#ux fraction settings. E Changes in internal #ows always result in changes in external #ows. Therefore, the short-term time-constant becomes not manifest. E Usually, the re#ux #ow is used to control the composition trajectory. Vapour #ow adjustments are only used to control the maximum tray load. In the article, "rst, the composition dynamics of the batch column will be experimentally determined by means of step and frequency responses. Next, for an interesting case, the dominant time constant will be calculated from the static distributions derived from detailed column simulations. Finally, the variations of the time constant during a batch run will be discussed and the consequences for constant quality control and constant re#ux control will be examined. 1.2. Dominant time constant The dominant time constant is extensively discussed in the literature (Skogestad & Morari, 1987). Actually, two di!erent methods can be distinguished to estimate the dominant time constant from static design values: the
B. H. L. Betlem / Chemical Engineering Science 55 (2000) 3187}3194
`change of inventorya time and the retention time. The `change of inventorya time corresponds to the time required to change the total column hold-up of each component from an initial to a "nal composition distribution. This global method uses the observed result of a composition distribution change in the column, but does not explain why this change, in this extent, occurs. The idea has been introduced by Moczek, Otto and Williams (1963). However, all studies are theoretical and based on simulations. The retention time corresponds to the average delay time of a response. In this approach, all mutual tray interactions are considered. The time constant is calculated from the mean column-stripping factor that determines the internal column composition gain. This detailed method starts from the tray driving forces causing the composition distribution change. The analytical solution is found by frequency or Laplace transformation. Armstrong and Wood (1961), and Edwards and Jassim (1977) determined experimentally the transient response for re#ux and feed disturbances, respectively. The responses appeared to be almost entirely dominated by the major time constant and the agreement between theory and experiment was quite good. 1.3. Application in control Batch operation can be carried out by a hierarchical multi-layer control system. Then, di!erent dynamic models have to be developed well "tted to the level of detail and the time horizon of the particular operational layer. Fig. 1 shows for the operation of batch distillation a hierarchy of models (Betlem, Krijnsen & Huijnen, 1998b). At
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the highest layer, distillation runs are scheduled to separate a multi-component mixture. At the middle layer, the dynamic optimal control is determined for a distillatecut, whereas at the lowest layer this optimal quality trajectory is realised by adjusting the re#ux ratio. The dynamic optimal control can be realised by imposing either the quality or the re#ux trajectory. However, imposing a re#ux trajectory directly is rather sensitive to model errors (Betlem, 1996). The time horizons of the three operational layers di!er from each other. At the scheduling layer, only the time to exhaust the feed amount plays a role. Models based on speci"c measures will satisfy, because the production rate is linear with the `di$culty of separationa (Betlem et al., 1998b). At the dynamic optimisation layer, also the dominant time constant is determinative. The distillate-cut has a smooth character dominated by the feed exhaustion and the correlated slowly moving composition distribution due to the decreasing light component contents. Interventions to compensate for the top composition change have to deal with the relatively long-term settling periods, which can be described by the dominant time constant which can be superposed on the feed exhaustion. Already in 1971, based on experimental studies, Robinson emphasised the importance of considering the tray hold-up e!ects in this dynamic region.
2. Experimental and simulated dynamic observations For the experiments, a batch distillation column with 21 bubble cap trays has been used with an internal diameter of 76 mm and a Murprhee vapour tray
Fig. 1. Control and model hierarchy of batch distillation. Abbreviations: dyn."dynamics, 5"and, 6"or.
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e$ciency of about 53%. It concerns a binary separation of ethanol and 1-propanol. The vapour rate is kept constant just below the #ooding constraint, which results in the maximum production rate. The only manipulated variable is the re#ux fraction RH. The tray mixing time q is 9}11 s, whereas the hydraulic time constant q is V * 2.5}3 s. The maximum exhaustion time is about 4 h. The dominant time constant has been measured at di!erent bottom fractions with step changes in the re#ux fraction and sinusoidal #uctuations in the re#ux ratio around an operating point. The continuing exhaustion of the bottom hold-up complicates the composition response measurement. Therefore, a stationary situation has been created by returning the top product into the bottom (see Fig. 2). Since the separation is binary and the pressure is nearly constant, the composition behaviour can be derived from the temperature responses. For the column simulations a rigorous dynamic model has been used, consisting of mass, component and energy balances per tray (Betlem et al., 1998a). Fig. 3 shows the Bode plot of the temperature for x "0.15 at two di!erent re#ux fractions: RH"0.85 and RH"0.75. Since the change in the top temperature was often not measurable, the temperature at tray 17 near to the top was taken. This will only give a slight deviation in the time constant. The Bode plot shows that a separation exists between the composition dynamics at the low frequencies (until 0.002 s\) and at the higher frequencies (from 0.002 s\). At the low-frequency range, the phase shift shows clearly a "rst-order response. At higher frequencies, the phase lag increases strongly without e!ect on the amplitude ratio. This dead time is mainly caused by the #uid dynamics. From other measurements and simulations, not shown here, it appeared that above a certain frequency, the #uctuations do not penetrate into
trays lower in the column anymore. In the simulations, this e!ect begins at cycle periods below n.q , which is the * time necessary for a liquid change to reach tray n. In the measurements, the dead time is larger. Apparently, other time lags that depend on the #ow rate also play a role. The dominant time constant measured with step responses is shown in Fig. 4. In addition, the two frequency responses are indicated. The time constant changes with a factor four and has a maximum at RH"0.85. 2.1. `Change of inventorya time From simulations it appeared that the following formulation of the `change of inventorya time showed a good agreement for batch distillation: q L>[(M x ) !(M x ) ] G G G G G " . 1/2[¸ (x !x )#< (x !x )] L> L> L>
(6) The batch `change of inventorya time formulation considers both ends of the rectifying column part, while omitting the bottom change. It is realistic to consider the bottom as an in"nitely large volume creating a distillation pinch point, as the composition remains unchanged in the time horizon considered. Further simpli"cation by means of a separation factor, such as proposed by Skogestad and Morari (1987), is not possible as for batch distillation this factor changes continuously. Fig. 4 shows the `change of inventorya time simulation results when x "0.15. The calculations are based on the simulated static behaviour. In the transition of convex to concave (RH"0.85) the molar fraction pro"le is nearly linear along a large part of the column causing a relatively large numerator. In addition, the top and bottom both behave like pinch points, which results in a relatively small denominator. Skogestad and Morari (1987) as well as Montesi and Brambilla (1995) studied the in#uence of the reboiler and condenser `non-activea hold-ups and found the same results. `Non-activea hold-ups are caused by `decouplinga which is associated with the presence of pinch zones in the column. 2.2. Retention time The retention time for tray i is de"ned by
q " G
Fig. 2. Simpli"ed diagram of experimental batch distillation set-up with product return pipe to study stationary behaviour.
[1!dx (t)/dx (in,nity)] dt. G G
(7)
An extra phase lag of 903 at the frequency of 0.007 s\ corresponds with a dead time of 25 s. At RH"0.75, q is about 4 s. The dead time * between the top tray (21) and tray number 17 where the temperature is measured explains: (21!17) ) q "16 s. *
B. H. L. Betlem / Chemical Engineering Science 55 (2000) 3187}3194
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Fig. 3. Measured Bode plot of tray 17, when x "0.15. The measurements were performed at RH"0.85 and 0.75 with an amplitude of dRH"0.05. The curves in the diagram correspond to the "rst-order estimations based plus dead time.
column behaviour. Notice that if RM approaches one and the number of trays is large, then the time constant is approximately proportional to the square of the trays number times the tray residence time
Fig. 4. Relation between the 17th tray composition dominant time constant and the re#ux fraction, when x "0.15. The dominant time constants are determined from 15 step response and 2 frequency measurements and estimated by the `change of inventorya time and retention time method.
The time constant can be derived by substitution of the solution of the second-order di!erence equation (5) for an average stripping factor. This leads to two answers (Rademaker et al., 1975, p. 143): a short and a long response. The dominant time constant corresponds with the long response: n q "q V (RM !1)(RM L>!1)
2RM (RM L!1) ; RM L>#1! . n(RM !1)
(8)
The approximation assumes that the average column operating values are a good approximation of the static
q Kq n. (9) V In the simulations, the arithmetic mean has been taken for RM . By using the simulated static values, the stripping factor is corrected for non-constant relative volatility, non-ideal equilibrium, internal re#ux, and tray e$ciency. Assuming an average stripping factor for the entire column section, the retention time calculations yield values, which are too low. The equilibrium slopes di!er considerably and the operating line has no constant slope due to internal condensation. Gilliland and Mohr (Rademaker et al., 1975, p. 152) found a relation between the largest time constant with non-uniform stripping factor and the largest time constant with uniform stripping factor. This relation can be used as a correction factor for the retention time constant. The correction factor is correction factor"1#pR bR +n, with 1 L> pR " (R !RM ). (10) G n G For the standard case shown in Fig. 4 the correction factor for our column is 1.1 at RH"1 and 2.1 at RH"0.7. The result in Fig. 4 corresponds rather well with the measurements. The stripping factor along the column varies minimal when the in#ection point of the molar fraction pro"le lies in the middle of the column. Then, the stripping factor is
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the composition distributions for the maximums in Fig. 7. For all three cases, the in#ection point of the molar fraction pro"le lies in the middle of the column. This corresponds to the results of Skogestad and Morari (1987, p. 614). For continuous distillation columns they found a relatively large time constant when the feed fraction is 0.5 and the top and bottom composition are reverse (x "1!x ). In that typical case, the in#ection " point also lays in the middle of the column.
3. Consequences for control strategies Fig. 5. Composition distributions with the in#ection point in the middle of the column.
Fig. 6. Measured development of temperature pro"les under re#ux control and constant quality control. A. Constant re#ux control: RH"0.75, B. Constant quality control: x "0.985. "
constant over a maximum number of trays. In that case, the tray interactions are minimal and, consequently, the column settling time will be maximal. Due to #ow variations, the pro"le changes from convex to concave inducing relatively large tray composition adaptations. Fig. 7 shows the time constant as a function of the re#ux at three bottom fractions. Fig. 5, approximately, shows
Let us consider the consequences for two common control policies. Fig. 6 shows the measured pro"le development during a batch for, respectively, constant re#ux control and constant quality control. Due to the bottom exhaustion, for both policies the bottom temperature will rise. However, the course of the dynamic behaviour is quite di!erent. In Fig. 7, the dominant time constant calculated by the `change of inventorya time, is depicted for three di!erent bottom fractions over a wide range of re#ux fractions. The time constant ranges from 3 to 30 min. During constant re#ux control (Fig. 6A) the shape of the composition distribution changes continuously, and the dominant time constant will vary accordingly. For instance (Fig. 7), at a re#ux fraction of 0.85 and a bottom fraction of 0.55 the initial time constant is 10 min. During exhaustion, at "rst, it rises to 23 min and next it falls to 3 min. For constant quality control, the dominant time constant varies only slightly. Fig. 6B shows that during the production run, after start-up, the quality controller enforces the composition pro"le to maintain a similar form. The in#ection point remains at the same position. These results are in conformity with the nonlinear wave theory (Hwang, 1991), which assumes that the top fraction is constant, when the composition wave does not propagate through the column. This condition is ful"lled when the in#ection point holds the same position. Fig. 7 shows that at a constant product quality of 0.95 and an initial bottom fraction of 0.55, the dominant time constant is 23 min. It will remain approximately the same value until the bottom becomes exhausted. At a bottom fraction of 0.02 the form of the pro"le changes, and "nally the time constant becomes 18 min. PID-controller settings depend on the process gain at high frequencies, which is the quotient between the static gain and the time constant (K /q ). From Figs. 7 and 8 follows that at a certain top fraction, the static gain as well as the dynamics varies only slightly. Consequently, constant quality control is easy to implement. In the experimental column, it has been realised by a partial least squares-based quality estimator combined with a PID-based quality controller. No gain scheduling was necessary. During constant re#ux control, the process gain and time constant changes continuously.
B. H. L. Betlem / Chemical Engineering Science 55 (2000) 3187}3194
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Fig. 7. Dynamic batch column relation. The `Change of inventorya time as a function of the re#ux ratio at x "0.05, 0.15, and 0.5. The "gures along the curves are the associated top fraction x . "
Fig. 8. Static batch column relation. Static gain between re#ux ratio and the top composition at di!erent bottom fractions.
However, this has no consequences for the controller itself. Constant quality control performs as good as dynamic optimal control when slop recycling is applied, since during the product-cut the exhaustion is restricted and during the slop-cut the separation is relatively easy (Betlem et al., 1998b). If a setpoint change is required to switch from product-cut to slop-cut distillation, then an adaptation of controller settings is inevitable as the process gain may di!er much (see Fig. 8). However, when a deep exhaustion is required and optimal re#ux control is applied, the changes in the column behaviour during the cut have to be considered in calculating the trajectory. For the realisation of the quality trajectory, a model-based controller can be used. Such a controller derived from input}output linearisation, which considers
the changes in the column time constant and the gain, only performed better in a simulation study (Betlem, 1996).
4. Conclusions The dominant time constant was estimated according to two methods. The "rst estimation was based on the `change of inventorya time and the second on a solution in the frequency domain based on the stripping factor. A series of time constants were calculated as a function of the re#ux fraction at a bottom fraction of 15% and were compared with the time constants derived from temperature measurement. The results were promising. The "rst method gives only a slight underestimate. For the second
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method, a correction is required known from the literature that corrects for the variations in the stripping factor. The dominant time constant was calculated for three bottom compositions and a large range of re#ux fractions. The magnitude of the time constant depends on the composition distribution. The results are especially interesting for batch distillation control. Although the bottom composition drifts continuously, constant quality control forces the molar fraction pro"le to maintain a similar form. Consequently, the dominant time `constanta remains nearly constant. However, under constant re#ux control the form of the pro"le changes continuously and can even go from convex to concave. Consequently, the dominant time constant can take values ranging from 3 to 30 min. Therefore, optimal re#ux policies have to account for internal pro"le changes.
References Armstrong, W. D., & Wood, R. M. (1961). The dynamic response of a distillation column to changes in the re#ux and vapour #ow rates. Transactions of the Institution of Chemical Engineers, 39, 65}90. Betlem, B. H. L. (1996). Hierarchical control of batch distillation. Ph.D. Thesis, University of Twente, Netherlands. Betlem, B. H. L., Rijnsdorp, J. R., & Azink, R. F. (1998a). In#uence of tray hydraulics on tray column dynamics. Chemical Engineering Science, 53(23), 3991}4003.
Betlem, B. H. L., Krijnsen, H. C., & Huijnen, H. (1998b). Optimal batch distillation control based on speci"c measures. Chemical Engineering Journal, 71, 111}126. Edwards, J. B., & Jassim, H. J. (1977). An analytical study of the dynamics of binary distillation columns. Transactions of the Institution of Chemical Engineers, 55, 17}28. Hwang, Y. L. (1991). Nonlinear wave theory for dynamics of binary distillation columns. A.I.Ch.E. Journal, 37(5), 705}723. Kim, Y. H. (1999). Optimal design and operation of a multi-product batch distillation column using dynamic models. Chemical Engineering and Processing, 38, 61}72. Moczek, J. S., Otto, R. E., & Williams, T. J. (1963). Approximation models for the dynamic responses of large distillation columns. Proceedings second IFAC world conference. Basel, (pp. 238}246). Montesi M., & Brambilla, A. (1995). Prediction of the e!ect of reboiler and condenser hold-up on the dynamic behaviour of distillation columns. IFAC-symposium Dycord#'95. Helsing+r (pp. 495}501). Rademaker, O., Rijnsdorp, J. E., & Maarleveld, A. (1975). Dynamics and control of continuous distillation units. Amsterdam, NL: Elsevier. Robinson, E. R. (1971). Optimum re#ux policies for batch distillation. Chemical and Process Engineering, 5}71, 47}55. Sa gfors, M. F., & Waller, K. V. (1995). Dynamic low order models for capturing directionality in nonideal distillation. Industrial Engineering and Chemical Research, 34, 2038}2050. Skogestad, S., & Morari, M. (1987). The dominant time constant for distillation columns. Computers & Chemical Engineering, 11(6), 607}617. Skogestad, S., & Morari, M. (1988). Understanding the dynamic behaviour of distillation columns. Industrial & Engineering Chemical Research, 27, 1848}1862.
Batch distillation column low-order models for quality program control B. H. L. Betlem* Department of Chemical Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands Received 3 March 1998; accepted 19 November 1999
Abstract For batch distillation, the dynamic composition behaviour can be described by the dominant time constant and the bottom exhaustion. Its magnitude is determined by the change of the composition distribution and is maximal when the in#ection point of the molar fraction pro"le is located in the middle of the column. Then, the tray interactions are minimal. The distribution change during the batch run strongly depends on the applied control strategy. Under constant quality control, the dominant time constant proved to be nearly constant whereas under constant re#ux control the dominant time constant can vary by more than a factor of four. To calculate the dominant time constant from static design calculations, two di!erent methods are discussed: the retention time and the `change of inventorya time method. These methods are experimentally veri"ed. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Batch distillation; Dominant time constant; Distillation control; Distillation modelling
1. Introduction 1.1. Dynamic behaviour A signi"cant proportion of the industrial production by volume and a much larger production by value is made in batch plants. Batch production is #exible and particularly suitable for low volume products with a high added value. The interest on batch distillation in industry as well as in academia is growing fast. However, analysis studies about process dynamics still lag behind in comparison with the continuous counterpart. Research in the area of continuous distillation dynamics has a rather long history. Relatively recent overviews describing loworder models are Sa gfors and Waller (1995), and Skogestad and Morari (1987). Usually, batch distillation studies restrict themselves to optimal design or optimal operation, using quasi-steady-state models or short-cut models (Kim, 1999). However, this may lead to solutions that are not accurate enough. This article focuses on the composition dynamics of batch distillation. Principles about inter-tray dynamics and the dominant time constant developed for continuous distillations are adjusted
* Tel.: 0031-53-4893043; fax: 0031-53-4893849. E-mail address: [email protected] (B. H. L. Betlem)
for batch operations. This leads to a few interesting notions. In addition, the consequences for control are discussed. Roughly speaking, the composition dynamics of continuous distillation columns due to manipulation of the re#ux fraction or the vapour rate can be divided into two time (or frequency) regimes. Both regimes can best be illustrated by the linear discrete tray composition equation. If the pressure is assumed to remain constant, then the tray relation reads (Rademaker, Rijnsdorp & Maarleveld, 1975, pp. 35, 127, 283): !dx
G>
ddx G !R dx # (1#R )dx #q G G V dt G G\
"(x !x ) G> G
d¸ d< G> ! G\ , ¸M
(1)
in which
MM dy < G G . q " , and R " (2) V G ¸M dx ¸ G G In the equations dx indicates a perturbation around the mean value x , q is the tray mixing time, and & is the V stripping factor incorporating the tray e$ciency. The relation between the composition of the sequential trays is described by the stripping factor. The di!erence between the terms (dx #R dx ) and (dx #R dx ) G G G G> G G\
0009-2509/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 5 8 4 - 9
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B. H. L. Betlem / Chemical Engineering Science 55 (2000) 3187}3194
Nomenclature M ¸ n RH < x y R q
liquid mass mol liquid #ow mol/s number of trays, dimensionless re#ux fraction (L/V), dimensionless vapour #ow mol/s liquid molar fraction, dimensionless vapour molar fraction, dimensionless stripping factor, dimensionless time constant s
can be considered to be a measure for the composition interaction between adjoining trays. The short-term response of the tray composition dx G depends on d¸ and d< , which behaviour is determined G G by the relatively fast liquid and vapour dynamics. This can be described by three #ow dynamic parameters: the tray hydraulic time constant q , the sensitivity of the * liquid over#ow to the vapour tray load, and to some smaller extent, the column pressure time constant (Betlem, Rijnsdorp & Azink, 1998a). Since the compositions change slowly and the terms (dx #R dx ) and G G G (dx #R dx ) tend to cancel each other, all tray G> G G\ compositions react equally for a considerable long time and Eq. (1) can be reduced to
ddx d¸ d< G "(x G> ! G\ . q !x ) V dt G> G ¸M
(3)
The initial response is linear and correlates closely with the local gradient (x !x ). However, the "nal reG> G sponse is nonlinear. When the average stripping factor of the column section is used, the equation for a rectifying Eq. (3) can be simpli"ed to !q
V
d ln(1!x ) d¸ d< G "(1!RM ) G> ! G\ . dt ¸M
(4)
The logarithmic transformation corrects for nonlinear behaviour (see Skogestad & Morari, 1988, pp. 1855). According to Eq. (4), all tray compositions react according to the same time constant. This pattern is disordered at the top, bottom, and feed tray. When the average stripping factor remains in tact along the column, this short-term time constant will become apparent. This condition is only ful"lled, when the #ow variations are restricted to internal #ows (d¸"d<) and #ow dynamics can be neglected. However, then the slope of the operating line remains nearly constant and, consequently, the tray compositions will hardly change. Therefore, in practice, this time constant becomes poorly detectable or identi"able (Skogestad & Morari, 1988, p. 1860; Sa gfors & Waller, 1995, p. 2047). The medium-term response results from the relatively slow column settling time for a new composition distri-
q* qV
hydraulic tray time constant *MG /*¸G tray mixing time M M /¸M
Indices B D F i 0 n#1
bottom distillate feed tray number bottom accumulator
bution pertaining to the new stationary #ows. This can be described by the left-hand side of Eq. (1) !dx
G\
ddx G !R dx "0. # (1#R )dx #q G G V dt G G>
(5)
The settling time is determined by all composition interactions in the column. When the #ow dynamic e!ects are neglected, the overall behaviour becomes "rst order (Rademaker et al., 1975, p. 140), which can be described by the dominant time constant. For batch distillation, three di!erences compared to continuous operation in dynamic behaviour apply. E In addition to the short-term and the medium-term dynamic regimes mentioned, the bottom exhaustion causes a continuously transient behaviour by gradually shifting the operating point. This long-term behaviour can be accelerated or slowed down by the production rate, which is the result of the vapour and re#ux fraction settings. E Changes in internal #ows always result in changes in external #ows. Therefore, the short-term time-constant becomes not manifest. E Usually, the re#ux #ow is used to control the composition trajectory. Vapour #ow adjustments are only used to control the maximum tray load. In the article, "rst, the composition dynamics of the batch column will be experimentally determined by means of step and frequency responses. Next, for an interesting case, the dominant time constant will be calculated from the static distributions derived from detailed column simulations. Finally, the variations of the time constant during a batch run will be discussed and the consequences for constant quality control and constant re#ux control will be examined. 1.2. Dominant time constant The dominant time constant is extensively discussed in the literature (Skogestad & Morari, 1987). Actually, two di!erent methods can be distinguished to estimate the dominant time constant from static design values: the
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`change of inventorya time and the retention time. The `change of inventorya time corresponds to the time required to change the total column hold-up of each component from an initial to a "nal composition distribution. This global method uses the observed result of a composition distribution change in the column, but does not explain why this change, in this extent, occurs. The idea has been introduced by Moczek, Otto and Williams (1963). However, all studies are theoretical and based on simulations. The retention time corresponds to the average delay time of a response. In this approach, all mutual tray interactions are considered. The time constant is calculated from the mean column-stripping factor that determines the internal column composition gain. This detailed method starts from the tray driving forces causing the composition distribution change. The analytical solution is found by frequency or Laplace transformation. Armstrong and Wood (1961), and Edwards and Jassim (1977) determined experimentally the transient response for re#ux and feed disturbances, respectively. The responses appeared to be almost entirely dominated by the major time constant and the agreement between theory and experiment was quite good. 1.3. Application in control Batch operation can be carried out by a hierarchical multi-layer control system. Then, di!erent dynamic models have to be developed well "tted to the level of detail and the time horizon of the particular operational layer. Fig. 1 shows for the operation of batch distillation a hierarchy of models (Betlem, Krijnsen & Huijnen, 1998b). At
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the highest layer, distillation runs are scheduled to separate a multi-component mixture. At the middle layer, the dynamic optimal control is determined for a distillatecut, whereas at the lowest layer this optimal quality trajectory is realised by adjusting the re#ux ratio. The dynamic optimal control can be realised by imposing either the quality or the re#ux trajectory. However, imposing a re#ux trajectory directly is rather sensitive to model errors (Betlem, 1996). The time horizons of the three operational layers di!er from each other. At the scheduling layer, only the time to exhaust the feed amount plays a role. Models based on speci"c measures will satisfy, because the production rate is linear with the `di$culty of separationa (Betlem et al., 1998b). At the dynamic optimisation layer, also the dominant time constant is determinative. The distillate-cut has a smooth character dominated by the feed exhaustion and the correlated slowly moving composition distribution due to the decreasing light component contents. Interventions to compensate for the top composition change have to deal with the relatively long-term settling periods, which can be described by the dominant time constant which can be superposed on the feed exhaustion. Already in 1971, based on experimental studies, Robinson emphasised the importance of considering the tray hold-up e!ects in this dynamic region.
2. Experimental and simulated dynamic observations For the experiments, a batch distillation column with 21 bubble cap trays has been used with an internal diameter of 76 mm and a Murprhee vapour tray
Fig. 1. Control and model hierarchy of batch distillation. Abbreviations: dyn."dynamics, 5"and, 6"or.
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e$ciency of about 53%. It concerns a binary separation of ethanol and 1-propanol. The vapour rate is kept constant just below the #ooding constraint, which results in the maximum production rate. The only manipulated variable is the re#ux fraction RH. The tray mixing time q is 9}11 s, whereas the hydraulic time constant q is V * 2.5}3 s. The maximum exhaustion time is about 4 h. The dominant time constant has been measured at di!erent bottom fractions with step changes in the re#ux fraction and sinusoidal #uctuations in the re#ux ratio around an operating point. The continuing exhaustion of the bottom hold-up complicates the composition response measurement. Therefore, a stationary situation has been created by returning the top product into the bottom (see Fig. 2). Since the separation is binary and the pressure is nearly constant, the composition behaviour can be derived from the temperature responses. For the column simulations a rigorous dynamic model has been used, consisting of mass, component and energy balances per tray (Betlem et al., 1998a). Fig. 3 shows the Bode plot of the temperature for x "0.15 at two di!erent re#ux fractions: RH"0.85 and RH"0.75. Since the change in the top temperature was often not measurable, the temperature at tray 17 near to the top was taken. This will only give a slight deviation in the time constant. The Bode plot shows that a separation exists between the composition dynamics at the low frequencies (until 0.002 s\) and at the higher frequencies (from 0.002 s\). At the low-frequency range, the phase shift shows clearly a "rst-order response. At higher frequencies, the phase lag increases strongly without e!ect on the amplitude ratio. This dead time is mainly caused by the #uid dynamics. From other measurements and simulations, not shown here, it appeared that above a certain frequency, the #uctuations do not penetrate into
trays lower in the column anymore. In the simulations, this e!ect begins at cycle periods below n.q , which is the * time necessary for a liquid change to reach tray n. In the measurements, the dead time is larger. Apparently, other time lags that depend on the #ow rate also play a role. The dominant time constant measured with step responses is shown in Fig. 4. In addition, the two frequency responses are indicated. The time constant changes with a factor four and has a maximum at RH"0.85. 2.1. `Change of inventorya time From simulations it appeared that the following formulation of the `change of inventorya time showed a good agreement for batch distillation: q L>[(M x ) !(M x ) ] G G G G G " . 1/2[¸ (x !x )#< (x !x )] L> L> L>
(6) The batch `change of inventorya time formulation considers both ends of the rectifying column part, while omitting the bottom change. It is realistic to consider the bottom as an in"nitely large volume creating a distillation pinch point, as the composition remains unchanged in the time horizon considered. Further simpli"cation by means of a separation factor, such as proposed by Skogestad and Morari (1987), is not possible as for batch distillation this factor changes continuously. Fig. 4 shows the `change of inventorya time simulation results when x "0.15. The calculations are based on the simulated static behaviour. In the transition of convex to concave (RH"0.85) the molar fraction pro"le is nearly linear along a large part of the column causing a relatively large numerator. In addition, the top and bottom both behave like pinch points, which results in a relatively small denominator. Skogestad and Morari (1987) as well as Montesi and Brambilla (1995) studied the in#uence of the reboiler and condenser `non-activea hold-ups and found the same results. `Non-activea hold-ups are caused by `decouplinga which is associated with the presence of pinch zones in the column. 2.2. Retention time The retention time for tray i is de"ned by
q " G
Fig. 2. Simpli"ed diagram of experimental batch distillation set-up with product return pipe to study stationary behaviour.
[1!dx (t)/dx (in,nity)] dt. G G
(7)
An extra phase lag of 903 at the frequency of 0.007 s\ corresponds with a dead time of 25 s. At RH"0.75, q is about 4 s. The dead time * between the top tray (21) and tray number 17 where the temperature is measured explains: (21!17) ) q "16 s. *
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Fig. 3. Measured Bode plot of tray 17, when x "0.15. The measurements were performed at RH"0.85 and 0.75 with an amplitude of dRH"0.05. The curves in the diagram correspond to the "rst-order estimations based plus dead time.
column behaviour. Notice that if RM approaches one and the number of trays is large, then the time constant is approximately proportional to the square of the trays number times the tray residence time
Fig. 4. Relation between the 17th tray composition dominant time constant and the re#ux fraction, when x "0.15. The dominant time constants are determined from 15 step response and 2 frequency measurements and estimated by the `change of inventorya time and retention time method.
The time constant can be derived by substitution of the solution of the second-order di!erence equation (5) for an average stripping factor. This leads to two answers (Rademaker et al., 1975, p. 143): a short and a long response. The dominant time constant corresponds with the long response: n q "q V (RM !1)(RM L>!1)
2RM (RM L!1) ; RM L>#1! . n(RM !1)
(8)
The approximation assumes that the average column operating values are a good approximation of the static
q Kq n. (9) V In the simulations, the arithmetic mean has been taken for RM . By using the simulated static values, the stripping factor is corrected for non-constant relative volatility, non-ideal equilibrium, internal re#ux, and tray e$ciency. Assuming an average stripping factor for the entire column section, the retention time calculations yield values, which are too low. The equilibrium slopes di!er considerably and the operating line has no constant slope due to internal condensation. Gilliland and Mohr (Rademaker et al., 1975, p. 152) found a relation between the largest time constant with non-uniform stripping factor and the largest time constant with uniform stripping factor. This relation can be used as a correction factor for the retention time constant. The correction factor is correction factor"1#pR bR +n, with 1 L> pR " (R !RM ). (10) G n G For the standard case shown in Fig. 4 the correction factor for our column is 1.1 at RH"1 and 2.1 at RH"0.7. The result in Fig. 4 corresponds rather well with the measurements. The stripping factor along the column varies minimal when the in#ection point of the molar fraction pro"le lies in the middle of the column. Then, the stripping factor is
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the composition distributions for the maximums in Fig. 7. For all three cases, the in#ection point of the molar fraction pro"le lies in the middle of the column. This corresponds to the results of Skogestad and Morari (1987, p. 614). For continuous distillation columns they found a relatively large time constant when the feed fraction is 0.5 and the top and bottom composition are reverse (x "1!x ). In that typical case, the in#ection " point also lays in the middle of the column.
3. Consequences for control strategies Fig. 5. Composition distributions with the in#ection point in the middle of the column.
Fig. 6. Measured development of temperature pro"les under re#ux control and constant quality control. A. Constant re#ux control: RH"0.75, B. Constant quality control: x "0.985. "
constant over a maximum number of trays. In that case, the tray interactions are minimal and, consequently, the column settling time will be maximal. Due to #ow variations, the pro"le changes from convex to concave inducing relatively large tray composition adaptations. Fig. 7 shows the time constant as a function of the re#ux at three bottom fractions. Fig. 5, approximately, shows
Let us consider the consequences for two common control policies. Fig. 6 shows the measured pro"le development during a batch for, respectively, constant re#ux control and constant quality control. Due to the bottom exhaustion, for both policies the bottom temperature will rise. However, the course of the dynamic behaviour is quite di!erent. In Fig. 7, the dominant time constant calculated by the `change of inventorya time, is depicted for three di!erent bottom fractions over a wide range of re#ux fractions. The time constant ranges from 3 to 30 min. During constant re#ux control (Fig. 6A) the shape of the composition distribution changes continuously, and the dominant time constant will vary accordingly. For instance (Fig. 7), at a re#ux fraction of 0.85 and a bottom fraction of 0.55 the initial time constant is 10 min. During exhaustion, at "rst, it rises to 23 min and next it falls to 3 min. For constant quality control, the dominant time constant varies only slightly. Fig. 6B shows that during the production run, after start-up, the quality controller enforces the composition pro"le to maintain a similar form. The in#ection point remains at the same position. These results are in conformity with the nonlinear wave theory (Hwang, 1991), which assumes that the top fraction is constant, when the composition wave does not propagate through the column. This condition is ful"lled when the in#ection point holds the same position. Fig. 7 shows that at a constant product quality of 0.95 and an initial bottom fraction of 0.55, the dominant time constant is 23 min. It will remain approximately the same value until the bottom becomes exhausted. At a bottom fraction of 0.02 the form of the pro"le changes, and "nally the time constant becomes 18 min. PID-controller settings depend on the process gain at high frequencies, which is the quotient between the static gain and the time constant (K /q ). From Figs. 7 and 8 follows that at a certain top fraction, the static gain as well as the dynamics varies only slightly. Consequently, constant quality control is easy to implement. In the experimental column, it has been realised by a partial least squares-based quality estimator combined with a PID-based quality controller. No gain scheduling was necessary. During constant re#ux control, the process gain and time constant changes continuously.
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Fig. 7. Dynamic batch column relation. The `Change of inventorya time as a function of the re#ux ratio at x "0.05, 0.15, and 0.5. The "gures along the curves are the associated top fraction x . "
Fig. 8. Static batch column relation. Static gain between re#ux ratio and the top composition at di!erent bottom fractions.
However, this has no consequences for the controller itself. Constant quality control performs as good as dynamic optimal control when slop recycling is applied, since during the product-cut the exhaustion is restricted and during the slop-cut the separation is relatively easy (Betlem et al., 1998b). If a setpoint change is required to switch from product-cut to slop-cut distillation, then an adaptation of controller settings is inevitable as the process gain may di!er much (see Fig. 8). However, when a deep exhaustion is required and optimal re#ux control is applied, the changes in the column behaviour during the cut have to be considered in calculating the trajectory. For the realisation of the quality trajectory, a model-based controller can be used. Such a controller derived from input}output linearisation, which considers
the changes in the column time constant and the gain, only performed better in a simulation study (Betlem, 1996).
4. Conclusions The dominant time constant was estimated according to two methods. The "rst estimation was based on the `change of inventorya time and the second on a solution in the frequency domain based on the stripping factor. A series of time constants were calculated as a function of the re#ux fraction at a bottom fraction of 15% and were compared with the time constants derived from temperature measurement. The results were promising. The "rst method gives only a slight underestimate. For the second
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method, a correction is required known from the literature that corrects for the variations in the stripping factor. The dominant time constant was calculated for three bottom compositions and a large range of re#ux fractions. The magnitude of the time constant depends on the composition distribution. The results are especially interesting for batch distillation control. Although the bottom composition drifts continuously, constant quality control forces the molar fraction pro"le to maintain a similar form. Consequently, the dominant time `constanta remains nearly constant. However, under constant re#ux control the form of the pro"le changes continuously and can even go from convex to concave. Consequently, the dominant time constant can take values ranging from 3 to 30 min. Therefore, optimal re#ux policies have to account for internal pro"le changes.
References Armstrong, W. D., & Wood, R. M. (1961). The dynamic response of a distillation column to changes in the re#ux and vapour #ow rates. Transactions of the Institution of Chemical Engineers, 39, 65}90. Betlem, B. H. L. (1996). Hierarchical control of batch distillation. Ph.D. Thesis, University of Twente, Netherlands. Betlem, B. H. L., Rijnsdorp, J. R., & Azink, R. F. (1998a). In#uence of tray hydraulics on tray column dynamics. Chemical Engineering Science, 53(23), 3991}4003.
Betlem, B. H. L., Krijnsen, H. C., & Huijnen, H. (1998b). Optimal batch distillation control based on speci"c measures. Chemical Engineering Journal, 71, 111}126. Edwards, J. B., & Jassim, H. J. (1977). An analytical study of the dynamics of binary distillation columns. Transactions of the Institution of Chemical Engineers, 55, 17}28. Hwang, Y. L. (1991). Nonlinear wave theory for dynamics of binary distillation columns. A.I.Ch.E. Journal, 37(5), 705}723. Kim, Y. H. (1999). Optimal design and operation of a multi-product batch distillation column using dynamic models. Chemical Engineering and Processing, 38, 61}72. Moczek, J. S., Otto, R. E., & Williams, T. J. (1963). Approximation models for the dynamic responses of large distillation columns. Proceedings second IFAC world conference. Basel, (pp. 238}246). Montesi M., & Brambilla, A. (1995). Prediction of the e!ect of reboiler and condenser hold-up on the dynamic behaviour of distillation columns. IFAC-symposium Dycord#'95. Helsing+r (pp. 495}501). Rademaker, O., Rijnsdorp, J. E., & Maarleveld, A. (1975). Dynamics and control of continuous distillation units. Amsterdam, NL: Elsevier. Robinson, E. R. (1971). Optimum re#ux policies for batch distillation. Chemical and Process Engineering, 5}71, 47}55. Sa gfors, M. F., & Waller, K. V. (1995). Dynamic low order models for capturing directionality in nonideal distillation. Industrial Engineering and Chemical Research, 34, 2038}2050. Skogestad, S., & Morari, M. (1987). The dominant time constant for distillation columns. Computers & Chemical Engineering, 11(6), 607}617. Skogestad, S., & Morari, M. (1988). Understanding the dynamic behaviour of distillation columns. Industrial & Engineering Chemical Research, 27, 1848}1862.