Modeling of the Recycle Effects in Distillation Systems RP. Hajare', P.R. Krishnaswamy', C.A. Plank, and P. B. Deshpande' Universify of Louisville, Louisvi//e,Kenfucky 40292, USA and D.E. Ralston Boden Chemicals Co., Louisville, Kentucky 402 16, USA The possible existence of recycle effects in distillation have been investigated and tested using a commercial phenol-toluene column. The observed column response wasfound to be much more sluggish than the theoretical response predicted b y dynamic mathematical modeling and simulation studies. This discrepancy was accounted for by incorporating the features of recycle into the dynamic mathematical model. The model now allows more accurate prediction of the operating column transients, and it enables the engineer to design reliable control systems based on this computer model. Introduction Considerable research has been directed towards examining the influence of the recycle on process response characteristics [l-71. Examples include composition dynamics of a reactor with recycle [8], an activated sludge process with large recycle [9],and a reactor-heat exchanger recycle system [lo]. The recycle concept has also been used in the context of general model behavior to account for, say, partial bypass and initial dehy in response [I], or departure from perfect mixing [I 11. In the context of process control, it has been observed [12] that the overall response of industrial distilIation columns could be significantly slower than that predicted from individual tray dynamics. That is, the open-loop response of a controlled variable (e.g. distillate composition) to a change in the manipulated variable (e.g. reflux flow) may appear "sluggish" when compared to the response obtained via a simulation based on the idealized differential equations. This discrepancy may be accounted for by assuming internal recycling on each tray within the column [12]. It is well-known [7lthat the effect of recycle on plant dynamics is to render the output response more sluggish. Qualitatively, the different phenomena which could contribute to the overall recycle effects in distillation might include : (i) entrainment of liquid in the vapor phase; (ii) the associated flow of the vapor with the liquid in the downcomer; (iii) effect of mixing lags: and (iv) weeping flow of the liquid onto the lower tray. The objective of this work was to incorporate the previously unaccounted multiple-recyclingeffect into the dynamic model of a distillation column, to enable 1. Currently at Enron Chemical Co., Baton Rouge, Louisiana, USA. 2. Visiting Professor: permanent afiliation: Department of Chemical Engineering, National University of Singapore. *Author for correspondence. 29 R.P.Hajme. P.R. K r k h a m y , CA. Plank, P.B. Deshpande, and D.E. Ralston a more realistic column response to be predicted. This would provide a more flexible general model for distillation. In particular, it would also allow more efficient design of model-based control systems for the column 1131. Therefore, the experimental response data from a 37-tray industrial column separating a mixture of phenol and toluene was used. Analysis of Recycle Streams The basic recycle system shown in Fig. 1 can be defined by [l]: where Gm represents the model of a given process. The recycle dynamics, for simplicity, are assumed to be negligible. Defining a recycle fraction as r = R/(F + R), then Equation (1) can be expressed as: Figure 1.Block diagram of a process with recycle. For distillation, the relevant variables representing XF and XR (see Fig. 1) are temperature or composition. As a first approximation, the open-loop transfer function Gm(s) could assume the form of a fmt-order lag plus dead time model: Gm(s) = Ke-eS / (7s + 1) (3) Combining Equations (2) and (3): G ~ ( s= K ) (1-r) e-eS/ 17s + 1 - K r e -9~1 (4) For small relative values, the dead time in the denominator of Eqn. (4)may be approximated by: e-es = 1 - 8s Upon rearrangement of Eqn. (4): G ~ ( s= K ) e-8S / (7's + 1) where K =K(l-r)/(l-Kr) and ~ ' = ( ~ + K r e ) / ( l - K r ) 30 (5) Modeling of the Recycle Effects in Distillation Systems Considering these equations, the system will be open-loop stable only if (1-Kr) and (7 + K18) are both greater than zero. Based on physical reasoning, r has to be a nonnegative fraction.Under these conditions, a comparison of Equations (3) and (6) reveals that recycle would always cause an increase in the process time constant. The gain of the process would also be affected. For k K 4 , the process gain with recycle would be greater than that without recycle. The reverse is true for the case: kKc1. Thus the effect of recycle in distillation would be to slow down the output response, and to alter its final steady-statedeviation. Application of Recycle Effects in Distillation A distillation column can be considered as a sequence of units or trays. Assuming negligible vapor holdup, the differential equations representing the unsteady-state overall and component balances associated with each tray are: [ ~ " , / d ] = h + 1 - L n + vn-1 - V n (7) If the phenomenon of recycle is hypothesized on each tray, then the effect of multiple recycle would be equivalent to making the overall column response more sluggish, compared to the prediction based on the idealized expressions given above. A simple way to achieve the required increase in the response time is by introducing a factor a into the dynamic balances. The term a , as employed in Eqns. (9) and (10) below, is a fractional multiplier chat is equivalent to creating a reduction in throughput. This amounts to increasing the mean residence time and hence the time constant of a given stage. For many flow processes, time constants are often related to the mean holdup time, Thus the tray equations for recycle systems may be modified by: [dM,/dt]=~+l-Ln+V,-1-V,)a (9) [mnXn /dt] = Gn+1Xn+1 -Ln (10) Xn + Vn-1 Yn-1-vnyn )a Correct choice of the a-value would then account for the recycle effect, and would make the simulated transient response fit the observed response. As flow conditions are normally different in the stripping and enriching sections, a could also be different for the two sections. To study the recycle effects on an industrial size distillation column, a phenoltoluene separating column operating at Borden Chemicals was selected. The column consisted of 37 sieve trays, 0.75 m diameter, spaced 0.45 m apart, each having a downcomer area of 6.8% of the column area and a straight 2.5 cm weir of length 40 cm. Under steady-state operating conditions, a feed of 4%wt phenol and 96%wt. toluene was introduced onto tray 17 (numbered from bottom to top) at the rate of 660 cm3/s. This feed was separated to produce toluene as distillate at a rate of 630 cm3/s and with an impurity level of 20 ppm phenol. The bottoms had a composition of 99% phenol. The top product was totally condensed at a pressure of 105 kN/m2, and 31 R.P. Hajam. P.R. Krkhnanuamy, C A . Plank,P . B. Deshpande. and D.E. Ralston reflux was introduced at a rate of 315 cm3/s and subcooled at a temperature of 353 K. The reboiler heat duty was 550 kW. The steady-state operation of the column was modeled to yield the temperatures and compositions on each tray in the column, using a rigorous tray-by-tray procedure [ 141. The model included component and total material balances, equations comprising energy balances, phase equilibria, and a Murphree tray-efficiency factor. The results of the steady-state simulation are represented in Fig. 2, and these served as a starting point for the dynamic simulation. The very low bottoms flow rate required good accuracy from the steady-state simulation. A computer program [13] was adapted to account for non-ideal VLE data (as supplied by Borden Chemicals) for the system under consideration. L70 650 J" 1.0 0.8 I - 0.4 - 0.2 n, stage number (from bottom) Figure 2. Steady state simulation results for the phenol-toluene column: (a) total flowrate profiles; (b)temperatureprofile; (c) composition profilesfor toluene. To identify process dynamics, while causing minimal disruption to normal plant operations, a closed-loop pulse test was perfoxmed on the column. This involved keeping the column temperature loop on automatic control, and introducing a rectangular pulse of ten minutes duration at the temperature set point. Analysis of the resulting pulse test data gave a closed-loop column transfer function in the frequency 32 Modeling of the Recycle Effects in Distillatwn S y s t m domain. A model was fitted to the resulting Bode plot. The open-loop transfer function %(s) (relating temperature on tray 21 to reflux rate) was developed for the enriching section of the column, with the aid of the generalized Black-Nichols Chart r.151. An identical pulse test was introduced into the process model obtained fiom the idealized my dynamic equations. It was observed that the experimental transient response was much more sluggish than the response obtained through simulation. The recycle hypothesis was considered and the effect of recycle was incorporated into the dynamic tray equations by using the factor a. The a-values in the stripping and enriching sections of the distillation column were varied to yield a simulated transient response which closely matched the experimental transient response. Results and Discussion The experimental open-loop response of the temperature on my 21 (T21) to a step change in the reflux rate is compared in Fig. 3 with the simulated response based on tray dynamics with no recycle. The multiple lags representing the trays between reflux and tray 21 were replaced by a time-delay/time-constanttransfer function. Therefore, for the experimentalcolumn: GE =T21(s)/Ro(s)= 1.678e4-5s/(111.6s+ 1) (1 1) For the model simulation without recycle effect: Gm = T2l(s) / Ro(s) = 1.392 e-2.0s / (66s + 1) '*" I /._ (12) ----- -- - ---- -L I -- Simulation response (without recyde) - Simulation response (with recycle) Experimental response 0 200 I I I so0 600 000 lime tmin) Figure 3. Temperature (T21) response to a 5% step change in refluxflow. 33 R.P. Hajme, P.R. Krkhnaswamy, C A . Plank,P . B. Deshpande, and D.E. Ralston The discrepancy between experimental data and the model results without recycle is obvious. The model was repeated with consideration of the recycle effects. The numerical a-values in the stripping and enriching sections were varied to obtain a simulation which matched the experimental transient response: where GmR is defined in Eqn. (2). Knowledge of the two transfer functions (Gm and GF) facilitated calculation of r, and provided a match between experiment and simulation. Substitution of Eqn. (12) into Eqn. (2), and approximating the e-2s term as (1-2s) in the denominator of the resulting expression, yields: where K = 1.392 (1 - r) / ( 1 - 1.392 r) and 5' = [66 + (1.392) (2) (r)] / (1 - 1.392 r). By equating the gains and time constants of the transfer functions (GmR and %, i.e. Eqns. 14 and l l ) , the fraction r was estimated to be approximately 0.29. The corresponding a-values for the stripping and enriching sections were found to be 0.74 and 0.4, respectively. The ratio of the a-values in the enriching and stripping sections was calculated to be nearly equal to the ratio of liquidvapor flow in these sections of the column. The prediction arising from Eqn. (14) is also included in Fig. 3, and it can be seen that there is now a significantly better match between theory and experiment. Conclusions The results of this study indicate that the experimental response of a commercial distillation column could be more sluggish than the predicted response obtained by simulation based on dynamic modeling. The experimental evidence also shows that the discrepancy between theory and experiment could be eliminated by incorporating the features of recycle into the dynamic analysis. Acknowledgement The cooperation and assistanceof Mr. James Szofer at Borden Chemical Company is appreciated. Nomenclature F Mass flow rate of the feed (kgjmin) GE G, G d K K L, 34 Experimental process transfer function Transfer function of the process model Transfer function of the process model including effects of recycle Steady state gain Steady state gain in the recycle transfer function GmR Liquid flow rate from the nth tray (moYmin), Modeling of the Recycle Effects in Distillation System Liquid holdup on the n* tray (moles) n* tray (from bottom) Recycle fraction Mass flow rate of the recycle stream (kg/min) Reflux flow rate (moVmin) Temperatureon nth tray (K) Temperatureon my 21 (K) Vapor flow rate from the n* tray (mournin) Composition of the feed stream (moVkg) Mole fraction in liquid from the nth tray Composition of the recycle stream (mol/kg) Mole fraction in vapor from the n* tray Multiplier in Eqns. (9) and (10) Dead-time (min.) Time constant (min.) Time constant in the recycle transfer function, GmR (min.) References 1. Gibdaro. L.G. 1971.The recycle flow-mixingmodel. Chem. Eng. 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Attk, U. and Denn, M.M. 1978.Dynamics and control of the activated sludge wastewater ~ ~ O C ~ SAIChE S. J., 693-698. 10. Silverstein, J.L. and Shinnar, R. 1982. Effect of design on the stability and control of fixed bed catalytic reactors with heat feedback. Id.Eng. Chem. Process Des. Dev., 2, 241-256.. 11. Denn, M.M. 1979. In: Control and Dynamic Systems, C.T. Leondes (Ed.), pp. 8. a u, a, Academic Press, New York., 12. h a r d , I. and Benjamin, B. 1982. ContTol of recycle systems. I: Continuous Control, Paper WAS. Presented at the American Conhol Cog., Arlington, Virginia,U.S.A.. 13.Deshpande. P.B. 1985. Distillation Dynamics and Control. pp. 391-492. ISA, Research Triangle Park.North Carolina. 35 R P . Hajare. P R . Krishnanuamy. C A . Plank, P . B . Deshpande, and D.E. Rakton 14. Luyben, W.L. 1990. Process Modeling, Simulation and Control for Chemical Engineers, 2nd Ed. pp. 132-141. McGraw Hill, New York.. 15. Srinivas,S.P., Deshpande. P.B. and Krishnaswamy. P.R. 1984. Extension of Nichols chart for identificationof open-loop unstable systems.MChE J.. 684-686.. u, Received: 29 January 1993;Accepted after revision:2 August 1993. 36

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