Ind. Eng. Chem. Res. 1987,26, 1041-1043
1041
Design Algorithm for Periodic Cycled Binary Distillation Columns Bjarne Toftegard and Sten Bay Jargemen* Instituttet f o r Kemiteknik, The Technical University of Denmark, DK-2800 Lyngby, Denmark
An algorithm for the design of periodic cycled distillation columns is presented. T h e algorithm is applicable t o binary mixtures with nonlinear thermodynamic equilibrium relationships. The design algorithm is noniterative and determines the necessary number of trays and a feed location. Application of the algorithm is illustrated for a benzene-toluene separation. Periodic cycled distillation was first introduced by Cannon (1961). A period consists of two parts: the vapor flow part (VFP) and the liquid flow part (LFP). In the VFP, the liquid stays on the trays and the vapor flows up through the column and is collected in the condenser. In the LFP, bottom product is taken out of the reboiler, the liquid is shifted one tray down without being mixed, part of the condensor holdup is entered on the top tray as reflux, and top product is taken out. This operation form gives ideally a much higher column efficiency than conventional tray distillation columns. The performance of a periodic cycled distillation column with equal tray holdups and infinite reboiler holdup is exactly the same as for a Lewis case I1 distillation (Lewis, 1936) where the liquid flows in plug flow across the trays in the same direction on every tray without mixing and the vapor also flows in plug flow without mixing. Periodic cycled distillation is thus one method for performing a Lewis case I1 distillation. For calculation of steady-state profiles in periodic cycled distillation columns, Sommerfeld et al. (1966) performed a dynamic simulation until a stationary periodic solution was reached. In the general case, however, this procedure may require simulation for several hundred cycle periods. Even though special integration routines are available (Tofteglrd and Jerrgensen, 1987a) where it is sufficient to integrate a few of the cycle periods, these methods are mainly suited for dynamic simulation. Baron et al. (1981) simulated a single cycle period and iteratively adjusted the initial value until conditions at the end of the cycle matched the initial conditions using Shanks diagonal transformation. In the work a t our institute, we are successfully applying the same principle but with a Newton-Raphson algorithm with an analytical Jacobian. This method requires only integration of a few cycle periods. However, when applied for design purposes, the above-mentioned methods all lead to an iterative design procedure. The purpose of this paper is to present a noniterative method for designing a periodic cycled distillation column for a desired binary separation with an arbitrary vaporliquid equilibrium relationship. The method is in principle based upon a tray-to-tray calculation procedure. Starting from a specified bottom product concentration, i.e., reboiler liquid concentration a t the end of the VFP, the reboiler liquid concentration a t the beginning of the VFP can be calculated. Knowing this concentration, the liquid concentration on tray one at the end of the VFP can be calculated. Knowing this concentration and the variation of the vapor concentration from the tray below, the liquid concentration a t start of the VFP can be calculated and similarly for the following trays until the feed concentration is reached, at which point the feed is introduced. The calculations are then continued tray by tray until the specified distillate concentration is reached. I t is shown that the calculations can be performed by integrating a 0888-5885/87/2626-1041$01.50/0
Table I. Distillate and Reboiler Concentrations for Periodic Cycled Distillation of Benzene and Toluene as Described in the ExamDle stages in column
if
xc
xb
8 8 9 9 9
4 5 4 5 6
0.9480 0.9419 0.9550 0.9556 0.8445
0.0520 0.0581 0.0450 0.0444 0.0555
single differential equation. This equation is obtained by introducing a dimensionless independent variable which combines a dimensionless time within the VFP and the tray number. At each integer value of the independent variable, corresponding to the LFP, a change in liquid composition may occur. An example is given to demonstrate the applicability of this algorithm.
Methods The general model formulated (Tofteglrd and Jerrgensen, 1987b) is used here with the same normalization and the incorporation of a general equilibrium relationship yi* = f ( x i )
(1)
where a point efficiency is used: yi = cyi*
+ (1- 4yi-1
(2)
The dimensionless model is reboiler (i = 0) (3)
VFP:
(5)
Define a dimensionless variable 6 = (1 -
+)+ i
(7)
where the tray number i is zero for the reboiler, one for the lowest tray, etc. The variable 6 decreases on tray i over a cycling interval from (i + 1)-to i+. When changing the tray, a discontinuity may occur in the concentrations due 0 1987 American Chemical Society
1042 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987
to feed, reflux, or nonideal liquid draining. This change is indicated by using the superscript + for variable values just before carrying out the LFP and - for the resulting values. Substituting t by 0 in eq 3-6 gives reboiler (i = 0) oIoI1-
-dX(@ - - T(y(0)- y(0 - l))/hi d0
X(e+)
i+ I 0
= (hix(&) - K X ( 0 - l)+)/hi+l 0=i
d e + ) = ((h,x(&) - K X ( 0
I
(i + 1)-
+1
- 1)+)- fXf)/hi+*
(11)
i # if (12a)
+1 (12b)
0 = if
The necessary number of trays are found by integration of eq 8 from 0 to 1 and then eq 11 from 1 and upward. When an integer value of e is reached, the LFP is carried out. The molar fraction must be changed if any mixing occurs using eq 10 and 12 to fulfill the mass balance for the LFP. Note that it is not necessary to define if and N in advance. When the value of xfis reached, the simplest action is to use this tray as the feed tray; i.e., when 0 reaches the next integer value, use eq 12b. Another possibility is to feed in the middle of a period and then eq 12b is not used; instead: hi = 1 + f
hi = 1
x ( 0 ) I Xf
(134
> Xf
(13b)
x(0)
+
Since x((N 1)+)is equal to x,, the integration is stopped when the acual value of x , is exceeded. The integration routine used is a fourth-order RungeKutta method with variable step length. The delayed values are determined by using up to fourth-order Lagrangian interpolation on saved integration values.
Results and Discussion The design example is the binary separation of benzene and toluene used by Rivas (1977). The relative volatility is assumed constant, a = 2.25. The other parameters are K = 0, bo = *, xf = 0.5, f = 1.2739, xC = 0.95, d = 0.6369, xb = 0.05, and e = 1.00. The calculation results are shown in Figure 1 where the feed is entered when x ( 0 ) reaches xf and (13) is used. The total number of trays needed in the column is 8.2 when feeding on “tray” 5.4. The actual distillate and reboiler concentrations, when feeding in the LFP, are shown in Table I with different number of stages and feed locations. These results show that nine trays are sufficient, and the feed should be introduced on tray 5. The general rule for the design algorithm is that the total number of trays should be rounded up and the feed tray number down. The optimal procedure is to feed during the VFP, i.e., in the above example on tray 5, four-fifths into the VFP. Rivas (1977) found that the number of trays needed for this separation is 5.27. Figure 1 demonstrates that one of the assumptions for the approximate analytical method presented by Rivas does not hold. Rivas assumed that the concentration profile over the VFP could be described as
0.05
i + 0.0
it05
i+1.0
Figure 1. Concentration as a function of B for the benzene-toluene periodic cycled distillation (i is the stage number).
a sum of a linear and a exponentially decaying term. Note that the 0 and t axis are in opposite directions. This assumption is clearly not valid for the present example, where a nonlinear thermodynamic equilibrium relationship is used. The necessary number of trays in the periodic cycled column is 513‘% of the number needed for a conventional distillation column, where 16.0 trays are needed. Conclusion A design algorithm for periodic cycled distillation with a nonlinear thermodynamic equilibrium relationship is presented. The algorithm is noniterative and determines the necessary number of trays and a feed location. Due to the similarity with a Lewis case I1 distillation, this algorithm may easily be adapted for design of such columns as well as for periodic cycled stripping and absorption columns. If desired, the actual concentration profiles may be determined by another method, once the number of trays and the feed location are determined by the present algorithm. The design algorithm is demonstrated on a binary separation. This example also shows that one should be very careful in using approximate methods when the thermodynamic relationship is nonlinear. Nomenclature b = dimensionless reboiler holdup (BIH,) c = dimensionless condenser holdup ( C / H N ) d = dimensionless distillate amount ( D / H N ) f = dimensionless feed amount ( F / H N ) h = dimensionless tray holdup ( H / H N ) H = tray holdup i = stage number N = number of trays in column t = dimensionless time (t’ V / H N ) T = dimensionless cycle period duration ( T ’ V / H N ) V = vapor flow
Ind. Eng. Chem. Res. 1987,26, 1043-1045 - = limit from lower values
r = liquid molar fraction y = vapor molar fraction
* = equilibrium
' = dimensioned variable
Greek Symbols a = relative volatility t = vapor point efficiency
e = new independent variable; see (7) K
1043
= part of liquid staying on tray
Subscripts b = bottom = reboiler c = condenser f = feed N = top tray 0 = initial condition
Literature Cited Baron, G.; Wajc, S.; Lavie, R. Chem. Eng. Sci. 1981,36, 1819-1827. Cannon, M. R. Ind. Eng. Chem. 1961,53, 629. Lewis, W. K. Znd. Eng. Chem. 1936,28, 399-402. Rivas, 0. R. Ind. Eng. Chem. Process Des. Deu. 1977,16,400-405. Sommerfeld, J. T.; Schrodt, V. N.; Parisot, P. E.; Chien, H. H. Sep. Sei. 1966, 1, 245-279. ToftegArd, B.;Jerrgensen, S. B. submitted for publication in Comp. Chem. Eng. 1987a. Tofteghrd, B.; Jmgensen, S. B. submitted for publication in Ind. Eng. Chem. Res. 198713.
Superscripts + = limit from higher values
Receiued for review March 18, 1985 Accepted February 13, 1987
COMMUNICATIONS Robustness with Respect to Integral Controllability A multivariable closed-loop process may become unstable for small changes in steady-state gains if integral action is used. The results of this work show that the Relative Gain Array (RGA) is related quantitatively t o the amount of change (or error) allowed in each individual steady-state gain before the system becomes unstable. The derived criterion is applicable for any type of feedback control system as long as integral action is used. Controllers with integral action are extensively used in chemical process control. As pointed out by several authors (Koppel, 1985; Grosdidier et al., 1985), the integral controllability of a multiple-input-multiple-output (MIMO) process depends largely on the steady-state information of the process. If a process is not integrally controllable, it will eventually go unstable no matter what the controller settings are (usually evidenced by a slow drift into instability). Due to the complexity and strong nonlinearity of many processes, the model used in controller design is usually only an approximation of reality. The ability to remain integrally controllable (stable) in the face of plant/model mismatches is called Robustness with respect t o integral controllability. This is a unique feature of MIMO processes. In a single-inputsingle-output (SISO) process, the system will lose integral controllability only when the steady-state gain changes sign (since a system with positive feedback is not integrally controllable). However, in a MIMO process, positive feedback may arise for small changes in process gains. Robustness for integral controllability is an inherent property of the system, and it can be determined from the steady-state gains, G(0). Grosdidier et al. (1985) related the relative gain array, RGA (Bristol, 1966), to the optimally scaled condition number of G(O),which gives an approximate bound for the allowable modeling error. Their approach dealt with simultaneous perturbations in all the giis. Furthermore, they also pointed out that the RGA is a measure of system sensitivity, i.e., the bigger the ijth element of the RGA, the more sensitive the gij will be to errors. 0888-5885/87/2626-1043$01.50/0
The purpose of this work is to show that each RGA element (Pij) is related quantitatively to the amount of perturbation (Ag,) allowed in each individual element of the steady-state gain matrix before the system loses integral controllability. Integral Controllability Integral controllability has been studied by Morari and co-workers (Grosdidier et al., 1985; Morari, 1985). If multiloop SISO controllers are used, integral controllability is an important criterion in variable pairing. Yu and Luyben (1986) eliminated pairings with negative Morari indexes of integral controllability (MIC) to ensure integral controllability. The MIC's are the eigenvalues of the G'(0) matrix (the plant steady-state gain matrix with the signs adjusted so that all diagonal elements have positive signs). If all of the individual loops are integrally controllable, a negative value of any of the eigenvalues of G+(O)means that the variable pairing will produce an unstable closedloop system if each loop is detuned a t an arbitrary rate. It should be noticed that for 3 X 3 or higher order systems, there are instances for which no variable pairing will give MIC's that are all positive.
RGA a n d Robustness The RGA gives some information on variable pairing. Grosdidier et al. (1985) gave an updated summary of the uses of the RGA. Each element of the RGA is defined as
p., v = g$.. 1 11
(1)
where pi, = the ijth element of RGA, gij = the ijth element 0 1987 American Chemical Society
