Ind. Eng. Chem. Fundam. 1980, 19, 379-384
K' = reduced kinetic constant of the reverse reaction K A = acidity equilibrium constant of nitromethane, mol m-3 Kw = water equilibrium constant, mol2 md I = aggregate dimension, m lo = initial dimension of big aggregates, m I , = dimension of small aggregates, m Ls = Corrsin's macroscale of mixing, m M = CBO/CAO = ratio of initial concentration of the reactants N = rotation speed of the stirrer, mn-l Q1, Q2 = respective flow rate of the two feed streams, m3 s-l r = reaction rate, mol m-3 s-l S c = v / D = Schmidt number t = residence time, s tD = diffusion time constant, s t , = erosion time constant, s t~ = l/KlCAO = reaction time, s u = shrinking aggregate volume (big aggregates), m3 V = reactor volume, m3 w = age of the small aggregates, s YA, YB, ys = reduced concentration of A, B, S G r e e k Symbols a = W / T = reduced age of the small aggregates p = fraction of the total reactor volume in the shrinking phase y , q5 = constants t, el, e2 = power per unit mass (efficient, input by jet stirring, input by mechanical stirring), m2 s - ~ q, q', ql, q2 = efficiencies (relative, of the jet stirring, of the
mechanical stirring)
0 = t / T = reduced time 0, = t , / r = reduced erosion time constant = tD/r = reduced diffusion time constant v = kinematic viscosity, m2s-l
8d
379
T = V/(Q1 + Q 2 ) = space time, s subscript 0 = initial conditions overbar = average value
L i t e r a t u r e Cited Angst, W., Bourne, J. R., Kozicki. F., Roc. Eur. Conf. Mixing, 3rd, A4. (1979). Aubry, C., Villermaux, J., Chem. Eng. Sci., 30, 457 (1975). Beek, J., Jr., Miller, R. S., Chem. Eng. Prog. Symp. Ser., 55(25), 33 (1959). Calderbank, P. H., Moo-Young, M. B., Chem. Eng. Sci., 16, 39 (1961). Corrsin, S., AIChE J.. 10, 870 (1964). Danckwerts, P. V., Chem. f n g . Sci., 8, 93 (1958). David, R., Villermaux, J., Chem. Eng. Sci., 30, 1309 (1975). Holland, F. A., Chapman, F. S.. "Liquid Mixing and Processing in Stirred Tanks", Reinhold, New York, 1966. Kattan, A., Adler, R. J., AIChE J., 13, 580 (1967). Nabholz, F.. Ott, R. J., Rys, P., Helv. Chim. Acta, 60, 2926 (1977). Nauman, E. B., Chem. f n g . Sci., 30, 1135 (1975). Ng, G. Y. C.,Rippin, D. W. T., "European Symposium on Chemical Reaction Engineering, 3rd", p 161, Pergamon Press, Oxford, 1965. Pearson, R. G., Dillon, R. L., J. Am. Chem. SOC.,75, 2439 (1953). Plasari, E., David, R., Villermaux, J., Chem. Eng. Sci., 32, 1121 (1977). Plasari, E., David, R., Villermaux, J., ACS Symp. Ser., 65, 11 (1978). Rao, D. P., Edwards, L. L.. Chem. Eng. Sci., 26, 1179 (1973). Ritchie, B. W., Tobgy, A. H., Adv. Chem. Ser., No. 133, 376 (1974). Spielman, L. A., Levenspiel, O., Chem. Eng. Sci., 20, 247 (1965). Treleaven, C. R., Tobgy, A. H., Chem. Eng. Sci., 26, 1259 (1971). Treleaven, C. R.. Tobgy, A. H., Chem. Eng. Sci., 27, 1497 (1972). Treieaven, C. R., Tobgy, A. H., Chem. Eng. Sci., 28, 413 (1973). Truong, K. T., Methot, J. C., Can. J . Chem. f n g . , 54, 572 (1976). Turnbull, D., Maron, S. H., J. Am. Chem. SOC.,65, 212 (1943). Villermaux, J., Zoulalian, A., Chem. Eng. Sci., 24, 1413 (1969). Villermaux, J., Devillon, J. C., Int. Symp. Chem. React. Eng., Znd, Amsterdam, BI-13 (1972). Weinstein, H., Adler, R. J., Chem. Eng. Sci., 22, 65 (1967). Wheland, G. W., Farr, J., J. Am. Chem. SOC.,65, 1433 (1943). Zoulalian, A., Villermaux, J., Adv. Chem. Ser., No. 133, 348 (1974).
Received for review January 14, 1980 Accepted June 6 , 1980
Feasibility of Decoupling in Conventionally Controlled Distillation Columns K. Weischedel and T. J. McAvoy" Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1003
Recent steady-state work on decoupling in distillation column control has raised a number of questions about the feasibility of such control. The work of Jafarey et al. suggests that at most one-way decoupling seems to be achievable in distillation columns. The work of McAvoy further indicates that no decoupling will be possible in columns with large gains. Several other studies, both simulational and experimental, have reported the successful implementation of two-way decoupling. In this paper the results of dynamic simulations are presented which clarify this controversy. These results show that decoupling cannot be achieved in some columns.
Introduction Due to increasing fuel and feed stock costs, recent trends in distillation have been toward higher throughputs, smaller and fewer holdups, smaller reflux ratios, and more energy integration. Energy costs are particularly important t o the economics of distillation column operation. It has been estimated that distillation columns consume 40% of the energy used in chemical plants (Shinskey, 1979). Luyben (1975) has shown that minimum energy consumption for a column is achieved when both product compositions are controlled and the column is subjected to feed composition changes. Such a control scheme has
* Department of Chemical and Nuclear Engineering, University of Maryland, College Park, MD 20742. 0196-4313/80/1019-0379$01 .OO/O
been called dual composition control or two-point control. Although dual composition control results in minimum energy usage, in the past industry has generally avoided such a control scheme because of the interation between the loops. A common industrial approach is to control one product composition and over-reflux the column to more than ensure product specifications. In the future energy costs may well make such an approach less and less attractive. The columns where there should be the greatest incentive to find better approaches to dual quality control are those with the highest throughputs and the highest purities since these columns will consume the most energy. Decoupling is one approach which industry has used to achieve dual composition control. Its use, however, is not nearly as common as simply controlling one composition and paying an energy penalty. With decoupling the two
0 1980 American Chemical Society
380
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
Table I. Parameters for Distillation Columns Studied ( q = 1;xf = 0 . 5 )
a h
column
components
approx a
product split
no. of trays
A B C
benzene-toluene benzene-toluene methanol-ethanol
2.4 2.4 1.65
0.07-0.93 0.02-0.98 0.01-0.99
13 17 27
ha reflux and reflux ratio boilup manipulated
1.2 X Rmin 1.3 X Rmin 1.5 X Rmin
2.3 5.0 18.4
values are approximate. They were determined from step tests.
composition loops are isolated from one another and problems with interaction are avoided. There have been several academic studies of decoupling both simulational and experimental (Luyben, 1970; Luyben and Vinante, 1972; Toijala and Fagervik, 1972; Schwanke et al., 1977; Wood and Berry, 1973). As discussed by McAvoy (1979), the results of these studies generally support the conclusion that decoupling improves control system performance. Three recent studies (Jafarey and McAvoy, 1978; McAvoy, 1979; Jafarey et al., 1980), however, have questioned whether decoupling can be achieved in certain columns. All of these studies are based upon considering steady-state column performance. Jafarey and McAvoy (1978) and Jafarey et al. (1980) showed that in practice it may not be feasible to completely decouple both composition loops in a column. Although two decoupling relationships exist, they turn out to be almost identical for high purity towers and thus it may not be possible to implement both of them simultaneously. McAvoy (1979) further questioned whether any decoupling can be achieved in practice for some towers. By using the relative gain array (RGA) proposed by Bristol (1966) and basing his analysis on an earlier analysis by Shinskey (1977),he defined a decoupling sensitivity. This sensitivity indicates how much error in a decoupler gain can be tolerated by a decoupling control system. McAvoy concluded that high purity columns, and/or columns separating close boiling components would have the greatest sensitivity problems. He further concluded that past studies on decoupling have essentially treated low sensitivity columns. Thus, these past studies may not have uncovered all of the practical problems inherent in decoupling industrial columns. Since the results of Jafarey and McAvoy (1978), McAvoy (1979), and Jafarey et al. (1980) were based on steady-state considerations only, a series of dynamic column simulations was run to verify their conclusions. Three columns having a low, moderate, and high sensitivity as defined by McAvoy (1979) were chosen for study. The results of these dynamic simulations are presented here. It is shown that perfect decoupling cannot be achieved in high and moderate sensitivity columns, thus confirming the steady-state predictions discussed above. Conventional control where reflux and boilup are manipulated is treated in this paper. A future paper will discuss decoupling material balance control systems where distillate and vapor boilup are manipulated. Columns Studied Three binary columns were chosen for study based on their decoupling sensitivity. As defined by McAvoy (1979), decoupling sensitivity is given as s=l-X (1) where X is the relative gain of the undecoupled system. By using approximate analytical expressions for X (Jafarey et al., 1979), it is possible to relate X to column conditions such as feed composition, feed quality, product split, relative volatility, reflux ratio, and number of column trays. In choosing which three columns to study, relative gains in the neighborhood of 2 (lows), 10 (moderate s), and 50 (high s) were sought for the case where reflux and boilup
Table 11. Column Specifications ~
diameter = 1.8 f t weir height = 2.0 in. weir length = 1.26 f t tray area = 2.32 f t 2 pressure drop per plate = 0.0085 atm pressure a t reboiler = 1.000 atm reboiler holdup = 10 times average tray holdup condenser holdup = 10 times average tray holdup
were the two manipulated variables. Several distillation systems with components of different relative volatilities were examined and a column designed for each. Table I gives the important parameters for the three columns chosen. As can be seen in Table I, columns with the largest reflux ratios, highest product purities, and the lowest a’s have the largest relative gains and therefore are most sensitive to decoupler errors. Digital Simulation A summary of the approach taken for simulating the three columns on a digital computer is presented here. Complete details can be found in the thesis of Weischedel (1980). In developing the dynamic column models the following assumptions were made: vapor holdup is negligible in comparison with liquid holdup, each plate acts as an ideal stage, the column operates adiabatically, and there is perfect mixing on each plate and in the reflux accumulator and the reboiler. Using these assumptions dynamic mass and energy balances are written for each stage in the column as overall mass balance dHn - = Vn-1 L,+, dt component mass balance
+
-
L,
-
v,
energy balance d(HnhnL) = Vn-lhn_lV+ L,+lh,+lL - V,hnV - L,hnL (4) dt In order to simplify the calculations of enthalpies and vapor-liquid equilibrium, binary systems which exhibit essentially ideal solution behavior were chosen. Correlations of physical properties were obtained from the literature and these correlations are given in the thesis of Weischedel (1980). The Francis Weir formular (Smith, 1963) was used to relate liquid holdup to liquid flow from a tray. Perfect pressure control at the top of the tower was assumed as well as a constant pressure drop per plate. It was also assumed that the dynamic response of the energy balance, eq 4, was so fast that the right-hand side could be set equal to 0. The resulting set of equations was integrated using the ODE program described by Shapine and Gordon (1975). The column specifications such as column diameter, weir height, etc., are the same for all three columns studied and are given in Table 11.
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
381
Figure 3. One way decoupling. Figure 1. Ideal decoupling.
I
-
I
-
PI I
L
XD-
IDEAL DECOUPLING
SIMPLIFIED DECOUPLING
Figure 4. Resulting distillate loops.
Figure 2. Simplified decoupling.
Several other aspects about the digital sim$ation should be noted. First it was assumed that perfect level control was achieved in both the reflux accumulator and the reboiler. Secondly, it was assumed that ho sub-cooling of the reflux occurred. Thirdly, a 1-min dead-time was used to model the composition nheasurement system. This value for a dead-time is reasonable and probably conservative if one considers the speed of response of composition measuring devices in service today (Gtiffin et al., 1979). Lastly, reboiler heat transfer dynamics were neglected. Thus, the bottoms composition controller directly manipulated the vapor from the reboiler, VB. Decoupling Schemes Studied The three decoupling control schemes studies are shown in Figures 1to 3. These schemes are simplified, ideal, a d partial decoupling. Luyben (1970) has discussed the difference between simplified and ideal decoupling. To explain this difference as well as to determind the D,'s, the P,'s are taken as transfer functions. It should Be emphasized, however, that the simulation results bresented here are for the cases where the linear decouplers are applied to the nonlinear column models discussed above. The D,,'s in Figures 1 to 3 are given by the expression
tions relating f gand gcZ, In the discussion that follows only the distillate loop is considered. However, the conclusions reached hold for the bottoms loop as well. As eq 6 and 7 show, Oc2 has no effect on XD and thus the distillate composition loop is decoupled from the bottoms composition loop. For iddal decoupling the effective process transfer function is identical with that gotten if the bottoms composition loop is oh manual. For simplifiea demultiplies coupling an extra factor of [l- (P12P21/PllP22)] the Pll transfer function to, give the effective process transfer function. If perfect ideal or simplified decoupling is achieved then the effective distillate loop for both types of decoupling is shown in Figure 4. For ideal decoupling one can use single loop tuning to set Gcl. For simblified decoupling the factor [ l - (P1zPz1/PllP22)] complicates the problem of designing Gcl. This factor contains both steady-state and dynamic eiemerlts. The subject of the is treated in a future dynamic effect of [l- (P12p21/PllP22)] paper (McAvoy, 1980). In this paper a less general approach which only considers the steady-state value of this factor will be taken in order to give a simple explanation of decoupling. The steady-state value of [l - (Pl2PZl/ PI1Pz2)] can be related to X as (Witcher and McAvoy, 1977) ).""A=
1-KllK22
P
then the effective process gain given by eq 8 is The difference between simplified and ideal decoupling can be seen by deriving the open loop transfer functions relating f D and e,, for both cases. These transfer functions are fD _ - Pll (ideal) 8c1
RD
(
:: )
r = P l 1 I-Ocl
(simplified)
(7)
Similar expressions can be derived for the transfer func-
X
Equation 9 shows that simplified decoupling of eolumns with large relative gains will result in a large reduction in the effective process gain relative to Kll. If a single lodp approach is used to design Gcl then the resulting distillate loop will be extremely sluggish. Alternatively, one can increase the controller gain to compensate for the reduction in effective process gain and thereby achieve a fast control system response. A disadvantage of this latter approach is that if a decoupler should fail one is left with a feedback
382 Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980 Table 111. Gains of Column C 1%Increase in L and V
1%Decrease in L and V
Ed
=
E]
0.719 -0.200 [0.136-1.531
system which has too high a gain. Thus decoupler failure could produce instability. In addition, it was found that increasing the feedback gains for columns B and C actually resulted in a deterioration in control system performance. Thus, in the simulations on simplified decoupling the single loop tuning parameters were used. Partial decoupling, shown in Figure 3, is halfway between simplified and ideal decoupling. The effective open loop process transfer functions for both the distillate and bottoms loops for this case are different in form from one another and are given by the expressions RD = P&l (10) fB = P21ecl + ~cZPZZ[l- (p12p2l/p11p22)1 (11) As can be seen from eq 10, the distillate loop is unaffected by changes in the bottoms loop. Equation 11 shows, however, that BC1 does affect xB. For the XB loop the factor 11 - (P1&'21/PllP22)1again appears in the effective transfer function. Thus, the discussion given above for designing feedback controllers for simplified decoupling applies to the XB loop. Control System Design 1. Decouplers. The approach taken to designing the various decouplers is the same as that which would be taken in industry. Transfer function models for the three columns were developed by step forcing the manipulative variables in the nonlinear column simulation. Both positive and negative steps were made. An alternative industrial approach would be to step force the actual column and record the XD and xB responses. First-order transfer functions with dead-time and second-order transfer functions both with and without dead-time were then fitted to each response using a least-squares approach. The gains in the transfer functions were not fitted but were determined from the difference between the initial and final compositions. In choosing which transfer function model to use, it was desired to have a model with the smallest number of fitted parameters which adequately
represented the column dynamics. The F distribution method discussed by Law and Bailey (1963) was used to discriminate between the various transfer function models which were fitted. The level of significance chosen for the F test was 99.5%. For columns A, B, and C step changes of lo%, 2%, and 1% respectively were made in the manipulated variables. The time constants and dead-times for each transfer function were similar regardless of whether a positive or negative step was made. The gains for column A were also similar for both positive and negative forcing. However, the gains for columns B and C differed depending on the forcing. In Table I11 the gains for both positive and negative step changes are given for column C. As can be seen depending on the direction of the forcing some gains can differ by an order of magnitude. Column B shows similar but less severe gain changes. The higher the product purities are then the more nonlinear are the column gains. In arriving at final transfer function models for columns B and C the average of the gains was used. The final transfer function models for all three columns are given in Table IV. By substituting the various P,,and P,, transfer functions into eq 5, transfer functions for the decouplers tan be derived. Once the D,transfer functions are known they can be transformed into the time domain and solved together with the nonlinear differential equations for a column. Details of this procedure are given by Weischedel (1980). It should be reemphasized here that the decoupling approach taken parallels that which would be followed in industry. The resulting decouplers are at least as accurate as, if not more accurate than, those that would be used industrially. This is particularly true for columns B and C where step changes of 2% and 1%, respectively, were made. Thus, using the approach taken it should be possible to make a realistic evaluation of decoupling conventionally controlled columns. Even though the approach taken here is realistic, some error has been introduced into the decouplers, particularly in the gain terms for columns B and C. This error will affect the feasibility of decoupling through the sensitivity analysis discussed earlier. 2. Feedback Controllers. All of the feedback controllers used in this study are proportional integral controllers. The Ziegler Nichols rules were used to determine individual loop settings from the P,, transfer functions. As discussed previously, these settings were also used in studying decoupling control. The Ziegler Nichols settings are given in Table V.
Table IV Column A
#-
I I--
+ O.58e-l.Os
+ 1)(1.96s + + 0.35e-1.28s
(6.3s
1
(5.0s
+
-.I
- 0.45e-l.0s
1)(5.68s
+
1)(0.67s + 1)(4.7s
+
I r.1
1)(3.0s + 1)
- 0.48e-l.0s
1)(0.36s + 1)
Column B 0.56 2e- 0.516e-l.5s (7.74s + 1)(7.74s + 1) (7.1s + 1)(7.1s + 1) 0.33e-1.5s - - 0.394e-l.0S
+ 1) (13.8s + 1)(0.4s + -(15.8s + 1)(0.5sColumn C c
]E
0.471e-1.0s 0.49 5e- 2.0s (30.7s + 1)(30.7s + 1)(28.5s + 1)(28.5s + 1) 0.749e-1.7s - 0.832e-1.0s (57.0s + 1)(57.0s + 1) (50.5s + 1)(50.5s + 1)
-
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980 383
Table V. Controller Settings Calculated from t h e Ziegler Nichols Rules (L,V Manipulated)
distillate loop gain distillate loop reset bottoms loop gain bottoms loop reset
,075 x0
,070
,065
column A
column B
column C
7.1 6.7 - 6.4 4.0
12.9 10.4 - 20.7 4.4
59.7 20.5 - 55.7 26.2
P1
d
l j.
-
2______.___
015
0
IO
20 30 40 50 t, min
I
Figure 6. Column B (simplified decoupling).
H
1, min.
Figure 5. Column A (simplified decoupling).
Results In order to study decoupling feasibility, step changes in the set point of either the XD loop or the X B loop (but not both loops simultaneously) were made. For column A a 0.01 step was made while a 0.005 step was made for columns B and C. The reason for using only single loop forcing is as follows. If the two loops are perfectly decoupled, then one should see a transient response only in the loop whose set point is changed. For perfect decoupling the nonforced loop should remain at steady state and show absolutely no transient response. Also, even if the feedback controllers are over- or under-designed this fact will not affect one's ability to judge decoupling feasibility. The loop that is forced may respond too sluggishly or show too oscillatory a response. However, the nonforced loop should show no response whatsoever. In the results which follow changes in the xB set point are made. Similar results are achieved if the XD set point is changed. Figures 5 to 7 present results for simplified decoupling for columns A to C, respectively. As can been seen in Figure 5, the XD loop is effectively decoupled from the XB loop. The maximum deviation between xD and its set point, 0.93, is approximately 5 X lo4. Thus, the maximum response of the XD loop is only 5% of that of the X B loop. For columns B and C the ability to decouple the loops becomes less and less. For column B a maximum deviation of X D of +0.003 occurs for an X B set point change of -0.005. For column C a maximum deviation in XD of +0.002 occurs for an XB set point change of -0.005. Thus, the XD loop for columns B and C responds 40% to 60% as much as the X B loop, indicating that the two loops are not decoupled. The nonlinear gain characteristics of columns B and C no doubt contribute strongly to the inability of decoupling to work in these columns. Figures 5-7 also show that as h increases the response of a simplified decoupling scheme becomes increasingly sluggish, as eq 9 indicates. For column C the approach to steady state is particularly slow. Figure 8 shows the response of column C with no decou-
005 L _
0
_
_
_
IO
20
30
~
_ 40
~
~
50
1 , min
Figure 7. Column C (simplified decoupling)
t x ,990 o rl ,992
,988
i"------Il
0
___---
,(-'-,'---
IO
20
30
40
50
1 , min
Figure 8. Column C (no decoupling).
pling. As can be seen by comparing Figures 7 and 8, the addition of decouplers actually deteriorates the control system responses for column C. By contrast, in Figure 9 the response of column A with no decoupling is shown. As can be seen, the response is underdamped and considerably poorer than that achieved with decoupling and shown in Figure 5. For no decoupling the response of column B is
384
Ind. Eng. Chem. Fundam., Vol. 19,
No. 4, 1980
confirm the predictions of Jafarey and McAvoy (1978), McAvoy (19791, and Jafarey et al. (1980) that decoupling high sensitivity columns is not feasible. Secondly, it has been shown in one case that the addition of decouplers can lead to a deterioration in control system performance. Thirdly, in studying the applicability of any control strategy to distillation it is important to consider a spectrum of columns. A technique which seems to work well on one column may fail on another. Column sensitivity or equivalently the relative gain is one means of choosing the spectrum of columns to study. Fourth, it can be noted that the sensitivity analysis of Shinskey (1977) and McAvoy (1979) is not limited to distillation control systems but it applies to any arbitrary control system. The dynamic results presented here add credibility to this steady-state sensitivity analysis and they show that decoupling probably cannot be applied to systems with large relative gains.
,925
T
.070 ,065
f1
0
yl I ,
10
20
30
40
50
t, rnin
Figure 9. Column A (no decoupling).
similar to that shown in Figure 9. For column A, decoupling definitely improves the response of the control system. The results achieved with one-way decoupling for columns A and C for the case where the DZl decoupler is active are essentially the same as those given in Figures 5 and 7 . For column B, use of only DZlresulted in a highly oscillatory control system. When ideal decoupling was tried, the results were even more definitive. Ideal decoupling worked for column A but its response was poor compared to simplified decoupling. However, for columns B and C ideal decoupling produced a very oscillatory response of large amplitude. The oscillations appeared to be growing in time when fhe simulations were stopped because of excessive computer usage. These results on ideal decoupling agree with those of Luyben (1970), who found ideal decoupling to be unstable for a high-purity tower. In summary, none of the three decoupling schemes (ideal, one-way, and simplified) actually produced decoupling in the moderate and high purity towers studied. Since a number of investigators (Luyben, 1970; Luyben and Vinante, 1972; Toijala and Fagervik, 1972; Schwanke e t al., 1977; Wood and Berry, 1973) bave reported success with simplified decoupling, the above conclusions need to be put into perspective. First, most of the past studies treated low sensitivity columns or used computer simulations where no decoupler errors were present. For such cases complete decoupling appears to be feasible. Secondly, although simplified decoupling may not produce decoupling, it can produce an acceptable control system response. In some columns (Luyben, 1970; Tqijala and Fagervik, 1972; Wood and Berry, 1973) the use of controller parameters determined with one loop off results in an unstable or highly oscillatory response when both loops are active. As eq 9 shows, simplified decoupling lowers the individual loop gaips. This lowering should help to stabilize the resulting 2 X 2 system. While the resulting response may be acceptable in practice, the results presented here show that it will not be a decoupled response. Also as figures 8 and 9 show, for high sensitivity columns the resulting response should be compared to that which can be achieved with no decoupling or by simply detuning the two feedback loops. Conclusions There are several important conclusions which one can draw from the results presented here. First, the results
Nomenclature a = relative volatility Di. = decoupler transfer function h k .= enthalpy of liquid on tray n h," = enthalpy of vapor from tray n H, = holdup'on tray n, mol Kij = transfer function gain X = relative gain element L = reflux flow, mql/time L, = liquid flow from tray n, mol/time Pij = transfer function q = feed quality Rmin= minimum reflyx ratio s = decoupler sensitivity 6',i = controller output V = vapor flow from reboiler, mol/time V , = vapor flow from tray n, mol/time XB = bottoms mole fraction X D = distillate mole fraction X F = feed mole fraction x , = mole fraction of liquid on tray n yn = mole fraction of vapor from tray n - = Laplace transform
Literature Cited Bristol, E. H. I€€€ Trans. Autom. Control 1966, AC- 11, 133. Gdffin, D.; Parson, J.; Sjmth, D. ISA Trans. 1979, 18(1), 23-31. Jafarey, A.; McAvoy, T. J. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 485-460.' Jafarey, A,; McAvoy, T. J.; Douglas, J. M. Ind. Eng. Chem. Fundam. 1979, 18, 181. Jafgrey, A.; McAvoy, T.; Doughs, J. I d . Eng. Chem. Process Des. Dev. 1960, 19, 114-117. Law,'V.; Bailey, R. Chem. Eng. S d . 1963, 18, 189-202. Luyben, W. L. AIChE J. 1970, 16, 198. LLlyben. W. L.: Vinante, C. D. Kem. Teollisuus 1972, 29, 499-514. Luyben, W. L. Ind. Eng. Chem. Fundam. 1975, 14, 321-325. McAvoy, T. J. Ind. Eng. Chem. Fundam. 1979, 18, 269-273. McAvoy, T., submitt@ for publicatlon in AIChE J. Schwanke, C. D.; Edgar, T. F.; Hougen. J. 0. ISA Trans. 1977, 16(4), 69-81. Shampine, L.; Gordon, M. "Computer Solution of Ordinary Differential Equations", W. H. Freeman 8 Co.: San Francisco, 1975. Shinskey, F. G. "The Stability of Interacting Control Loops With and Without 'Decoupling", Proceedings, IFAC Multivariable Technological Systems Conference, 4th International Symposium, University of New Brunswick, July 4-8, 1977, pp 21-30. Shinskey, F. G. "Process Control Systems", 2nd ed., p 277; McGraw-HIII: New York, 1979. Smith, 8. "Design of Equilibrium Stage Processes", McGraw-Hill: New York, 1963, p 486. Toijala, K.; Fagervik, K. Kem. Teollkuus 1972, 29, 1-12. Weischedel, K., M.S. Thesis, University of Massachusetts, Amherst, 1980. Wltcher, M. F.; MaAvoy, T. J. ISA Trans. 1977, 16(3), 35-41. Wood, R. F.; Berry, M. W. Chem. Eng. Sci. 1973, 28, 1707-1727.
Received for review February 4 , 1980 Accepted August 11, 1980
This work was supported by the National Science Foundation under ENG-76-17382.
