Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
181
Analytical Relationships for the Relative Gain for Distillation Control Azmi Jafarey, Thomas J. McAvoy,’ and James M. Douglas Deparfment of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1003
In this paper new, more accurate analytical expressions for the relative gain array (RGA) for dual composition control of binary columns are presented. Both material balance control and conventional control achieved through manipulation of reflux and boilup are examined. The new expressions, which can be derived from the rigorous Smoker equation, compare very well with published numerical resutts. A consistency relationship for material balance and conventional RGA’s is presented.
Introduction One of the most useful tools that has appeared in the control literature is the relative gain array, RGA (Bristol, 1966). For multivariable control problems the RGA can be used to pair manipulated and controlled variables, to assess the extent of interaction present, and to reverse a process model for decoupling (Shinskey, 1967). Shinskey (1977a,b) has presented an extensive discussion of the application of the RGA to dual composition control. The expressions for the RGA which he has presented will be considered in detail in this paper and their accuracy assessed. Nisenfeld and Stravinsky (1968) have applied the array to the control of an azeotropic distillation column. McAvoy (1977) has presented a numerical approach to rigorously calculating the RGA for dual composition control. Recently, Jafarey and McAvoy (1979) have used McAvoy’s approach to show that complete decoupling does not seem feasible in distillation systems. In this paper approximate analytical expressions for the RGA are presented. For material balance control an excellent approximation to the RGA is derived from a version of the exact Smoker (1938) equation discussed by Strangio and Treybal (1974). For columns with high reflux ratios this approximation simplifies to Shinskey’s expression (1977a). For conventional control achieved through manipulating reflux and boilup, a recently published, accurate approximation to the Smoker equation (Jafarey et al., 1979) is used to calculate the RGA. All of the new approximations presented here compare quite favorably with exact RGA results published by McAvoy (1977). The new relationships predict all of the trends in the RGA as feed composition, product split, relative volatility, and reflux ratio are varied. It is shown that the RGA for material balance control can be related to the RGA for conventional control. This relationship provides a means of testing the consistency of numerically calculated RGA’s. Lastly, a brief discussion of the extension of the results to multicomponent columns is given. Relative Gain Array (RGA) The RGA expressions developed here apply to dual composition control of binary columns with arbitrary feed quality, fixed relative volatility, and constant molal overflow. For manipulative variables m, and m,, the RGA is given as mi
mi
(
XD
(1-
hml,m,
(1- hml,m2)
hml,m,
where 0019-7874/79/1018-0181$01.00/0
It is assumed that ml is paired with XD and m2 with xw. Two types of material balance control are treated. These are m, = D with m2 = P and ml = L with m2 = W . Conventional control with ml = L and m2 = is also treated. As discussed by Witcher and McAvoy (1977), A,, values which are either large and positive or close to 0.5 signify that a particular control system is highly is always interacting. For material balance control A,, a positive fraction while for conventional control A,, is always positive and greater than 1.0 (Shinskey, 1977a,k). Relative Gain for Material Balance Variables-L, W Shinskey (1977a) has presented the following expression for AL,W
Equation 3 can be derived from a column’s material balance equations and the approximate empirical separation factor expression (4) where V is the vapor flow in the rectifying section of the column. In Table I values of AL,w for 24 columns studied by McAvoy (1977) are shown. In Table I 6 is the ratio of reflux to minimum reflux. Exact AL,w’s were calculated using a numerical iteration scheme to solve the Smoker (1938) equation. Also shown in Table I are AL,w values calculated from eq 3. The average absolute error produced by eq 3 is 0.0913, which corresponds to a 13.3% average error. This degree of accuracy is good considering the simplicity of eq 4 upon which eq 3 is based. However, in some cases eq 3 gives an erroneous impression of the degree of interaction present in a column. For example, for the third entry in Table I, eq 3 predicts a AL,w of 0.778, indicating mild interaction. The exact AL,w is 0.592, which indicates that interaction will be strong (Witcher and McAvoy, 1977). A much more accurate expression for XL,w can be derived using a version of the Smoker (1938) equation, discussed by Strangio and Treybal (1974). The derivation, which 0 1979 American Chemical Society
182
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
Table I. Values of h ~w ,( q = 1 ) Considered by McAvoy (1977)‘ exact eq 3 eq 5 eq 1 3 01 XF 6 0.614 0.760 0.592 0.676 0.825 0.672 0.573 0.724 0.578 0.549 0.734 0.552 0.677 0.795 0.655 0.716 0.845 0.709 0.673 0.795 0.686 0.624 0.785 0.634
0.760 0.898 0.778 0.760 0.898 0.778 0.500 0.721 0.500 0.500 0.721 0.500 0.760 0.898 0.778 0.760 0.898 0.778 0.500 0.721 0.500 0.500 0.721 0.500
0.646 0.807 0.623 0.686 0.836 0.674 0.572 0.760 0.581 0.548 0.743 0.553 0.716 0.850 0.698 0.727 0.860 0.716 0.672 0.828 0.697 0.624 0.797 0.640
0.611 0.797 0.611 0.667 0.833 0.667 0.500 0.714 0.500 0.500 0.714 0.500 0.611 0.797 0.611 0.667 0.833 0.667 0.500 0.714 0.500 0.500 0.714 0.500
1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
0.25 0.25 0.25 0.25 0.25 0.25 0.50 0.50 0.50 0.50 0.50 0.50 0.25 0.25 0.25 0.25 0.25 0.25 0.50 0.50 0.50 0.50 0.50 0.50
1.3 1.3 1.3 1.75 1.75 1.75 1.3 1.3 1.3 1.75 1.75 1.75 1.3 1.3 1.3 1.75 1.75 1.75 1.3 1.3 1.3 1.75 1.75 1.75
For columns with high reflux ratios, eq 6 and 7 show that
0.98- 0.02 0.98 - 0.05 0.95- 0.05 0.98- 0.02 0.98- 0.05 0.95- 0.05 0.98- 0.02 0.98- 0.05 0.95- 0.05 0.98- 0.02 0.98- 0.05 0.95- 0.05 0.98- 0.02 0.98- 0.05 0.95- 0.05 0.98- 0.02 0.98- 0.05 0.95- 0.05 0.98- 0.02 0.98 - 0.05 0.95- 0.05 0.98- 0.02 0.98 - 0.05 0.95- 0.05
= 0.685; average error, eq 3 = a Average exact h L 0.0913 or 13.3% of 6.685; average error, eq 5 = 0.0163 or 2.4% of 0.685; average error, eq 13 = 0.0561 or 8.2% of 0.685.
-
-
is somewhat lengthy and is given in the Appendix, assumes that high purity columns, where XD 1.0 and xw 0, are considered. The resulting expression for XL,w is 1 (5) XL,W (xF)(1 - X D ) ( ~ D - k l R ) ( X i ) ( k l S - Xi) 1+ (1 - x F ) ( 1 - x i ) ( x i - k l R ) ( X W ) ( k l S - x W ) where 1
(7) X F / ( ~-
4) - ~
D / ( +R 1)
-
(8)
xi= Equation 5 is very accurate. In Table I values of X L , w calculated from eq 5 are shown. The agreement between eq 5 and the exact results is excellent. The average absolute value of the error produced by eq 5 is 0.0163 or 2.4%. Similar accuracy has been found for the same 24 cases for vapor feeds ( q = 0). One of the important uses of an analytical expression for XL,w is to predict trends as such parameters as product split, a , xF, and R change. McAvoy (1977) has discussed such trends for the exact values of XL,w. Equation 5 correctly shows: that as cy increases interaction decreases; that symmetric product splits give rise to more interaction than asymmetric splits; and that XF’S close to 0.5 give rise to more interaction than XF’S different from 0.5. The effect of increasing R is discussed below. Equation 3 also correctly predicts the trends for product symmetry and xF. Equation 5 is as simple to use as eq 3. Equation 5 can be used to analyze interaction either in an existing column or one at the design stage. For such use all that is required is knowledge of a column’s compositions, cy, and R.
(9)
klR = 0 lim R m k1s = 1.0 lim R m
XD - X W
-
Substitution of eq 9 and 10 into eq 5 gives eq 3, Shinskey’s expression for XL,w Since Shinskey’s expression for XL,w is a limiting form of eq 5, it is most accurate for columns with large reflux ratios and high purities. As can be seen in Table I, eq 3 does become more accurate as 6 and therefore R increases. It can be noted that this alternate derivation of eq 3 does not rely on any empirical postulates. One of the interesting points about interaction in material balance control is that increasing R sometimes results in more interaction and sometimes less. For the first entry in Table I, XL,w = 0.614 for 6 = 1.3. As 6 m, XL,w approaches Shinskey’s value of 0.760. Thus increasing R decreases interaction in this case and for all cases where X F = 0.25. For all cases where X F = 0.50, increasing R results in more interaction as the results in Table I show. Relative Gain for Material Balance Variables-D,
-
v
The development of an expression for XD,p exactly parallels that for XL,w If one carries out the derivation the result is 1 (11) hD,V (1- x F ) ( 1 - x i ) ( x i - k l R ) ( X W ) ( k l S - X W ) 1+ ( x F ) ( 1 - x D ) ( x D - k l R ) ( x i ) ( k l S - xi) The accuracy of eq 11 has been found to be equal to that of eq 5 for both liquid and vapor feeds.
Properties of the Relative Gain for Material Balance Control Equations 5 and 11 exhibit two properties which Shinskey (1977a) has noted for his expressions for the RGA. As discussed in the Appendix, all of the terms in the denominators of both eq 5 and 11 are positive. Thus, XL,w and X p,, will always be positive fractions. This result is important since the sensitivity of a decoupler to errors depends upon the magnitude of, , , ,X, (Shinskey, 1977b; McAvoy, 1979). Systems with fractional Xml,,,’s are relatively insensitive to decoupler errors. A second property of eq 5 and 11 can be gotten by adding the two equations. Noting that the right-hand sides sum to 1.0, one obtains XL,W
+ XD,V = 1.0
(12)
Equation 1 2 is highly accurate as will be shown later on. If ,a, , ,X, is less than 0.5, the pairing XD - ml and xw - m 2 cannot be used since it will be unstable (Shinskey, 1967). Since both XL,w and XD,v are positive fractions, eq 12 shows that one material balance pairing will always be stable will be 10.5. since its,,,,X, Material Balance Control of Columns Carrying Out Difficult Separations (a 1.0). For columns where cy 1.0, eq 5 and 11 simplify considerably. As shown in the Appendix, XL,w for such columns is
-
-
-
Equation 12 is also valid for columns where cy 1.0. Values of hL,Wcalculated from eq 13 are shown in Table I. The average absolute error produced by eq 13 is 0.0561
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
Table 11. Values of by McAvoy (1977)“
183
h ~ ,(vq = 1)Considered
exact
eq14
eq 1 6
cy
XF
6
Xn - Xw
21.4 15.8 12.1 58.5 34.0 26.5 17.9 11.6 9.36 46.7 27.8 21.3 7.48 5.55 4.46 16.2 10.4 8.71 6.20 4.12 3.54 14.3 8.50 7.18
17.0 7.92 7.81 22.2 10.4 10.2 19.2 11.8 9.43 24.6 15.1 12.0 6.78 3.16 3.19 8.52 3.98 3.97 8.72 5.34 4.41 10.5 6.42 5.21
30.8 20.0 15.3 56.9 34.2 27.8 25.6 18.2 10.6 49.8 33.8 21.1 11.9 7.78 6.05 20.0 12.2 9.90 9.78 7.07 4.05 17.2 11.9 7.10
1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
0.25 0.25 0.25 0.25 0.25 0.25 0.50 0.50 0.50 0.50 0.50 0.50 0.25 0.25 0.25 0.25 0.25 0.25 0.50 0.50 0.50 0.50 0.50 0.50
1.3 1.3 1.3 1.75 1.75 1.75 1.3 1.3 1.3 1.75 1.75 1.75 1.3 1.3 1.3 1.75 1.75 1.75 1.3 1.3 1.3 1.75 1.75 1.75
0.98- 0.02 0.98- 0.05 0.95 - 0.05 0.98- 0.02 0.98 - 0.05 0.95- 0.05 0.98- 0.02 0.98- 0.05 0.95- 0.05 0.98- 0.02 0.98- 0.05 0.95- 0.05 0.98- 0.02 0.98- 0.05 0 . 9 5 - 0.05 0.98- 0.02 0.98 - 0.05 0.95- 0.05 0.98- 0.02 0.98- 0.05 0.95- 0.05 0.98- 0.02 0.98- 0.05 0.95- 0.05
Equation 15 closely approximates the exact Smoker (1938) equation and it is more accurate than Shinskey’s (1977a) expression for separation (Jafarey et al., 1978). For the 24 columns shown in Tables I and 11, eq 15 gives theoretical tray predictions within an accuracy of two trays in 22 of the cases (an average error of 5 % ) and in the remaining two cases it is off by three trays (less than 9 % error). For theoretical tray predictions Shinskey’s (1977a) expression had an average error of 15.3% for the 24 columns. Similar accuracies result when eq 15 is used for vapor feeds. Equation 15 takes into account feed quality and composition, and it is particularly accurate for high-purity columns. Since the derivation of XL,p is somewhat lengthy it is given in the Appendix. The resulting expression is
a Average exact h L , =~ 16.7; average error, eq 1 4 = 7.36 or 44.1% of 16.7; average error, eq 1 6 = 3.05 or 18.3% of 16.7.
or 8.2% for the 24 cases considered. For the 12 columns with CY = 1.5, the average absolute error is 0.0296 or 4.3%. Similar accuracy has been found for vapor feeds. Thus, it should be possible to use eq 12 and 13 to accurately predict XL,w and XD,p values for columns where CY I1.5. Relative Gain for Conventional Variables-L, V Shinskey (1977) has presented the following expression for XL,+pwhich can be derived from eq 4 and a column’s material balance 1+ XL,V
b(xD
-
XW)*
x ~ ( -1 X D ) ( X F
= 1+
(W)(XW)(l
-
-
xw)
XW)
(14)
( D ) ( x D ) ( 1 - XD) (If desired, Wand D can be eliminated in favor of X F using eq A-25 and A-26 in the Appendix.) In Table I1 values of XL,p for the same 24 cases treated in Table I are shown together with the exact values. The average absolute value of the error produced by eq 14 is 7.36, which corresponds to 44.1% of the average exact XL,p. Thus eq 14 is considerably less accurate than eq 3 and unlike eq 3, eq 14 is particularly poor for columns with large reflux ratios. The chief problem which arises in trying to develop expressions for XL,p is that its denominator, a x D / a L l X w is small. Although reasonably accurate estimates can be gotten for axD/aLlXw the errors associated with these estimates are greatly magnified when one takes the inverse of the expressions. The approach which was taken for determining XD,pand X L , ~based on Strangio and Treybal’s (1974) equations was attempted for XL,p. However, only a slight improvement in accuracy was achieved over eq 14 (36.9% error vs. 44.1% error). In addition, the resulting expressions for XL,p were complicated since not as many terms could be neglected as was possible for hD,p and XL,w. In order to develop more accurate expressions for XL,p an alternate approach was taken. This approach was based on using a new approximation for separation which has been recently published (Jafarey e t al., 1979).
where g is given by
( R + q)(BRxF + XF + Q ) - (R + 1)(RXF + 9 ) (17) (RXF+ q)((R + ~ ) ( R x + F 4 ) - (R + 4 ) ) (In eq 16, as in eq 14, W and D can be eliminated in
g=
favor of xF using eq A-25 and A-26 in the Appendix.) Values of XL,p calculated from eq 16 are given in Table 11. The average absolute value of the error produced by eq 16 is 3.05 or 18.3%. For liquid feeds eq 16 gives high estimates for XL,p in 21 of the cases in Table 11. For the same cases, but for a vapor feed, eq 16 gives low estimates for 19 of the 24 columns (16.7% average error). Equation 16 requires the same information for its use as eq 14 does. As can be seen in Table 11, eq 16 is much better than eq 14 in predicting high values of hL,p (see entries 4 and 10). As discussed by McAvoy (1979), the columns which will exhibit the highest sensitivity to decoupler inaccuracies are those with the highest values of .,,,,X, Thus, for use in predicting decoupler sensitivities eq 16 is superior to eq 14. McAvoy (1977) has discussed the trends in the exact XL,p’s. Equation 16 correctly predicts that XL,p is a weak function of X F and that an X F = 0.5 produces less interaction than an X F = 0.25. Equation 16 also predicts that total product purity has more effect on interaction than the symmetry of the product split. Thus, for a fixed a , xF, and 6 a 0.98 - 0.02 split has the highest XL,p and a 95 - 0.05 split the lowest XL,p and a 0.98 - 0.05 split is in between. Lastly, eq 16 correctly predicts that higher values of CY will produce less interaction than smaller valaues of a.
Equation 14 also predicts some of the trends in XL,p correctly. One of the chief differences between eq 14 and 16 concerns their ability to predict the effect of R on XL,p. In eq 14 b is proportional to R + 1. Thus, eq 14 predicts a linear variation of XL,p with R. In order to gain more insight into how R enters into eq 16, an approximation to eq 16 will be considered. Of the four terms which result when the numerator in eq 16 is multiplied out, the dominant one by far is (xD- xW)/D. Using only this term,
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Ind. Eng. Chern. Fundarn., Vol. 18, No. 2, 1979
eq 16 can be approximated for q = 1 as 2(xD - XW)(R)(RXF+ 1) (1 - XD)(NTXD) AL,P
=
(18)
merically (McAvoy, 1977). Equation 26 can be used to check the consistency of numerically generated Aml,mz’~thus providing a check on the numerical schemes used. Equation 26 can also be used to evaluate the accuracy of the approximation given by eq 12. If t is the error associated with eq 12, then AL,W
A similar approximation can be gotten for q = 0. As can be seen in eq 18, AL,p is predicted to be a quadratic function of R. For columns where R is large, doubling R will result in a quadrupling of AL,p and therefore the extent of interaction. The numerical results given in Table I1 show that a quadratic rather than linear variation of AL,p with R is more correct. A quadratic increase in AL,p is also predicted and observed for columns with vapor feeds. A second point about eq 18 concerns the limit of A L , p as X D 1.0 and xw 0. If these limits are approached and the ratio x w / ( l - xD) remains fixed, then because NT is only a logarithmic function of xw and (1 - xD) the numerator in eq 18 becomes infinite because of the presence of the 1 - X D term. Thus, very high purity columns will show the severest interaction when L and 7 are manipulated. It can be noted here that A,, ’s for material balance control, derived using eq 15, are less accurate than even eq 3. Consistency Relationship for Material Balance and Conventional Am,,mz’~ A relationship exists between A L , w , AD,p, and AL,p and therefore these three Aml,mz’~are not independent. If xD and xw are taken as functions of L and V , then -+
-
Shinskey (1977b) has shown that from eq 19 and 20 AL,p can be calculated as 1 (21) AL,P = a12a21 1-allas2 If dL is replaced in eq 19 and 20 with d V - dD one obtains dXD = -a11 dD + (all + a1z)dV (22) dxw = -a21 dD (a22 + az1)dV (23) From eq 22 and 23, AD,p is given as (Shinskey, 197713)
+
Similarly d P can be replaced in eq 19 and 20 with dL d W and AL,w can be determined as
The aij’s can be eliminated from eq 21, 24, and 25 to give the expression
Equation 26 is valid in general. Since A,, involves the ratio of two partial derivatives, considerable errors can arise when it is calculated nu-
+ XD,V = 1 + t
(27)
If eq 27 is substituted into eq 26, t can be determined as AL,W(l t =
-
XL,W)
(AL,P - AL,W)
(28)
From the exact results given in Tables I and 11, t can be calculated for the 24 columns considered using eq 28. The average value of t is 0.0235 and maximum value is 0.0755 (entry 21). Thus, on average the approximation given by eq 12 is accurate to within 2.35% for the 24 columns treated in Tables I and 11. The reason that t is so small is that AL,p in the denominator of eq 28 is large. Also since X L , w is a fraction, the maximum value of the numerator of eq 28 is 0.25. Both of the facts result in being small. Multicomponent Columns This paper has considered relative gain relationships for binary columns. However, the results presented may also prove useful for multicomponent columns. Strangio and Treybal (1974), Hengstebeck (1969), and Geddes (1959), have presented methods of approximating a multicomponent column as a pseudo-binary column. In addition, Shinskey (1977a) has presented a method of calculating multicomponent RGA’s which is equivalent to Geddes’ (1959) method. Equation 15 has been shown to give excellent results in terms of predicting column operability and control for a nine-component de-ethanizer (Douglas et al., 1979) using Hengstebeck’s (1969) method. While excellent design results are gotten using a pseudo-binary approach, it is not known whether the same holds true for the calculation of A,, values. It is planned to compare the results given in this paper with exact numerical results for a variety of multicomponent columns using the pseudo-binary approaches referenced above. Equation 26 is valid for multicomponent as well as binary columns. From a degrees of freedom argument (Robinson and Gilliland, 1950) it can be shown that the composition of the light key in the distillate XDl, and the heavy key in the bottoms, XWh, are functions of L and 7 only. Thus, X D in eq 19 can be replaced with ”91 and xw in eq 20 can be replaced with XWh. The derivation of eq 26 for multicomponent columns is then identical with that for binary columns. For multicomponent columns, Shinskey’s (1977a) expression for AD,pis a positive fraction. It is also probable that XL,p will be large for multicomponent columns. Given these two facts then, t in eq 28 will be small and the approximation given by eq 12 will also be valid for multicomponent columns. Conclusions New, accurate approximate expressions for the relative gain for dual composition control of binary columns have been presented. The expressions apply to columns of arbitrary feed quality, and they can be derived from the rigorous Smoker equation. When compared to exact results for 24 columns, the new expressions show an average error of 2.4% for material balance control (L and W manipulated) and 18.3% for conventional control ( L and V manipulated). For the same 24 columns, Shinskey’s empirical RGA expressions show average errors of 13.3% (material balance control) and 44.1 70 (conventional
Ind. Eng. Chem. Fundam., Vol. 18, No. 2 , 1979
185
These roots are shown schematically in Figure 1. The number of ideal trays in each part of the column is then given by
I.
Y
(rectifying section)
[
In ns = -
4)
In 0
X
Figure 1. McCabe-Thiele diagram.
control). The new expressions account for all of the trends in the exact relative gains when relative volatility, feed composition, product split, and reflux ratio are varied. It has been shown that the relative gains for the two versions of material balance control ( L and W or D and manipulated) and the relative gain for conventional control are related. This relationship can be used to check the accuracy of the approximation that the relative gains for the two versions of material balance control sum to 1.0. Lastly, a brief discussion of the extension of the results in this paper to multicomponent columns has been given. Acknowledgment This work was supported by the National Science Foundation under Grant ENG-76-17382. Appendix RGA f o r Material Balance Variables. In order to derive an equation for XL,w, the numerator, a x w / a m L will be considered first. Following Strangio and Treybal, a slope and intercept is defined for each section of the column
v
(stripping section) where xl, shown in Figure 1, is given by the intersection of the q line and the operating lines X F / ( ~- 9 ) - ~ D / ( R + 1) x, = (A-9) q R R + l 1-q
+-
k,s
N
If eq A-10 to A-13 are substituted into eq A-7 and A-8 and xi is eliminated between the resulting equations, the following approximate operating equation for the column is obtained
(1 - x D ) U k l R N
s=
L L +D
+ qF -
(1 - q ) F
(A-2)
where (A-1) and (A-2) represent the rectifying section and (A-3) and (A-4) the stripping section. Two roots, k l and kz,which result from the intersection of the operating line and equilibrium are calculated for each part of the column k, = -(s + b(a - 1) - a ) + b(a - 1) -a)' - 4bs(a - 1)
d(s
2s(a - 1)
(-4-5) b kls(a
-
1)
XD -
-xwk1s (C(xw -his) - XW)
(A-14)
)
(A-16)
where
(A-3) (A-4)
k, =
(A-11)
0.0
(A-12)
(a(1 - XD)
L
-
Substitution of eq A-10 and A-11 into eq A-5 and A-6 and approximating X F as D / F gives approximate values for klR and kLs as
(XD -
SXDD b=-
-
For high purity columns, where X D 1, and xw 0, it is clear from Figure 1that the following approximate values for k 2 R and k,s are obtained h2R 1.0 (A-10)
(A-6)
1 c = ( 1 + ( a - 1)kls
Equation A-14 can be differentiated with respect to W, holding L constant. For high purity columns, the contribution of the resulting terms involving aa/awL, ac/awL, aklR/aWIL, and ah,s/aWIL is much smaller than that of axD/amLand these four terms can be neglected. Carrying out the differentiation gives
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Ind. Eng. Chem. Fundarn., Vol. 18, No. 2 , 1979
It is possible to approximate a and c from eq A-7 and A-8 as
A-13 and taking the limit as cy of kIR,kls, and xi as k1s
(1 - ZD)(XD - k l R ) ( X i ) ( k l s -xi) (1 - xi)(xi - k l R ) ( X W ) ( h l S - X W )
1
---*
1+ xi
(A-19) Substituting eq A-18 and A-19 into eq A-17 gives
-
k1R
(A-18)
-
1gives the limiting values
xF/6
-
(XF -
(A-30) 1)/6
XF
(A-31) (A-32)
These limits are independent of q. Substitution of eq A-30 to A-32 into eq A-27 and approximating kls - x w with kls and XD - k l R with 1 - k l R gives XL,w as
(A-20)
An additional relationship between dxD/aW(L and anw/a WIL can be gotten by differentiating the column material balance X$ x ~ = Wx F F (A-21)
+
Differentiation of eq A-21 with respect to W, holding L constant, and noting that aD/a WIL = -1, gives waxw -aI L x D = x D - x W - -(A-22) aW D D awlL By eliminating axD/aWIL between eq A-20 and A-22, the numerator of XL,w can be calculated as
RGA for Conventional Control. To calculate XL,p, eq 15 is used together with eq A-21. To determine the numerator of XL,v,a~D/aLlp,the left-hand side of eq 15 is first differentiated with respect to R, holding V constant giving a LHS ,.+R
Iv=
~
axW -
(XD - XW)
/w
(A-23) ( D ) ( 1- XD)(XD - k l R ) ( X i ) ( h l S - xi) 1+ (w)(1 - x i ) ( x i - k l R ) ( x W ) ( k l S - X W ) To develop an expression for the denominator of Xk,w, eq A-21 can be differentiated with respect to W holding X D constant
.awl'
(A-24)
For high purity columns D and W can be approximated as D N xFF (A-25) N (1 - X F ) F (A-26) Dividing eq A-23 by A-24 and substituting eq A-25 and A-26 gives XL,w as
w
XL,W
1 (xF)(1 - X D ) ( X D - h l R ) ( X i ) ( k l S - Xi) 1+ (1- xF)(1 - xi)(xi - h l R ) ( X W ) ( h l S - X W ) (A-27)
By an exactly parallel derivation it can be shown that XD,pis given by eq 11. All of the individual terms in the denominators of both eq A-27 and 11 are the same. Clearly, xF, (1- xF), x i , (1-xi), (1- xD), and XU' are positive. An examination of Figure 1 shows that the remaining four terms, (XD - k l R ) , (kls - xi), ( x , - kid, and (k1s - XW) are also always positive. Thus XL,w and XD,v will always be positive fractions. Material Balance Control of Columns Carrying Out Difficult Separations ( a 1.0). For columns where a 1.0, eq A-27 simplifies considerably. For a column of arbitrary feed quality and high purity, the general expression for minimum reflux ratio is R , = 2q/{[(a - 1 ) x F + a(q - 1) - 41 + [ ( 4 + a(1 4 ) + (1- a ) X F ) ' + 4xF(cy - l)q]"'] (A-28)
-
-
From the definition of 6, R can be expressed as R = 6R, (A-29) Substitution of eq A-28 and A-29 into eq A-9, A-12, and
(A-34) where f(R,xF,q) =
(R
+ q ) ( 2 R x ~+ X F + 4 ) - (R + ~ ) ( R x +F 4 ) (R + ~)'(RxF+ 4)' (A-35)
The aR/aLJp can be determined by differentiating the definition of R, (R = LID), and noting that aD/aLIv = -1. Carrying out the differentiation gives (A-36) Differentiation of the right-hand side of eq 15 with respect to L, holding V constant gives a RHS -1 XD axw (1 - x w ) axD Iv = -___Iv + -IY aL xw2(1 - XD) aL xw(l - xD)2 aL (A-37) Carrying out the same differentiation on eq A-21 gives
Equation A-38 can be used to eliminate axw/aLlp in eq A-37. Further, eq 15 can be used to eliminate the terms in eq A-34 with NT in their exponents. Carrying out these eliminations and equating aLHS/aLlv with aRHS/aLIp gives
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
constant. Equations A-34 and A-35 again result but with
aR/aLIv replaced with aR/aLI,,. Differentiation of R with respect to L, holding xw constant gives aR 1 RaD (A-41) -IaL xw = - D - - - I D a L xw The aD/aLIxwcan be calculated by differentiating eq A-21 giving
Differentiating the right-hand side of eq 15 with respect to L holding xw constant gives
Equating aLHS/aLI,, with aRHS/aLIxwand substituting eq A-41 and A-42 gives axD/aLlXwas
187
k,S = root defined by eq 7 and A-13 L = reflux flow LHS = left hand side ml = variable manipulated to control xD ( L or D ) m2 = variable manipulated to control x w (W or V) n R = number of theoretical trays in rectifying section ns = number of theoretical trays in stripping section N T = total number of theoretical trays q = fraction of feed which is liquid (feed quality) R = reflux ratio R , = minimum reflux ratio RHS = right hand side s = slope, defined by eq A-1 and A-3 = vapor flow in rectifying section ( V + (1 - q)F) V = vapor flow in stripping section W = bottoms flow xD = distillate composition xD1 = light key composition in distillate XF = feed composition xi = defined by eq 8 and A-9 x w = bottoms composition XWh = heavy key composition in bottoms
Greek Letters a = relative volatility = constant in eq 4 6 = R/Rm t = error defined by eq 27 A,,, = relative gain element
Literature Cited Bristol, E. H., I€€€ Trans. Autom. Control, AC-11, 133 (1966). Douglas, J. M., Jafarey, A,, Seeman, R., Ind. Eng. Chem. Process Des. Dev., 18, 203 (1979). Geddes, R. L., AIChE J . , 4, 389-392 (1958). Hengstebeck, R. J., "Distillation Principles and Design Procedures", Chapter 7, Reinhold. New York, N.Y., 1961. Jafarey, A., Doughs, J. M., McAvoy, T. J., I d . Eng. Chem. ProcessDes. Dev., 18, 197 (1979). McAvoy, T. J., ISA Trans., 16 (4), 83-90 (1977) McAvoy, T. J., Ind. Eng. Chem. Fundam., in press (1979). Nisenfeid, A. E.,Stravinski, G.. Chem. Eng., 227, 227-236 (Sept 23, 1968). Robinson. C. S..Gillihnd. E. R.. "Elements of Fractional DD 214-219. . ~ . Distillation". . ~. . r r McGraw-Hill; New York, N.Y., 19507 Shinskey, F. G., "Process Control Systems", Chapter 7, McGraw-Hill, New Ywk, N.Y., 1967. Shinskey, F. G., "Distillation Control for Productivity and Energy conservation", Chapter IO, McGraw-Hili, New York, N.Y., 1977a. Shinskey, F. G., "The Stability of Interacting Control Loops With and Without Decoupling", Proc. IFAC Multivariable Technological Systems Conf. 4th International Symposium, U. of New Brunswick, 21-30 (July 4-8, 1977b). Smoker, E. H., Trans. AIChE, 34, 165 (1938). Strangio, V. A., Treybal, R. E., Ind. Erg. Chem. ProcessDes. Dev., 13,279-285 (1974). Witcher, M. F., McAvoy, T. J.. ISA Trans., 16 (3), 35-41 (1977). ~~
~
Nomenclature a = defined by eq A-15 aij = gains defined by eq 19 and 20 b = intercept defined by eq A-2 and A-4 c = defined by eq A-16 D = distillate flow f = function defined by eq A-35 F = feed flow g = function defined by eq 17 and A-40 kl, k 2 = roots defined by eq A-5 and A-6 k l R = root defined by eq 6 and A-12
-.
Received f o r review September 18, 1978 Accepted January 31, 1979
