Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979 (36) Beuther, H.. Ondrey, J. A . , Swift, H. E., U.S. Patent 3644550 (Feb 22. 1972). (37) Ondrey, J. A . , Swift, H. E., U.S. Patent 3644551 (Feb 22, 1972). (38) Ondrey, J. A., Swift, H. E.,U.S. Patent 3769237 (Oct 30, 1973). (39) Kaliberdo, L. M., Vaabel’, A . S., Suprin, R I . . Kinet. Katal., 17(6), 1612 (1976).
197
(40) Gruia, M., Nowak, S.S., Vieweg, H. G., Anders, K.. Feldhaus, R., Rev. Chim. (Bucharest),26(8), 635 (1975).
Keceiced f o r reuieu: June 12, 1978 Accepted December 1, 1978
ARTICLES
Short-Cut Techniques for Distillation Column Design and Control. 1. Column Design A. Jafarey, J. M. Douglas,” and T. J. McAvoy Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1003
An approximate analytical solution of Smoker’s equation can be used to develop simple models for predicting the design and control of distillation columns. The design equation
I-.>
In (I:DxD)(’ -N=
--
gives predictions that are usually within two or three trays of the exact solution, which exceeds the accuracy associated with plate efficiency calculations.
It is common practice to use short-cut design procedures for preliminary project evaluations or process designs, where the primary objective of the calculations is to get some idea of the sign and the magnitude of the potential profit (or loss). Since distillation is perhaps the most common type of unit operation encountered in these analyses, it is not surprising that a considerable effort has been directed toward developing short-cut procedures for predicting both the design and the control of these units. Most of these short-cut methods are empirical correlations that are simple to use and yet give essentially the same predictions as either the exact analytical solutions for binary systems or tray-by-tray calculations. On the other hand, the empirical expressions that are used for design are quite different than those that are used to predict column operability and control. In these papers a new short-cut procedure is developed based on an approximate analytical solution of Smoker’s equation. This approach has the advantage that the problems of column design, quick estimates of the optimum reflux ratio, and the determination of column operability and steady-state control can be approached using a consistent set of relationships, rather than disparate empirical correlations. The accuracy of the predictions for design purposes is not quite as good as a computer fit of Gilliland‘s graphical plot, but the predictions for operability and control are superior to existing results. Previous Work Gilliland’s empirical correlation is a widely used procedure, even in computer-aided design programs, for obtaining first estimates of the number of trays required 0019-7882/79/1118-0197$01.00/0
to accomplish a given separation in a distillation column. Also, Happel and Jordan (1975)used Gilliland’s correlation to show that the ratio of the optimum t o the minimum reflux ratio could be expected to fall within the range of 1.03 to 1.37, although their analysis required a tedious graphical differentiation of Gilliland’s plot. Eduljee (1975) noted that Gilliland’s graph is well represented by the equation
which is simpler to use for many applications, but it is not reliable near minimum reflux conditions. A similar empirical correlation for column design has been proposed by Westerberg (1976)
(3) These expressions were created by least-squares fitting the data from a small set of example problems and were intended only as a means to estimate quickly ball park figures, without requiring estimates of R , or N,. Once a column has been designed and built, then questions concerning the column operability and control become important. For example, it is common practice for operators to increase the reflux rate above the design value and thereby increase the purity of the overhead stream (we are assuming that the desired product goes 0 1979 American
Chemical Society
198
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979 F
overhead). With this approach, disturbances in the feed composition seldom cause the top product to fall below the product quality specification, but this advantage is balanced by the excess steam supply needed to overreflux the column. In order to solve problems of this type and to develop better control systems for columns, Shinskey (1976) developed an empirical correlation between the boilup ratio and the separation factor of a column
Nm
N
-
where
Ins - 0.286X if X < 1.825 Nlna = l / d 8 . 3 5 / X 2 + 1.163 if X > 1.825 (4)
p--pi+D
v x=-p . 3 2 F a1.68
Of course, these expressions could be used for column design, but Shinskey states that they are not intended to be accurate enough for this purpose. At the other extreme of sophistication, Smoker (1938) developed an exact analytical solution for binary distillation columns for the case of constant relative volatility. Perhaps the simplest way of understanding Smoker’s procedure is by analogy to Fenske’s analytical derivation for the minimum number of trays at total reflux N m = -In S (5) In 01 For Fenske’s problem, both the rectifying and the stripping lines correspond to the 45’ diagonal, so that tray-by-tray calculations on a McCabe-Thiele diagram can be carried out analytically, leading to the result above. For the case of finite reflux, Stoppel (1946) developed an analytical solution based on extending the operating and equilibrium lines outside the normal McCabe-Thiele diagram, and then creating new coordinate systems so that each of the operating lines become the diagonals of rectangles (see Figure 1) which makes the calculation procedure similar to Fenske’s solution for infinite reflux. These are the same coordinate transformations that were used earlier by Smoker (1938) in his analysis, and they require finding the intersections of the operating lines and the equilibrium line. For the rectifying section, the equations of interest are
R
XD
operating line: y = R + l X + K equilibrium line: y =
CYX
1
+ (a
-
1)x
(6)
McCABE THIELE DIAGRAM
I $0
ABCDA:ORDINARY EFGHE: MODIFIED I JKLI :MODIFIED EO :RECTIFYINQ W L I N E 1K:STRIPPING OF! LINE
Figure 1. Ordinary and modified versions of the McCabe-Thiele diagram for a column.
same difficulty, because each of them requires finding a solution of eq 8. It would be advantageous, a t least in terms of our basic understanding of distillation operations, to have a single relationship that could be used for preliminary studies of distillation column design, operability, and control, as well as estimates of the optimum reflux ratio, that would be as simple to apply as the empirical correlations. An expression of this type is developed below by starting with an approximate solution of Smoker’s equations. Approximate Analytical Solution of Smoker’s Equations For preliminary design calculations we are primarily interested in the most expensive columns, which often correspond to high-purity separations. Thus, if we assume that xD = 1.0 in eq 8, the roots are simple to find 1.0 R ( a - 1)’ For Smoker’s analysis we are interested in the first root. With this simplification, Smoker’s equation for the rectifying section reduces to the expression x=-
(7)
so that after eliminating y and rearranging, we find that the intersections are given by the roots of the quadratic equation
R +1 For column design problems, x, = xF, and since X D = 1.0 we can modify the result slightly so that is resembles Fenske’s equation with reflux correction terms in the numerator and denominator
Unfortunately, the roots of this expression are complicated, and therefore Smoker’s final set of equations are not normally written as explicit functions of the design variables. The other analytical solutions developed, using finite difference equations or other procedures (see, for example, Underwood, 1932; King, 1971, for a review of several procedures; Jenson and Jeffreys, 1963; Mickley et al., 1967; and Strangio and Treybal, 1974) suffer from this
A similar analysis for the stripping section where xw = 0.0, leads to the result
N s = (ln
Ind. Eng. Chern. Process Des. Dev., Vol. 18, No. 2, 1979
(z) X
where
Rs =
R ( X F- x,) xD
+ ~ ( X D- x,) -
(13)
xF
For the case of a saturated liquid feed where q = 1.0, for high-purity separations where XD = 1.0 and x, = 0, and for a design problem were x, = xF, we can modify eq 1 2 to give
Exactly the same results can be obtained from Underwood’s analysis (1932) simply by adding an expression for the top plate to the set of expressions following eq 35 in his paper, and then proceeding in the manner which he describes. Also, an identical result is obtained by solving the finite-difference equations fo? the high-purity case (see Jenson and Jeffreys, 1963; Mickley et al., 19571, since the transformation required to linearize the Riccati difference equation is exactly the same as that used by Smoker (1938). We see that as the reflux ratio becomes very large, eq 11 and 14 reduce to Fenske’s equation for total reflux. Similarly, for high-purity separations, where Underwood’s minimum reflux expression (1932) simplifies to
the results can be written as
so that the expressions predict that an infinite number of trays are required in both the rectifying and stripping sections at minimum reflux. Thus, the equations exhibit the correct behavior for a column a t the limiting reflux flows, and they resemble Fenske’s equations with reflux correction factors. We would expect that the term l / [ ( R / R , ) - 11 which appears in the numerator of the expressions for both N R and Ns should dominate the behavior of the solutions in the neighborhood of minimum reflux operation. Therefore, it should be fairly easy to use eq 16 and 17 to develop a
199
simple procedure for estimating the optimum reflux ratio. An analysis of this type is presented in the Appendix. It should also be recognized that the equations are written for the number of theoretical stages in each section of the column, which includes the reboiler (and may include a partial condenser). However, for preliminary designs we are primarily interested in the most expensive columns, which are those containing a large number of theoretical trays. For these cases there will be little error introduced by assuming that NT + 1 = NT, Le., that the total number of trays plus the reboiler is equal to the total number of trays. Thus, for the sake of simplicity, we will use this assumption, although it is possible to modify the analysis to account for the reboiler explicity. If we refer again to Figure 1, we see that the approximate solution forces the operating line to pass through the point x = y = 1.0, rather than intersecting the 45O diagonal a t x = xD, as shown. Hence, the upper end of the operating line (and the lower end of the stripping line) is pushed closer to the equilibrium line, and this additional pinching condition means that we should predict too many plates, Le., a conservative design. In order to test this expression we considered 24 examples with a saturated liquid feed and various combinations of the design parameters; a = 1.5 and 2.5, x F = 0.25 and 0.5, R / R m = 1.3 and 1.75, xD = 0.95 and 0.98, and x, = 0.02 and 0.05 (see Table I). The average of the absolute value of the error between the total number of trays predicted by eq 11 and 14 and the exact solutions of Smoker’s equations was 25.6% (see Table I), while the maximum error was 74.5%. Also, for every case the approximate solution was conservative, as expected. In contrast, to the results above, for the same cases with Eduljee’s (1975) version of Gilliland’s correlation the average error was 2.4% and the maximum error was 6.0%, with Westerberg’s correlation the average error was 16.6% and the maximum error was 43.370, and with Shinskey’s correlation the average error was 12.4% and the maximum error was 38.1%. Thus, we see that the approximate solution given by eq 11 and 14 is more complicated than the other empirical correlations and it does not give as good predictions because it is too conservative. The analysis is therefore continued further in the next section, where a simple, reliable equation for design is developed.
Simplified Approximate Solution for Total Theoretical Trays While the reflux correction terms in the numerators of eq 16 and 17 are very significant in the neighborhood of minimum reflux (see the Appendix), the denominator correction terms are more significant in the region of normal operation. These denominator correction terms are the absorption and stripping factors for the rectifying and stripping columns, respectively, that we would obtain if we replaced the equilibrium line by a straight-line approximation a t each end of the column. For expensive columns we expect most of the trays to be located near the ends, and therefore in order to simplify the approximate solution we merely drop the correction terms in the numerator. With this simplification we can write eq 11 and 14 as
and
200
Ind. Eng. Chem. Process Des. Dev., Vol.
18, No. 2, 1979
Table I. Comparison of Predictions for Total Theoretical Tray speci ficat ionsa no.
(cy, XF, XD, x w , R/Rnl)
Smoker
Gilliland (Eduljee)
Westerberg
Shinskey
eq 11 and 1 4
eq 22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
aaaaa aaaba aabba aaaab aaabb aabbb abaaa ababa abbba abaab ababb abbbb baaaa baaba babba baaab baabb babbb bbaaa bbaba bbbba bbaab bbabb bbbbb
35.27 29.73 26.60 28.33 24.22 21.35 36.47 32.52 28.25 29.28 25.88 22.41 16.73 14.24 12.77 13.39 11.47 10.16 17.05 15.36 13.30 13.72 12.21 10.60
35.09 30.91 26.78 28.03 24.67 21.34 36.11 31.82 27.65 28.76 25.32 21.97 16.86 14.92 13.03 13.30 11.74 10.23 18.05 15.97 14.10 14.21 12.55 11.06
32.85 28.57 28.36 32.85 28.57 28.36 35.66 32.57 30.85 35.66 32.57 30.85 16.17 14.06 13.96 16.17 14.06 13.96 17.56 16.03 15.19 17.56 16.03 15.19
42.40 41.06 33.49 31.96 29.49 24.72 36.31 31.66 26.98 30.07 26.26 22.49 15.16 14.18 11.77 12.88 11.79 9.95 12.95 11.31 9.70 11.89 10.25 8.83
42.60 36.89 36.71 31.87 27.53 25.62 44.07 40.28 41.71 33.23 29.88 28.14 20.07 17.64 17.84 15.17 13.19 12.44 21.65 20.05 23.21 16.10 14.63 14.31
32.64 28.67 25.43 28.16 24.73 21.74 33.05 29.03 26.37 28.41 24.95 22.29 14.89 13.08 11.63 13.01 11.43 10.08 15.18 13.33 12.30 13.20 11.59 10.51
Key: For a, cy = 1 . 5 ; xp R, = 1.25.
=
0.25; x n = 0.98, x,
= 0.02; R/R,=
1.3. For b, cy = 2.5; XF = 0 . 5 ; =~ 0.95; ~ xw = 0.05; R/
Then, if we assume that N R = N s = N / 2 , and multiply the two expressions together we find that
It is interesting to note that if we let R = 1.2R,, and use eq 15 to approximate Rm, then eq 23 becomes
or
However, the right-hand side of this expression only changes from 0.547 to 0.518 as Q increases from 1.3 to 5.0, so that we expect
In S
Nm N
- = 0.5
which provides a simple model for the column. We can also rearrange this expression to obtain 1nS --
Nlna
- -Nm = 1 - y * N
In (1 + l / R x f ) In a
(22)
Both of these expressions are simpler than most of the other empirical correlations, and when we test them with our 24 examples we find that the average error is only 5.3% and that the maximum error is 13.2%. After an examination of the results and observing that negative errors were obtained in almost every case, we found that we could improve the approximation by writing
1.05 In S Nm = 1.05 -= N In a N
1-
‘/2
In (1
+ l/RxF) In a
(23)
and with this expression the average error was 3.9% and the maximum error was 8.9%. In some cases eq 21 may give very poor predictions of the column size; i.e., when a is very small, the required reflux ratio is very large and the denominator may become negative. The difficulty arises because the correct solution for these cases is very sensitive to the difference between R and R , and because our assumption that most of the trays are located near the ends of the column is no longer valid. However, Gilliland’s correlation also may give misleading predictions for similar problems (see Rolles, 1966).
which is a common rule of thumb based on Gilliland’s correlation. In the development of the simplified design models above we assumed that q = 1.0 in eq 13. If we relax this assumption and proceed with the analysis, we find that In S (26) I lna 1N= (R
(
so that it is easy to account for the effect of feed quality.
Conclusions It is a relatively simple matter to derive an approximate solution for Smoker’s equations, which properly reduces to Fenske’s equation at total reflux and predicts an infinite number of trays for each section of the column a t minimum reflux conditions. The approximate solution is very sensitive to certain terms near the minimum reflux condition, and therefore a simplified expression can be developed to estimate the optimum reflux ratios. Alternately, the solution is most sensitive to other terms at normal reflux conditions, and an alternate simplification can be developed for this region. The simplified equation for estimating column designs normally agrees with Smoker’s exact analytical solution within *5%, or two or three trays. The accuracy is not quite as good as Gilliland’s correlation, but it seems to be
Ind. Eng. Chem.
better than most other empirical correlations. Also, the error associated with this prediction is probably smaller than the error involved in estimating plate efficiencies. Hence, the results provide simple, useful tools for estimating the optimum reflux ratios and column designs for preliminary design purposes. An even more significant advantage of the simplified column approximation is that it also provides a procedure for studying column operability and column control, as is shown in part 2 of this series. Appendix. Optimum Reflux Ratio A simplified analysis of the economics of distillation was developed by Colburn (1943) and is discussed in some detail by Happel and Jordan (1975). The cost per mole of distillate is given by
c,
ClNT
= -(1
EYG,
c2 + R ) + -(l
YGb
+ R ) + C3(1 + R )
Process Des. Dev., Vol. 18,
order-of-magnitude arguments we also assume that (A-7) is a constant and that [(RIR,) - 1 + (YXF] is a constant. With these assumptions
]
T o find the optimum reflux ratio, we let the total number of plates Nrr be the sum of the plates in the rectifying and stripping sections, and let dC,/dR = 0
Thus, eq A-3 can be written as
(A-10) According to Happel and Jordan (1975), a reasonable range for the ratio of cost factors, K 2 / K 1 ,is between 40 and 180, and we expect the number of theoretical trays in a column for many separations to be between 30 and 50. Thus, the left-hand side of eq A-10 will have a magnitude of around 100, and in order for eq A-10 to be satisfied, we see that the term ( R / R m- 1) in the denominator on the right-hand side of eq A-10 must be very small; i.e., R / R , must be close to unity. Since the solution to eq A-10 will be most sensitive to this value of RIR,, we write
R Conceptually, we can use the approximate analytical solutions given by eq 16 and 17 to describe the relationships between NT,NR,Ns, and R needed for eq 21, rather than the graphical procedure described by Happel and Jordan (1975). However, the results obtained in this way are too complex to be of much value, and therefore we want to look for additional simplifications of the column equations. We note that reflux correction in the denominator of eq 16
R/Rm - -_..-____ (-4-4) R + 1 (R/Rm) + (l/’Rm) for a case where R , = 1, will only vary between one-half and unity as R / R , varies between unity and infinity; the maximum variation for larger values of R , is even less. R
1
In order to obtain a quick estimate of the optimum reflux ratio, we assume that R / R m = 1.2, NR + Ns = 2 In S / l n N (see eq 5 and 2 5 ) , and that R, = l / ( a - l)xF (see eq 15) on the right-hand side of this expression. With these assumptions, we obtain
R Rm
- -
1=
~
1.2 +
((Y
-
1)x*
Kz Ins -+2K1 In (Y
XF
(A-5) - ~-I
Rm
which, for a reasonably well-balanced column, i.e., xF = 0.5, changes from infinity to unity as R / R mchanges from unity to infinity, we see that the denominator is relatively insensitive t o changes in R. Similarly, even in the range of R / R mfrom 1.05 to 1.2, the numerator correction terms are much more sensitive than the denominators, and, in fact, the only term which varies significantly is ( R / R , - 1). Therefore, to siniplify the analysis, we assume that
(A-6) is a constant and ( R / R m- x F ) is a constant. Using similar
[
1 In
1.2
+ (a
- 1)XF
1+
1
When we compare this change to the variation of the reflux correction term in the numerator n
(A-9)
(A-1)
where the first term is related to the capital cost of the column, the second term includes the capital cost of both the condenser and reboiler, and the third term includes the operating costs of both steam and cooling water. Obviously, we can rewrite this equation as c, = (K1N.I.+ KJ(1 + R ) (A-2)
-K_
No. 2, 1979 201
In (Y
+ 0.2
For a particular case, where we use Happel and Jordan’s (1975) values for the cost parameters, K1 = 4.18 x IO-* $/mol and K 2 = 2.204 x lo-’ $/mol, and we let cy = 2 , xF = 0.5, and XD = 1 - x, = 0.99, we find that
+ 0.5 52.7 + 26.6[
R 1.2 -~ 1=
1‘
1 + = 0.109 (A-13) 0.345 0.453 Rm Alternately, we can obtain a more consistent estimate by solving eq A-11 in an iterative fashion. Using the same design parameters as given above and starting with R / R m = 1.2 on the right-hand side, we get
R R .,..
- -
1=
52.7
1.2 + 0.5 + 16.9 + 14.7
(0.345 + 0.4353
= 0.105
(A-14) Now, if we use this value to recalculate the right-hand side of eq A-11, we obtain
202
-R_ Rm
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
1=
+
(
1.105 0.5 1 + 52.7 + 19.8 + 16.5 0.320 0.422
2) = 0.099 (A-15)
We see that none of the values on the right-hand side of the equation has changed very much, and if we iterate once more, we obtain essentially the same result. Thus, we have established a simple procedure for calculating the optimum reflux ratio. We could also use our simplified analysis to show that the optimum value of the total cost is fairly flat above the minimum cost point (see Peters and Timmerhaus, 1968) so t h a t it is normally desirable to operate with values of R / R m 20 or 25% greater than the actual minimum. With this procedure we can add flexibility to the design without adding very much cost to the system. I t is interesting to note that the magnitude of the value of [(RIR,) - 11 is essentially fixed by the ratio of K 2 / K l , which is related to the ratio of the annual operating costs to the annual capital costs. This ratio is quite large because it contains a factor of 8400 h/year of operation in the numerator. Hence, the optimum reflux ratio depends more on the relationship between capital and operating costs than it does on the characteristics of a particular distillation operation, which is the reason for the existence of the rule of thumb that R / R , ;= 1.2. A similar result for gas absorbers has also been published (Douglas, 1977). Nomenclature C,, Cs,C3 = cost factors E = overall plate efficiency F = feed rate f = fractional recovery of key components G,, G b = vapor capacity factors of column, and the condenser and reboiler K1, K 2 = cost factors N = NT = number of theoretical trays N , = number of theoretical trays at total reflux NR = number of theoretical plates in rectifying section N s = number of theoretical plates in stripping section R = reflux ratio R, = minimum reflux ratio Rs = see eq 13 S = separation factor = XD (1 - x w ) / ( l - x&, V = boilup rate
X = see eq 4 xf = liquid composition XD = distillate composition XF = feed composition x, = composition at bottom of rectifying section or top of stripping section x, = bottom composition Y = h/year of operation Y = vapor composition Greek Letters a = relative volatility &, &, = see eq A-6 and A-7 Literature Cited Boiies, W. L., "Economic Evaluation of Poly-Grade Propylene Production", Washington University Design Case Study No. 1, B. D. Smith, Ed., Washington University, St. Louis, Mo., 1966. Colburn, A. P., "Division of Chemical Engineering Lecture Notes," University of Delaware, Newark, Del., 1943. Coiburn, A. P., Ind. Eng. Cbern., 33, 459 (1941). Douglas, J. M., Ind. Eng. Cbern. fundarn., 16, 131 (1977). Eduljee, H. E., Hydrocarbon Process., 120 (Sept 1975). Fenske, M. R., Ind. Eng. Chern., 24, 482 (1932). Gilliiand, E. R., Ind. Eng. Cbern., 32, 1220 (1940). Happel, J., Jordan, D. G., "Chemical Process Economics", 2nd ed, p 384, Marcel Dekker, New York, N.Y., 1975. Jenson, V. G., Jeffreys. G. V., "Mathematical Methods in Chemical Engineering", p 336, Academic Press, New York N.Y., 1963. King, C. J., "Separation Processes", Chapter 8, McGraw-Hill, New York, N.Y., 1971. Mickley, H. S., Sherwocd, T. K., Reed, C. E., "Applied Mathematics in Chemical Engineering", pp 324, 326, McGraw-Hill, New York, N.Y., 1957. Peters, M. S., Timmerhaus, K. D., "Plant Design and Economics for Chemical Engineers", 2nd ed, p 312, McGraw-Hill, New York, N.Y., 1968. Robinson, C. S., Gilliland, E. R., "Elements of Fractional Distillation", 4th ed, p 347, McGraw-Hill, New York, N.Y., 1950. Shinskey, F. G., "Distillation Control", p 43, McGraw-Hill, New York, N.Y., 1977. Smith, B., "Design of Equilibrium Stage Processes", p 165, McGraw-Hill, New York, N.Y.. 1963. Smoker, E. H., Trans. AICbE, 34, 165 (1938). Stoppel, A . E., Ind. Eng. Cbern., 38, 1271 (1946). Strangio, V. A., Treybal, R. E., Ind. Eng. Chem. Process Des. Dev., 13, 279 119741. Underwood, A. J. V.. Trans. Inst. Cbem. Eng. (London), I O , 112 (1932). Westerberg, A. W., personal communication, Carnegie-Mellon University, Pittsburgh, Pa., 1976.
Received for reuieu: July 22, 1976 Accepted November 17, 1978
The authors are grateful to the National Science Foundation for partial support of this work under Grants ENG 76-21923 and ENG '76-17382.
