ARTHUR ROSE, ROBERT
F.
SWEENY, and VERLE
N.
SCHRODT
The Pennsylvania State University, University Park, Pa., and Applied Science Laboratories, Inc., State College, Pa.
Continuous Distillation Calculations by Relaxation Method This is an entirely
new method for calculating product compositions in multicomponent continuous distillation, when feed conditions, flow rates, plate efficiency, number of plates, and reflux ratio are established
THE relaxation method has been developed for calculating the product compositions in multicomponent continuous distillation, when feed conditions, flow rates of product streams, plate efficiency, number of plates, and reflux ratio are established. By obtaining several solutions for various numbers of plates and various reflux ratios, optimum operating conditions can be chosen with certainty for a particular set of product purities and yields. Although based on material balance and phase equilibrium relations and plate calculations, the method is entirely different from any other. Each choice of operating conditions leads directly to an answer in terms of product compositions, regardless of the complexity of the problem, number of components, extent of nonideality, irregularity of thermal relations, or number of side streams. Thus successive solutions lead directly to the desired optimum solution. No simplifying assumptions or approximations are needed. Other methods often lead to no solution, unless a great deal of experience and judgment are applied in choosing trial values. The relaxation method is not universally preferable to conventional methods, but is particularly advantageous for complicated design problems involving multiple columns or components, nonideal equilibrium and thermal behavior, multiple feed or take-off streams, extremes of concentration, and automatic control applications (6). I t requires an automatic computer, but this need not be fast, nor large, unless many components are involved. Digital computers are indicated because analog computers are limited to problems involving relatively few plates and few components. The method can deal
with any type of nonideality for which data are available. I t is not subject to the difficulties in conventional methods when plate compositions change widely and in a complex manner with slight changes in end compositions. Some of these difficulties are particularly serious when automatic computers are used, because of the difficulty of preparing adequate instructions for automatic choice of successive trial values. In the relaxation method calculations are made for the gradual change in all the plate and product compositions that occur in a column from initial startup until steady state is reached. The equations previously used for batch distillation ( 5 ) with appreciable holdup are used. The calculations may be started with all plate compositions equal to the feed composition, but other starting compositions may be used. Use of the material balance equations once for each plate gives a new set of plate compositions slightly different from those at the start. Repetition of this kind of calculation gives one set after another of plate compositions, which gradually approach the steadystate compositions. When these are reached, no further composition changes occur upon repetition of the calculating cycle. The steady-state solution is independent of starting compositions and holdup, but if the real starting composition, holdup, and corresponding flow rates and time lags are used, the calculations describe the approach to steady state. Basic Equations
The basic equation is that for expressing the change in the moles of a component present in the holdup on a particular plate, during any brief interval during the distillation operation. This equation is, in general terms, when usual simplifying assumptions of distillation are applicable,
+
+
(VY(~-I).< Lxta+~),d- Vyn.6 - LXn.i)
Hxn(i+l) = H ~ n i
(1)
where n refers to the plate number, and i refers to a particular interval of time, and (i 1) refers to the next succeeding interval. Hrefers to the total moles 03 holdup on the plate; and V and L refer to the vapor and liquid flow rates.
+
The equation is merely a material balance stating that the quantity of the component present in the holdup on plate n during interval i 1 is equal to that during interval i, increased (or decreased) by the net quantity added and removed as the result of the flows of the two vapor and two liquid streams to and from the plate. The basic equation can be immediately solved to give composition x , ( ~ + ~ )in terms of the composition and flows for interval i. This working equation is
+
Xfl(i+l)
=
Xni
+ 1 (VY(n.-,),i +
Lx(n+
-
Vya,i
- Lxn.i)
(2)
An equation of this kind can be written for each plate and each component, and similar but slightly different equations can be written for the top plate, the feed plate, and the still pot. The equation takes a more complex form when simplifying assumptions are not applicable (4). If the compositions and flows at any time,i are known, Equations 1 and 2 may be used to calculate compositions at 1, and the process repeated time i 2, and so to obtain values for time i on. The necessary values of vapor composition, y, are obtained from the corresponding value of liquid composition, x, by use of the applicable vaporliquid equilibrium relation. This same general procedure has Keen used in predicting the course of composition changes and effect of operating variables in batch distillation with appreciable holdup ( 5 ) ,and more recently in studying the effects of column upsets in continuous distillation and related control problems (6). Use of the procedure for continuous distillation design calculations is simplified by the fact that the desired steady-state compositions are independent of the starting compositions, and of the path toward steady state. Thus any convenient set of starting compositions may be used, without concern as to the path followed.
+
+
Basic Calculation Procedure for Continuous Distillation Design
I t is most convenient to start the calculations using feed composition values for all the plate liquid compositions,
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VOL. 50, NO. 5
M A Y 1958
737
These are taken as the compositions for time zero. The first step is the use of the basic equation for the top plate to calculate the composition of the liquid from this plate a t the end of the first time interval. The second step is to use the basic plate equation to calculate the composition of the second plate from the top, a t the end of the first time interval. I n the same manner the compositions at the end of the first time interval are calculated for each of the other plates and the still pot or reboiler. Alternatively the calculations can begin at the still pot and proceed upward. In fact, the calculations for the various plates (at any one interval) may be done in any desired or random order, as the calculation for each plate is independent of that for all other plates. When all the compositions for time interval one have been obtained, the procedure is reported over and over to obtain compositions for time intervals two, three, and so on. I n general, most of the compositions change rapidly at first and then gradually approach steady-state values. When the latter are reached, the compositions no longer change from one interval to the next, and these compositions represent the desired steady-state solution to the particular continuous distillation problem under consideration.
flow rates V and L should be one fifth to one tenth of the plate holdup, H, to avoid distortions and instability in the calculations. Arithmetical errors must be eliminated by making an overall material balance of each component after each new set of compositions is obtained. I t is also desirable to check first and secondary differences in the compositions for successive intervals, as appreciable changes in these are indicative of arithmetic errors. Simplifying Assumptions
The procedure as explained involves some of the usual simplifying assumptions of distillation calculations, but these are not inherent in the method. One advantage is the ease with which these assumptions can be eliminated to make the method more rigorous, when desirable. Constant molal overflow Perfect plate efficiency Negligible vapor holdup Composition of liquid on a plate identical with that of liquid leaving the plate The methods of modifying the calculations to take these complexities into account have been described in detail ( 5 ) . Requirements for Calculation
Arithmetical Precautions
The calculation procedure is purely arithmetical, but requires special precautions. The time interval must be sufficiently small to avoid distortions that result when the quantities flowing during an interval are large compared with the holdup on a plate. In general,
To carry out the calculation procedure for a specific case, the following quantities must be known, specified, or fixed by choice of a trial value: Total number of theoretical plates, reflux ratio, and relative quantities of overhead and bottoms. Location of feed plate (or feed plates
The curves show how the mole per cent benzene on various plates would change from initial startup until the column compositions teveled off when steady state was reached. The steady-state compositions are the desired solution to the continuous distillation problem
\ Bottoms 20
40
NUMBER
738
Application to BenzeneToluene-Xylene Distillation
The application of new method may be illustrated by checking the solution to the benzene-toluene-xylene distillation described by Robinson and Gilliland ( 3 ) . This involves a 16-theoretical plate fractionating column with a reboiler which also acts as a theoretical plate. The column is to be used to obtain a purified benzene overhead from a feed containing 60 mole % ' benzene, 30 mole % toluene, and 10 mole % xylene. The feed, which is a liquid at its boiling point, enters the column on the ninth plate from the bottom. The reflux ratio is set at 2 to 1, and 60.17, of the feed is to be taken overhead. The moles of feed per interval are arbitrarily fixed as 10. The corresponding overhead rate is then 6.01 moles with 3.99 moles as the bottom rate. Thus, V , is 18.03 moles, L, is 12.02 moles. V , is 18.03 moles, and L, is 22.2 moles. The holdup on each plate and in the reboiler is taken as 50 moles. The calculation is started with liquid of feed composition on each plate and in the reboiler. All the above data were entered into a digital computer, together with a program for repeated calculation according to Equation 2. The curves at the left show the variation of typical plate compositions from the initial interval to steady state. At steady state the top composition is 99.5% benzene, 0.5% toluene, and no xylene. and the bottom composition is 0.5% benzene, 74.4%> toluene, and 25.1% xylene. Robinson and Gilliland (3) obtained the same result, using the familiar Lewis and ,Matheson method. General Use of Relaxation Method
I 10o[
' 0
in the case of multiple feeds) and any side product streams. All flow rates (vapor and liquid in the column, overhead, bottoms, and feed and side streams). Absolute quantities are not necessary, but correct relative values must be used, and the feed rate per interval must be one fifth to one tenth of the holdup per plate. Holdup on each plate and in reboiler. The final steady-state compositions are independent of the choice. but arithmetic relations require that the holdup be at least five times the feed rate per interval. Initial composition of liquid on each plate and in the reboiler. The final steady-state compositions are independent of the choice of initial compositions. I t is convenient to use feed composition as the initial plate compositions.
60
80
100
120
140
OF INTERVALS
INDUSTRIAL AND ENGINEERING CHEMISTRY
If the Robinson and Gilliland solution to the benzene-toluene-xylene problem was not available, and it was desired to solve the general problem by the relaxation method described, the number
MACHINE COMPUTATION IN PETROLEUM RESEARCH of plates and the feed plate location would not be known. I t would therefore be necessary to choose trial values for these items and carry out a series of trial calculations. The first might be for ten plates with feed introduced on plate 5 from the bottom. This would give inadequate separation. Twenty plates would give better separation than that specified. Fifteen plates with feed on plate 8 would not quite achieve the desired separation. The relaxation method does not avoid repeated trials for a general distillation design problem. However, each trial does give the exact answer for distillation with the chosen conditions. Repeated trials give an assembly of correct solutions for various conditions of operation and a true over-all picture and understanding of the effect of variation in conditions on the results. In using conventional methods, such as the LewisMatheson, all the trials except the last give no answer at all, but only a discrepancy in material balances that aids in choosing conditions for the next trial. The successive trials thus provide but little over-all information about the problem. The relaxation method is especially useful in more complicated problems where the conventional method encounters serious difficulties in that selection, of trial starting values is so complex that it is difficult or impossible to find an answer even after many trials. This situation is most likely to arise in problems whose solution is most worth while-Le., for cases of operation near minimum reflux, and where very small changes in composition at one end of a column are associated with very large differences at the other end. Such problems are easily solvable by the relaxation method, but difficult or impossible to solve by conventional methods. This is particularly true of nonhydrocarbon mixtures.
a - ~ - ~ - ~ - ~ - ~ - aBOTTOMS ~ a , ~ ~ oPRODUCT ~o~a
/
OoZ4F
0.0
I
I
20
/a-
1
I
40
60
Computer Program
The relaxation type of calculation is not easy without any automatic computer. The sample benzene-toluene-xylene problem was programmed for solution on Pennstac, a medium-speed digital computer similar to the IBM 650. The simplified block diagram for the
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100
120
I40
160
INTERVAL NUMBER
The curves show how the mole fraction of component B of the ternary mixture on the various plates changes from its value of 0.01 2 at initial startup, to steadystate values that persist during continuous distillation TOP PRODUCT ,o-o---rcJ-cJ-o-o--o--o-o-n-u~~~o~cr
4
c
a
0 BO
4-cp-c-p-o-~-~-a-~-~-4-~-~ FEED P L A T E
H
*-*C-*-e-.-*-
5
P L A T E BELOW FEED P L A T E
i P
0
a
?
0 60
050-
\
Relaxation Solution for Side Stream Problem
The curves at the right give the solution to a problem in which a ternary mixture consisting of two close boiling less volatile components (B and C) are separated from a single more volatile component (A). Side streams enter and leave an upper plate, and feed is near the bottom.
I
80
o/o~-o-o-o-o -0-
I
BOTTOMS PRODUCT
I
INTERVAL+
I
I
I
NUMBERS
The mole fraction of component A of the ternary first decreases and then rises on the lower plates, as the progression occurs from initial startup to the steadystate compositions that are indicated b y the flat right-hand portions of the curves, Component C curves are not shown but would appear as approximately the inverse of those for component A program is given on page 740. This is applicable to any constant molal overflow three-component fractionation problem for any number of theoretical plates above and below the feed plate, and any reflux ratio. The various flow rates in the enriching and stripping section, the holdup on the plates and in the reboiler, th'k feed composition, and various constants, such as those found in the expressions for relative volatility, are entered as data. The machine does
the interval by interval calculations automatically and prints the answers as it proceeds after each interval, or after any specified number of intervals. The overhead and bottoms compositions are printed, but any other compositions in the column may also be printed as desired. l i m e Required for typical Solutions
The rate of approach toward steadyVOL. 50, NO. 5
*
MAY 1958
739
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E n t e r Initial Data I n t o C a l c u l a t e Storage
I_
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A Equilibrium Vapor S u b m u t i n e (EVSR) Calculate r e l a t i v e v o l a t i l -
Reset t r a f f i c c o n t r o l s an6 c y c l e e s c a p e s f o r next i n t e r v a l .
EVSR T r a f f i c Control. ReguLates t h e EVSR so it Vi
Tolerance Check
c e r t a i n p l a t e and s t o r e answers i n proper p l a c e .
See i f s t e a d y s t a t e has
been reached.
EVSR Cycle Escape.
P r i n t i n t e r v a l number and d e s i r e d
Determines when a l l t h e
I
1
I
Escape Vateerial Balance Subroutine
MBSR CYCIE Escape Determines when a l l t h e new l i q u i d compositions have been c a l c u l a t e d .
*
(MESR) Calculate the liquid c o m o s i t i o n on a p l a t e a t t h e end o f an i n t e r v a l .
I MESR T r a f f i c Control.
R e g u l a t e s t h e hBSR so i t w i l l o b t a i n c o r r e c t v a l u e s f o r a given p l a t e and s t o r e answers in t h e c o r r e c t p l a c e .
state compositions is rapid a t first, but extremely slow toward the end. Thus a n approximate solution can be obtained quickly, but a precise answer requires an unduly long period of computation. The latter can be greatly reduced by determining the form of the first portion of the curve, and then extrapolating this to steady state. This extrapolation can be included i n the overall computer program, so that computation time is reduced to a nominal value. Evaluation of Method The method presented here is basically different from the Lewis-Matheson method in that it determines overhead and bottoms compositions, given the flow rates, number of plates, feed plate location, feed composition, and relative volatility relations. The Lewis-Matheson method essentially determines the number of plates necessary to produce overhead and bottoms compositions which are specified or desired. If a problem involves an existing column with a known number of theoretical plates, the relaxation method has a definite advantage. Using this method is equivalent to having a pilot plant column and actually operating it to obtain the answers. Such a problem can be solved using a Lewis--Matheson type method only by trial and error. For multicomponent problems the choice
740
as zero. This in no way upsets the calculations of the relaxation procedure, but does cause difficulty in LewisMatheson and other types of conventional calculations. The relaxation method can be adapted for multiple feed and for side stream products, and can be used to solve multiple column problems where the columns are interrelated-for example, if bottoms from column one is the feed to column two and part of the overhead from column two is refluxed to column one. In every case the method proceeds positively and directly to the correct product compositions for the conditions chosen, and when successive trials are necessary, there is no uncertainty in selection of new starting values. With some modification, the method can be used to determine the time to reach steady state from an empty column start, and to study control problems.
of successive trial values can many times become very complex, particularly if there is but little experience with the system. Shelton and McIntire (7) have developed automatic computer programs for Lewis-Matheson type solution called the feed mesh methods. Bonner (2) has developed a similar method. However, difficulties are sometimes encountered in the choice of successive trial values and these become critical when they must be completely systematized in advance in the form of program instructions for the computer. Amundson’s (7) method avoids these difficulties by setting up simultaneous equations for the steady-state conditions, and solving these by a matrix inversion technique which requires a very large, fast computer. Another difficulty in using the LewisMatheson method is that of extremes in composition. For example, the final solution to a problem might indicate that the overhead is almost pure light component and the bottoms contains only parts per million of the same component. Under these conditions, the top composition will sometimes be vey sensitive to minute changes in bottoms composition. I n the relaxation method the only effect of very small quantities of a component is that this component might disappear entirely from the computer registers (usually seven or more decimal places) and appear
INDUSTRIAL AND ENGINEERING CHEMISTRY
Acknowledgment The investigations underlying this work would not have been possible without financial support from the h’ational Science Foundation in the form of a grant to the Pennsylvania State University. Nomenclature
H = total holdup per plate, moles L = liquid flow rate, moles per time interval vapor flow rate: moles per time interval x = liquid composition, mole fraction of any component y = vapor composition, mole fraction of any component
V
=
SUBSCRIPTS
i
=
m
= to identify a vapor or liquid rate
number of time interval
below feed plate to identify a vapor or liquid rate above feed plate 1, 2 . .16 = to identify a plate by number, counting from bottom up
n
=
Literature Cited (1) Amundson, N. R., Pontinen, A. J., IND.ENG.CHEM.50, 730 (1958). (2) Bonner, J. S., Pittsburgh Meeting, Am. Inst. Chem. Engrs., September 1956. (3) Robinson, C. S., Gilliland, E. R., “Elements of Fractional Distillation,” 4th ed., pp. 219-29, McGrawHill, New York, 1950. (4) Rose, Arthur, Johnson, R. C., Chem. Eng. Prugr. 49, 15-21 (1953). (5) Rose, Arthur, Johnson, R. C. Williams, T. J., Zbid., 48, 549-56 (1952). (6) Rose, Arthur, Williams, T. J., Harnett, R. T., IND. ENG.CHEW48,1008-19 (1956). (7) Shelton, R. O., McIntire, R. L., Pittsburgh Meeting, Am. Inst. Chem. Engrs., September 1956. RECEIVED for review November 21, 1957 ACCEPTED March 7, 1958 Division of Petroleum Chemistry, Symposium on Application of Machine Computation to Petroleum Research, 132nd Meeting, ACS, New York, N. Y., September 1957.
