Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 3, 1978 313
On Optimal Steady-State Operation in Distillation Kurt V. Waller’ and Tore K. Gustafsson Deparlment of Chemical Engineering, Abo Akademi, 20500 Abo 50, Finland
Optimal steady-state operation in distillation is treated. The criterion used for optimality is to minimize a weighted sum of losses or costs due to incomplete recovery, off-specification products, and energy needed for separation. From a number of computed optimal steady states some simple rules are obtained. The rules concern, e.g., the relations between the feedplate location and the optimal internal and external streams of the column both when the feedplate is optimally and nonoptimally located. The effects of various cost changes on the optimal operation variables are illustrated. The conditions for feed preheating to be profitable are also treated, and a simple rule is obtained.
Introduction
The operation of a chemical plant involves control on several levels. On the lowest level the separate units are kept close to predetermined steady states or run according to predetermined time schedules. On the second level the units are optimized so as to give, e.g., the steady states a t which the units are to be operated. Higher levels are concerned with optimizations of larger objectives, such as plants and companies. The present paper is focused on the second level of control. The aim is to obtain simple rules-of-thumb for optimum steady-state operation of distillation columns. Although computer programs for detailed column simulation are becoming more common all the time, it is felt that there is a need in everyday industrial practice for rough rules-of-thumb which give a better understanding of and feeling for the process than does an ordinary computer program. To achieve this goal, it is necessary to make a number of simplifications. In the present investigation, binary separations in plate columns are treated, plate efficiency is assumed to be independent of tray loading, and pressure is not included in the optimization. These assumptions are discussed later. There are numerous papers and textbooks available on optimum design of distillation columns, but there is not much guidance for optimum operation of existing columns in nondesign situations, although the latter problem is more frequent than the former. Considering the recent interest in optimal connection of several columns (Rathore et al., 1974a,b; Freshwater and Henry, 1975), this is somewhat astonishing. There are many reasons for the operating situation to differ from the one used in the column design. Examples are changed prices on products and/or energy as well as use of columns for separation of other mixtures than those used for the design of the column. Except for case studies permitting almost no generalizations, there seem to be very few reports on overall optimizations of distillation operations. The recent text by Shinskey (1977) on distillation control partly fills the gap and should be mentioned in this context. On the other hand, there has been a number of reports on partial optimization, starting with the classical paper of McCabe and Thiele (1925).McCabe and Thiele state that for a chosen reflux “the feed should be introduced on that plate on which the composition of the liquid is just less than that of the feed”. The statement gives rise to several questions, such as: what about the case when the reflux is not chosen a priori, Le., nonoptimally, but rather included in the optimization? And further: if the feed plate is chosen in the first place, thus nonoptimally, instead of the reflux, should the reflux be such that the statement of McCabe and Thiele is true? 0019-7882/78/1117-0313$01,00/0
Several other papers have also treated feed plate location, but the criteria for optimum have been very restricted. The commonly used criterion (Gilliland, 1940; Scheibel and Montross, 1948; Floyd and Hipkin, 1963; Maas, 1973) might be formulated: the location is optimal if it results in maximum separation for a fixed number of plates. There are, however, a number of other factors than feedplate location, such as reflux rate, affecting the degree of separation, and they should certainly be included in the optimization. Furthermore, often also the degree of separation is a variable that should be included in the optimization. The significance of the problem has been expressed, e.g., by Rosenbrock (1969) in a rapporteur’s lecture on distillation column control when commenting on a paper by Rijnsdorp and Maarleveld (1969). Rosenbrock states: “In passing, the authors make the remark that if costs of heat are the same for reboiler and feed it is always better to supply the heat in the reboiler. This is one of those simple, yet powerful generalizations which are of the greatest value in evaluating different control schemes”. Criterion for Optimality One of the main problems in an investigation aiming at general conclusions concerning processing optimality is to formulate an optimality criterion general enough to give results which are optimal or nearly optimal for a wide range of process conditions. Naturally, a rule of maximum separation per plate cannot be used as a criterion for optimality. Firstly, this would imply an infinite reflux ratio. Secondly, the price relations between the two products can very well be such as to favor an average separation smaller than the maximum possible one, even with a fixed finite reflux ratio. Optimal conditions in distillation are discussed by Shinskey (1967), who suggests the following criterion for separation into two product streams. In a distillation column part of the light component is leaving the column with the bottoms. The economic loss due to this is expressed by B X B Uwhere ~, u1 expresses the difference between the value of the light component in the top product and the value of this component in the bottoms product. The economic loss because part of the heavy component leaves the column with the top product can be analogously expressed by D ( l - X D ) U ~The . costs for vapor generation in the reboiler are taken to be Vsuo.The criterion suggested by Shinskey (1967) is to minimize the sum of these costs counted per unit of feed, written in dimensionless form, e.g., as
0 1978 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 3, 1978
The criterion is directly applicable to binary separations into two product streams, but extensions to multicomponent and complex separations as well as to separations with sidestreams are straightforward. The same goes for extensions to optimization of systems of several columns. The criterion described is well suited for separations where the quality of neither product needs to be kept on specifications, as often, e.g., in refineries. In many separations, however, there are demands on the purity of one or both products. In the present study, the loss function (1)has been generalized by inclusion of such purity demands. Further, reflux cooling (and pumping) costs, although usually small, have been included. The purity demands have been quantitatively expressed through costs (= not obtained income) for off-specification product. These costs are expressed by a function utilizing a sum of a linear and a quadratic term PD’ = D
[ u ~ D1 o o ( X D ’ - XD)
PB’ = B[ulB *
100(aB - XB’)
+ UqD
- XD)2] + UqB * io4(xB - XB’)2] *
io4(XD’
Here U1D and U ~ J J(resp. U1B and U ~ Bexpress ) the decrease in value of the top (bottom) product if the mole fraction XD(XB) is 0.01 unit lower (higher) than the mole fraction XD’(XB’), at which the price for the product is specified. The costs for reflux cooling (and pumping), PL, are taken to be proportional to the vapor flow to the top condenser
+
P,’ = Veu3 = [V, (1 - q ) F ] ~ 3 Then, the optimality criterion used in this work is
P* = min (P)
(2)
The independent variables in the optimization are taken to be DIF, V,IF, and UF. In eq 2 P is given by
. ”
Thus, the criterion used to define optimum operation means minimizing a weighted sum of losses or costs due to incomplete recovery, off-specification products, reboiler heat input, and reflux cooling. As shown by eq 4 and 5, there are no rewards or losses for making a purer product than the one set by the specifications. This is probably the usual situation (Shinskey (1977) p 4). Distillation Systems Studied One of the crucial points in process modeling concerns complexity; in each application there is an optimal amount of modeling complexity to be used and hence accuracy obtained. When modeling the distillation process in a study aiming a t general conclusions, it is of vital importance not to
obscure the properties typical of most distillation systems by a number of specific details. In this work the following simplifications have been used: (1) binary separations in plate columns, steady states calculated according to McCabe and Thiele (1925); (2) plate efficiencies independent of tray loading; (3) pressure constant, not included in the optimization. The first two assumptions are used for convenience only. The third assumption, however, needs some comments. Pressure is actually one of the main free variables to be used in the optimization. It can, however, be taken into account separately, since it is in most cases determined by the constraints set by reboiler heat effect and overhead condensor effect. There are, of course, other effects that cannot be taken into account by such a simple superposition, such as change of relative volatility with pressure. A number of steady states have been computed, optimal in the sense defined in the previous section. Data for the steady states treated in this paper are collected in Tables I and 11. The optimization gives the optimal product streams, the optimal internal streams, the optimal composition profile in the column, and the optimal feedplate location. Further, the value of possible feed preheating has been studied (see below). The following variables have been varied in the following ranges in the computations: 10 5 number of plates s 150; 0.6 5 - plate efficiency (Murphree liquid) s 1.0; 1.1 5 relative volatility 5 2.5. Mixtures with nonconstant relative volatility, such as ethanol-water and trichloromethane-benzene, have also been studied. The optimal steady states have been calculated on a digital computer. A two-dimensional steep descent partan method (Wilde, 1964) is used to optimize DIF and V J F for a fixed feed plate location. The total optimization algorithm tests different feed plate locations, optimizing D I F and V,IF for these locations, until the optimum according to criterion (2) is found. Properties of Optimal Steady States For q FX 1 the rule of McCabe and Thiele cited in the introduction is equivalent to the rule that the change between operating lines in the McCabe-Thiele diagram should be made at the intersection with the q-line. In contrast to the former rule, the latter has been found valid in most cases computed in this investigation (see, e.g., Figures 1 and 2). Exceptions from the latter rule have been found in few cases only. Even in the exceptional cases, however, the optimal feed plate has been very close to that predicted by the rule implying change at operating lines at the q-line intersection. Thus, for the optimality criterion and distillations studied, this rule (and for q = 1 also the cited rule of McCabe and Thiele) can be considered suitable as a rule-of-thumb also when the reflux is not chosen a priori but included in the optimization. In most cases, however, the reverse of the rule is not true: when the feed plate location is not the optimal one, the (sub-) optimal column operation is not generally such that the change between operating lines is made a t the q-line intersection. An illustration is provided in Figure 3, where the feed is introduced at plate No. 8. The steady state shown is suboptimal, Le., in this case optimal for the (nonoptimal) choice U F = 8. Figure 4 shows the optimal steady state, the optimal feed plate location being a t plate No. 4. For the separation in question, Table I11 quantitatively shows the change in the loss function P for some values of UP. Effects of Various Parameters on the Optimal Steady States. Optimal feed plate location is shown in Figure 5 as a function of the relative volatility for various feed compositions for Systems 2 and 9. As seen from Table I, there are no specific
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 3, 1978
315
Table I. Data for Distillation Systems and Performance Criteria
System
Y
XF
cy
w2
w1
w3
W1B
WqB
WlD
0.039 1.05 146.5 0.0031 0.037 0 0 7.08 var 1.0 100 100 0.05 0 0 0 0.3 1.0 100 100 0.05 0 0 0 4 0.5 var 100 100 0.05 0 0 0 5 10 0.5 var 100 100 0.05 0 0.5 0 6 10 1.5 0.5 var 100 100 0.05 0 0 0 0 7 10 1.5 0.5 -0.5 100 100 0.05 0 0 8 10 1.5 0.5 3.0 100 100 0.05 0 0 0 9 50 var var 1.0 100 100 0.05 0 0 0 Equilibrium curve for water-ethanol, plate efficiency in the column = 0.8, in the reboiler = 1.0. 1 2 3
a
UT
wqD
XD'
x B'
0.894
...
25
a
0
20 20 10
var 1.3 1.5 1.5
0 var 0 0 0.5
...
1.0
... ...
1.0
0
...
0 0
...
...
...
... ... 0.0
...
... ...
...
Table 11. Data for Steady States Obtained through Optimization
System Case 1 1 7 8
1 2
UF
UF*
8
...
... .. . ...
(V,IF)* (DIF)*
R*
9.59 0.524 9.21 0.502 10.1 3.95 6.7 5.89
4
4 6
XD*
XB*
0.0447 0.870 0.00008 0.0442 0.876 0.0003 0.491 0.863 0.150
0.505
Y
0.855 0.139
Figure 3. Suboptimal steady state for System 1, Case 1. 1.0
Y
Figure 1. McCabe-Thiele diagram for System 7 (optimal steady state). The feed is superheated vapor, y = -0.5. Dashed line shows liquid composition on feed plate. 0.0 0.0
1 .o
X
Figure 4. Optimal steady state for System 1, Case 2. Table 111. Comparison between the Values of the Loss Function for System 1 with Different Feed Plate Locations
"I 0.0V' 00
UF I
'
'
'
(
X
'
'
'
'
'
'
1.0
8 5
( P - P*)/P*
14.2%
4
2.2% 0.0%
3
0.2%
Figure 2. McCabe-Thiele diagram for System 8 (optimal steady state). The feed is liquid below boiling point, q = 3.0.
purity demands on the products, and the recovery is equally important for both components. The figure illustrates the effect of varying separation per plate on the optimal feed plate location. A small relative volatility and a modest total separation mean that the average separation per plate is of the same magnitude both in the stripping and the enriching section. Anyhow, this is the case when the feed concentration is near 0.5. If, on the other hand, the feed concentration is not near 0.5, and the relative volatility is high, the average separation per plate will be of different magnitude in the stripping section and the enriching section, especially if the product purities are very high. This reasoning is easily verified from McCabe-Thiele diagrams for the different cases.
Some general conclusions concerning the optimal feed plate location can be drawn from Figure 5. If the average separation per plate in both column sections is of the same magnitude, the ratio UF*.YT should lie between the value of xF and 0.5. If the average separation per plate in the column sections is of very different magnitude, the optimal number of plates in the section with larger separation per plate may be less than in the section with a smaller separation per plate. Especially is this the case when a very high total separation is achieved, as in Figure 5 for UT = 50 and CY = 2.0-2.5. In this case a lower feed concentration means that the feed should be introduced higher in the column, the exact opposite of what is the case when the relative volatility is small, or when the total separation is modest. Similar results have been obtained by Luyben (19751, who determined the optimum feed plate location
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Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 3, 1978
0 2/ : I ( ,
,
10
,
v, = 50 , , , , , , , , ,
j
2a
25
15
(1
Figure 5. Optimal feed plate location as a function of relative volatility for different feed compositions. System 2 (UT= 20) and System 9 (UT= 50). (The points shown in Figure 5,6, and 7 have been obtained through parabolic interpolation between the discrete values for W / U T obtained through optimization.)
Figure 6. Optimal feed plate location for System 3 as a function of losses due t o top product composition below 1.0.
0.8
-" 151
r
"
j
02 -05
9
-
/"
00
05
~
10
15
20
Figure 7. Optimal feed plate locations for Systems 4 to 6 as a function of the feed enthalpy.
for fixed purity of the products and varying feed compositions. The importance of introducing the feed a t the optimal location depends on the extent of separation on the feedplate and on the adjacent plates. If the separation is high in the vicinity of the feed plate, the location of the feed plate is quite important, as, e.g., in System 1 (see Table 111).If, on the other hand, the separation is small in the vicinity of the feed plate, the optimal location of the feed plate is not important. For unequal demands on purity or recovery and for q # 1, the optimal location of the feed plate will change in the following way. When the purity demands on one of the components is (relatively) increased, this component should pass through a larger part of the column than devised above. Concerning recovery this means that if the value of the heavy component is relatively high in the bottoms product, Le., L U ~>> w1,the column should be run so as to have a high recovery of the heavy component, which means a pure top product. Then, the feed should be introduced relatively low in the column. A quantitative illustration is given in Figure 6, which shows that increased purity demands on the top product moves the optimal feed plate down the column. The same fact is illustrated also in Figure 7 for Systems 4 to 6. Compared to System 4,System 5 has increased purity demands on the bottom product and System 6 on the top product. The figure further illustrates the effect,of the feed enthalpy (temperature) on the optimal feed plate location. As seen, a colder feed should be introduced higher up in the column, Le., in a colder section. I t might be mentioned that this
means a lower entropy generation due to mixing of streams of different temperatures. A criterion for optimal feedplate location based on the entropy-of-mixing-concept has been used, e.g., by Shipman (1972). In addition to the change in optimal feed plate location, a change in feed enthalpy mainly affects the reboiler heat input. The optimal concentrations on the various plates, however, remain almost unchanged. The optimal feed plate location alters, so that the shift between operating lines in the McCabe-Thiele diagram usually is made a t (or close to) the intersection with the q-line, regardless of q . A quantitative illustration is given in Figures 1and 2. T h e results presented above have given some rough and simple rules-of-thumb for the location of the proper point of feed introduction to the column. Considering that most columns are designed so as to have only a small number of possible feed inlets, the accuracy obtained is believed to be satisfying for many everyday industrial decisions. With an (close to) optimal feed inlet location, the compositions of the product streams can be adjusted by the external streams, e.g., through D I F , and by the internal ones, e.g., through R . It would be desirable to have simple rules for the trade-off between these manipulators. Since we are concerned with how to operate existing columns for nondesign separations, there cannot, of course, exist any general rule-of-thumb for R*lRmin,as in column design. The following qualitative comments on R and DIF are more or less self-evident. Top product purity is increased by an increase in R andlor a decrease in DIF. DIF is the natural variable to adjust differences in demands on recovery, so that a low DIF (Le., D / F < XF) is used if the recovery of the heavy component is relatively important. The trade-off between energy costs and losses due to product composition below specifications as well as the sum of recoveries is then controlled by R. Analogously, a low BIF (i.e., B I F < 1- XF) is used if the recovery of the light component is relatively important Increasing the relative price on energy decreases R. An easy separation (high relative volatility andlor a column with many plates) means that relativeiy low reflux should be used. Optimal Feed Preheating Feed preheating can be included in the optimization through an additional term in the objective function. In the present study, the costs for feed preheat have been taken to be proportional to the resulting change in q-value, Le., proportional to the heat added. The feed preheating costs are then
where q u is the q-value of the feed before preheating. Before adding eq 6 to eq 3, it is divided by Fuo to give
The optimaiity criterion then takes the form
Ptot= min(P
+ PF)
(8)
where P and PF are obtained from eq 3 and 7 . The independent variables in the optimization are now D I F , V,IF, U F , and 4.
The optimization problem expressed by eq 8 is solved in the following indirect way, which permits more general conclusions than a direct optimization. The optimal value for P , Le., without any feed preheating, is plotted in Figure 8 for the same systems as used in Figure 7. For our purposes the relationships might well be expressed by linear relations
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 3, 1978
::m 190
'905
00
05
10
15
20
q
F i g u r e 8. Optimal values o f t h e f o r Systems 4 t o 6.
System 4: System 5: System 6:
loss f u n c t i o n w i t h o u t feed p r e h e a t i n g
+ + +
P* = 18.8 0.48q P* = 30.3 0.21q P* = 29.9 0.749
(9)
The ratio between U F h O and dP*ldq determines whether feed preheating is profitable or not. If U F / U 0 > dP*ldq, feed preheating is not profitable, but as long as UF/uO < dP*/dq, the feed should be heated as much as possible. For the system studied, the following approximate conditions are thus obtained for feed preheating to be profitable System 4: System 5: System 6:
UF UF VF
< 0.48~0 < 0.21~0 < 0.74~0
(10)
The inequalities (10) show that for equal feed preheat and reboiler heat costs, feed preheating is not profitable. As also done by Rijnsdorp and Maarleveld (1969) and recently by Shinskey (1977, p ZO), this might be explained in the following way. With increased feed preheating the vapor and liquid streams in the enriching section of the column are increased whereas the streams in the stripping section are unaffected (for fixed D I F and V , / F ) . This means that the separation is increased in the enriching section, whereas it is increased in the whole column if heat is supplied in the reboiler. The goodness of this explanation is quantitatively illustrated by a comparison of Figure 7 with eq 10. The comparison reveals that the largest ratio U F / U O a t which feed preheating is profitable is approximately equal to (UT- Y F ) / U T . The following rule-of-thumb may be given: if the ratio between the number of enriching section plates and total number of plates is larger than the ratio between costs for feed preheating and reboiler heating, then the feed should be
317
preheated. If not, the reverse is true. If, in the former case, the cost of feed preheating ( U F )is not constant but increases with increasing heat supply, e.g., with increasing temperature, the feed should be heated until the ratio UF/uO equals the ratio (UT
- UF)/UT. Nomenclature bottomsflowrate subscript, bottoms distillate flow rate subscript, distillate feed flow rate liquid flow rate in enriching section = loss function q = fraction of feed leaving the feed plate as liquid R = reflux ratio, R = L I D Rmin = minimum reflux ratio needed to obtain a specified product purity, with a fixed total number of plates u = cost, value w = normalized cost, value (see ea 3-5) V , = vapor flow rate'in enriching section V , = vapor flow rate in stripping section x = mole fraction more-volatile component in liquid X F = mole fraction more-volatile component in feed x ' = mole fraction, parameter in eq 4 and 5 y = mole fraction more-volatile component in vapor LY = relative volatility U F = feedplate number (number of plates in stripping section) UT = total number of plates * = superscript, denotes optimum
B = B = D = D = F = L = P , P'
Literature Cited Floyd, R. B., Hipkin, H. G., Ind. Eng. Chem., 55, 34 (1963). Freshwater, D. C., Henry, B. D., The Chem. Eng., 533 (Sept 1975). Gilliland, E. R., Ind. Eng. Chem., 32, 918 (1940). Luyben, W. L., Ind. Eng. Chem. fundam., 14, 321 (1975). Maas, J. H., Chem. Eng., 96 (April 1973). McCabe, W. L., Thiele, E. W.. Ind. Eng. Chem., 17, 605 (1925). Rathore, R. N. S., Van Wormer, K. A., Powers, G. J., AIChE J., 20, 491 (1974a). Rathore, R . N. S.. Van Wormer, K. A., Powers, G. J., AIChE J., 20, 940 (1974b). Rijnsdorp, J. E., Maarleveld, A., I. Chem. E. Symp. Ser., No. 32, 6, 33 (1969). Rosenbrock. H. H., I. Chem. E. Symp. Ser., No. 32, 6, 3 (1969). Scheibei. E. G., Montross, C. F., Ind. Eng. Chem.. 40, 1398 (1948). Shinskey, F. G., "Process Control Systems", McGraw-Hill, New York, N.Y., 1967. Shinskey, F. G., "Distillation Control", McGraw-Hill. New York. N.Y., 1977. Shipman, C. W., AlChEJ., 18, 1253 (1972). Wilde, D. J., "Optimum Seeking Methods", Prentice-Hall, Englewood Cliffs, N.J., 1964.
Received {or reuieu! J u n e 13, 1977 Accepted J a n u a r y 1 3 , 1 9 7 8
