Some Practical Aspects of Optimal Batch Distillation Design William 1. Luyben Department of Chemical Engineering, Lehigh C‘niuersity, Bethlehem, P a . 18015
Various design and operating policies for binary batch distillation columns are studied with the objective of achieving a maximum production rate. Significant increases in number of capacity can be realized by both optimum design of the column-i.e., trays, tray holdup, reflux drum holdup, and amount of initial charge to the stilland optimum operation-i.e.,
reflux ratio and startup procedure. The time consumed
in starting up the column is taken into account. This equilibration time can be extremely significant in close-boiling or high holdup systems. The time required to purify the bottoms product left in the still is also included. Digital simulation of the system equations permitted study of a range of conditions for binary batch distillation.
Results show
that reflux drum holdup should be minimized, but some tray holdup may be beneficial. The optimum combination of reflux ratio and number of trays i s usually in the direction
of many trays and low reflux ratios.
B a t c h distillation was studied extensively in the 194050 period. Lacking present-day analog or digital computers, workers in this period concentrated on approximate methods for predicting results. Tray and reflux drum holdups had to be neglected, and it was usually assumed that the column had attained equilibrium, total-reflux conditions. The problem of equilibration time was recognized, however (Huffman and Urey, 1937). The principal design objective was to improve the “sharpness of separation,” the amount of distillate drawn off between some specified compositions-e.g., between X D= 0.95 and X n = 0.05). This intermediate cut represents off-specification material that is lost or must be redistilled. Pigford et ai. (1951) in presenting one of the first analog computer applications in the process dynamics area, explored the effects of tray holdup on “sharpness of separation” and showed that a small amount of tray holdup was beneficial in some cases. Archer and Rothfus (1961) reviewed the work prior to 1960. Since the early 195O’s, batch distillation has received little attention, owing to the emphasis during this period on continuous processes. However, batch distillation columns have survived in many chemical plants because of their inherent advantages in some systems. The advent of the process computer may well increase the popularity of batch distillation columns because their operation can be fully automated. Recent work in batch distillation has been concerned with the numerical integration aspects of obtaining transient solutions (Distefano. 1968; Holland, 1966: Meadows. 1963). practical control schemes (Block, 1967). and optimization (Barb, 1967; Bowman and Clark. 1963: Converse and Gross, 1963: Converse and Huber, 1965; Coward. 1967; Price, 1967; Robinson, 1969. 1970; Simacek, 1968: Speight, 1968). These optimization studies have been limited to fairly idealized systems, usually with no tray holdup, no consideration of startup time. and a fixed number of trays. The usual problem considered is one of finding the optimum reflux ratio policy. cia the calculus 54
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
of variations or Pontryagin’s principle. that minimizes the time required to produce a given amount of acceptable product. Results of these optimization studies have been far from spectacular. Comparisons of batch times with “optimal,” “constant reflux ratio,” and “constant overhead composition” control strategies have usually shown only slight differences. Since the fixed reflux ratio policy is the easiest to implement, there appears to be little advantage in going to time-optimal policies. The purpose of this paper is to show that batch distillation cycle time, or capacity, can be much more significantly affected by other design and operating conditions: tray holdup. reflux drum holdup, initial still charge. number of trays, and reflux ratio (assumed constant throughout). An approach to practical optimization of batch distillation is presented. System
The design objective was to maximize capacity. Capacity was defined as the amount of total on-specification products produced per unit time-Le., the sum of distillate and bottoms. Only binary systems were considered in this study. but the basic ideas appear to be readily extendable to multicomponent cases. For simplicity, a constant relative-volatility, equimolal-overflow system was studied. Tray and reflux drum holdups were constant. Reflux was saturated liquid. Vapor boilup was constant at 100 moles per hr in all runs. A complete batch distillation cycle includes: Startup-the time needed to bring the column to steady state conditions on total reflux ( t d . Overhead Product Withdrawal-the period during which on-specification distillate product is produced (t,). Final Bottoms Product Purification-the time required to remove enough low boiler from the still pot and column trays so that the bottoms product will be on-specification when the liquid in the column drains down into the still pot (ti).
Product Removal and Recharging-the time needed to empty the still pot and recharge a fresh batch. The last period was not considered in this study because it is usually fixed by off-site facilities and is often negligible compared to the processing time. The amount of material in the still pot, H H ,was equal to the initial amount of charge, HRO, less the amount held up on all the trays and in the reflux drum. At the beginning of the startup period ( t = 0), compositions on all trays and in the reflux drum and still pot were assumed equal to the initial charge concentration (0.50 mole fraction low boiler). I t would perhaps be more rigorous t o assume that if the feed were charged into the still pot, the initial compositions of the trays and reflux drum would be equal to the composition of the vapor in equilibrium with the feed. The column will not produce any rectification until the vapor boilup has worked its way u p the column and condensed to form enough liquid to fill the reflux drum and the trays. During the period of establishing liquid flows, the still would represent only one equilibrium stage. This difference between initial liquid and vapor compositions was not considered in this study to simplify the calculations. The column was run a t total reflux until steady state was attained. Equilibration time, t E , was defined as the time when the rate of change of reflux drum composition ( d X i l / d t ) was less than 0.01 mf per hour. Justification for this operating policy is discussed later. Distillate product was then withdrawn a t a constant rate as dictated by the desired reflux ratio, RR.
D = V/(l
+ RR)
down into the still pot a t the end of the batch ( t = t ~ ) For . columns with many trays, there may be considerable advantage in diverting the liquid from the bottom of the column to avoid mixing with the material in the still pot. But this operation is not considered in this paper. The average composition of bottoms product was computed: L
xst
H B X~ B +~
H.vXnt n = l
=
(3)
L
H B +~
HN n = 1
All tray holdups were assumed equal so:
HBI = Hgo - HD
- NH,v
- D (t - tE)
(4)
Material was removed overhead a t the same rate as during
(1)
The instantaneous value of the composition of material in the reflux drum was Xo,and this was the instantaneous composition of distillate withdrawn. The average value of overhead product produced was computed with time:
5
4
TIME
6
(HOURS)
Figure 1 . Typical batch distillation curves of X D and X B . (Effect of tray holdup) N = 20, 01 1.5, RR = 5 , H n = 10, H ~ r = i 100
Distillate was withdrawn to product tankage for as long as its average composition, X D , was greater than 0.95. The cutoff time was called t,. The instantaneous value of X n was, of course, lower than 0.95 a t t,. Some typical batch curves are given in Figures 1 and 2. At this point as much on-specification distillate product had been drawn off as possible. I t s amount and composition were known. Therefore, the amount of material left in the still pot was known. However, the still pot composition, X H , was not uniquely fixed because of the variable compositions of the material held up on the trays and in the reflux drum. To make on-specification product in the still pot, it was sometimes necessary to continue taking material from the reflux drum (and send it t o rework tankage) until enough low boiler had been removed from the still pot and from the column. The material in the reflux drum, Hn,was included in the material to be reworked. This bottoms purification period varied with the number of trays, the sharpness of separation, and the holdups. I t was assumed that all the material in the column drained
TIME
(HOURS)
Figure 2. Typical batch distillation curves of X D and X g . (Effect of number of trays, reflux drum holdup, and reflux ratio) 01
N 10 2
40
HD
RR
5 0
2
0
1.5, H.v = 1
-
-
10
5
=
+
+
+
-
-
+
+
+
+
10
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
55
Table I. Effect of Tray, Reflux Drum and Still Holdup
V = 100
N = 20
RR
a = 1.5 Capacity, moles total
Batch times, hr
Holdups, moles
Run
=5
no
HRO
HD
H\
tE
tD
1 2 3 4 5 6 7 8 9 10 11
100 100 100 100 100 100 100 100 80 300 1000
10 10 10 10 0 0 0 0 10 10 10
0.25 0.50 1.0 2.0 0.25 0.5 1.0 2.0 1.0 1.0 1.0
1.80 1.93 2.23 2.85 0.36 0.65 1.11 1.87 2.24 2.22 2.22
3.71 4.05 4.57 5.21 2.55 3.06 3.77 4.61 4.06 8.81 22.14
product withdrawal (reflux ratio was held constant) until X B = 0.05. The capacity. C, of the system would then be calculated in moles of total products per hour:
ti
praducts/hr
5.08 5.06 5.13 5.73 4.21 4.31 4.46 5.06 4.41 13.02 41.26
13.2 14.5 15.7 14.2 17.2 18.3 19.9 18.3 14.6 16.9 16.3
C = [HBO- D (t, - tF) - HD]'tF
(5)
Production rate of a batch distillation column is affected by many design parameters. Equations describing the dynamic behavior of the column/still potireflux drum system were:
Table II. Pigford Column"
H D =0
N =7
Troy holdup,
Reflux rotio,
Run
H B O =100
V = 100
t,
ti
Capacity, moles total products/hr
4.70 5.01 5.34 5.57 3.22 3.62 5.78 6.06 6.26 6.50
4.70 5.01 5.34 6.00 3.54 3.95 5.78 6.06 6.26 6.50
21.3 20.0 18.7 15.9 26.0 23.7 17.3 16.5 16.0 15.4
Batch tirnes,hr
iih,
no.
a
RR
moles
fE
12 13 14 15 16 17 18 19 20 21
2.23 2.23 2.23 2.23 2.23 2.23 3.0 3.0 3.0 3.0
8.0 8.0 8.0 8.0 3.2 4.0 10.0 10.0 10.0 10.0
0.25 1.0 2.4 4.5 4.5 4.5 0.25 1.0 2.0 4.0
0.12 0.40 0.81 1.32 1.32 1.32 0.12 0.27 0.49 0.85
'Pigford et al. (1951).
These equations were simulated on the Lehigh University CDC 6400 digital computer. Using a simple Euler integration algorithm with a step size of 0.002 hr, one complete batch run of about five process hours required approximately 0.4 sec of computer time per tray-e.g., a 20-tray column ran in 8 sec. Results
Figures 1 and 2 show typical profiles of X D and Xa with time. Figure 1 shows the effect of increasing tray holdup, H y , from one to two moles per tray for a 20-tray column with relative volatility a = 1.5, reflux ratio, RR = 5 . reflux drum holdup, HD= 10, and initial still charge, HBo = 100 moles. The time required to come to steady state, t ~ is, increased, and capacity is therefore reduced from 16 t o 14 moles per hr. For still lower tray holdups,
0
r \
I-
o
Iv)
-1
W -I
420I-
15
*I
O
Iv)
-
18-
0
I
>
E
16-
o
2a 0
I4
-
0
N = 7
I
I
IHo=l0
I I -A -
0 I
0
HN
2
TRAY
3
4
HOLDUP (MOLESITRAY)
Figure 3. Effect of tray holdup on capacity
56
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
2
4
HD
6
8
IO
REFLUX DRUM HOLDUP (MOLES)
Figure 4. Effect of initial still charge and reflux drum holdup on capacity n = 1 5 , N = 20, RR = 5 , H \ = 1
v)
>.
2 50 IJ
a
a
0
40 I-
-I
a
W
I-
(L
;20.
0 W
I I-
30
(L
W
rn 5 3
2c
2
z IO
0 51
0
L
2
0
RR
4
6
8
IO
RR
REFLUX
RATIO
( M O L E S REFLUX /MOLE D I S T I L L A T E )
8
01
=
1.5, H v = 1, H D = 10, HAO= 100
1.5,H.y= 1 , H o = 1 0 , H ~ , = 100
however, capacity can decrease (Table I, Figure 3). This is apparently owing to the “fly wheel” effect (Pigford et al., 1951). Holdup in the column tends to make compositions change more slowly. Overhead composition can stay richer longer under conditions of low reflux ratio where the still contents are removed rapidly. Equilibration time and batch time are both decreased as tray holdup is lowered below one mole per tray, but the time during which on-specification distillate product is withdrawn (t, - t i ) is also reduced. The relative importance of these opposing effects varies from case to case. Some of the cases considered by Pigford et al., were studied (Table 11. Figure 3), and no drop in capacity with tray holdup was observed, probably owing to fewer trays and larger reflux ratios. Figures 2-4 and Table I show that capacity is significantly increased by reducing reflux drum holdup. Two factors contribute t o this capacity increase. First, equilibration time is reduced; t E appears to be a strong function of reflux drum holdup, but is practically unaffected by the number of trays, N (probably because of the corresponding decrease in H B ) .Second, the amount of material t o be reworked is reduced since all of the reflux drum material is off-specification. This can be seen explicitly in Equation 5 . Figure 4 and Table I also show the effect of changing the amount of initial still charge. Capacity drops off rapidly if H x ~is, less than 100 moles (on a vapor boilup basis of V = 100 moles per hr) since the startup time becomes a large fraction of the total batch time. For HHi, above 200, capacity drops off slowly due to the relatively longer period needed to purify the bottoms product -
6
Figure 6. Capacity contours
Figure 5. Effect of number of trays and reflux ratio on capacity
(tF
4
REFLUX RATIO (MOLES REFLUX /MOLE DISTILLATE)
N =
IO
2
tp).
Figure 2 and Table 111 show the effects of changing
the number of trays. Equilibration time is not changed. Capacity is reduced as fewer trays are used because the time during which distillate is on-specification is reduced and the time required to purify the bottoms product is increased. Figure 5 gives results for various reflux ratios and numbers of trays. For a given system and number of trays there is an optimum reflux ratio that maximizes capacity. Running a t higher reflux ratios sacrifices time by decreasing the rate of distillate withdrawal. Running a t lower reflux ratios sacrifices column separating capability and hence product purities. The optimum reflux ratio
Table 111. Effect of Number of Trays and Reflux Ratio a = 1.5
HD = 10
H.s
= 1
H x o = 100 Capacity, moles totol
Number
Reflux
Run no.
trays,
ratio,
N
RR
ti
t,;
ti
productsihr
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
10 10 10 10 20 20 20 20 30 30 30 30 40 40 40
2.5 5.0 7.5 10.0 2.5 5.0 7.5 10.0 2.0 3.0 4.0 6.0 1.5 2.0 3.0
2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22
2.62 3.05 3.73 4.66 3.35 4.57 5.82 7.00 3.41 3.89 4.37 5.33 3.28 3.52 4.00
4.62 5.71 6.76 7.84 4.29 5.13 6.00 7.00 3.79 4.10 4.42 5.33 3.38 3.54 4.00
7.1 8.0 8.0 7.8 14.7 15.7 14.6 12.8 20.4 20.7 20.1 16.9 25.4 25.2 22.5
Butch times, hr
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
57
_ _ --- 7c
I
I
I
I
90 VJ
2 6C
v)
>
2I- a o
I
\
a II
-I
\
o l i
< 2 70
\
d
50
0 I-
I-
W
W
a W
a
\
I
0
4c
\I
60
I I-
: 30 rn
a W
50
5
3 2
2
z 2c
40
z
z
0 : V
’
IO
I
30 I
RR
2
REFLUX R A T I O
I
4
3
RR
( M O L E S REFLUX /MOLE D lS T l L L A T E )
2
REFLUX
N
naturally varies with the number of trays since the column separating capability is changed and lower reflux ratios can be tolerated. Figures 6-8 give similar results for several systems in the form of contour maps. The solid lines are “isocapacity” lines in this “N-RR” plane (number of trays-reflux ratio). By generating these types of curves for a given system the design engineer can readily determine the optimum design policy. Notice that the operable region in Figure 6 is bounded on the bottom by the minimum number of trays to get the required separation, on the lower left by the minimum reflux ratios needed to achieve the separation, and on the upper left and top by the condition of having so many trays that the holdup in the still goes to zero before X, is purified down to the required 0.05. Increasing the initial still charge H x o relieves this constraint and gives higher capacity while using more trays. Comparison of Figures 6 and 7 shows the capacity raised from 25 moles per hr in a 40-tray column with HBO= 100 moles to 33 moles per hr in an 85-tray column. This increase in capacity may well justify the increased capital expenditure. Figure 8 illustrates the effect of changing relative volatility from O( = 1.5 to 2. The easier separation permits lower reflux ratios and higher capacities. The increase in equilibration time with decreasing relative volatility is summarized below. 3 2.33 1.5
Equilibration time,
fF,
REFLUX DI S T I L L A T E )
= 2, HI = 1, H D = 10, HHO= 200
Table IV. Effect of Nonequilibrated Start
H ~ = 1 0 Hy=l -
HBI,,
Batch times hr
Run no.
moles
n
N
RR
fc.
t,
tF
37 38 39 40
100 100 200 200
1.5 1.5 2.0 2.0
20 20 70 70
5 5 1 1
2.23 1.15 1.36 0.63
4.57 3.23 3.21 1.59
5.13 4.11 3.34 2.82
Distillste withdrawal begun when
Capacity, moles total products/hr
15.7 18.3n 54.8 45.7
XD= 0.95,
Meadows (1963) suggested starting distillate withdrawal before the initial steady state is attained to speed up the startup. In Runs 38 and 40 in Table IV product withdrawal was begun as soon as X D came up to 0.95, thus reducing tE. Capacity was increased in the system of Runs 37 and 38. However, for the system of Runs 39 and 40, capacity was decreased. This is owing to the shorter period (t, - t E ) of on-specification distillate product and the longer time to purify still product. Therefore a nonequilibrated startup may or may not be advantageous. I t appears to be helpful for higher reflux ratios where product is withdrawn slowly, and harmful for low reflux ratios where product is drawn off rapidly. Conclusions
hr
0.27 0.40 1.11
The above results are based on H,v = 1 and H D = 0. 58
5
4
(MOLES / M0L E
Figure 8. Capacity contours
Figure 7. Capacity contours n = 1.5, H\ = 1, H D = 10, HHO= 200
Relative volatility, a
3
RATIO
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
Batch distillation capacity can be significantly affected by design and operating parameters. Startup time can be significant in batch distillation optimization, particularly for close-boiling separations and systems with large holdups.
Reflux drum holdup should be minimized since it increases both equilibration time and the amount of off-specification material that must be redistilled. For a given separation, tray holdup. still charge, and vapor boilup rate, there is an optimum choice of reflux ratio and number of trays that maximizes capacity. Nomenclature
c = D = HH = HHii
=
Hi, Hi
= = = =
s
RR
t = ti =
ti: =
t,, =
v = XH
XR
= =
xi, = XO =
x,,= v* = ?’.\ =
capacity or production rate, moles of total products ’ hr distillate withdrawal rate, moles / hr still pot holdup, moles initial amount of feed charged to the column, moles reflux drum holdup. moles tray holdup. moles per tray number of theoretical trays in the column reflux ratio; flow rate of reflux./flow rate of distillate time, hr startup or equilibration time, hr time when the batch is finished, hr time when all on-specification distillate has been withdrawn, hr vapor boilup rate. moles hr still pot composition, mole fraction low boiler average composition of still pot plus column trays, mole fraction low boiler reflux drum composition, mole fraction low boiler average composition of distillate product withdrawn from the reflux drum, mole fraction low boiler liquid composition on nth tray, mole fraction low boiler vapor composition in still pot. mole fraction low boiler vapor composition from top tray, mole fraction low boiler
y,, = vapor composition on nth tray, mole fraction low boiler (1 = relative volatility Literature Cited
Archer, D. H., Rothfus. R. R., Chem. Eng. Progy. Symp., 37 (36). 2 (1961). Barb, D. K., Ph.D. Dissertation, Texas A&M University, College Station, Tex., 1967. Block, B., Chem. Eng., 74 ( 2 ) , 147 (1967). Bowman, W. H., Clark, J. B., Chem. Eng. Progr., 59 (5), 54 (1963). Converse, A. O., Gross, G. D., Ind. Eng. Chem. Fundam., 2, 217 (1963). Converse, A. O., Huber, C. I., ibid., 4, 475 (1965). Coward. I., Chem. Eng. Sei., 22, 1881 (1967). Distefano, G. P.: AIChE J . , 14, 190 (1968). Holland. C. D., “Unsteady State Processes with Applications in Multicomponent Distillation,” Prentice-Hall, S e w York, 1966. Huffman, J. R., Urey, H. C.. Ind. Eng. Chem., 29, 531 (1937). Meadows. E. L., Chem. Eng. Progr. S j , m p . . 59 (46), 48 (1963). Price. P. C., Inst. Chem. Eng. Symp. Series, 23. 96 (1967). Pigford, R . L., Tepe, J. B., Garrahan. C. J., Ind. Eng. Chem., 43, 2592 (1951). Robinson, E. R., Chem. Eng. Sei., 24, 1661 (1969); 25, 921 (1970). Simacek, P. E., Ph.D. Dissertation, Montana State University, Bozeman, Mont., 1968. Speight, C. E.. Ph.D. Dissertation. Xortheastern University, Boston. Mass.: 1968.
RECEIVED for review December 11, 1969 ACCEPTED September 10, 1970
An Improved Redlich-Kwong Equation of State in the Supercritical Region David G. Skamenca’ and Dimitrios P. Tassios’ Newark College of Engineering, Newark, N. J . 07102 T h e equation of state as originally introduced by Redlich and Kwong (1949) contains only two constants, expressed in terms of the critical temperature and critical pressure:
P = RT (V-b,,) - a x i i l T ” ” V ( V + b H I C(1) ) Where
ai:, = 0.4278 R’ T,” / P, b K X = 0.0867 RT, J P,
(2) (3)
A comparison of the Redlich-Kwong equation t o other equations of state was conducted by Shah and Thodos
-
Present address. Lrnion Carbide Corp., Houston Texas T o whom correspondence should be addressed.
(1965). They concluded that the R-K equation contains a “combination of advantages in simplicity and accuracy over wide ranges of temperature and pressure” and is especially attractive for predicting the behavior of compounds where PVT data are not known. However, the R-K equation does have drawbacks: I t predicts a critical compressibility of 0.33 for all compounds; and. in the reduced volume range of 0.5 to 1.0, the R-K equation predicts compressibilities with large deviations for all gases, polar and nonpolar. Considerable emphasis was placed by previous workers on modifying the R-K equation for nonpolar vapor mixtures (Ackerman, 1966; Chueh and Prausnitz, 1967, 1968: Gray et al., 1969; Joffe and Zudkevitch, 1966; Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
59
