Ind. Eng. C h e m . Res. 1995,34, 3027-3036
3027
Effect of Feed Characteristics on the Controllability of Binary Distillation Columns Joel G. Cantrell, Timothy R. Elliott, and William L. Luyben* Chemical Process Modeling and Control Research Center and Department of Chemical Engineering, Lehigh University, Iacocca Hall, 111 Research Drive, Bethlehem, Pennsylvania 18015
The design and control of binary distillation columns have been the focus of a great deal of industrial and academic research. However, the effect of the feed characteristics on the controllability of the column has remained unclear. This paper first examines how feed tray location affects the inherent controllability of a binary distillation column. It is shown t h a t controllability can be significantly improved by feeding on a nonoptimum tray. The effects of feed thermal condition are also explored. Generic rules are proposed to improve the controllability of the column by changing the thermal condition of the feed.
Introduction The literature contains numerous papers discussing the design and control of distillation columns. Simple iterative calculations can be used to compute the optimum feed tray from the standpoint of steady-state economics (the feed tray location that minimizes energy consumption). The impact of feed tray selection on dynamic controllability should also be weighed against its effect on energy consumption. This paper examines the tradeoffs encountered between steady-state process design and process control when the feed tray location is varied for a distillation column with a fixed number of total trays. Both rigorous simulation and linear frequency analysis are used to show that for small increases in capital and energy expenditure significant improvements in process controllability are easily achieved for (1) columns perturbed by load disturbances and (2) columns subjected to servo changes on one end, i.e., changes in product composition controller setpoints. The thermal condition of the feed is typically governed by the energy-conservationaspects of the column design. This paper demonstrates that modification of the feed thermal condition may be used as a design tool to improve the inherent controllability of the column.
Process Studied
A binary mixture of constant relative volatility (a = 2.5) is fed to a column with both stripping and rectifying sections (Figure 1). The feed has a composition ( 2 ) of 0.5 mole fraction light component and a rate (F)of 250 lb mol/h. For the first part of the paper, the feed is saturated liquid (q = 1.0). This parameter will be varied in the second part of the paper. The distillate composition and bottoms compositions were fixed at 0.98 and 0.02 mole fraction light component, respectively.
Steady-State Design Procedure In this study we assume equimolal overflow, theoretical trays, a total condenser, constant average molecular weight, and a partial reboiler. Tray holdups and the liquid hydraulic time constants are calculated from the Francis weir formula assuming a 1-in. weir height. The
* To whom correspondence should be addressed. Phone: (610) 758-4256, Fax: (610) 758-5297, Internet E-mail:
[email protected]. 0888-5885/95/2634-3027$09.00/Q
Xd=0.98
F=UO lbmovhr q=1.0
I
Figure 1. Schematic of a binary distillation column.
reflux drum and column base were sized to provide 5 min of holdup based on their respective steady-state flow rates. The following steady-state design procedure was used to size the equipment and obtain the operating conditions necessary to achieve the desired separation. 1. Select the total number of trays in the column NT (note: NT > 2. Calculate the distillate (D) and bottoms ( B )flow rates.
v).
B=F-D
(2)
3. Select a feed tray location ( N F )and use a rating
program to determine the vapor boilup and reflux necessary to achieve the desired separation. 4. The diameter of the column was calculated assuming an F factor of 1 (Henley and Seader, 1981).
F factor = lg(qv)1’2 (3) where ev = vapor density in the column = MWPlRgasT, MW = molecular weight = 50 lb/(lb mol), P = pressure = 44.7 psia, Rgas= perfect gas constant = 15151144 psia 0 1995 American Chemical Society
3028 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995
+
ft3/(lb mol OR),T = average temperature = 300 460 OR, and 1 9 = vapor velocity in the column (Ws). 5 . The diameter of the column (Dc)is then calculated,
where V = vapor boilup (lb mol/h). 6. The height of the column (Lc) is calculated assuming 2-ft tray spacing and allowing 20% more height for base level volume.
Lc = 2"
(5) 7. The liquid heights over the weir for the stripping (/tows)and the rectifying (how,)sections are calculated from the Francis weir formula for q = 1.
(
(F
+ R)MW
213
NF V(1b m o m )
Dc (fi) M B (lb mol)
M D (lb mol) M,, (lb mol) M,, (lb mol) Ps
where F = liquid feed flow rate (lb mom), R = reflux flow rate (lb mom), and QL = liquid density (lb/ft3). 8. The liquid tray holdups for the stripping Mns and rectifying M,, sections are calculated using the height over the weir and a l-in. weir height.
410 383 381 383 391
Table 2. Results of Steady-State Designs for NT = 22
(SI
AR(ft2) Ac (ft2)
(7)
8 10 11 12 14
15 20 22 25 30
Mt,t (lb mol) P R (s)
RMW 213 h o W ~ = 96O0eLD,)
bows = ( 96O0eLDc )
Table 1. Total Annual Cost (TAC) for a Range of Total Trays (NT). All Costs Are Multiples of $1000
4 456.4 3.93 48.5 38.0 1.52 1.76 120.9 3.74 3.10 1141 1902
7 314.8 3.26 36.6 26.2 0.973 1.18 85.7 3.51 2.66 786 1312
15 316.3 3.27 36.8 26.4 0.979 1.19 87.8 3.52 2.66 790 1318
11 291.4 3.14 34.7 24.3 0.885 1.89 80.7 3.49 2.57 728 1214
Table 3. Results of Steady-State Economic Analysis for NT = 22. All Costs Are Multiples of $1000 ~
NF
column cost ($1 tray cost ($1 exchanger cost ($) energy cost ($/year) total annual cost ($/year)
4 252 8 618 250 543
~~
7 207 6 485 172 405
15 207 6 486 173 406
11 199 6 461 159 381
18 244 8 592 234 515
&)
(
energy cost = V 8760-y,h,r)"Vap(
(9) MW where Mn = holdup on tray n, (lb mol). 9. The liquid hydraulic time constant in hours is calculated for the stripping (PSI and rectifymg (PR) sections of the column using a weir length equal to 80% of the diameter of the column (Papastathopoulou and Luyben, 1990).
18 427.9 3.80 35.7 35.7 1.41 1.64 116.9 3.69 3.02 1069 1783
(14)
12. The capital costs of the column, trays, and heat exchangers were estimated using a M&S index of 950 (Douglas, 1988). column cost =
+
M&S101.9Dc1~066L,0~s02(2.18 (3.67 280
+
+ 1.20)) (15)
tray cost = M&S4.7Dc1.55Lc(1,0 0.0 280
+ 1.7)
(16)
+
(10)
heat exchanger cost = M&S101.3(A2.65 x 280 (2.29 (1.35 0.10)3.75) (17)
(11)
The annual capital cost was assumed to be one-third of the capital cost (3-year payback). Materials of construction were stainless steel, and the design pressure was taken as 300 psig.
10. The heat transfer areas of the reboiler and the condenser were calculated using the steady-state vapor of 250 boilup rate and a heat of vaporization (AHvap) B t d b . The reboiler was assumed to have a heat transfer coefficient UR = 100 Btu/(h O F ft2) and a temperature differential ATR = 50 O F . The condenser was assumed to have a heat transfer coefficient Uc = 150 Btu/(h OF ft2) and a temperature differential ATc = 20 OF. (12)
(13) 11. Energy cost is based on the vapor boilup rate, using a steam cost of $5/106 Btu
+
+
Results of Steady-State Design The optimum steady-state column design was determined to have 22 total trays. Table 1 shows how this design compares to others. Tables 2 and 3 show that the design with a feed tray of 11 minimizes the vapor boilup in the column. This consequently minimizes energy consumption and the diameter of the column and thus the total cost of the column. However, it should be noted that the steady-state designs with NF = 7 and N F = 15 appear economically reasonable as they incur only slightly higher total annual costs, $405 000/year and $406 OOOIyear, respectively, as compared to the economic optimum ofNF = 11($381000/year). If these designs were to provide significant controllability improvements, the additional capital investment and energy costs may result in increased profits due t o less
Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 3029 Table 4. Controller Tunings for a Column with NT = 22. All Gains Are Dimensionless, Using Flow Transmitter Spans Equal to Twice the Steady-State Value and Comuosition Transmitter Suans of 0.1 mole fraction NF
KIJXB-V loop WIJxB-vlOOp (radih) KIJXB-R loop WUXD-Rloop (radih) detuning factor for BLT
4 2.00 33.6 0.427 10.8
7 0.975 25.7 0.820 11.8
11 0.695 21.0 1.37 15.7
15 0.610 19.3 1.80 19.8
18 0.550 19.0 3.15 25.5
1.17
1.58
1.97
1.88
1.76
+20% Feed flow Rale DisNrbance: dotted4,dot-dash-7,slld=l l,dasbd=l8 0.055
variability in the product (Downs et al., 1994). The optimum column has 2.6 times the minimum number of trays and operates at 1.07 times the minimum reflux ratio. Designs using other multipliers were examined and found to support the results presented in this paper.
Dynamics and Control: Effect of Feed Tray Location on Controllability Having examined the steady-state characteristics of the five proposed feed tray locations for the 22-tray column, the dynamic aspects of each design will be explored. One might anticipate that a column with a nonminimum vapor rate would exhibit a larger time constant (Mt,JF) because its diameter is larger, giving larger tray holdup in the column. A larger time constant may suggest that the dynamics of the larger diameter columns are slower. Dual-Ended Composition Control. The compositions at both ends of the column were controlled using the widely used R-Vcontrol structure (i.e., the bottoms composition is controlled using the vapor boilup rate and the distillate composition is controlled using the reflux flow rate). Perfect reflux drum and bottoms level control is assumed. Other control structures, such as controlling the distillate composition by manipulating the ratio of the distillate to the reflux flow rate and controlling the bottoms composition by manipulating the ratio of the vapor boilup to the bottoms flow rate, were investigated and exhibited similar dynamics to those found using the R-V structure. Since the objective of this work is to assess controllability, conventional proportional-integral (PI) controllers were implemented to provide fair dynamic comparisons between the designs. The ultimate gain and frequency for each loop (Table 4) was obtained using relay feedback tests on each pairing and verified through frequency analysis on a linearized model of the process. Three 1-min lags were assumed on each composition measurement. The two composition controllers were tuned using the BLT algorithm (Luyben, 1990)for multiple interacting singleinput single-output control loops. The criterion for the BLT procedure was a peak multivariable log modulus of f 4 dB. This tuning method requires transfer function relationships between the two inputs and the two outputs. Approximate transfer functions (Luyben, 1987) were used initially for the BLT procedure and later verified using a full linear model of the process. Regulatory Control. Simulations of the nonlinear process models describing the five column designs were performed. First, a load disturbance of 20% increase in feed flow rate F t o the column was simulated. Figures 2 and 3 show the dynamic response for four of the five systems (the fifth was left off for clarity). In the presence of a feed flow rate disturbance, bottoms composition control improves significantly as the feed tray is moved down the column; however, the control remains about the same for columns fed above the
0.95;
1
0.5
1.5
2
2.5
3I
Time (hours)
Figure 2. Bottoms composition XB control for a 20% increase in feed flow rate F with dual composition control. Vapor boilup rates are scaled by steady-state values. 40% Feed flow Rate Dislurbance: dotted~,~t-dash=7.oolid~1 1,dashed-18
I
I
0.95 0.945,
0.5
1
1.5
2
2.5
3
Time (hours)
+20% F d flow Rate Dishubanca: dolled4.dot-dash-7,~lld=l1,dashed-I8
1
0.95'
.
0.5
2
1
1.5
2
5
2.5
3I
Time ( b r a )
Figure 3. Distillate composition XDcontrol for a 20% increase in feed flow rate F with dual composition control. Reflux flow rates are scaled by steady-state values.
optimum. Distillate composition control degrades as the feed tray is moved down from the top of the column (Figure 3). To verify these results, Bode plots of the closed-loop regulator transfer functions relating bottoms
3030 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 A: +20% Feed CmporiUon Dlrtwbance
00
0 t
0
.-
Frequency [radlhr]
NF
NF
C:Step in XdrM (-0.01 mole Irwlon)
0:Step in Xbret (+0.01 d o kaclion)
45
Figure 4. Bode plot of closed-loop regulator transfer function relating X d F with dual composition control. Ratio is made dimensionless by a flow transmitter span of 2 times the steadystate feed flow rate and a composition transmitter span of 1.0 mole fraction.
t
OOOO 0000000 0
5 NF
10 NF
15
0
20
Figure 6. Integral square error versus feed tray for q = 1.0 and NT = 22. 0 denotes bottoms composition X B and + denotes distillate composition XD (dual composition control).
Frequency (radhr]
Figure 5. Bode plot of closed-loop regulator transfer function relating X d F with dual composition control. Ratio is made dimensionless by a flow transmitter span of 2 times the steadystate feed flow rate and a composition transmitter span of 1.0 mole fraction.
and distillate compositions to feed flow rate were examined using a linear model of the process. Figures 4 and 5 demonstrate that the same controllability trends continue over a wide range of input frequencies. To further verify the trends of the simulations, the integral squared errors of both distillate and bottoms compositions were calculated for many more feed tray possibilities. The results (Figure 6b) show that these trends are rather smooth across a wide range of feed tray locations. These trends were also found to persist for a decrease in feed flow rate. One reason for the improvement in bottoms control as the feed is moved lower in the column is the speed of disturbance detection. If the disturbance enters the process near the bottoms composition measurement, the controller can take action sooner and keep the composition on target. A high feed tray would lengthen the time to detect the disturbance, and the composition profile of the column would be pushed further from steady state. Increased liquid flow rates a t high and low feed tray locations also filter out the disturbance. These two effects are superimposed to generate the controllability trend seen in Figure 6. A column with a low feed tray
has the two effects added beneficially. At high feed locations, the two effects seem to cancel each other out. The distillate control is a different matter. Because the feed is all liquid, the feed flow rate disturbance is primarily seen in the bottom of the column. The disturbance to the distillate composition is caused by the bottoms control action. In Figure 2 the normalized vapor rate is shown rising the steepest for a low feed tray. This quick rise and large magnitude (due t o higher steady-state flow rates) causes a large disturbance in the distillate end. When the feed tray is located higher in the column, the disturbance on the bottoms end is not as dramatic, and the resulting bottoms control action is not as drastic, so the distillate stays at its setpoint. Similar controllability trends were found to exist when a load disturbance in feed composition was introduced. The integral square error USE) data are shown in Figure 6a. Bottoms composition control was found to improve as the feed tray is moved away from the steady-state optimum in both directions with the best control coming when the feed tray is positioned toward the bottom. It is important to note that for a feed composition disturbance, bottoms control is very good when the feed tray is very high or very low. This is again due to the fact that the liquid holdup is greater in these columns (Table 21, so the effect of a feed composition change is greatly reduced. Distillate composition was found to be most controllable when the feed tray is positioned near the top of the column. For a plant design in which the operability of the process requires tight composition control on one end of the column, moving the feed tray up or down several trays will not result in an excessive increase in capital expenditure (Table 3) but will improve process controllability and reduce process variability.
Ind. Eng. Chem. Res., Vol. 34,No. 9, 1995 3031 Another way to look at the change in controllability is in the time constants relating the inputs t o the outputs. The transfer function matrices approximation (Luyben, 1987) used in the BLT tuning method are shown in eq 18.
4.01 XD Stpoint C h a n g e : d o n e d ~ 4 . ~ l ~ d - l l . d o t - d s s h - l 5 , d a ~ l 8 0.025 A
I 0.5
112.3e-0.14S-19.1e-0.09s1
1.5 Tlme (hwrs)
1
2
2.5
3
-0.01 XD Setpoint Cm:doned-4,SOild-l l,dot-dsBh-l5,da~h18 1
,,
.. .... ...,,, ,.,.. . .,.. ' ., .... , .. ......, ,,., ,,, ,,,,, ,,,. ,,, .,,. ,.,.. ....,
0.5 ,
o0.8
8
1.5
2
2
0 2.5
~
3
Time (hour$)
Figure 7. Bottoms composition XB control for a setpoint change in X P t (98-97 mol %) with both ends controlled. Vapor boilup rates are scaled by steady-state values. -0.01 XD Setpalnt Change: d0ned=4,soM-1l.dot-daoh=15.daohed-18
[3.06s
+1
1.86s
+1 1
The gains are in [mole fractionMlb moVh1, and the time constants and dead times are in hours. For the lower feed trays, the time constants for the bottoms are much smaller than the time constants for the distillate. In the region of the steady-state optimum (NF= 111, the time constants for the distillate and bottoms are very similar. At higher feed tray locations, the distillate time constants are smaller than the bottoms. This would suggest that at N F= 11,the control for both ends would be about equal with noticeable interaction, which is the case. For a higher feed tray, one would expect better distillate control, but worse bottoms control, and the opposite to be true for a lower feed tray. These results are verified by the simulations. Both higher and lower feed tray situations also show an off-diagonal element with a time constant somewhat larger than the other three. This would indicate reduced interaction between the two loops, allowing for smaller detuning factors and more aggressive controller tuning. The high and low feed tray cases do have smaller detuning factors, although the direct connection has not been proven. A generic rule for dual-ended regulatory control can then be proposed: For the best regulatory control of bottoms composition, feed the column below the steadystate optimum feed tray; for the best regulatory control of distillate composition, feed the column above the steady-state optimum feed tray. Servo Control. Next we explored the effect of feed tray location on setpoint changes made to either composition controller. Figures 7 and 8 show the dynamics of four different designs for a 1mol % decrease in X D ~ ~
0.9651 0
0.5
1
1.5 Tlme (hwrs)
2
3
2.5
-0.01 XD Stpaint Change: daned.4,solld-1 I , d o l - d s d r - l 5 , d a ~ 1 8
- -.-. -.- . . - . _ . -
1
__
jO.88-
0.88
-
0.82 0.84
0.8:
*--
0.5
1
-- -------_ 1.5 Tlme (hwrs)
-------------: 3I
2
2.5
Figure 8. Distillate composition XDcontrol for a setpoint change inXDSet(98-97 mol %) with both ends controlled. Reflux flow rates are scaled by steady-state values.
The ISE data for a broad range of feed tray locations are shown in Figure 6c. Clearly, both bottoms and distillate composition control improve significantly as ~the . column is fed closer to the bottom. A composition
3032 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995
B
Figure 11. Lever analogy for servo control (A, SS optimum; B, far from setpoint change; C, near setpoint change). 10'
1oz
Figure 9. Bode plot of closed-loop servo transfer function relating XBjXDsetwith both ends controlled. Ratio is dimensionless. 1,
,
,
I
,
I
Figure 10. Steady-state composition profiles for different feed tray locations.
setpoint change is not only a servo control problem but also a regulator control problem because the end maintaining its initial setpoint is subjected to an internally-generated disturbance. We would then expect the generic rule for regulatory control to still be valid, that is the control of the regulator end is improved as the feed tray is located nearer to it. With the simulation results supporting this claim, the Bode plot Of Xg/XDSet(Figure 9) generated from the linear process model shows that the generic rule holds for a wide range of frequencies. Figure 6d shows the ISE data for a +0.01 mole fraction setpoint change in Xgset. The opposite controllability trends were found to occur while still upholding the generic rule for regulatory control of distillate composition. Bottoms and distillate composition control improves as the feed tray is moved from the bottom of the column to the top. If the setpoint change is on an end far from the feed tray, the flat region of the composition profile (Figure 10)can absorb the load, with little effect on the constant end. The disturbance propagation in the column is analogous to a lever with the feed tray acting as a fulcrum. Figure 11shows the relationship of a setpoint change on end i to the load on end j . In case A, the fulcrum is in the middle, so the effect is balanced on each end. In case B, the fulcrum is far from the setpoint change, so the setpoint change causes a very
small disturbance on the opposite end. For case C, the fulcrum is near the setpoint change, and the load on the opposite end is amplified. A generic rule for dualended servo control can then be proposed: For the best control of bottoms and distillate composition in the presence of servo changes to the bottoms, feed the column above the steady-state optimum feed tray; for the best control of bottoms and distillate composition in the presence of servo changes to the distillate, feed the column below the steady-state optimum feed tray. Results obtained from parametric sensitivity tests performed on other columns with different relative volatilities a, feed compositions z , and bottoms and distillate purity levels showed similar results and supported the generic controllability rules stated above. Single-Ended Composition Control. There are many cases in which tight control of one product is more important than another. The results above show that one reason for the degradation of control is control loop interaction. If we value one product more than another we could more tightly tune that end and detune the other. Taken to an extreme, we would have singleended control. This case was also examined. The composition at the end of interest was controlled using the XB-V or XD-R control structure (i.e., the bottoms composition is controlled using the vapor boilup rate and the distillate composition is controlled using the reflux flow rate). The PI controllers were tuned using the TL settings (Tyreus and Luyben, 1992) for integratorldead time processes. Regulatory Control. The generic rules for dualended regulatory control were also found to apply for columns having only one end controlled. Instead of the distillate control being affected by the bottoms action, it was influenced by the holdup filtering and speed-ofdetection effects. Servo Control. The performance of the column with one controller undergoing a setpoint change was the opposite of that for dual-ended control. Columns with the feed tray near the control end rose to the new setpoint and settled out more quickly than designs with the feed tray farther from the control end. Figure 12 shows a distillate transition from 0.97 mole fraction light component to 0.98 for several feed tray locations. These underdamped responses are much simpler than those for dual-ended control. This behavior can be explained by the process model, shown in eq 18. The time constant relating reflux flow rate to distillate composition becomes larger as the feed tray is located lower in the column. This suggests that the singleended control system is exploited the inherent column dynamics, as opposed to the loop interaction effect seen when both ends were controlled. A similar dynamic
Ind. Eng. Chem. Res., Vol. 34,No. 9, 1995 3033 +M% Feed Composition Disblrbanm 0.05
0.0451
4
p,q-1.0
.
0.025
.-.-.:.-.-.-.-------:
0.02
0.015A
0.5
1
1.5
n m (houri)
2
2.5
I 3
40% Feed Caposkion Dlslurbanm
1.14-
Figure 12. R-XD control for setpoint change inXDSet(97-98 mol %). Reflux rate scale by steady-state value.
_ _ - _ _ _ _ _ _ _ - _ - _ _ - _ i
...... ................................
effect was seen in the bottoms setpoint changes, improved servo performance for columns fed near the bottom. The time constants relating the vapor boilup rate to the bottoms composition show the appropriate trend. A generic rule for single-ended servo control can then be proposed: for the best control of servo changes t o the distillate, feed the column above the steady-state optimum feed tray; for the best control of servo changes to the bottoms, feed the column below the steady-state optimum feed tray.
0.S
0.5
2
2.5
+a%Feed Composition Disturbanfa
A study was also conducted t o test the effect of the feed thermal condition (4)on the controllability of a binary column. The five test columns were redesigned using q values of 0.5 and 0.0, corresponding t o a split vaporkquid feed and a saturated vapor feed, respectively. For each feed thermal condition, the columns were redesigned and the control loops retuned. The designs change because the liquid and vapor profiles of the column change. The vapor flow rate in the rectifying section becomes
+ (1- q)F
1.5
nm (hours)
Figure 13. Bottoms composition XB control for a 20% increase in feed composition z for NT = 22 and NF = 11 with both ends controlled.
Feed Thermal Condition Study
VR = v
1
(19)
a
0.9750
1.5
0.5
2
2.5
Tim (hours)
Because VR L V for all values-of q between 0 and 1,VR should be used in eq 4 to calculate the column diameter. Another necessary change is in eq 7 to calculate the height of the liquid over the weir in the stripping section. It becomes
(20) Regulatory Control. The two new sets of columns were simulated with the same four disturbances used in part one of the paper. The simulation results of a feed composition disturbance on a column with N F= 11for q values of 0.0,0.5, and 1.0 are shown in Figures 13 and 14. The simulations show that the control of the distillate end becomes worse as the q value is decreased, while the bottoms control shows the opposite trend. This might be expected because the disturbance propagates primarily up the column with an all-vapor feed but primarily down the column with an all-liquid feed. The impact of the feed thermal condition across a wide range of feed trays
+M% Food C ~ m p o ~ i tDiSblrbBnoo i~n
3
E
.__---_________ ...........................
0.91
0
I 0.5
1
1.5
n m (hours)
2
2.5
3
Figure 14. Distillate composition XD control for a 20%increase in feed composition z for NT = 22 and N F = 11 with both ends controlled.
is shown in Figures 6a, E a , and 16a. For a pure vapor feed, the worst distillate control was found for N F = 11 and the worst bottoms control for NF = 15. The distillate control improves as the feed tray goes up the
3034 Ind. Eng. Chem. Res., Vol. 34,No. 9, 1995 2
A +20% Feed ComporiUon Dirtwbonee 0 0 7 ,
180.
-Y C
B
140.
+
+
m
~ 1 0 0 .
*
80.
$
60.
v,
+
+ +
160.
f 120.
6
+ +
9001
+
+,?U% Feed flow Rate DlsUrbanca
8: +20% Feed Flow k t e Dimurbmce +-+ ' I
1000,
+
+
+
+
40.
oO09,
20.
0O0
+
. +
0O0
0.015l 0
I 0.5
1
1.5
2
2.5
2
2.5
Tim (hours) C: Step in Xdret (-0.01 mole frsclion)
I
501
D: Step in Xbrat ( ~ 0 . 0 1mole haion) 501
I
O.%l0 5
NF
10 NF
15
4 0 % Feed Composiflon DiWrbanr
180
8: +20% Feed Flow Raw Dlrturbance
looor---l 900
C: Step In Xdret (-0.01 mole franioi
q-----
NF
NF
Figure 16. Integral square error versus feed tray for q = 0.5 and NT = 22. 0 denotes bottoms composition XB and distillate composition XD(both ends controlled).
-I 0.5
20
Figure 15. Integral square error versus feed tray for q = 0.0 and NT = 22. 0 denotes bottoms composition XB and + denotes distillate composition XD(both ends controlled).
2oi
3
+ denotes
column, and the bottoms control improves as the feed tray goes down the column, which follows the generic rule for regulatory control. For a q value of 0.5 (Figure
1
1.5
"e(hours)
Figure 17. Bottoms composition XB for a 20% increase in feed flow rate F for NT = 22 and NF = 11 (both ends controlled).
161, the magnitude of the integral square error for the composition disturbance is balanced much more equally on the two ends because the disturbance is propagating about equally due to the split feed. Again we see the trend of better distillate control when the feed tray is higher and better bottoms control when the feed tray is lower. The simulation results for three values of q in the presence of a feed flow rate disturbance are shown in Figures 17 and 18. In the case of a feed rate disturbance, distillate control is found to be better for a liquid feed (high q ) and bottoms control is better for a vapor feed (low q). From Figures 15b and 16b we also see the same trends as before: distillate control is improved at a higher feed tray, and bottoms control is improved at a lower feed tray. Two more generic rules for dual-ended control can be suggested from the feed thermal condition study: To minimize distillate disturbances, a liquid feed is preferred. To minimize bottoms disturbances, a vapor feed is preferred. The generic rules suggested for improving dual-ended regulatory control by varying the feed thermal condition also apply for columns having only one end controlled. Servo Control. The control of bottoms and distillate setpoint changes for single- and dual-ended control were very similar in trend and magnitude for all the values of q tested (Figures 15c and 16c for dual-ended distillate and Figures 15d and 16d for dual-ended bottoms). This might be expected because the disturbance is not coming from the feed.
Conclusions The effect of feed tray location on steady-state economics and controllability for a binary distillation
Ind. Eng. Chem. Res., Vol. 34,No. 9, 1995 3036 +20% Feed flow Rata DlrtJrbanca
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Figure 18. Distillate composition XD for a 20% increase in feed flow rate F for NT = 22 and N F = 11 (both ends controlled). Table 5. Summary of the Generic Rules Proposed (B = Bottoms Product More Important, D = Distillate Product More Important, Both = Both Products Important, 2 = Dual Ended, 1 = Single Ended)
tion ( q ) had the greatest impact on the controllability tests. Again, a potential physical explanation was provided and a generic rule was proposed for this effect. The generic rules proposed in this paper for improving the controllability of binary distillation columns are summarized in Table 5. In this table, B indicates when the bottoms is the important product and D indicates when the distillate is the product of greater interest. Both in the first column shows the result when both products are equally important. The numeric value indicates whether the result is for single- or dual-ended control. Obviously for single-ended control, the controlled end is also considered the more significant product.
Nomenclature Ac = condenser heat transfer area (ft2) AR = reboiler heat transfer area (ft2) B = bottoms flow rate (lb m o b ) D = distillate flow rate (lb mom) DC = diameter of column (ft) F = column feed flow rate (lb mom) how, = height of liquid over weir in rectifymg section (ft) how,= height of liquid over weir in stripping section (ft) KU = ultimate gain LC = length of column (ft) M,, = holdup on tray n (rectifylng section) (lb mol) MflS= holdup on tray n (stripping section) (lb mol) Mtat = total column holdup (lb mol) MW = molecular weight (1bAb mol) N F = feed tray location W$t = optimum steady-state economic feed tray NT = total number of trays in column N;F'" = minimum number of trays in column P = pressure (psia) q = fraction of the heat of vaporization (AHvap) required to turn the feed into saturated vapor R = reflux flow rate (lb mom) Rgas= perfect gas constant (psia ft3/(lbmol OR)) T = temperature UC= heat transfer coefficient for the condenser (Btu/(h O F
Regulatory Control feed below optimum feed below optimum feed above optimum feed above optimum feed at optimum Servo Control, Bottoms Changes feed above optimum feed below optimum feed above optimum feed below optimum Servo Control, Distillate Changes feed below optimum feed above optimum feed below optimum feed above optimum Feed Thermal Condition, Regulatory Control vapor feed preferred vapor feed preferred liquid feed preferred liquid feed preferred
V = vapor boilup (lb mom) X B = bottoms composition (mole fraction) X B = ~setpoint ~ ~ for bottoms composition controller (mole fraction) XD = distillate composition (mole fraction) X D = ~setpoint ~ ~ for distillate composition controller (mole fraction) z = column feed composition (mole fraction)
column have been explored. Each column was designed and tuned separately using consistent methods. The optimum steady-state design positioned the feed tray to minimize vapor boilup. For a small penalty in capital and energy costs, the controllability of the column was improved by moving the feed tray. This increase in controllability was demonstrated by both nonlinear simulation and linear frequency analysis for four typical dynamic situations. A possible explanation was suggested, and generic rules were proposed to take advantage of this behavior. The effect of this phenomenon was tested for several values of column parameters. The feed thermal condi-
Greek Symbols a = relative volatility ATc = condenser temperature differential (OF) ATR = reboiler temperature differential ("F) AHvap= heat of vaporization (Btdlb) PR = liquid hydraulic time constant for the rectifymg section (h) PS = liquid hydraulic time constant for the stripping section (h) eL = liquid density in column (lb/ft3) ev = vapor density in column (lb/ft3) 19 = vapor velocity in the column (ft/s) wu = ultimate frequency (radh)
B12 Bll Dl2 Dl1 BotW2 B12 B11 D12 Dl1 B/2 Bll Dl2 Dl1 B12 Bl1 Dl2 Dl1
( O R )
ft2))
UR= heat transfer coefficient for the reboiler (Btu/(h "F ft2))
3036 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995
Literature Cited Douglas, J . M. Conceptual Design of Chemical Processes; McGrawHill Publishing Company: New York, NY,1988. Downs, J.; Hiester, A,; Miller, S.; Yount, K. Industrial viewpoint on desigdcontrol tradeoffs. In ZFAC Workshop on Integration of Process Design and Control, Baltimore, Maryland, 1994. Henley, E. J.; Seader, J. D. Equilibrium-Stage Separation Operations in Chemical Engineering; John Wiley & Sons: New York, 1981. Luyben, W. L. Derivation of transfer functions for highly nonlinear distillation columns. Znd. Eng. Chem. Res. 1987,26(12), 24902495. Luyben, W. L. Process Modeling, Simulation and Control for Chemical Engineers; McGraw-Hill Publishing Co.: New York, 1990.
Papastathopoulou, H. S.; Luyben, W. L. Tuning controllers on distillation columns with the distillate-bottoms structure. Znd. Eng. Chem. Res. 1990,29(9), 1859-1868. Tyreus, B. D.; Luyben, W. L. Tuning PI controllers for integfatod 2625dead time processes. Ind. Eng. Chem. Res. 1992,31(11), 2628.
Received for review December 16, 1994 Revised manuscript received May 4, 1995 Accepted May 23, 1995@ IE940743R @Abstractpublished in Advance A C S Abstracts, August 1, 1995.
