Ind. Eng. Chem. Res. 1995,34, 3907-3915
3907
Capacity-Based Economic Approach for the Quantitative Assessment of Process Controllability during the Conceptual Design Stage Timothy R. Elliott and William L. Luyben* Chemical Process Modeling and Control Research Center and Department of Chemical Engineering, Lehigh University, Iacocca Hall, 11 1 Research Drive, Bethlehem, Pennsylvania 18015
This paper addresses what is unquestionably the most important unsolved problem in process control: how to quantitatively incorporate dynamic controllability into conventional steadystate economic design. Historically, plant design was the precursor to control system design. In recent years it has finally been widely recognized t h a t much can be gained by simultaneously designing the plant with its control structure. Reduced capital costs, improved process dynamics, and tighter product specifications are advantages of this approach. This paper outlines a generic methodology called the capacity-based economic approach that can be used to compare or screen preliminary plant designs by quantifying both steady-state economics and dynamic controllability. The method provides a n analysis tool t h a t explicitly considers variability in product quality. The design of a simple coupled reactor/stripper system is used to demonstrate the method.
Introduction
plant Design 111
Probably the most important unsolved problem in the area of process control is how to quantitatively incorporate dynamics into the economics of process design. If we consider three plant designs that meet the same production requirements and exhibit the total annual cost (TAC) relationship shown in eq 1, steady-state
TAC 1 < TAC2 < TAC3
1
(1)
Plant Design #2
I
2,
economic analysis would lead us to build the plant that minimizes costs, plant design 1. On paper this choice appears to be excellent. We have designed a plant that will meet the market needs and in doing so minimized costs. What we have failed t o do is investigate the operability and controllability of the three designs. If the product variability is quite high, the product will become less desirable to the customer and may incur additional reprocessing or disposal costs, resulting in decreased profitability. Figure 1 depicts closed-loop dynamic responses for the three different plant designs. With the specification limits on the product set at f l , plant design 1 produces significantly more off-spec product than plant design 2, while plant design 3 produces all on-spec product. If this behavior could be predicted and quantified, justification for building a plant design which exhibits a higher total annual cost (plant design 2 or 3) may be warranted. Several approaches in assessing process controllability can be found in the literature: (1)Gannavarapu and co-workers (1990) developed a constraint-based approach which uses the “perfect control assumption” as an upper bound for which all designs are to be compared. The controlled variables are fixed in the presence of disturbances so that all variation in the system is absorbed by the remaining state variables. In order for the process to remain within its state constraints, the plant’s steady-state operating point must be moved away from its design optimum t o accommodate the disturbances. Plant ~
* To
whom correspondence should be addressed. Phone: (215)758-4256. Fax: (215)758-5297. Internet E-mail: w11W 1ehigh.edu. 0888-588519512634-3907$09.00/0
Plant Design X3 2
., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ’0
0.1
0.2
03
0.4
0.5
0.6
07
0.8
0.9
_1 1
t
Figure 1. Closed-loop dynamic responses for three hypothetical plant designs.
designs are compared on their ability to minimize the move from the optimum. Although time consuming, this approach provides a means for screening plant designs. In practice however, plants will never operate under the “perfect control assumption”, and the feasibility of operating a t the modified steady-state remains in question. (2) Brengel(1991) combines two nonlinear programs into a coordinated design and control optimization. Several types of disturbances are simulated, and the design objective function is penalized for poor controllability. The solution of this optimization problem is a plant that remains subject to process constraints and minimizes the deviation from the set point. The inte0 1995 American Chemical Society
3908 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995
gration of design and control is facilitated through the use of a weighting factor called the design and control integration parameter. Venture profit is examined as a function of the design and control integration parameter t o determine the best design. However, the physical significance and means of selecting the design and control parameter are unclear. (3) Luyben and Floudas (1994) formulate a multiobjective optimization problem that incorporates open-loop controllability measures with steady-state economic design. This approach constructs a mixed-integer nonlinear (MINLP) optimization problem that minimizes cost (capital and utility) and the steady-state condition number. The combined design and control optimization is facilitated through the selection of trade-off weights, and therefore the results obtained from this method are strongly dependent on the weights used. The capacity-based economic approach described in this paper incorporates both steady-state economics and dynamic controllability into a simple methodology that allows conceptual plant designs t o be compared quantitatively. This method is an assessment tool that may be used to screen alternative plant designs on their ability to minimize costs (capital and operating) and process variability. The method can be applied at the stage in the conceptual design where we want to compare alternative flow sheets and various choices of design parameters. It should be emphasized that this is not a synthesis tool but an analysis tool. The dynamics of the process are approximated by linear models. Closed-loop frequency response techniques are used to determine the “ p e a k or worst disturbance conditions. In general, Bode plots of closedloop regulator transfer functions exhibit a maximum value of log modulus a t some peak frequency where the magnitude ratio of the controlled variable to load disturbance is greatest. By noting the peak frequency and phase angle for each load disturbance to the plant, we can perturb the plant with load disturbances entering the process as sine waves occurring at their respective peak conditions. Although sinusoidal disturbances do not resemble actual disturbance sequences that may occur in the plant, they do represent a worst case disturbance scenario that is inherent to the plant. If desired, nonlinear simulation of expected disturbance sequences may be incorporated within the framework of this methodology. As we demonstrate in this paper, the results using the worst case sinusoidal disturbances were the same as those obtained when the disturbances were a sequence of typical load changes. Having specified limits for product quality variability, plant designs are screened on their ability to maximize annual profit in the presence of their associated peak disturbances. Plant capacity is reduced by the fraction of time that the product is outside the specified upper or lower specification limits. When product is off-spec, profits are being lost and reprocessing or disposal costs are incurred. Thus the method deals explicitly and quantitatively with the question of product quality variability, which is an increasingly important criterion of control performance (Downs et al. (1994)).
process flow sheets that differ in several design parameters. The process flow sheet must be sufficiently defined in order to generate a process model necessary for this approach. Design alternatives are compared on the basis of their ability to maximize annual profit for a given product quality specification range: (1)Perform steady-state design. (a) Select processing equipment configuration. (b) Determine operating conditions required to achieve the desired steady state. (c) Identify likely disturbances entering the process and their magnitudes. (2) Linearize the nonlinear dynamic process model, identifying controlled, manipulated, and load variables along with the process outputs for which product quality is to be judged. S =Ax
+ Bm + DL
(2)
y = cx
(3)
(3) Select a control structure for the process. Derive open-loop transfer functions relating the controlled variables to the manipulated variables and to the load disturbances
a(.).
(4)Tune controllers B(+ In this paper we will use conventional PI controllers in a multiloop SISO structure. However, this method can be applied for any type of controller (multivariable, model predictive, etc.). (5) Derive closed-loop regulator transfer functions. (6) (6) Obtain a magnitude ratio, phase angle, and frequency that correspond t o the peak log modulus of the Bode plot of the closed-loop regulator transfer function for each load disturbance. (7) Obtain a closed-loop linear time response to “peak sinusoidal disturbances. (8)For a desired specification range, estimate the percent capacity of the design from the linear time response.
specification range = USL - LSL where USL = upper specification limit
LSL = lower specification limit (9) Calculate annual profit accounting for capital costs, operating costs, and the reprocessing of off-spec material. ‘ann
= Mprod,,,Pprod
- Mrawcraw
- Mprod,RCprod,ff ]cap
Capacity-BasedEconomic Approach The goal of the capacity-based economic approach is to quantitatively measure both steady-state economics and dynamic controllability. The methodology can be used to compare alternative process flow sheets and
(7)
-
- C o p e r (8)
where
Pa,, = annual profit Mprod,,= mass of on-spec product produced per year
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3909 Pprod
= market price per mass of product
Mraw= mass of raw materials fed per year Craw= cost per mass of raw materials kfprodofl Cprod,,
= mass of off-spec product produced per year
= disposal and/or reprocessing
t i l l l
er,
cost per mass of off-spec product
Icap = annual capital investment cost
Caper = annual operating cost for the plant Process Studied In order to demonstrate the capacity-based economic approach, a reactor/stripper system was studied. Reactant A is fed t o the process and is reacted to form product B. The reactor/stripper system is one of several conceptual designs for this processing step. Other alternative designs include a single continuous stirredtank reactor (CSTR), several CSTRs in series, a plugflow reactor, and a reactor/full column system. For purposes of illustrating the method, we only consider in this paper the effects of design parameters on the controllability of a fured flow-sheet process (the reactor/ stripper system). The process is that studied by Luyben (1993). A firstorder, irreversible liquid phase reaction A B occurs in a single nonisothermal CSTR of constant holdup, VR. The reactor is operated at 140 O F with a specific reaction rate k of 0.34086 h-l. An activation energy Eact of 30 000 Btu/(lb mol) was used. Fresh feed with a flow rate F of 239.5 lb m o m and a composition zo of 0.9 mole fraction is fed into the reactor along with a recycle stream. The reactor effluent is assumed to be saturated liquid and contains a binary mixture of both A and B because some A remains unreacted. This binary mixture of constant relative volatility (a = 2.0) is fed to the stripper (Figure 2) that takes the product B out the bottom and the reactant A out the top t o be recycled back t o the reactor.
-
Steady-State Design Procedure In this study we assume constant density and molecular weight, equimolal overflow, theoretical trays, a total condenser, and a partial reboiler. Tray holdups and the liquid hydraulic time constant are calculated from the Francis weir formula, assuming a l-in. weir height. The reflux drum and column base were sized to provide 5 min of holdup resulting from the respective steady-state flow rates into each. The system has two degrees of freedom. For this study, the reactor holdup VR and the number of trays in the stripper NT will be taken as design parameters. The following steady-state design procedure was used to obtain the necessary operating conditions for the reactor and the stripper to achieve a bottoms purity XB of 0.0105 mole fraction of component A. Given F,20, and k : (1)Select a reactor holdup VR. (2) Calculate the concentration of A in the reactor z .
I'lOlW
Figure 2. Schematic of reactor/stripper recycle system.
(3) Select the number of trays in the stripper NT. (4)Use a rating program to determine the vapor boilup rate Vs needed when NT trays exist in the stripper. First guess VS and then calculate Fout.
Fout= Vs
+F
(10)
Using Vs and Foutcalculate tray-by-tray from XBup NT trays to obtain the vapor composition on the top tray Y N ~ . Compare this value with that obtained from a component balance around the reactor (eq 11).
(11) If the value of Y N ~is not the same as that calculated from the reactor component balance, a new value of Vs is guessed. (5) Calculate the diameter of the reactor DR. (12) where MW = molecular weight = 50 lb/(lb mol) and p~ = liquid density = (50 lb/ft3). The reactor was assumed to have a diameter-to-height LR ratio of 0.5. (6)The diameter of the column was calculated assuming an F factor of 1.
F factor = O(&)'''
(13)
where pv = vapor density in the column = M W PIRgasT, P = pressure = 44.7 psia, R g a s = perfect gas constant = 1545/144 psia ft3/(lbmol OR), T = temperature = 300 + 460 OR, and d = vapor velocity in the column (fvs). (7) The diameter of the column (Dc)is then calculated.
(8) The height of the column (Lc) is calculated assuming 24% tray spacing and allowing 20% more height for base level volume. Lc = 2.4NT
(15)
(9) The liquid height over the weir (how)is calculated from the Francis weir formula.
3910 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 Fout how
Mw
-
Optimum Volume I WOO 0 , Optimum Trays I 9
2/3
(16)
= (9600pLDc)
Saturated liquid feed to the stripper is assumed. (10) The holdup on each tray M , is calculated using the height over the weir and a 1-in. weir height.
where M , = holdup on tray n (lb mol). (11)The liquid hydraulic time constant (p) in hours is calculated for the column using a weir length equal to 80% of the diameter of the column. Reaclor Holduo llbmolsl
(18) (12) The heat transfer areas of the reboiler and the condenser were calculated using the steady-state vapor boilup rate and a heat of vaporization (AHvap)of 250 Btdlb. The reboiler was assumed t o have a heat transfer coefficient UR = 100 Btu/(h OF ft2) and a temperature differential ATR = 50 O F . The condenser was assumed t o have a heat transfer coefficient Uc = 150 Btu/(h O F ft2) and a temperature differential ATc = 20 OF.
Figure 3. Total annual cost versus reactor holdup and number of trays. Table 1. Results of Steady-State Economic Analysis for
NT = 15a VR
2
VS
reactor cost
energy cost
1000 3000 5000 7000 10000
0.625 0.208 0.125 0.089 0.063
1056.9 383.9 308.5 271.3 236.6
355 704 967 1193 1489
579 210 169 149 130
a
(19)
All costs are multiples of $1000.
1000 lb mol < VR < 10 000 lb mol
10 trays (20)
NT < 20 trays
(25) (26)
The reactor holdup and total number of trays were varied by increments of 1000 lb mol and 1 tray, respectively, giving a total of 110 different designs. The (13) The capital costs of the reactor, column, trays, optimum steady-state design for this process configuand heat exchangers were estimated using an M&S ration (total annual cost minimized) was determined to index of 950 (Douglas (1988)). have a reactor holdup VR of 3000 lb mol and a stripper reactor cost = with 19 total trays NT(Figure 3). Since the concentraM&S . ~ D R ~ ' ~2.18 in the reactor z is inversely proportional to reactor 1.5-101 ~ ~ (3.67 & ~ ' 1.20)) ~ ~ ~ ( tion volume (eq 9),the mole fraction exiting the reactor z is 280 quite high for small reactor holdups. With more A (21) entering the stripper, more vapor boilup is required to column cost = perform the separation. Energy costs and column capital costs increase because the diameter of the M~S101.9DC1~066L~~802(2,18 (3.67 1.20)) (22) column and the required heat transfer area increase as 280 reactor capital cost decreases (Table 1). Increasing the number of trays in the stripper decreases energy and tray cost = w 4 . 7 D c 1 . 5 5 L c(1.0 0.0 1.7) (23) heat exchanger costs while adding tray and column costs 280 (Table 2). heat exchanger cost = M&S Application of the Capacity-Based Economic -101.3 ( ~ 4 ~Ac0.65)(2.29 " ~ ~ (1.35 0.10)3.75) Approach to the Reactor/Stripper System 280 (24) Disturbances were assumed to occur in the fresh feed flow rate F and fresh feed composition 20. The magniThe annual capital cost was assumed to be one-third of tude of each disturbance was set at 10% of its steadythe capital cost (3-yr payback). Materials of construcstate value. Since the product stream for this process tion were stainless steel, and the design pressure was is the bottoms of the stripper (Figure 2), the controltaken as 300 psig. Energy costs were calculated from lability of each design will be evaluated on the basis of the vapor boil-up rate using a cost of $5/106 Btu, and its ability to keep the bottoms composition XB within the cost of cooling water was $O.OOOl/gal. the desired specification range. A nonlinear model of the process was linearized using Results of Steady-State Design Taylor series expansion about the steady-state operating The range of the two design parameters (VRand NT) point, giving UVT 5 states. In order to maintain the specific reaction rate used in the designs, reactor selected for this study is given below.
+
+
+
+
+
+
+
+
+
+
Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995 3911 Table 2. Results of Steady-State Economic Analysis for V R = 50ooa
NT 10 13 15 17 20 Q
column cost 117 137 151 164 185
tray cost 0.295 0.355
0.397 0.441 0.509
heat exchanger cost 524 49 1 479 471 463
energy cost 194 175 169 165 161
All costs are multiples of $1000.
temperature T was controlled by manipulating the cooling water flow rate Fj to the jacket. The bottoms composition of the stripper is controlled using the vapor boil-up rate. Two 1-min lags were assumed on the temperature measurement, and three 1-min lags were assumed on the composition measurement. Conventional proportional-integral (PI) controllers were implemented t o provide fair dynamic comparisons between the designs. The ultimate gains and frequencies for each loop were found using linear frequency analysis. Both controllers were tuned using the Tyreus-Luyben settings (Tyreus and Luyben (1992)). Perfect reactor, reflux drum, and bottoms level control were assumed. Obviously, the control structure selected for this study is not the only possibility for this process configuration. The variable-volume structure proposed by Luyben (1993b)is one alternative. A model predictive controller is another. We have used this methodology to compare alternative control structures, but we do not include details of these studies in this paper. Comparisons of PI and model predictive controllers were performed to examine the effects of controller type on this analysis. We found that the type of control does not significantly affect the dynamic comparisons of alternative plant designs. Bode plots of the closed-loop regulator transfer functions relating the output variable XB (variable for which each plant design’s controllability will be evaluated) to the two specified load disturbances (F and 20) were generated. The ratio Xdzo is made dimensionless by composition transmitter spans of 0.1 mole fraction. The ratio X$F is made dimensionless by a flow transmitter span of two times the steady-state feed flow rate and a composition transmitter span of 0.1 mole fraction. As suggested and verified through nonlinear simulation by Luyben (1993), the dynamics of the reactorhtripper system will be dominated by the reactor, since its holdup is significantly larger than that of the column, and the controllability of the system will improve as the reactor holdup is increased. As the reactor holdup is increased, the reactor becomes more controllable because the larger heat transfer area gives better temperature control and because the larger volume does a better job in filtering disturbances to the stripper. Figures 4 and 5 show Bode plots for a fixed number of trays in the stripper (NT = 15). The Bode plots for both load disturbances (F and 20) demonstrate that reactor/ stripper systems with larger reactor holdups (fixed NT) are more controllable over most frequencies. For feed flow-rate disturbances, decreasing the number of trays in the stripper for a fixed reactor holdup improves the controllability of the reactorhtripper system (Figure 6). We attribute this departure from conventional wisdom to the fact that feed rate changes have less effect on shorter columns because these columns require more vapor boil up t o perform the separation, and therefore a change in F has proportion-
10
10’
’
1OD Frequency [radlhrl
10’
10’
Figure 4. Bode plot of closed-loop regulator transfer function relating X$ZO for different reactor holdups and NT = 15.
10’
10
’
1oo Frequency [rad!hr]
10‘
lo?
Figure 5. Bode plot of closed-loop regulator transfer function relating X$F for different reactor holdups and NT = 15. M-20 .5
-
U
Frequency [rod/hr]
Figure 6. Bode plot of closed-loop regulator transfer function relatingX$F for different numbers of trays in the stripper NT and VR = 5000 lb mol.
+
ally less effect on Fout(FOut = VS F). This trend is less pronounced for larger reactor holdups because there is less variability in the vapor boil-up rate as the total number of trays in the stripper is changed (Table 3).
3912 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 Table 3. Steady-Stateand Controller Parameters for Different Numbers of Trays in the Stripper NT and Three Reactor Holdups VR=
NT
vs
Mtot
Ku Xb-V
13 15 17 20
1422.6 1114.3 1056.9 1034.4 1023.7
VR = 1000 320.7 268.7 264.7 268.3 278.9
1.72 1.50 1.51 1.54 1.56
10 13 15 17 20
354.5 320.5 308.4 300.6 293.2
VR = 5000 93.3 90.1 89.9 90.5 92.3
4.53 4.05 3.99 4.05 4.27
255.4 241.5 236.6 233.3 230.3
VR = 10000 72.7 72.8 73.6 74.7 76.8
10
10
13 15 17 20
0 02 0
E! $001
h’ 0‘
5
10
15
20
25
30
35
40
45
50
H
; “01
0 02 0 0
5001 D
0
0
0 02
6.99 6.51 6.49 6.60 6.85
The ultimate gain on the bottom controller is made dimensionless using a flow transmitter span equal to twice the steady-state vapor boil-up rate and a composition transmitter span of 10 mol a
%.
Varying the number of trays in the stripper has little effect on the controllability of the reactor/stripper system in the presence of feed composition disturbances because they are filtered by the CSTR. Results obtained from the linearized model of the process were verified through nonlinear simulation. Relay feedback tests were performed on each loop, and closed-loop sine wave tests were performed on each load variable. Step tests for both load variables supported the controllability trends stated above. While the controllability trends are prevalent over most frequencies, the use of the peak disturbance conditions for this analysis provides a worst case prediction of the overall controllability for each design. The magnitude ratio and phase angle corresponding to the peak frequency for each load variable were used to generate closed-loop linear time responses to the two load disturbances occurring simultaneously. Figure 7 shows time responses of the linear model for five different reactor holdups and a fixed number of trays in the stripper (NT= 15). Reactor/stripper systems with smaller reactor holdups are found t o exhibit more variability in their product streams in the presence of the two feed disturbances (Figure 7). Discrete data were taken over a sufficient period of time (50 h for this study) to ensure that at least several periods of each peak frequency were able to affect the output. For each specification range, the data were then scanned to determine the percent of product that was off-spec for the simulation period. The capacity of the plant was reduced by periods of off-spec production. The results were then prorated to encompass one year’s time, and the annual capacity of the plant was estimated. Plant designs were then compared on their ability to maximize annual profit (eq 8) for a given specification range. Initially, the capacity-based economic approach was performed for a large specification range of 0.03 mole fraction. For this case, all 110 designs were at 100% capacity (i.e. no off-spec product was produced). As expected, with all designs operating at 100%capacity the design that minimizes conventional steady-state
8 h
$001
d
1 ‘0
5
10
15
20
25
30
35
40
45
50
4
10
1’5
2b
;5
d0
d5
d0
45
50
0 02
X 0 0
I
no01
> R
Figure 7. Closed-loop linear response to peak load disturbances occurring in feed flow rate F and feed composition zo for NT = 15. X B is mole fraction, and time is in hours. Spec Range = 0 0300 Opt Volume 3 m 0 Opi Trays. 19
Number 01 Trays
10
-
Reanor Holdup [Ibmols]
Figure 8. Specification range of 0.03 mole fraction.
total annual cost (VR= 3000 lb mol and NT = 19) is the preferred design. Several specification ranges were analyzed to determine if a process controllability trend existed for the two design parameters (VRand NT)and if it could be quantified. As the specification range decreases, the amount of off-spec product increases (annual profit decreases) because the reactorlstripper system is unable to hold the bottoms composition within the desired limits. Less controllable systems will produce more offspec product for a given specification range because their peak disturbance conditions produce more variability in the stripper’s bottoms stream. Figures 9-11 demonstrate the controllability trend for the reactor/ stripper system. Clearly the capacity-based economic approach demonstrates a trend in controllability for the two design parameters. For peak disturbances in feed composition and flow rate occurring simultaneously, the controllability of the reactor/stripper system improves as the
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3913 Spec Range = 0 0100 Opl Volume = 30000 Opl Trays = 19
2
-1
I
5
d t
, ""
-
0
5 ' 2
5
10
15
10
15
lime [hr]
$05
0 20 1o
m I
Number 01 Trays
0
5
lime IhrJ
Reanor Holdup [lbmols]
Figure 12. Disturbances in feed composition zo and flow rate F generated as integrated moving average (IMA) time series.
Figure 9. Specification range of 0.01 mole fraction. Spec Range = 0 0052, Opt Volume = 5wo 0,Opt Trays = 12
2.
.
c
proach demonstrates how incurring slightly higher total annual costs can significantly improve the profitability of the process by decreasing process variability (Downs et al. (1994)). This approach may be applied to all possible conceptual designs, allowing design alternatives (flow sheets and design parameters) to be quantitatively compared on the basis of both steady-state economics and dynamic controllability. For this study, 110 different plant designs were screened and compared. The following steps were performed for each design: (1) Steady-state values for UVT 5 states were obtained. (2)Linearized state space models for the stripper and the reactorlstripper system were generated. (3) Ultimate gains and frequencies were obtained, and the controllers were tuned. (4) Closed-loop regulator Bode plots were generated for each load disturbance, and peak disturbance conditions were identified. (5) Closed-loop linear responses were generated. (6) Discrete data for 100 different specification ranges were scanned. (7) Annual profit was calculated for each specification range. The following economic factors were used in this analysis: market price for on-spec product = $0.03/lb, raw material cost = $0.006/lb, and disposal cost for offspec product = $0.006/lb. These calculations required about 2 min of computation time per case on a RISC 6000 IBM Workstation model 340 using MATLAB.
+
10000
Number 01 Trays
Reador Holdup (Ibmols]
Figure 10. Specification range of 0.0052 mole fraction. Spec Range = 0 0032.Opl Volume = loo00 0 Opl Trays
-
10
F 5 t 0
f' 7 $05
0 20
10000
Number01 Trays
Reanor Holdup [lbmols]
Figure 11. Specification range of 0.0032 mole fraction.
reactor holdup is increased. However, defying conventional wisdom, controllability was found t o improve as the number of trays in the stripper decreases. The trend is largely due to the feed flow-rate disturbance, which has a much more significant effect on the system than the feed composition disturbance. A change in the feed flow rate permeates through the reactor and hits the stripper immediately whereas a change in feed composition must filter through the reactor before it affects the stripper. By quantifying both steady-state economics and dynamic controllability, the capacity-based economic ap-
Simulation of Nonlinear Process Model To verify the controllability trends found from the linear dynamic capacity-based economic approach, nonlinear simulations were performed in the presence of the two feed disturbances. The disturbances were assumed t o be integrated moving average (IMA) time series (Figure 12). Each disturbance L was generated discretely by the following expression (Box and Jenkins (1974)).
L(t)= L(t - 1)
+ (1 - O&(t - l))~,,
(27)
where at,is a normally distributed random number with a mean of zero and a variance 021,and 6% is a number between zero and one. The parameters used to generate the IMA feed disturbances were azz0= 0.000 005,O,, = 0.8,0 2=~0.000 000 25, and OF= 0.99. Figures 13 and
3914 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 IMA Disturbances: V=3000,NT=IS
- ’-‘; V=5000,NT=lZ -
I--”
0 0095
0 009
io Urne [hr]
Figure 13. Closed-loop nonlinear response for bottoms composition to IMA disturbances occurring simultaneously: (solid line) VR = 3000, NT = 19; (dashed line) VR = 5000, NT = 12.
14 show the closed-loop responses in XBand Vs for two different plant designs. The nonlinear simulation results support the controllability trends predicted by the capacity-based economic approach. It should be noted that the IMA disturbances, while being similar in magnitude, affect each reactor/stripper system less than their respective peak disturbances (worst possible case).
Conclusions and Future Work
5
V
I
io
15
t h e [hr]
Figure 14. Closed-loop nonlinear response for vapor boilup rate to IMA disturbances occurring simultaneously: (solid line) VR = 3000, NT = 19; (dashed line) VR = 5000, NT = 12. c = observability matrix D = coefficient matrix of the load variables DC = diameter of the column (ft) D R= diameter of the reactor (ft) Eact= activation energy (Btu/(lb mol)) F = fresh feed flow rate (lb mom) Fout = flow rate out of the reactor (lb mom)
= open-loop process transfer function matrix = open-loop load transfer function matrix how= height of liquid over weir (ft) I = identity matrix K = specific reaction rate (h-l) KU = ultimate gain L = vector of the load variables LC = column height (ft) LR = reactor height (ft) m = vector of the manipulated variables M,,= holdup on tray n (lb mol) Mtot = total column holdup (lb mol) MW = molecular weight (lb/(lb mol)) NT = total number of trays in the stripper P = pressure (psia) q = fraction of the heat of vaporization (AI-Ivap)required to &(,
The capacity-based economic approach has been introduced and demonstrated on a simple process. We believe that this analysis method has the potential to be a viable screening tool for a rapid and quantitative assessment of both steady-state economics and process controllability. The approach was applied to 110 different design parameter alternatives for a reactor/ stripper recycle system. The controllability of the system was found to improve as the holdup in the reactor was increased and the number of trays in the stripper was decreased. The results of the linear dynamic model were verified through simulation of the system’s rigorous nonlinear model. We are continuing t o explore the general applicability of this methodology. More complex process flow sheets are currently under study. Results will be reported in future papers. We believe this method represents a major breakthrough in quantifying the interaction between design and control.
Nomenclature A = coefficient matrix of the states B = coefficient matrix of the manipulated variables B(#) = controller transfer function matrix
turn the feed into saturated vapor Rgas= perfect gas constant (psia ft3/(lbmol OR)) T = reactor temperature (OR) UC= heat transfer coefficient for the condenser (Btu/(h ”F ft2)) UR = heat transfer coefficient for the reboiler (Btu/(h OF ft2)) VR = reactor holdup (lb mol) VS = vapor boilup (lb mom) x = vector of the states XB = bottoms composition (mole fraction) y = vector of the output variables
Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995 3915 = vapor composition on the top tray of the stripper (mol fraction) z = reactor composition and column feed composition (mole fraction) zo = fresh feed composition (mole fraction) Greek Symbols
a = relative volatility ,f3 = liquid hydraulic time constant (h) ATc = condenser temperature differential ( O F ) ATR = reboiler temperature differential ( O F ) AHvap= heat of vaporization (Btdlb) p~ = liquid density (lb/ft3) pv = vapor density in stripper (lb/ft3) 8 1 = IMA time series parameter for disturbance L t? = vapor velocity in the stripper (Ws) 01 = IMA time series parameter for disturbance L wu = ultimate frequency (radiansh)
Downs, J.; Hiester, A.; Miller, S.; Yount, K. Industrial viewpoint on desigdcontrol tradeoffs. In IFAC Workshop on Integration of Process Design and Control, Baltimore, Maryland, 1994. Gannavarapu, C.; Barton, G.; Perkins, J.; Lear, J . Economic choice of control structures for continuous processes. In Fourth Conference on Control EngineeringThe Institution of Engineers Australia, 1990. Luyben, W. L. “Dynamics and control of recycle systems. 2. Comparison of alternative process designs”. Znd. Eng. Chem. Res. 1993a,32 (31, 476-486. Luyben, W. L. “Dynamics and control of recycle systems. 3. Alternative process designs in a ternary system”. Znd. Eng. Chem. Res. 1993b,32(61, 1142-1153. Luyben, M. L.; Floudas, C. A. “Analyzing the interaction of design and control. 1. A multiobjective framework and application to binary distillation synthesis”. Comput. Chem. Eng. 1994,18 (lo), 933-969. Tyreus, B. D.; Luyben, W. L. “Tuning PI controllers for integrator/ dead time processes”. Znd. Eng. Chem. Res. 1992,31 (111, 2625-2628.
Received for review October 6, 1994 Revised manuscript received February 14, 1995 Accepted April 4, 1995@
Literature Cited Box, G. E. P.; Jenkins, G . M. ”Some recent advances in forecasting and control”. Appl. Stat. 1974,23(2), 158-179. Brengel, D. D. Design for the Controllability and Operability of Chemical Processes. Ph.D. dissertation, University of Pennsylvania, 1991. Douglas, J. M. Conceptual Design of Chemical Processes; McGrawHill Publishing Company: New York, 1988.
IE9405807
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Abstract published in Advance ACS Abstracts, October 15,
1995.
