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Dynamic controllability and resiliency diagnosis using steady state process flowsheet data
Computers them. Engng
Vol. 20, No. 4, pp. 325-335. lYY6 Copyright @ 19% Elsevier Science Ltd Printed in Great Britain. All rights reserved
Pergamon 0098-1354(95jBBO23-2
098-1354196
$15.00+0.00
DYNAMIC CONTROLLABILITY AND RESILIENCY DIAGNOSIS USING STEADY STATE PROCESS FLOWSHEET DATA 0. WEITZ and D. R. LEWIN~ Department of Chemical Engineering, Technion IIT, Haifa 32000, Israel Abstract-A simple procedure for assessing the controllability and resiliency of a process flowsheet is presented. The approach relies on a simplified modeling strategy, whereby an approximate linear dynamic model of the process is derived from steady-state flowsheet information. The resulting linear approximation can then be tested using the wealth of available linear controllability and resiliency measures available. In this paper, the procedure is used to screen the disturbance resiliency of alternative heat-integrated distillation column schemes reported in the literature. The results confirm the conclusions obtained by others largely relying on closed-loop simulations using rigorous dynamic models. The overall approach appears to be very promising as a short-cut diagnostic and screening tool, which in theory could be integrated into commercial flowsheeting software.
1. INTRODUCTION
The design of continuous chemical processes is usually carried out for a given operating point, assuming that an appropriate controller will succeed in regulating the process. It is possible that under certain conditions, the final design will be unable to satisfy the process specifications, or that they will only be attainable if a sophisticated (expensive) control system is augmented to compensate for the (bad) design. Usually, alternative designs are judged on economics alone without taking operability into account. This may lead to the elimination of easily-controlled but slightly less economical alternatives in favor of slightly more economical designs which may be extremely difficult to control. Moreover, plants which are difficult to control will be inflexible and possess poor disturbance rejection properties, which will themselves incur extra cost. These problems call for the introduction of controllability and resiliency measures as diagnostic tools within the design procedure. Controllability can be defined as the ease with which a continuous plant can be held at a specific steady state. An associated concept is switchability, which measures the ease with which the process can be moved from one desired stationary point to another. Similarly, resiliency measures the degree to which a processing system can meet its design objectives despite external disturbances and uncertainties in design parameters. Clearly, it wold be greatly advantageous to be able to predict how well a given flowsheet meets these dynamic performance requirements as early as possible in the design. t To whom all correspondence
should be addressed.
This paper describes a simple procedure which can be used to investigate the degree to which a process flowsheet is resilient to external disturbances. The approach involves the derivation of an approximate linear dynamic plant model, generated solely from steady-state flowsheet information and independent of control system design. As such, it is intended to be used as a rapid screening method which will eliminate design alternatives which cannot meet the basic requirement of being able to maintain the process at the required steady state operating specifications. Resiliency to parametric uncertainty can in principle be handled in the same framework (see Naot and Lewin, 1995). The issue of switchability should in general be addressed using more detailed dynamic modeling. Since this will invariably involve dynamic simulation requiring contol system design, it should be implemented only after significantly screening has already weeded out poorer designs. The paper is structured as follows. Firstly, we briefly review commonly used controllability and resiliency measures. Next, we outline the steps involved in the overall procedure. Finally, as an example, the proposed method is used as a screening tool in the selection of the appropriate flowsheet for the design of a heat integrated distillation column system.
2. CONTROLLABILITYANDRESILIENCYMEASURES
Quantitative assessment of controllability and resiliency (C&R) for process plants has naturally considerable interest industrially. generated
326
0.
WEITZ and D. R. LEWIN
Methods
have been developed, based on linear mathematical analysis of plant models, that can assist the designer in rejecting designs which may have the desired steady-state properties but unacceptable dynamic ones. Commonly used controlfability measures include the relative gain array [RGA Bristol (1966)] and the minimized condition number [MCN, Nguyen et al. (1988)], both of which rely on a linear model describing the effect of control variables on the process outputs P(s). Typical resiliency measures are the disturbance condition number (Skogestad and Morari, 1987), the disturbance cost (Lewin, 1995) and the relative distrubance gain array (Chang and Yu, 1992), which in addition to P(s), also require a disturbance model, Pd(s), which describes the effect of disturbances on the process outputs. These linear process models are not usually readily available at the early stages of the process design. Thus, a crucial step in the analysis is the generation of adequate linear approximations which can be derived using the minimal information available in preliminary flowsheeting of a potential design. An alternative C&R diagnosis approach relies on closed-loop simulations using rigorous dynamic process models. This is impractical for diagnosis of controllability and resiliency problems in the early stages of process design for two reasons: (a) it is both time and resource consuming and may require unavailable data; and (b) any conclusions obtained will depend on the control scheme adopted in the simulations, and as such will compromise the objectivity of the diagnosis. In this study, the most significant resiliency screening method was found to be the disturbance cost (DC), which is a direct measure which accounts for the degree to which the controlled process is insensitive to disturbances, d defined as:
In other words, we assume perfect control and then measure the integral square error of the control action u which satisfies this assumption. On the one hand, perfect control is in general impossible, either because of noncausality, instability or realizability problems. On the other hand, perfect control defines an upper bound to performance, and is independent of any particular controller design or tuning. These advantages make this measure an ideal one to use at an early stage of the design. If perfect control cannot give adequate disturbance rejection, or if such a control action is limited by process constraints, then clearly one should question the selection of the process design itself.
3. SHORT-CUT CONTROLLABILITY
AND RESILIENCY
DIAGNOSIS
The proposed steps:
approach
involves the following
(I) Conceptual analysis of the process flowsheet and the implied control problem. This involves the definition of process outputs to be controlled y(t) and process inputs available as control variables u(t) and those pertaining to disturbances, d(t). A conceptual linear approximation is therefore: Y(S)= p(s)+)
+ P&W(s).
(2)
(2) The definition of component parts, their inputs and outputs. Component parts are defined as the process sub-sections which can be approximated by simple lags. (3) Steady state simulation of the complete process, decomposed into its component parts, using a process simulator. (4) Computation of the steady state gains of the component parts of the process by perturbing each input, one at a time. (5) Computation of time constants and delay times on the basis of either mixed tank or plug flow assumptions, as appropriate. The flow rates and capacities are available from the steady state design data. In this paper, we will limit the discussion to the development of approximations to the time constant and time delays for distillation columns and heat exchangers. Similar expressions can be derived for other processing equipment (see Naot and Lewin, 1995). of approximate overall input(6) Generation output transfer function matrices P(~w) and P&u), for the complete flowsheet. This involves the re-combination of the component parts of the process as dictated by plant topology, and the generation of the frequency response of each element of the transfer function matrices. (7) Computation of frequency dependent C&R measures directly using the approximate linear model {P(~w), P&w)}. The analysis of disturbance resiliency is carried out for steps in the vector a. The expected magnitudes of each disturbance variable must be defined a priori. All the possible direction combinations are tested. In our study, we have used PRO/II to carry out steady state calculations and have found it convenient to carry out all frequency and time domain
Dynamic
controllability
using MATLAB (1991) and computations SIMULINK (1992). Clearly, the main task involved is the generation of adequate dynamic process models. Thus, before illustrating the potential impact of this simple method on representative flowsheet screening problems, we shall first detail the modeling approach taken.
and resiliency
diagnosis
321
change Au, is made small enough such that the gain computed is relatively insensitive to perturbation direction. In order to ensure that this is done accurately, it is important that the fowsheet convergence tolerances on the computed outputs be set appropriately. 4.2. Computing the dynamics matix W(s)
4. GENERATING SIMPLE DYNAMIC PROCESS MODELS
Following conceptual analysis of the flowsheet, and the definition of component parts, the flowsheet is represented as a connected set of component parts, each expressed in the following linear form: Y(s)=Kx’P(s).U(s),
(3)
where U(s) and Y(s) are input and output component part vectors in (em and %“, respectively, K is a matrix of steady state gains in ijn”‘“’ and Y(s) is a matrix describing the dynamics in %nx’n, each element of which is of the form:
There has been recent interest in the literature on the generation of simple dynamic models for processing equipment, such as heat exchanger networks (Woolff et al., 1991) and simple binary distillation columns (Skogestad, 1987). Our approach requires the representation of dynamic elements as time constants and delay times, and these must be computable using only flowsheet data. The following constitutes the model equations utilized in this study for distillation columns and heat exchangers. 4.2. I. Distillation columns. Time constants: The single dominant time constant is computed as:
y,,&)=G. ‘I The term x denotes the Schur product. As will be shown, in the case of distillation columns, each component part is characterized by a single time constant r. For heat exchangers, two separate time constants can be identified, associated with the tube and the shell side fluids, respectively. It will now be shown that the parameters of the models for each component part can be almost entirely computed using flowsheet data. 4.1. Computing the steady state gain matrix K The steady state gains of the gain matrices between inputs and outputs for each component part are generated using the following procedure:
where rc and rR are the time constants (in min) associated with the condenser and reboiler, respectively, and t, is the time constant (in min) for the column itself, computed according to:
(6)
where Mj is the volumetric holdup in each plate i of the column (m’) and L, is the liquid flowrate (m’lmin) in plate i, and N is the number of plates in the column. The liquid holdup can be expressed as:
M, = A,(hw + h,,) = 4
(hw+ h,,),
(7)
(4 The heat and mass balances for the complete flowsheet at the nominal operating point are solved using the flowsheet program. (b) For convenience, the heat and mass balances for each component part are solved separately, with the component part inputs values obtained from step (a). Cc)Each gain element in equation (3) K,, is defined as the derivative: K,, = 8y,l&q = Ay,/Au,. This is computed on the basis of flowsheet data generated by imposing small positive and negative perturbations on each input Au, of each component part, one at a time, and recording the corresponding change on each component part output Ay,. This procedure is repeated until the selected
where DC is the column diameter (m) and h, and h,,, are the weir heights and the level of fluid above the weir (m), respectively; the latter can be expressed in terms of the weir length 4,(m) using the Francis equation:
(8) Delay times (Shinskey, 1984): There is a delay time associated with a change in the internal liquid flow in the column, caused by a change in the nominal fluid holdup above the weir. For a single tray, this can be estimated by considering the time
0.
328
WEITZ and
for the over-weir fluid hold up to stabilize after a change in liquid flow, given by: taken
D. R. LEWIN exchangers: UHE (in PRO/II, the product UHEA is given and A must be estimated) u the average fluid velocity in the tubes (m/min) and tube and shell diameters (the latter taken from industrial standards). 5.
XD; = 666*e,.h:,5
Thus, the overall delay experienced by the bottoms product to change in the flows, temperatures or compositions of either the feed or the reflux will depend on the number of trays effected. The top product will respond immediately to changes in flow rates, but will experience a considerable delay when faced with feed concentration or temperature changes. In these cases, the delay time is assumed to be the sum of the residence times of all trays between the feed and the top of the column, since such a change is assumed to proliferate by affecting the entire tray holdup rather than just the over-weir fluid. Typical design parameters: The following values were taken as being typical: rc= tK = 0.57,; e, = 0.650,; h,=2”. Apart from the flowsheet data, these were the only pieces of information required in order to compute Y(s). 4.2.2. Heat exchangers. In some cases, heat exchangers will be incorporated into the component parts of major processing units (e.g. condensers and reboilers are an integral part of a distillation column component part). However, heat exchangers are in general modeled explicitly. We assume that any time delays associated with heat exchangers are negligible in comparison to those of other processing units (e.g. fixed beds of distillation columns). Thus, the dynamics of heat exchangers will be lumped into single lags. For simplicity, we shall assume all heat exchangers to be one shell-one pass Tube and Shell. Time constants: Distinct values were computed for the fluid in the tubes and for that in the shell, given respectively by rt = V&q, and t, = k’,lq,. The total volumes of fluid holdup in the tubes and shell, V, and V, can be computed on the basis of the estimated heat transfer area, which is either given directly by the steady state simulator, or given in the form of UHEA (the product of the overall heat transfer coefficient with the heat transfer area). The fluid volumetric flow rates in both sides of the heat exchanger, q, and q,, are also flowsheet data. For more details, see Weitz (1994). Typical design parameters: Apart from the flowsheet data, the following design parameters are also required in order to compute Y(s) for heat
A CASE STUDY
Distillation is a process in which energy is invested in return for separation. It is therefore common to reduce energy costs of distillation by heat integration or thermal coupling of two or more columns. Chiang and Luyben (1988) considered the three alternative heat-integrated schemes shown in Fig. 1 for the continuous separation of a 50/50 mixture of methanol and water. Methanol must be delivered as a 96% pure product, while the bottoms product must contain at most 4% methanol. 45 kmollmin must be processed. The alternatives are: FS (feed split). The feed is optimally split (FL-_ F’) between the two columns. Energy is conserved by heat integration of the overhead condenser of the high-pressure column with the reboiler of the low-pressure column. LSF (light split/heat integration forward). The entire feed is fed into the high-pressure column, with heat integration as in FS, in the direction of mass flow. LSR (light split/heat integration reverse). The entire feed is fed into the low-pressure column, meaning that heat integration is in the opposite direction as the mass flow. Since all three heat integrated designs conserve about 40% of the energy demand of the single column, one should therefore select the appropriate design from operability considerations. Chiang and Luyben carried out detailed nonlinear dynamic modeling of the three heat-integrated systems, and then carried out C&R analysis on the basis of RGA and minimum singular values using linear approximations fitted to the transients obtained by perturbation simulations of a full-scale rigorous dynamic model of the process. Their findings based on the linear analysis were rather inconclusive, but on the basis of rigorous closed-loop simulations, they did show that the FS scheme is by far the worst. We shall now apply our method to analyze each of the alternatives from the point of view of C&R. In what follows, we will describe each step of the process using the LSF scheme as an example. 5. I. Conceptual analysis of the problem As shown in Fig. 2, for the LSF scheme, the process outputs are the compositions of the three
Dynamic
controllability
and resiliency
329
diagnosis
FS
F-
.:...:.: :.: :.:. :.:A:.:
:::::::::::: . . . . :.::.:. j:j j::: .j:j . \ . \ .. .:. .:.:.:.:.
I:: i::: :;:j . .. .:. .: : ..:. .. . .. :: :,.,:
LSR
LSF
xl Fig. I. A single binary
distillation
Fig. 2. Conceptual
column
analysis
(SC), with three LSF and LSR).
and component
alternative
heat-integrated
parts for the LSF scheme
schemes
(FS,
330
0. WEITZ and D. R. LEWIN
Table 1. The results from the steady-state simulation for the three heat-integrated configurations methanol-water mixture, in comparison with a single column FS Variable
SC COLI
F, feed flow (kmol/min) xF, feed concn (CHJOH mol fraction) D, distillate flow (kmol/min) xo, distillate concn (CH,OH mol fraction) El, bottoms Row (kmollmin) xB, bottoms concn (CH,OH mol fraction) N, number of trays Nf , feed tray (1~ top)
45.00 0.50 22.50 0.96 22.50 0.04 13 9
R, reflux ratio P, working pressure (mmHg)
QR, reboiler heat duty (lo6 kcal/min) Q,, condenser heat duty ( lOhkcallmin) TR. reboiler temperature (“C) Tc, condenser temperature (“C) DC. column diameter (m)
0.82
760 0.353 0.348 93.7 65.1 3.2
COLI
LSF COL2
22.04 0.50 11.02 0.96 11.02 0.04 16 12
COLl
22.96 0.50 11.48 0.96 11.48 0.04 13 9
1.12
45.00 0.50 11.05 0.96 33.95 0.35 16 13
I a6
0.82
3900 0.205 0.180 146.3 113.4 1.8
for the distillation of a
76x1 0.180 0.178 93.7 65.1 2.3
3900 0.222 0.175 126.5 113.4 2.0
LSR COL2
COLl
33.95 0.35 11.45 0.96 22.50 0.04 13 II
45.00 0.50 12.04 0.96 32.96 0.33 13 11
COL2
1.10
0.75
760 0.175 0.205 93.7 65.1 2.4
760 0.180 0.179 71.2 65.1 2.3
32.96 0.33 10.46 0.96 22.50 0.04 16 12 1.15
3900 0.205 0.180 127.2 95.9 2.0
Note: The number of trays N includes the condenser (tray I) and reboilcr (tray N).
product streams (.xuu, xn’ and xaJ. The inputs are the control variables (&, LL and QRu) and disturbances (F and xF). It has been demonstrated (Skogestad, 1987) that a reasonable approximation for distillation column dynamics is a first order lag. Thus, the LSF system is described in terms of two component parts, one for each column. This requires that intermediate variables be tracked, in order to completely model the information transfer between the two component parts. Thus, we account for both mass transfer and heat transfer between the two parts via the variables: ~au, modeling species accumulation: BH, modeling overall mass accumulation; and Tau and Qc’, ( = - Qa’.). modeling energy accumulation. Here we should note that temperature alone will not be an adequate representative of the energy accumulation for systems exhibiting partial vaporization. The control variables have been scaled between zero and twice the nominal values and disturbances at a given fraction above and below the nominal values. Thus the nominal scaled values of all control variables are 0.5, with the allowed variation of &OS. For this example, the disturbances are assumed to always be steps above and below the nominal value, limited in magnitude by 20% of the full-scale range. The scaling of the outputs is arbitrary, and we selected the scaling factors in order to get a reasonable match between steady state gains obtained and those obtained by Chiang and Luyben (these authors do not state explicitly the scaling they used). We should note that the output scaling has no effect of the overall result with respect to the C&R measures used (RGA, MCN and DC). 5.2. Steady state analysis For this case study we assumed no pressure drop in the columns, no heat losses to the surroundings
and tray efficiencies of 75%. The thermodynamic properties were computed using the UNIFAC system. These assumptions were the same as those used by Chiang and Luyben (except they did account for heat losses). The results for all four flowsheets, carried out using PRO/II, are given in Table 1. We note that the energy requirements of the LSR and FS schemes are the lowest (0.205 x lo6 kcal/ min) followed by the LSF scheme (0.222~ IO6kcahmin).
5.3. Computing K and Y(s) Data for K and Y(s) were computed following the procedure outlined in Sections 4.1 and 4.2. Example results are now given for the LSF configuration. For the high-pressure column, we obtained (with time units in minutes): /‘n”\
I
0.017 O.Olle-‘,-% x - 0.330e-’ ” 0.916e-“” 4e - 5e-‘,”
X
- 1.109 0.001 - 1.859 O.O06e-“” 59.0 - 0.2e-“,‘” - 123.7 l.l27e-“‘” -0.994 O.OOle-’ Is
0.090e-6.4” 1.296-O Is - 41.05e-“‘” -0.02e-0.‘” 0.003e-“~‘”
(9)
331
Dynamic controllability and resiliency diagnosis
FS
Y
I XF LSF
XF CCL1
ccc2 IL
XBL
c-cl
+, XBH
TBL
+cdzl__--
u
U’
XII&
xlla
il+!l+ XF
I QRLQCH -
Fig. 3. The combined SIMULINK models for the three heat-integrated
schemes.
332
0.
Chiang
WEITZ and D. R. LEWIN
Proposed
and Luyben
”
method
”
lo'2
i0.2
loo Frequency [radlmin]
IO0
Frequency [radlmin] FS
F2
2
D ____ -\._-l,i, _
_._
_.-
!.
-_-_-
- >.
E -----ml
\\\ ._
;’ \ If 1
\/
0
1Li*
10°
Frequency [rad/min]
Frequency [rad/min]
LSF
LSF
3-
31
_---___
T2
- -.-
d a :: ml
-1
\
_.
.._
- \ ‘1 I ,-,I
/i
Y, \
;’
it!
0’
I
0'
1U2
Frequency [radlmin]
iOO Frequency [rad/min]
LSR
LSR
10.*
loo
a
0
E.5 P ---_ -.-.-
-_ --_-
_. . .
\
0 ‘--
0 10” lo‘* Frequency [racYminI
1O’2
IO”
Fig. 4. Comparison of the diagonal RGA elements as a function of frequency for all four configurations obtained with the proposed method with those given by Chiang and Luyben (1988).
Dynamic
controllability
while for the low pressure column, the model is: 1
xBL =-
(
XDL
1
17s+l x
0.792e-” ”
- l.O29e-” ”
- 0.051 ( 0.790e-‘.” O.O07e-“.I” -2.161 - 3.291
0.003
O.O12e-’
”
0.038
>
XBH
&I
.
(10)
e KL
11 LL 5.4.
diagonal RGA values obtained (the optimal pairing for decentralized control for all four schemes is diagonal) as shown in Fig. 4. Clearly, the results are in close agreement. The resonant peaks exhibited by the results of Chiang and Luyben are the result of differences in the dynamics of the transfer function elements. It is possible that these differences could be the result of independent fitting of transfer functions to the transients obtained from rigorous modeling. 5.5. Computation of C&R Measures
T BH x
333
and resiliency diagnosis
Generation of transfer function matrices
The overall linear approximation for the flowsheet dynamics is obtained by recombining the component part models according to the plant topology. Figure 3 shows how this is achieved for all three heat-integrated schemes using SIMULINK( 1992). The transfer functions P(&J) and P&w) are obtained directly in the frequency domain by computing the Fourier transform of the simulated pulse response of the combined model. It is useful to quantitatively compare the dynamic model derived with that obtained by the linear approximations derived by Chiang and Luyben. This can be achieved conveniently by plotting the
a _._.-.-.-.-.-.-.-.-._
The MCN (Fig. 5) was found to be inconclusive as a screening measure for this example; all three heat integrated schemes scored equivalently in the frequency range of interest (0.1-l rad/min). Figure 6 shows DC contour maps computed for each control variable for three schemes: SC, FS and LSR. The ordinates indicate the direction of the disturbance [F, +IT, while the abscissae track the effect of frequency. Thus, for example, directions of 0”, 90” and 45” indicate positive disturbances in F alone, xF alone and both together, respectively. Since the contour value of 0.5 indicates the control variable constraint, it is apparent that steady state disturbance rejection is achievable by all designs. However, for a wide range of disturbance vector directions, the FS scheme will have difficulty with high-frequency components of the disturbance beyond 0.1 rad/min (see LH, QRH and F,/F,) and thus disturbance rejection is expected to be very
_ “.L,\sF 1
Fig. 5. The MCN
plotted
against frequency
for all four configurations, method.
as obtained
by the proposed
0. WEITZ and D. R. LEWIN
FS-FWFL
LSR. QRH
LSR.LL
I50
lm
100
n Fig. 6. DC Maps for the SC, FS and LSR configurations for the disturbances limited by Id,/ GO.2.
in this case. The LSR scheme, on the other hand, does about as well as a single column. Thus, from the point of view of disturbance rejection, the FS scheme shold be eliminated, and the LSR scheme selected on the basis of its low-energy demand. These conclusions are in line with the simulation results reported by Chiang and Luyben. sluggish
6. CONCLUSIONS
This study has been motivated by the need for a rapid screening tool that will hopefully enable the efficient elimination of some of the alternative designs with relatively little efort. The approach described in this publication is meant to be used in the early stage of a process design, where design decisions have the greatest economic impact. We have shown how the approach can be used to screen alternative simple heat-integrated distillation column schemes, based on data reported in the literature. Additional studies have shown this approach
to be also useful for more involved problems featuring extensive heat integration (Weitz, 1994) and a system including an exothermic reactor, distillation columns and material recycle (Naot and Lewin, 1995). Our short-cut methods are able to confirm the conclusions obtained by others largely relying on simulation studies using rigorous dynamic models. The overall approach appears to be very promising as a short-cut diagnostic tool, which in theory could be integrated into commercial flowsheeting software. Acknowledgement-This work was supported by the Fund for the Promotion of Research at the Technion. We are grateful to the reviewers for their helpful comments and suggestions, which significantly improved the final vesion of this paper.
NOTATION
A, = Distillation column cross-sectional area (m*)
335
Dynamic controllability and resiliency diagnosis A = Heat exchange heat transfer area (m*)
B = Bottoms flow rate (kmollmin) D, = Distillation column diameter (m) d= Disturbance vector D = Distillate flow rate (kmollmin) F= Feed flow rate disturbance (kmollmin) h,, = Height of fluid above weir (m) K = Component part steady state gain matrix L = Reflux flow rate (kmollmin) e, = Weir length (m) M,, = Hold-pup of tray fluid above the weir (m) N = Number of distillation colum trays P= Process model describing the effect of control actions on outputs Pd = Process model describing the effect of disturbances on outputs Qa, Qc = ztgy supplied to reboiler/condenser (kcal/ 4 = Volumetric flow rate (m’/min) R = Reflux ratio
T= Temperature (“C) f = Time (min) Un, = Heat transfer coefficient (kcallm “C min) ri = Control action vector V= Volume (m”) x = Concentration (mol fraction) ,‘= Output vector 19,,= Hydraulic delay time for a single distillation tray (min) t = Component part characteristic time constant (min) r$= Disturbance direction (degrees) Y = Matrix of component part dynamics w = Frequency (radlmin) Subscripts
B = Bottoms C = Condenser D = Distillate F = Feed H = High-pressure L = Low-pressure R = Reboiler s = Heat exchanger shell t = Heat exchanger tube
REFERENCES Bristol E. H., On a new measure of interactions for multivariable process control. IEEE Trans. Auf. Control AC-11, 133-134 (1966). Chang J. and C. Yu, Relative disturbance gain array. AIChE Jl38, 521-534 (1992). Chiang T. and W. L. Luyben, Comparison of the dynamic performances of three heat-integrated distillation contigurations. Ind. Engng Chem. Res. 27, 99-104 (1988). Lewin D. R., Simple tool for disturbance resiliency diagnosis and feedforward control design. Computers them. Engng, in press (1995). Ludwig E. E., Applied Process Design for Chemical and Petrochemical Plants, 2nd Edn. Gulf Pub. Co., Houston, TX (1977). MATLAB for Windows User’s Guide. The MathWorks, Inc., Natick, MA (1991). Naot I. and D. R. Lewin, Analysis of process dynamics in recycle systems using steady-state flowsheeting tools and simplified dynamic models. Presented at the 4th IFAC Symp. on Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes (DYCORD +
‘95)) Helsingor (1995). Nguyen T. C., G. W. Barton, J. D. Perkins and R. D. Johnston, A codition number scaling policy for stability robustness analysis. AIChE JI 34, 1200-1206 (1988). PROIII Keyword Input Manual, Ver. 3.1. Simulation Sciences Inc. (1991). The SlMULlNK Users Guide. The Math-Works, Inc., Natick, MA (1992). Skogestad S., Studies on robust control of distillation columns. Ph.D. Thesis, California Institute of Technology (1987). Skogestad S. and M. Morari, The effect of disturbance dcections on closed loop performance. Ind. Engng Chem. Res. 26. 2029-2035
(1987).
Shinskey F. G., Distillation ‘Control for Productivity & Energy Conservation, pp. 83-89 (1984). Weitz O., Integration of controllability measures into process design. M.Sc. Thesis, Technion (1994). Wolff E. A., K. E. Mathisen and S. Skogestad, Dynamics and controllability of heat exchanger networks. Computer-Oriented Process Engineering, Amsterdam, pp. 117-122 (1991).
Vol. 20, No. 4, pp. 325-335. lYY6 Copyright @ 19% Elsevier Science Ltd Printed in Great Britain. All rights reserved
Pergamon 0098-1354(95jBBO23-2
098-1354196
$15.00+0.00
DYNAMIC CONTROLLABILITY AND RESILIENCY DIAGNOSIS USING STEADY STATE PROCESS FLOWSHEET DATA 0. WEITZ and D. R. LEWIN~ Department of Chemical Engineering, Technion IIT, Haifa 32000, Israel Abstract-A simple procedure for assessing the controllability and resiliency of a process flowsheet is presented. The approach relies on a simplified modeling strategy, whereby an approximate linear dynamic model of the process is derived from steady-state flowsheet information. The resulting linear approximation can then be tested using the wealth of available linear controllability and resiliency measures available. In this paper, the procedure is used to screen the disturbance resiliency of alternative heat-integrated distillation column schemes reported in the literature. The results confirm the conclusions obtained by others largely relying on closed-loop simulations using rigorous dynamic models. The overall approach appears to be very promising as a short-cut diagnostic and screening tool, which in theory could be integrated into commercial flowsheeting software.
1. INTRODUCTION
The design of continuous chemical processes is usually carried out for a given operating point, assuming that an appropriate controller will succeed in regulating the process. It is possible that under certain conditions, the final design will be unable to satisfy the process specifications, or that they will only be attainable if a sophisticated (expensive) control system is augmented to compensate for the (bad) design. Usually, alternative designs are judged on economics alone without taking operability into account. This may lead to the elimination of easily-controlled but slightly less economical alternatives in favor of slightly more economical designs which may be extremely difficult to control. Moreover, plants which are difficult to control will be inflexible and possess poor disturbance rejection properties, which will themselves incur extra cost. These problems call for the introduction of controllability and resiliency measures as diagnostic tools within the design procedure. Controllability can be defined as the ease with which a continuous plant can be held at a specific steady state. An associated concept is switchability, which measures the ease with which the process can be moved from one desired stationary point to another. Similarly, resiliency measures the degree to which a processing system can meet its design objectives despite external disturbances and uncertainties in design parameters. Clearly, it wold be greatly advantageous to be able to predict how well a given flowsheet meets these dynamic performance requirements as early as possible in the design. t To whom all correspondence
should be addressed.
This paper describes a simple procedure which can be used to investigate the degree to which a process flowsheet is resilient to external disturbances. The approach involves the derivation of an approximate linear dynamic plant model, generated solely from steady-state flowsheet information and independent of control system design. As such, it is intended to be used as a rapid screening method which will eliminate design alternatives which cannot meet the basic requirement of being able to maintain the process at the required steady state operating specifications. Resiliency to parametric uncertainty can in principle be handled in the same framework (see Naot and Lewin, 1995). The issue of switchability should in general be addressed using more detailed dynamic modeling. Since this will invariably involve dynamic simulation requiring contol system design, it should be implemented only after significantly screening has already weeded out poorer designs. The paper is structured as follows. Firstly, we briefly review commonly used controllability and resiliency measures. Next, we outline the steps involved in the overall procedure. Finally, as an example, the proposed method is used as a screening tool in the selection of the appropriate flowsheet for the design of a heat integrated distillation column system.
2. CONTROLLABILITYANDRESILIENCYMEASURES
Quantitative assessment of controllability and resiliency (C&R) for process plants has naturally considerable interest industrially. generated
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Methods
have been developed, based on linear mathematical analysis of plant models, that can assist the designer in rejecting designs which may have the desired steady-state properties but unacceptable dynamic ones. Commonly used controlfability measures include the relative gain array [RGA Bristol (1966)] and the minimized condition number [MCN, Nguyen et al. (1988)], both of which rely on a linear model describing the effect of control variables on the process outputs P(s). Typical resiliency measures are the disturbance condition number (Skogestad and Morari, 1987), the disturbance cost (Lewin, 1995) and the relative distrubance gain array (Chang and Yu, 1992), which in addition to P(s), also require a disturbance model, Pd(s), which describes the effect of disturbances on the process outputs. These linear process models are not usually readily available at the early stages of the process design. Thus, a crucial step in the analysis is the generation of adequate linear approximations which can be derived using the minimal information available in preliminary flowsheeting of a potential design. An alternative C&R diagnosis approach relies on closed-loop simulations using rigorous dynamic process models. This is impractical for diagnosis of controllability and resiliency problems in the early stages of process design for two reasons: (a) it is both time and resource consuming and may require unavailable data; and (b) any conclusions obtained will depend on the control scheme adopted in the simulations, and as such will compromise the objectivity of the diagnosis. In this study, the most significant resiliency screening method was found to be the disturbance cost (DC), which is a direct measure which accounts for the degree to which the controlled process is insensitive to disturbances, d defined as:
In other words, we assume perfect control and then measure the integral square error of the control action u which satisfies this assumption. On the one hand, perfect control is in general impossible, either because of noncausality, instability or realizability problems. On the other hand, perfect control defines an upper bound to performance, and is independent of any particular controller design or tuning. These advantages make this measure an ideal one to use at an early stage of the design. If perfect control cannot give adequate disturbance rejection, or if such a control action is limited by process constraints, then clearly one should question the selection of the process design itself.
3. SHORT-CUT CONTROLLABILITY
AND RESILIENCY
DIAGNOSIS
The proposed steps:
approach
involves the following
(I) Conceptual analysis of the process flowsheet and the implied control problem. This involves the definition of process outputs to be controlled y(t) and process inputs available as control variables u(t) and those pertaining to disturbances, d(t). A conceptual linear approximation is therefore: Y(S)= p(s)+)
+ P&W(s).
(2)
(2) The definition of component parts, their inputs and outputs. Component parts are defined as the process sub-sections which can be approximated by simple lags. (3) Steady state simulation of the complete process, decomposed into its component parts, using a process simulator. (4) Computation of the steady state gains of the component parts of the process by perturbing each input, one at a time. (5) Computation of time constants and delay times on the basis of either mixed tank or plug flow assumptions, as appropriate. The flow rates and capacities are available from the steady state design data. In this paper, we will limit the discussion to the development of approximations to the time constant and time delays for distillation columns and heat exchangers. Similar expressions can be derived for other processing equipment (see Naot and Lewin, 1995). of approximate overall input(6) Generation output transfer function matrices P(~w) and P&u), for the complete flowsheet. This involves the re-combination of the component parts of the process as dictated by plant topology, and the generation of the frequency response of each element of the transfer function matrices. (7) Computation of frequency dependent C&R measures directly using the approximate linear model {P(~w), P&w)}. The analysis of disturbance resiliency is carried out for steps in the vector a. The expected magnitudes of each disturbance variable must be defined a priori. All the possible direction combinations are tested. In our study, we have used PRO/II to carry out steady state calculations and have found it convenient to carry out all frequency and time domain
Dynamic
controllability
using MATLAB (1991) and computations SIMULINK (1992). Clearly, the main task involved is the generation of adequate dynamic process models. Thus, before illustrating the potential impact of this simple method on representative flowsheet screening problems, we shall first detail the modeling approach taken.
and resiliency
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321
change Au, is made small enough such that the gain computed is relatively insensitive to perturbation direction. In order to ensure that this is done accurately, it is important that the fowsheet convergence tolerances on the computed outputs be set appropriately. 4.2. Computing the dynamics matix W(s)
4. GENERATING SIMPLE DYNAMIC PROCESS MODELS
Following conceptual analysis of the flowsheet, and the definition of component parts, the flowsheet is represented as a connected set of component parts, each expressed in the following linear form: Y(s)=Kx’P(s).U(s),
(3)
where U(s) and Y(s) are input and output component part vectors in (em and %“, respectively, K is a matrix of steady state gains in ijn”‘“’ and Y(s) is a matrix describing the dynamics in %nx’n, each element of which is of the form:
There has been recent interest in the literature on the generation of simple dynamic models for processing equipment, such as heat exchanger networks (Woolff et al., 1991) and simple binary distillation columns (Skogestad, 1987). Our approach requires the representation of dynamic elements as time constants and delay times, and these must be computable using only flowsheet data. The following constitutes the model equations utilized in this study for distillation columns and heat exchangers. 4.2. I. Distillation columns. Time constants: The single dominant time constant is computed as:
y,,&)=G. ‘I The term x denotes the Schur product. As will be shown, in the case of distillation columns, each component part is characterized by a single time constant r. For heat exchangers, two separate time constants can be identified, associated with the tube and the shell side fluids, respectively. It will now be shown that the parameters of the models for each component part can be almost entirely computed using flowsheet data. 4.1. Computing the steady state gain matrix K The steady state gains of the gain matrices between inputs and outputs for each component part are generated using the following procedure:
where rc and rR are the time constants (in min) associated with the condenser and reboiler, respectively, and t, is the time constant (in min) for the column itself, computed according to:
(6)
where Mj is the volumetric holdup in each plate i of the column (m’) and L, is the liquid flowrate (m’lmin) in plate i, and N is the number of plates in the column. The liquid holdup can be expressed as:
M, = A,(hw + h,,) = 4
(hw+ h,,),
(7)
(4 The heat and mass balances for the complete flowsheet at the nominal operating point are solved using the flowsheet program. (b) For convenience, the heat and mass balances for each component part are solved separately, with the component part inputs values obtained from step (a). Cc)Each gain element in equation (3) K,, is defined as the derivative: K,, = 8y,l&q = Ay,/Au,. This is computed on the basis of flowsheet data generated by imposing small positive and negative perturbations on each input Au, of each component part, one at a time, and recording the corresponding change on each component part output Ay,. This procedure is repeated until the selected
where DC is the column diameter (m) and h, and h,,, are the weir heights and the level of fluid above the weir (m), respectively; the latter can be expressed in terms of the weir length 4,(m) using the Francis equation:
(8) Delay times (Shinskey, 1984): There is a delay time associated with a change in the internal liquid flow in the column, caused by a change in the nominal fluid holdup above the weir. For a single tray, this can be estimated by considering the time
0.
328
WEITZ and
for the over-weir fluid hold up to stabilize after a change in liquid flow, given by: taken
D. R. LEWIN exchangers: UHE (in PRO/II, the product UHEA is given and A must be estimated) u the average fluid velocity in the tubes (m/min) and tube and shell diameters (the latter taken from industrial standards). 5.
XD; = 666*e,.h:,5
Thus, the overall delay experienced by the bottoms product to change in the flows, temperatures or compositions of either the feed or the reflux will depend on the number of trays effected. The top product will respond immediately to changes in flow rates, but will experience a considerable delay when faced with feed concentration or temperature changes. In these cases, the delay time is assumed to be the sum of the residence times of all trays between the feed and the top of the column, since such a change is assumed to proliferate by affecting the entire tray holdup rather than just the over-weir fluid. Typical design parameters: The following values were taken as being typical: rc= tK = 0.57,; e, = 0.650,; h,=2”. Apart from the flowsheet data, these were the only pieces of information required in order to compute Y(s). 4.2.2. Heat exchangers. In some cases, heat exchangers will be incorporated into the component parts of major processing units (e.g. condensers and reboilers are an integral part of a distillation column component part). However, heat exchangers are in general modeled explicitly. We assume that any time delays associated with heat exchangers are negligible in comparison to those of other processing units (e.g. fixed beds of distillation columns). Thus, the dynamics of heat exchangers will be lumped into single lags. For simplicity, we shall assume all heat exchangers to be one shell-one pass Tube and Shell. Time constants: Distinct values were computed for the fluid in the tubes and for that in the shell, given respectively by rt = V&q, and t, = k’,lq,. The total volumes of fluid holdup in the tubes and shell, V, and V, can be computed on the basis of the estimated heat transfer area, which is either given directly by the steady state simulator, or given in the form of UHEA (the product of the overall heat transfer coefficient with the heat transfer area). The fluid volumetric flow rates in both sides of the heat exchanger, q, and q,, are also flowsheet data. For more details, see Weitz (1994). Typical design parameters: Apart from the flowsheet data, the following design parameters are also required in order to compute Y(s) for heat
A CASE STUDY
Distillation is a process in which energy is invested in return for separation. It is therefore common to reduce energy costs of distillation by heat integration or thermal coupling of two or more columns. Chiang and Luyben (1988) considered the three alternative heat-integrated schemes shown in Fig. 1 for the continuous separation of a 50/50 mixture of methanol and water. Methanol must be delivered as a 96% pure product, while the bottoms product must contain at most 4% methanol. 45 kmollmin must be processed. The alternatives are: FS (feed split). The feed is optimally split (FL-_ F’) between the two columns. Energy is conserved by heat integration of the overhead condenser of the high-pressure column with the reboiler of the low-pressure column. LSF (light split/heat integration forward). The entire feed is fed into the high-pressure column, with heat integration as in FS, in the direction of mass flow. LSR (light split/heat integration reverse). The entire feed is fed into the low-pressure column, meaning that heat integration is in the opposite direction as the mass flow. Since all three heat integrated designs conserve about 40% of the energy demand of the single column, one should therefore select the appropriate design from operability considerations. Chiang and Luyben carried out detailed nonlinear dynamic modeling of the three heat-integrated systems, and then carried out C&R analysis on the basis of RGA and minimum singular values using linear approximations fitted to the transients obtained by perturbation simulations of a full-scale rigorous dynamic model of the process. Their findings based on the linear analysis were rather inconclusive, but on the basis of rigorous closed-loop simulations, they did show that the FS scheme is by far the worst. We shall now apply our method to analyze each of the alternatives from the point of view of C&R. In what follows, we will describe each step of the process using the LSF scheme as an example. 5. I. Conceptual analysis of the problem As shown in Fig. 2, for the LSF scheme, the process outputs are the compositions of the three
Dynamic
controllability
and resiliency
329
diagnosis
FS
F-
.:...:.: :.: :.:. :.:A:.:
:::::::::::: . . . . :.::.:. j:j j::: .j:j . \ . \ .. .:. .:.:.:.:.
I:: i::: :;:j . .. .:. .: : ..:. .. . .. :: :,.,:
LSR
LSF
xl Fig. I. A single binary
distillation
Fig. 2. Conceptual
column
analysis
(SC), with three LSF and LSR).
and component
alternative
heat-integrated
parts for the LSF scheme
schemes
(FS,
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Table 1. The results from the steady-state simulation for the three heat-integrated configurations methanol-water mixture, in comparison with a single column FS Variable
SC COLI
F, feed flow (kmol/min) xF, feed concn (CHJOH mol fraction) D, distillate flow (kmol/min) xo, distillate concn (CH,OH mol fraction) El, bottoms Row (kmollmin) xB, bottoms concn (CH,OH mol fraction) N, number of trays Nf , feed tray (1~ top)
45.00 0.50 22.50 0.96 22.50 0.04 13 9
R, reflux ratio P, working pressure (mmHg)
QR, reboiler heat duty (lo6 kcal/min) Q,, condenser heat duty ( lOhkcallmin) TR. reboiler temperature (“C) Tc, condenser temperature (“C) DC. column diameter (m)
0.82
760 0.353 0.348 93.7 65.1 3.2
COLI
LSF COL2
22.04 0.50 11.02 0.96 11.02 0.04 16 12
COLl
22.96 0.50 11.48 0.96 11.48 0.04 13 9
1.12
45.00 0.50 11.05 0.96 33.95 0.35 16 13
I a6
0.82
3900 0.205 0.180 146.3 113.4 1.8
for the distillation of a
76x1 0.180 0.178 93.7 65.1 2.3
3900 0.222 0.175 126.5 113.4 2.0
LSR COL2
COLl
33.95 0.35 11.45 0.96 22.50 0.04 13 II
45.00 0.50 12.04 0.96 32.96 0.33 13 11
COL2
1.10
0.75
760 0.175 0.205 93.7 65.1 2.4
760 0.180 0.179 71.2 65.1 2.3
32.96 0.33 10.46 0.96 22.50 0.04 16 12 1.15
3900 0.205 0.180 127.2 95.9 2.0
Note: The number of trays N includes the condenser (tray I) and reboilcr (tray N).
product streams (.xuu, xn’ and xaJ. The inputs are the control variables (&, LL and QRu) and disturbances (F and xF). It has been demonstrated (Skogestad, 1987) that a reasonable approximation for distillation column dynamics is a first order lag. Thus, the LSF system is described in terms of two component parts, one for each column. This requires that intermediate variables be tracked, in order to completely model the information transfer between the two component parts. Thus, we account for both mass transfer and heat transfer between the two parts via the variables: ~au, modeling species accumulation: BH, modeling overall mass accumulation; and Tau and Qc’, ( = - Qa’.). modeling energy accumulation. Here we should note that temperature alone will not be an adequate representative of the energy accumulation for systems exhibiting partial vaporization. The control variables have been scaled between zero and twice the nominal values and disturbances at a given fraction above and below the nominal values. Thus the nominal scaled values of all control variables are 0.5, with the allowed variation of &OS. For this example, the disturbances are assumed to always be steps above and below the nominal value, limited in magnitude by 20% of the full-scale range. The scaling of the outputs is arbitrary, and we selected the scaling factors in order to get a reasonable match between steady state gains obtained and those obtained by Chiang and Luyben (these authors do not state explicitly the scaling they used). We should note that the output scaling has no effect of the overall result with respect to the C&R measures used (RGA, MCN and DC). 5.2. Steady state analysis For this case study we assumed no pressure drop in the columns, no heat losses to the surroundings
and tray efficiencies of 75%. The thermodynamic properties were computed using the UNIFAC system. These assumptions were the same as those used by Chiang and Luyben (except they did account for heat losses). The results for all four flowsheets, carried out using PRO/II, are given in Table 1. We note that the energy requirements of the LSR and FS schemes are the lowest (0.205 x lo6 kcal/ min) followed by the LSF scheme (0.222~ IO6kcahmin).
5.3. Computing K and Y(s) Data for K and Y(s) were computed following the procedure outlined in Sections 4.1 and 4.2. Example results are now given for the LSF configuration. For the high-pressure column, we obtained (with time units in minutes): /‘n”\
I
0.017 O.Olle-‘,-% x - 0.330e-’ ” 0.916e-“” 4e - 5e-‘,”
X
- 1.109 0.001 - 1.859 O.O06e-“” 59.0 - 0.2e-“,‘” - 123.7 l.l27e-“‘” -0.994 O.OOle-’ Is
0.090e-6.4” 1.296-O Is - 41.05e-“‘” -0.02e-0.‘” 0.003e-“~‘”
(9)
331
Dynamic controllability and resiliency diagnosis
FS
Y
I XF LSF
XF CCL1
ccc2 IL
XBL
c-cl
+, XBH
TBL
+cdzl__--
u
U’
XII&
xlla
il+!l+ XF
I QRLQCH -
Fig. 3. The combined SIMULINK models for the three heat-integrated
schemes.
332
0.
Chiang
WEITZ and D. R. LEWIN
Proposed
and Luyben
”
method
”
lo'2
i0.2
loo Frequency [radlmin]
IO0
Frequency [radlmin] FS
F2
2
D ____ -\._-l,i, _
_._
_.-
!.
-_-_-
- >.
E -----ml
\\\ ._
;’ \ If 1
\/
0
1Li*
10°
Frequency [rad/min]
Frequency [rad/min]
LSF
LSF
3-
31
_---___
T2
- -.-
d a :: ml
-1
\
_.
.._
- \ ‘1 I ,-,I
/i
Y, \
;’
it!
0’
I
0'
1U2
Frequency [radlmin]
iOO Frequency [rad/min]
LSR
LSR
10.*
loo
a
0
E.5 P ---_ -.-.-
-_ --_-
_. . .
\
0 ‘--
0 10” lo‘* Frequency [racYminI
1O’2
IO”
Fig. 4. Comparison of the diagonal RGA elements as a function of frequency for all four configurations obtained with the proposed method with those given by Chiang and Luyben (1988).
Dynamic
controllability
while for the low pressure column, the model is: 1
xBL =-
(
XDL
1
17s+l x
0.792e-” ”
- l.O29e-” ”
- 0.051 ( 0.790e-‘.” O.O07e-“.I” -2.161 - 3.291
0.003
O.O12e-’
”
0.038
>
XBH
&I
.
(10)
e KL
11 LL 5.4.
diagonal RGA values obtained (the optimal pairing for decentralized control for all four schemes is diagonal) as shown in Fig. 4. Clearly, the results are in close agreement. The resonant peaks exhibited by the results of Chiang and Luyben are the result of differences in the dynamics of the transfer function elements. It is possible that these differences could be the result of independent fitting of transfer functions to the transients obtained from rigorous modeling. 5.5. Computation of C&R Measures
T BH x
333
and resiliency diagnosis
Generation of transfer function matrices
The overall linear approximation for the flowsheet dynamics is obtained by recombining the component part models according to the plant topology. Figure 3 shows how this is achieved for all three heat-integrated schemes using SIMULINK( 1992). The transfer functions P(&J) and P&w) are obtained directly in the frequency domain by computing the Fourier transform of the simulated pulse response of the combined model. It is useful to quantitatively compare the dynamic model derived with that obtained by the linear approximations derived by Chiang and Luyben. This can be achieved conveniently by plotting the
a _._.-.-.-.-.-.-.-.-._
The MCN (Fig. 5) was found to be inconclusive as a screening measure for this example; all three heat integrated schemes scored equivalently in the frequency range of interest (0.1-l rad/min). Figure 6 shows DC contour maps computed for each control variable for three schemes: SC, FS and LSR. The ordinates indicate the direction of the disturbance [F, +IT, while the abscissae track the effect of frequency. Thus, for example, directions of 0”, 90” and 45” indicate positive disturbances in F alone, xF alone and both together, respectively. Since the contour value of 0.5 indicates the control variable constraint, it is apparent that steady state disturbance rejection is achievable by all designs. However, for a wide range of disturbance vector directions, the FS scheme will have difficulty with high-frequency components of the disturbance beyond 0.1 rad/min (see LH, QRH and F,/F,) and thus disturbance rejection is expected to be very
_ “.L,\sF 1
Fig. 5. The MCN
plotted
against frequency
for all four configurations, method.
as obtained
by the proposed
0. WEITZ and D. R. LEWIN
FS-FWFL
LSR. QRH
LSR.LL
I50
lm
100
n Fig. 6. DC Maps for the SC, FS and LSR configurations for the disturbances limited by Id,/ GO.2.
in this case. The LSR scheme, on the other hand, does about as well as a single column. Thus, from the point of view of disturbance rejection, the FS scheme shold be eliminated, and the LSR scheme selected on the basis of its low-energy demand. These conclusions are in line with the simulation results reported by Chiang and Luyben. sluggish
6. CONCLUSIONS
This study has been motivated by the need for a rapid screening tool that will hopefully enable the efficient elimination of some of the alternative designs with relatively little efort. The approach described in this publication is meant to be used in the early stage of a process design, where design decisions have the greatest economic impact. We have shown how the approach can be used to screen alternative simple heat-integrated distillation column schemes, based on data reported in the literature. Additional studies have shown this approach
to be also useful for more involved problems featuring extensive heat integration (Weitz, 1994) and a system including an exothermic reactor, distillation columns and material recycle (Naot and Lewin, 1995). Our short-cut methods are able to confirm the conclusions obtained by others largely relying on simulation studies using rigorous dynamic models. The overall approach appears to be very promising as a short-cut diagnostic tool, which in theory could be integrated into commercial flowsheeting software. Acknowledgement-This work was supported by the Fund for the Promotion of Research at the Technion. We are grateful to the reviewers for their helpful comments and suggestions, which significantly improved the final vesion of this paper.
NOTATION
A, = Distillation column cross-sectional area (m*)
335
Dynamic controllability and resiliency diagnosis A = Heat exchange heat transfer area (m*)
B = Bottoms flow rate (kmollmin) D, = Distillation column diameter (m) d= Disturbance vector D = Distillate flow rate (kmollmin) F= Feed flow rate disturbance (kmollmin) h,, = Height of fluid above weir (m) K = Component part steady state gain matrix L = Reflux flow rate (kmollmin) e, = Weir length (m) M,, = Hold-pup of tray fluid above the weir (m) N = Number of distillation colum trays P= Process model describing the effect of control actions on outputs Pd = Process model describing the effect of disturbances on outputs Qa, Qc = ztgy supplied to reboiler/condenser (kcal/ 4 = Volumetric flow rate (m’/min) R = Reflux ratio
T= Temperature (“C) f = Time (min) Un, = Heat transfer coefficient (kcallm “C min) ri = Control action vector V= Volume (m”) x = Concentration (mol fraction) ,‘= Output vector 19,,= Hydraulic delay time for a single distillation tray (min) t = Component part characteristic time constant (min) r$= Disturbance direction (degrees) Y = Matrix of component part dynamics w = Frequency (radlmin) Subscripts
B = Bottoms C = Condenser D = Distillate F = Feed H = High-pressure L = Low-pressure R = Reboiler s = Heat exchanger shell t = Heat exchanger tube
REFERENCES Bristol E. H., On a new measure of interactions for multivariable process control. IEEE Trans. Auf. Control AC-11, 133-134 (1966). Chang J. and C. Yu, Relative disturbance gain array. AIChE Jl38, 521-534 (1992). Chiang T. and W. L. Luyben, Comparison of the dynamic performances of three heat-integrated distillation contigurations. Ind. Engng Chem. Res. 27, 99-104 (1988). Lewin D. R., Simple tool for disturbance resiliency diagnosis and feedforward control design. Computers them. Engng, in press (1995). Ludwig E. E., Applied Process Design for Chemical and Petrochemical Plants, 2nd Edn. Gulf Pub. Co., Houston, TX (1977). MATLAB for Windows User’s Guide. The MathWorks, Inc., Natick, MA (1991). Naot I. and D. R. Lewin, Analysis of process dynamics in recycle systems using steady-state flowsheeting tools and simplified dynamic models. Presented at the 4th IFAC Symp. on Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes (DYCORD +
‘95)) Helsingor (1995). Nguyen T. C., G. W. Barton, J. D. Perkins and R. D. Johnston, A codition number scaling policy for stability robustness analysis. AIChE JI 34, 1200-1206 (1988). PROIII Keyword Input Manual, Ver. 3.1. Simulation Sciences Inc. (1991). The SlMULlNK Users Guide. The Math-Works, Inc., Natick, MA (1992). Skogestad S., Studies on robust control of distillation columns. Ph.D. Thesis, California Institute of Technology (1987). Skogestad S. and M. Morari, The effect of disturbance dcections on closed loop performance. Ind. Engng Chem. Res. 26. 2029-2035
(1987).
Shinskey F. G., Distillation ‘Control for Productivity & Energy Conservation, pp. 83-89 (1984). Weitz O., Integration of controllability measures into process design. M.Sc. Thesis, Technion (1994). Wolff E. A., K. E. Mathisen and S. Skogestad, Dynamics and controllability of heat exchanger networks. Computer-Oriented Process Engineering, Amsterdam, pp. 117-122 (1991).