a generalized dynamic model for distillation columns-iii ... - CiteSeerX

Cornput. ckm. En@& Vol. 12, No. 1, pp. l-14, 1988 F’rintedin Great Britain. AI1 rights reserved

A GENERALIZED COLUMNS-III.

DYNAMIC MODEL FOR DISTILLATION STUDY OF STARTUP OPERATIONS

C. A.

Planta

0098.1354/88 $3.00 + 0.00 Copyright 0 1988 Pcrgamon Journals Ltd

Pilot0 de Ingenieria

Rurz,

I. T. CAMERON? and R. GANI$

Quimica,

(Received 4 September

UNS-CONICET,

8000 Bahia Blanca,

1986; received for publicohm

Argentina

3 June 1987)

Abstract-The dynamic behavior of distillation columns during startup operations have been studied and analyzed. A procedure is proposed for determining appropriate startup policies for distillation column operations based on the analysis of the dynamic behavior. The final startup policy consists of three strategies corresponding to the three characteristic stages of the startup operation. The proposed procedure has been successfully applied to a test problem. Numerical results of the dynamic behaviour and the test problem are presented. Scope--Startup operations of chemical processes and in particular, distillation columns, represent interesting problems in operability, control and simulation. Most research efforts on startup operations of distillation columns in the past have concentrated on actual plant experiences [I-3]. These papers have mainly described the problems related to plate hydraulics during the early stages of the startup period and operability problems. A good understanding of the dynamic behavior is necessary for the development of appropriate startup policies. Procedures by which these policies can bc deteimined could serve as useful tools both for the operators and designers of plants. For the operators, the information would prepare them better to diagnose and handle many possible operational problems and also give them a knowledge of the effects of the different actions taken during the startup operations. For the designers, the information could help them enormously in the screening of alternatives as well as avoiding as many operability and control problems as possible. Simulations of startup operations is difficult because of the complex plate hydraulics. For example, predictions of the sealing of downcomers, the sealing of the plate holes from the liquid flow, effect of pressure drop on the liquid holdup on plates and many more. These hydraulic effects can become important in the determination of appropriate startup policies. One way of attacking this problem is to model for distillation columns, verify the model against develop a reasonably reliable dyn experimental/plant data and then “r” hrough simulations of different startup operations, study the corresponding dynamic behavior. From this analysis, a procedure for obtaining the appropriate startup policies can be developed. Conclusions nnd Si-The ability of the dynamic model to predict, at least qualitatively, reasonably correct dynamic behavior during startup operations of distillation columns has been confirmed. Analysis of the simulated results have indicated trends in the dynamic behavior which enables the representation of the startup operation by the three characteristic stages. The first stage, called the discontinuous stage, is characterized by its short time period and the discontinuous nature of all the variables. The second stage, called the semi-continuous stage is characterized by the non-linear transients of the variables (but without any discontinuity) and approach of the hydraulic variables to their steady-state values. The time period of this stage is larger than the first stage. The third stage, called the continuous stage, is characterized by the linear transient responses of all the variables. At the end of this stage, all the variables reach their steady-state values. The proposed procedure for obtaining the appropriate startup policies has been successfully tested through a test problem. The method of solution employed to determine the strategies for the three stages were found to be satisfactory from a numerical stability point of view and in terms of robustness. The computing time, however, was found to be quite large. Present work is concentrating on this area and especially in the reduction of computing time during the discontinuous stage. With the developments of faster computers and parallel processors, this problem might not remain so significant as at present.

INTRODLKTlON

The behavior of a chemical process can be studied from the plant data or, in their absence, by a reliable model of the process. Recently, Gani et al. 141, have tPresent address: Chemical Engineering Department, University of Queensland, St Lucia, Australia. f&sent address: Instituttet for Kemiteknik, Danmarks Tekniske Hajskole, 2800 Lyngby, Denmark.

presented a dynamic model for distillation columns capable of simulating startup operations. The model has &en verified to give, at least qualitatively, the correct dynamic behavior of distillation columns during St&up operations. Also, the hydraulic submodel has been verified against plant/design data 149 51. The purpose

of this paper

is to thoroughly

study

2

C. A. Rtnz et al.

the dynamic behavior of distillation columns during startup operations and, based on this study, develop a procedure for obtaining appropriate startup policies. Emphasis is given on improving the understanding of the dynamic behavior during these operations and to simplify the problems related to startup operations. The dynamic model of Gani et al. [4] is used for the purpose. The dynamic behavior, obtained by simulation with different startup procedures, is first compared with known observations of other workers [l] and by visualizing the behavior during actual operations, Thus attempts have been made to confhm the validity of the simulated results. The scarcity of actual plant/experimental data however, makes this task quite difficult. The dynamic behavior of distillation columns are then studied to identify trends which could be used for characterizing the behavior and for simplifying and/or solving problems related to operability, control and design. Based on the analysis of the dynamic behavior, procedures for determining appropriate startup policies are proposed. Application of the proposed procedures are shown through a numerical example. SIMULATION

OF STARTUP

OPERATIONS

All simulations were performed by the generalized dynamic model of Gani et al. [4]. For the purposes of this work, an eight plate distillation column separating a binary mixture of i-butane and i-pentane has been considered. Details of the column is given in Table 1. The different startup operational procedures are generated from a basic sequence of actions whose time of employment, period of employment and nature can be varied. The sequence of actions which forms the basis for different startup operational procedures are listed below: at time t = 0, the column is empty and the liquid feed is introduced (only liquid feed is considered); t = t,, the liquid starts to “weep” to the plate below through the plate holes rather than through the downcomer. The liquid reaches the bottom of the column (or reboiler) thus increasing the liquid level there; t =f*, heat is introduced into the reboiler and the vapor starts to go up. t =t,, the condenser starts to operate as the vapor reaches the top plate and the reflux drum starts to fill up; t =t4, the reflux is introduced into the column and operation at total reflux begins; t =ts, the vapor flow through the plate holes seals the plates in terms of liquid weeping through the plate holes. This starts to increase the liquid holdup on plates; t =tb, all plates have enough liquid holdup so that the liquid can start to fall down the down-

comers. The downcomers are sealed and no vapor can go up through them; f = t,, column operation is changed ‘from total reflux. Distillate is taken out. Note: the shift from total reflux to specified reflux may be carried out in steps; t = t8. column operation continues and steadystate is attained. As can be seen from the list above, most of the actions in the beginning involve opening or closing of valves. That is, when should a;valve be opened or closed? During this period, some of the variables will vary from zero to nonzero and back to zero values (for example, liquid flow through the plate holes, vapor flow through the downcomers) while others will vary from zero values to nonzero values (for example, plate pressure drops, liquid tlow through the down-comers). At the beginning of the startup operation (at time, t = t,), all the different thermodynamic variables (for example, plate temperatures, plate pressures, plate compositions, etc.) will be the same on all plates while the hydraulic variables (for example, liquid and vapor flows, plate pressure drops, etc.) will have zero and nonzero values (depending on the startup procedure employed). From time t, until I,, therefore, “violent” changes in the column can be expected to take place. From time t = t,, conventional operation of the column may begin even though the column may be far from the desired steady-state. At time t = rg, the column is very near its desired steady-state and the control system designed to maintain the column operation around this state, may be employed. The generalized model [4] requires the user to specify the sequence of actions, the timings of different actions, length of period, valve position, etc. This allows the simulation of distillation dynamics for different startup configurations and procedures. Details of three different startup configurations and procedures are given in Fig. 1 and Table 2. These three operational procedures are representations of simulation problems of simple, intermediate and complex characteristics. The actual analysis of the dynamic behavior is made by using these procedures and variations of these procedures.

ANALYSIS OF THE DYNAMIC BEHAYIOR

For the distillation column described in Table 1 and the three startup procedures illustrated in Fig. 1 and Table 2, the transient responses of the hydraulic variables and thermodynamic variables are shown in Figs 2-6. Figure 2 shows the responses of the liquid flowrate through the downcomer of each plate for the operational procedure OPl. Figure 3 shows the response of the vapor flowrates through the plate holes and through the downcomers for 0P2. Figure 4 compares the responses of the plate pressures (thermodynamic variable) against plate pressure drops

A

generalized model for distillation column+III

Table 1. Details of the column studied Column description

:

sieve trays, lateral downcomer 8

liquid at bubble point i-butane 50% n-pentane 50% 113.4kmolh-’ 327.3 K 441.94 kPa (initial plate pressure)

Feed specilZ;cation :

Composition: Flowrate: Temperature: I4CSSUI-Z

Design pwametem

4.788 x 10skcal h-’ 3.780 x ltl’kcal h-’ 68.0 kmol h -’ 2.3 5

R&oiler heat duty (QB,: Condenser heat duty (PC): R&IX rate (R): Reflux ratio: Feed plate number: Plate

dimensions

Weir length Weir height Plate diameter Active area Plate spacing Downcomer clearance Holes area Plate thickness Hole diameter

0.532 0.030 0.762 0.366 0.457 0.030 0.037 0.003 0.002

m m m m* m m ml m m

(hydraulic variable) for OP3. Figure 5 shows the dynamic responses of the liquid holdup on plates and Fig. 6 shows the responses of the plate temperatures and liquid phase compositions for 0P3. It can be seen that the simulated results match qualitatively the startup hydraulics as described by Kister [I]. That is, at startup, vapor tends to flow through both the downcomer and the plate holes (Fig. 3), the liquid initially weeps through the plate holes and when the liquid height on the plate is high enough, it falls through the downcomers (Fig. 2), when the liquid seals the downcomer, the vapor flow through the downcomer stops (Fig. 3), when the vapor ffow becomes sufficiently large, the liquid flow through the plate holes become negligible (Fig. 2). If an actual startup operation is visualized, it can be poted that each plate is affected at different instants of time. That is, the sealing of the downcomer, the flow of vapor through the plate holes,

XF,

3

weeping of liquid etc. takes place on each plate at different instants of time. Figures 2-6 also show these phenomena. The behavior of the bottom plate as shown in Fig. 2 is due to the extra feed (Fig. 1) in the bottom plate which is stopped when the liquid starts to fall from the plate above through the downcomers. Also, in Fig. 2, the liquid flowrates for the plates above the feed plate are different from the others because the reflux rate is not the same as the feed flowrate. In Fig. 3, plate 7 is sealed before plate 3 because plate 7 receives the liquid entering as reflux, earlier than plate 3. In Fig. 4 it can be seen that DP for all the plates get nonzero values at t approximately equal to zero. This is because, initially, when the liquid rate through the plates are small and the downcomers are not sealed, the vapor flows up the column through the downcomers almost immediately. It is interesting to note the time constants for DP and the plate pressures in Fig.4. In Fig. 5a and 5b, the liquid holdups on each plate are not approximately the same (at steady-state) because the liquid and vapor flowrates for each plate are not the same. Figures 6a and 6b show that the thermodynamic variables do not start with zero values and that different plates are disturbed at different instants of time. Table 2. Different startup procedures details (corresponding Fig. 1) Option employed

OPI

Bottom dynamics included

No

YeS

Yes

Reflux drum dynamics included

No

No

YCS

Bottom liquid height controlled manipulating bottoms flowrate (P controller)

No

Yes

YCS

NO

NO

Yes

Extra feed introduced

Yes

No

NO

Reflex rate introduced progressively

No

No

Yes

I

Fig.

oP2

,

Reflux drum liquid height controlled manipulating distillate flowratc (P controller)

=XF, =0.5 OPl

P~OCCdUIC OP3 OP2

OP3

1. Configurations of the diRerent startup procedures.

to

4

C. A. RUE er al.

0

eomm

x

Feed pla(e

plate (I)

o

.Top

(5)

plawl6)

1.75

2.0

2.25

2.5

t (bin) Fig. 2. Responses of liquid flowrates through the downcomers (L) for different plates (OPI procedure).

64 Bottom

plate (I)

56

:

Feed

pbte(5)

26

t(min) Fig. 3. Responses of vapor flowrates through the downcomers (KS) and through the plate holes (V) for each plate (OP2 procedure). Note: solid symbols correspond to V and open symbols correspond to VS.

;;

0.36

a

55 a 0

0.24

.

1 0.00

Value

of

DP at (ims -0

I

I

I

I

I

I

I

I

I

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.6

0.9

1400 1.0

tl min) Fig. 4. Responses of plate pressure (P) and pressure drop (DP) of each plate for 0P3 procedure (all the plate pressures are lumped together because of the small DP between plates).

A generalized model for distillation columns-III

5

6.25

0

~Bottom

X .FBed 0

-TOP

pbate (1)

plate(S) plate

(81

0.00

t(min)

0.25

(b)

0

Bottom

x

-Feed

0

-Top

plate

plate

(I 1

(5)

ptote 18)

_I

0.00

a25

0.5

0.75

1.0

1.25 t(

1.5

1.75

2.0

2.25

2.5

min)

Fig. 5. (a) Responses of the liquid holdup (HOLD) on each plate (OPl procedure); (b) responses of liquid holdup (HOLD) on each plate (OP3 procedure). In all these figures (and many other responses not presented in this work) it can be seen that there is a very short period during which very “big” changes take place. The length of this period is not affected very much by the startup procedure. During this period, the hydraulic variables undergo drastic changes some taking on nonzero values at different instants of time from the initial zero values and others settling down to zero values after taking on nonzero values. This is also the period when the “sealing” of downcomers from vapor flow, “sealing” of plate holes from liquid flow occurs. The thermodynamic variables on the other hand, start with constant values on all plates and gradually form profiles (each plate variable changing at a different instant of time). As all the changes are discontinuous in nature, this period will be called the discontinuous stage of the startup operation. The end of this period also coincides with the time when all the criteria suggested by Kister [1], have been satisfied. It should be noted that during this period, it is more important to correctly

predict the hydraulic condition on the plate and their effect rather than the actual timing of their occurrence. Thus, a model that is qualitatively correct, is sufficient. Two other characteristic periods are also noticeable from the simulated dynamic behavior. The first of these starts at the end of the discontinuous stage. During this period, the thermodynamic variables undergo sharp changes (non-linear but not discontinuous) and the hydraulic variables undergo small changes (almost linear). Figures 2 and 3 show the behavior of the hydraulic variables while Figs 6a and 6b show the behavior of the thermodynamic variables. Figure 4 shows the behavior of both the hydraulic variable (pressure drop) and therrnodynamic variable (plate pressure). These figures clearly show that the hydraulic variables reach their steady-state values much earlier than the thermodynamic variables. The end of the second period is therefore marked by the time when the hydraulic variables are almost near their steady-state values.

C. A. Rum er al.

(a) 1 Bottom

plate

-

350

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

8

Top plate

3.0

3.5

4.0

4.5

5.0

4.0

4.5

5.0

t(min)

1.00

(b) 0.85

-

0.0

0.5

1.0

1.5

2.0

2.5

t(min)

Fig. 6. (a) Responses of plate temperatures (T) for OP3 procedure; (b) responses of the compositions of i-butane on each plate (liquid phase) for 0P3 procedure.

This period will be called the semi-continuous stage to distinguish from the last period which will be called the continuous stage. The start of the second period and the dynamic behavior during this period, is equivalent to the behavior of a column subjected to large disturbances. The validity of the predicted results for such simulations have already been established [4]. The operation during the continuous stage is similar to maintaining the operation of a distillation column around a desired steady-state subject to small disturbances. The control system designed to maintain.the column operation at a particular steady state can thus be employed at the start of this period. At the end of the continuous stage therefore, all the variables reach their steady-state values. Figure’7 compares the dynamic responses of plate pressure for these different startup operational procedures (OPl, OP2 and OP3). It can be seen that although the total length of the operation is indepen-

dent of the procedure employed, the transient paths followed can be quite different. Through an analysis of these responses different startup procedures can hence be screened. For example, the “kink” in the pressure response for OPl may cancel this option. Summarizing the analysis of above, the following characteristic dynamic behavior during startup operations were noted. The period comprising the empty column to the final steady state may be divided into three stages. The time period of the first stage is very short and independent of the operational procedure employed. The time period of the last stage is the largest. During the second stage the thermodynamic variables undergo bigger changes than the hydraulic variables. At the start of the third stage, almost all hydraulic variables are at their steady-state values. The different startup operational procedures may be screened taking into account the possible operability problems. The hydraulic stability conditions are statisfied during the first stage.

A

0.0

I 0.5

generalized model for distillation columns--III

I 1.0

I 1.5

I 2.0

I 2.5 t(

-

OPi

-

OP2

-x-

OP3

I 3.0

I

I

35

4.0

Based on the analysis of the dynamic behavior given above, a methodology for obtaining an appropriate startup policy is proposed. The policy is divided into three strategies each corresponding to the three stages of the dynamic behavior previously described. For the discontinuous stage, the strategy is to select a procedure which will have minimum operability problems. The procedure involves opening or closing of valves at specified instants of time. The procedure also ensures that no discontinuities exist after this stage. The second stage (i.e. the semi-continuous stage) is where an optimal control strategy may be employed. Because both the hydraulic variables and the thermodynamic variables are changing here, the dynamic responses are highly transient in nature and thus the column control system may not be applicable. The strategy here is to specify the values of the manipulative variables (for example, reboiler heat duty, reflux rate etc.) at different intervals of time, so that the end of this stage is reached at a minimum cost. The end of the stage may be verified by monitoring the values of the hydraulic variables and the profiles of the thermodynamic variables. The cost may be evaluated from the cost of energy, the product lost, etc. Changes in the manipulative variables should be bounded so that no discontinuities or hydraulic instabilities occur during this stage. At the beginning of the third stage, the control system designed to maintain the column operation around the specified steady-state is switched on. Since the open-loop responses during this stage showed linear behavior, this strategy seems reasonable. Thus to obtain the proposed startup policy, complex dynamic simulation problems (for the discontinuous stage), dynamic simulation coupled with optimization problems (for the semi-continuous stage) and simplified closed-loop dynamic simulation

I 4.5

5.0

min 1

Fig. 7. Comparison of the plate pressure (P) responses for three different

DESCRIPTION OF THE PROPOSED STARTUP POLICY

7

startup

procedures

(fed

plate).

problems will have to be solved. The generalized dynamic model used in this work allows all of this. The proposed solution procedure is described below. SOLUTION

PROCEDURE THE STARTUP

FOR OBTAINING POLICY

As can be seen from the description of the strategies for the three stages, a priori simulation is required for the solution of the problems of the discontinuous stage. A combination of physical insights, experience and analysis of the results of the dynamic simulation is proposed (for the present) to obtain the appropriate strategy. Examination of Figs 2-6 shows that alternative OP3 is better than the other two from an operability point of view. Also, from the description of the three alternatives (Table 2), alternative 0P3 seems more reasonable. Thus, the solution procedure for this stage is to simply perform simulations for the different alternatives and to select the one which shows the least operability problems. Many of the alternatives may be screened without simulation through physical insights and experience. The simulation in this case therefore serves the verify the advantages of a proposed alternative. Solution of numerical optimization problems coupled with dynamic simulation is required in order to obtain the strategy (for the startup policy) for the semi-continuous stage. The problem can be formulated thus. Let us consider the distillation column model of Gani ef al. [4] represented by a set of ordinary differential equations (ODES) and a set of procedures. The above set of equations are represented in the form: Y’ = f(Y, P.

where the procedures form:

t).

(1)

p can be written in the general

p = g(y, z, d, t)

(2)

The procedures are functions of the differential variables y, a set of algebraic variables z and the system parameters d. The set of algebraic variables z includes a sub-set of variables q which are designated as the manipulative variables (for example, reboiler heat duty, reflux rate, feed flowrate, etc.). The vector y ati t = to is known (i.e. the initial condition at the start of the semi-continuous stage). The performance of the startup policy (to be determined) can be evaluated through a scaYar objective function of the type: P,=dJIY(r,),t/l+

‘1 J I 10

KY 0)s u(t),

11dt,

(3)

where: tf is the final time of the semicontinuous stage, u(t) are the optimizing variables (for example, a set of manipulative variables). The startup policy to be selected should correspond to a minimum in P, subject to the following constraints: Umi”< u(t) < u,

,

t1 G &w., ah < a[yO), u(t), tl< G,, gmi. d g[u(tX

(4)

number of intervals is also unknown (see discussion below). The problem to be solved can therefore be defined thus: Determine the elements of each of the sub-vectors such that equation (3) is minimized and equations (4x6) are satisfied. In other words, specify the values of the manipulative variables at the different intervals of time so that the end of the semi-continuous period is reached with the minimum cost (including time) and without breaking any limiting conditions. As a special case of the minimization problem, the objective function may be defined to measure the difference between the desired steady state and the state at any instant of time. Thus, when this difference approaches a certain minimum value, the end of the semi-continuous stage would be reached and the switch to the linear controllers can be made. This special formulation of the objective function will therefore allow the specificatipn of the manipulative variables at different intervals of time in such a way that a particular “state” of the column is attained. The objective function in this case would be (considering only the temperature profile),

(5) PI= &T,--(TPFAT,,)]*,

(6)

(7)

ir; I

where: &u(t), t] is a set of linear constraints, ah(r), U(t), ?] is a set of non-linear constraints. Equation,(4) represents the limitations corresponding to the valve action for the manipulative variables. The linear constraints represent the restrictions imposed on the manipulative variables by the process. For example, the reflux rate or reboiler heat duty must satisfy a special criteria. The non-linear constraints represent the restrictions imposed on the process reaching a desirable steady state. For example, a specified product purity. Since these variables can only be determined through simulation, these are therefore non-linear functions of the manipulative variables. The problem formulated above can be defined thus: minimize equation (3) in terms of the variables u(t) subject to the constraints given by equations (4)-(6). A similar type of problem was solved by Sullivan [6] and Sargent and Sullivan [7] for the development of an optimal feed changeover policy. The above problem treats u(t) as continuous variables. By discretizing u(t) with respect to time the startup policy is coupled to this problem. That is, dividing the time period into intervals and for each interval, determining the values of u(t). Thus, each element of the vector u(t) will be represented by sub-vectors of dimension equal to the total number of intervals. It should be noted that the end of the semi-continuous period is not known and hence the

where: NP = r;J = T, = AT,, =

the total number of plates, the steady state temperature of plate i, the instantaneous temperature of plate i, the permitted deviation from the steady ’ state of the plate temperature.

A more general form of equation (7) which takes into account cost terms as well as other profiles (for example, compositions) may be detined thus: NT

NP

Pr = C C wkbw - (x;li + A~,~,ll* k-li-l

+

5 C&u,

k=l

21, (8)

where: NT = the total number of different variables to be considered in evaluating P,, wk = weighting terms for each type of variable, xkei= the variable of type k of plate i and is a function of u. If the period of time is discretized in NZ intervals of control, the vector of variables of optimization will be thus, u = [Ui, 21, uzr 12,. . . , 4,

ti, . . . , UN,,hlT.

(9)

where: u, is the vector of the optimizing variables at interval i, ti is the time at which a value is assigned to the variable ui at interval i.

A generalized model for distillation columns-111 Thus, the dimension of II will be (NC + 1)Nl and the trajectory of the control variables (with respect to time) will be given by: n(t) = uI

when

t,