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Dynamics and control of totally refluxed reactive distillation columns
Journal of Process Control 22 (2012) 1182–1197
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Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
Dynamics and control of totally refluxed reactive distillation columns Wei Liu a , Kejin Huang a,∗ , Liang Zhang a , Haisheng Chen a , San-Jang Wang b a b
College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China Department of Chemical and Material Engineering, Ta Hwa Institute of Technology, Chiunglin, Hsinchu 307, Taiwan
a r t i c l e
i n f o
Article history: Received 3 August 2011 Received in revised form 3 May 2012 Accepted 10 May 2012 Available online 15 June 2012 Keywords: Reactive distillation column Totally refluxed operation Process dynamics Inventory control Process operation
a b s t r a c t According to the mechanism of the reaction operation involved, reactive distillation columns are often designed to work in a totally refluxed operation mode. The totally refluxed operation mode makes the reflux drum interact solely with the reaction operation involved and retards considerably the dynamics of the latter. The resultant great difference in process dynamics between the reaction operation and the separation operation involved leads frequently to under-damped responses with the degree of underdampness closely dependent on the inventory control of the reflux drum. With the tight inventory control of the reflux drum, the degree of under-dampness can be suppressed and this presents a favorable effect to process dynamics and controllability of the totally refluxed reactive distillation columns. Two hypothetical ideal reactive distillation columns with and without a side reaction, respectively, and a high-purity ethylene glycol reactive distillation column are employed to examine the unique dynamics and controllability of the totally refluxed reactive distillation columns. The results obtained are in good accordance with the above interpretation. The current work reveals the general behaviors of the totally refluxed reactive distillation columns and can be particularly useful in control system synthesis and design. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction According to the mechanism of the reaction operation involved, reactive distillation columns are often designed to work in a totally refluxed operation mode. For example, in case of carrying out a reaction, A + B ↔ C, in a reactive distillation column, the desired product C should be withdrawn from the bottom of the process because it is the heaviest component. At the top of the reactive distillation column, the reactants, A and B, are usually accumulated and they should be recycled back to the reactive section through a totally refluxed operation mode in reflux drum. Since there is no rectifying section in the reactive distillation column, the reflux drum is directly connected to the reactive section and this leads to an intimate interaction between them. Such interaction affects considerably the reaction operation involved and can eventually present a strong impact to the dynamics and controllability of the totally refluxed reactive distillation columns. These should represent the unique behaviors of the totally refluxed reactive distillation columns and are worthwhile to be studied carefully. Recent years have seen increasingly more studies conducted on the dynamics and operation of reactive distillation columns [1–6]. For the totally refluxed reactive distillation columns, there have also appeared a number of papers dealing with their operation
∗ Corresponding author. Tel.: +86 10 64434801; fax: +86 10 64437805. E-mail address: [email protected] (K. Huang). 0959-1524/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jprocont.2012.05.007
issues. Luyben once compared the operation of two ternary ideal reactive distillation columns with and without chemically inert components, respectively [7]. He found that the presence of a chemically inert component could alter the process from a totally refluxed operation mode into a partially refluxed operation mode. In particular, a direct measurement of stage composition is required for the effective process control. Recently, Kaymak et al. further explored the inferential temperature control of the ternary ideal reactive distillation columns with and without chemically inert components [8,9]. They insisted that the direct measurement of stage composition might not be necessary provided that the control structure and control tray locations were properly chosen. Ethylene glycol reactive distillation column represents a typical example for the totally refluxed reactive distillation column and has received considerable attention in the aspects of process dynamics and controllability. Although Kumar and Daoutidis claimed that nonlinear controller was superior to linear PI controller in its operation [10], Al-Arfaj and Luyben showed that the process could be effectively controlled by a simple PI control scheme plus a feed-forward compensator [11]. Huang et al. addressed internal heat integration to a high-purity ethylene glycol reactive distillation column [12,13]. They found that seeking further internal heat integration between the reaction operation and the separation operation involved helped to improve the process dynamics and lessened the difficulties in process operation. Furthermore, the tight inventory control of the reflux drum could function to suppress the inherent process nonlinearity and
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
Nomenclature A Avp B Bot Bvp C CC D DEG E EG EO F FC H HR HV k kcon L LC nc NT PC R r T Ti V W x z
hypothetical component vapor pressure constant (Pa) hypothetical component bottom product flow rate (mol s−1 ) vapor pressure constant (Pa K) hypothetical component composition controller hypothetical component diethylene glycol activation energy of reaction (J mol−1 ) ethylene glycol ethylene oxide feed flow rate (mol s−1 ) flow controller stage holdup (mol) heat of reaction (J mol−1 ) latent heat of vaporization (J mol−1 ) controller gain controller gain of reflux drum level (s−1 ) liquid flow rate (mol s−1 ) level controller total number of components total number of stages pressure controller ideal gas law constant (J mol−1 K−1 ) reaction rate (mol mol−1 s−1 or mol m−3 s−1 ) temperature (K) reset time (s) vapor flow rate (mol s−1 ) water liquid composition feed composition
Greek symbols ˛ pre-exponential factor stoichiometric coefficient of a reaction Subscripts b backward reaction f forward reaction i component index stage index j m main reaction s side reaction
non-minimum phase behavior, favoring therefore process operation significantly. Although this conclusion was acquired from the high-purity ethylene glycol reactive distillation column, it invoked us to anticipate that the interesting phenomenon might be closely related to the particular structure of the totally refluxed reactive distillation columns. This consideration stimulated us to acquire the insights of the phenomenon. To the best of our knowledge, until now no pertinent work has been conducted on the relationship between process configuration and process dynamics and controllability for the totally refluxed reactive distillation columns. The primary objective of the current work is to investigate the dynamics and control of the totally refluxed reactive distillation columns. Two hypothetical ideal reactive distillation columns with and without a side reaction, respectively, and a high-purity ethylene glycol reactive distillation column are chosen as representative examples. Both open-loop process dynamics
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and closed-loop operation are examined with special attention focused on the effect of inventory control of reflux drum. The unique behaviors of the totally refluxed reactive distillation columns are analyzed and the acquired insights are then summarized.
2. Unique process dynamics and its implications on the control of the totally refluxed reactive distillation columns Unlike the conventional reactive distillation columns with distillate and bottom withdrawal, the totally refluxed reactive distillation columns contain a reactive section with a direct connection to the reflux drum (because there is no rectifying section). In an extreme point of view, it is even reasonable to treat them as an equivalent reactor with heat exchange in the condenser. Because of the large liquid holdup in the reflux drum, once the light and/or heavy reactants enter the reflux drum, they will generally need a long time to be driven back to the reactive section and this retards considerably the dynamics of the reaction operation involved. In the totally refluxed reactive distillation columns, the separation operation involved is solely carried out in the stripping section. Since the heat duty of reboiler (or the reboil flow rate) serves as the manipulated variable to control the purity of bottom product, it should be taken as the main variable to represent the dynamics of the separation operation involved. Because the bottom reboiler has much faster dynamics than the liquid hydraulics in the reflux drum and in the reactive section compounded further with the reaction kinetics, sharp difference exists in process dynamics between the reaction operation and the separation operation involved. This causes generally under-dampness in the open-loop dynamic responses of the bottom product. The greater the difference is, the severer the degree of the under-dampness becomes. Moreover, the sharp difference in process dynamics between the reaction operation and the separation operation involved can also induce a certain degree of degradation in system properties, e.g., in the aspects of process nonlinearity and non-minimum phase behavior. This inevitably gives rise to additional difficulties in process operation. In terms of the above analysis, one can now understand that the sharp difference in process dynamics between the reaction operation and the separation operation involved is the main reason for the occurrence of the under-damped behaviors in the totally refluxed ideal reactive distillation columns. To suppress the degree of the under-dampness, one must therefore rely on the acceleration of reaction kinetics and liquid hydraulics in the reactive section and in the reflux drum. Although operating pressure can be increased to accelerate reaction kinetics, it is generally determined by the match of operating conditions between the reaction operation and the separation operation involved. For the liquid hydraulics in the reactive section and in the reflux drum, it is decided by process capacity and is usually operationally infeasible to be enhanced through the reduction of their sizes. The only way left is to accelerate the dynamics of the reflux drum through tight control of liquid level. Remember, however, that this differs totally from the conventional design practice, i.e., the inventory of the reflux drum should be loosely controlled in order to avoid strong interaction between different control loops. In the remainder of this paper, we will employ three totally refluxed reactive distillation columns, including two hypothetical ideal reactive distillation columns with and without a side reaction, respectively, and a high-purity ethylene glycol reactive distillation column, to examine their unique dynamics and the effect of inventory control of reflux drum on process operation.
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Table 1 Physicochemical properties and operating conditions of Example I. Parameter
Value 8.0 0 9 5 20,000.0 1000.0 20,000.0 125,520 125,520 0.008 0.0004 12.63 12.82 10 2 1.0 1.0 4:2:1 0 29,053.7 98 12.3463/3862 11.6531/3862 10.9600/3862
Column pressure (bar) Rectifying section Reactive section Stripping section Condenser Column tray Reboiler Forward Backward Forward Backward A B A B A B
Number of stages
Liquid hodlup (mol) Reaction activation energy (J mol−1 ) Specific reaction rate at 366 K (mol s−1 mol−1 ) Feed flow rate (mol s−1 ) Feed location Feed thermal condition Relative volatility (A:B:C) Heat of reaction (J mol−1 ) Latent heat of vaporization (J mol−1 ) Bottom product specification (C, mol%)
A (Avp /Bvp ) B (Avp /Bvp ) C (Avp /Bvp )
Vapor pressure constants
3. Example I: A totally refluxed ideal reactive distillation system involving one reaction with two light reactants and one heavy product 3.1. Process description This hypothetical ideal reactive distillation system is taken from Sun et al. [14]. As is shown in Fig. 1, this reactive distillation column consists of a reactive section above the stripping section with 9 and 5 stages, respectively. The reactive distillation column is operated in a totally refluxed operation mode, and two pure reactant feeds, FA and FB , are fed to stages 10 and 2 (numbered from the top condenser
down to the bottom reboiler, and similarly hereinafter), respectively, with the latter in slight excess. Detailed physicochemical properties and operating conditions are listed in Table 1, and other relevant information can be found in the corresponding reference. In the reactive section of this totally refluxed ideal reactive distillation column, reactants A and B undergo a reversible liquid-phase reaction to form the desired product C. The detailed reaction mechanism is A+B↔C
(1)
The volatilities are such that the product C is the heavy key, A is the light key, and B is the intermediate key. The net reaction rate for component i on reactive stage j is given by ri,j = i Hj (Kf,j xA,j xB,j − Kb,j xC,j )
(2)
where Kf,j and Kb,j are the forward and backward specific reaction rate constants, respectively, described by the Arrhenius law as
2 FB = 12.82 mol·s–1
Kf,j = ˛f exp
Reflux = 34.41 mol·s–1
Kb,j = ˛b exp
−Ef RTj
−Eb RTj
(3.1)
(3.2)
Equimolar overflow is assumed in the nonreactive section. As the reaction reduces the molecular number, i.e., nc
FA = 12.63 mol·s–1
10
i = −1
(4)
i=1
the mass balance equations in the reactive section become
15
Vj = Vj+1 −
V[16] = 34.41 mol·s–1 NT=16
xA = 0.0026 xB = 0.0174 xC = 0.9800
Bot = 12.85 mol·s–1
Lj = Lj−1 +
rj HR
nc i=1
C
Fig. 1. Scheme of a totally refluxed ideal reactive distillation column with one reaction (Example I).
(5)
HV ri,j +
rj HR HV
(6)
Ideal vapor and liquid phase behaviors are assumed in the vapor–liquid equilibrium calculation. The steady-state profiles of temperature, vapor and liquid flow rates, liquid composition, and net reaction rates are calculated and shown in Fig. 2.
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
a
b
450
1185
60
L & V Flowrate (mol·s–1)
L
T (K)
430
410
390
V
40
20
370
0
1
4
7
10
13
16
1
4
7
Stage nu mber
1.0 A
B
C
x (mole fraction)
0.8 0.6 0.4 0.2
d
5
Reaction Rate (mol·s–1)
c
10
13
16
13
16
Stage nu mber
4 3 2 1 0
0.0 1
4
7
10
13
1
16
4
7
10 Stage nu mber
Stage number
Fig. 2. Steady-state profiles of Example I. (a) Temperature, (b) vapor and liquid flow rates, (c) liquid composition and (d) net reaction rates.
3.2. Evaluation of open-loop process dynamics
a
0.98 8 kcon = 0.2
kcon = 1.0
kcon = 2.5
x C (mole fraction)
0.98 6 0.98 4 0.98 2 0.98 0.97 8 0
2
4
6
8
10
12
Time (h)
b
0.99 kcon = 0.2
x C (mole fraction)
In Fig. 3, the open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon (i.e., kcon = 0.2, 1.0, and 2.5 s−1 ) are depicted, when it is subjected to a ± 5% step change in the reboil flow rate, respectively. Sharp differences can be found between the positive and negative responses, irrespective of the detailed parameters assigned to the inventory controller of the reflux drum. In the case of the positive change in the reboil flow rate, the totally refluxed ideal reactive distillation column exhibits under-damped responses with the degree of under-dampness inversely proportional to the controller gain of the reflux drum. In the case of the negative change in the reboil flow rate, the totally refluxed ideal reactive distillation column still displays under-damped responses with the degree of under-dampness inversely proportional to the controller gain of the reflux drum and this can clearly be observed before the time instant of 4 h. After that time instant, transition between the inherent multiple steady states occurs and this moves the totally refluxed ideal reactive distillation column to an unexpected steady state. This is why a high degree of asymmetry has been noticed between the positive and negative responses in this situation. The open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon are shown in Fig. 4, when it is confronted with a ±2% step change in the feed flow rate of reactant B, respectively. No under-damped responses are observed and the variation in the controller gain of the reflux drum shows almost no impact on the system responses. Regardless of the positive and negative changes in the feed flow rate of reactant B, the composition of the bottom product settles down to two values lower than the initial steady state, implying the possible transition between the inherent multiple steady states.
kcon = 1.0
kcon = 2.5
0.97
0.95
0.93 0
8
16
24
32
40
Time (h) Fig. 3. Open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±5% step change in the reboil flow rate, respectively (Example I). (a) Positive responses and (b) negative responses.
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Table 2 Controller parameters for Example I. Parameter
Control loop
k Ti (min)
Reflux drum level
Base level
0.2/1.0/2.5 –
0.2 –
3.3. Evaluation of closed-loop control performance Closed-loop control of the totally refluxed ideal reactive distillation column is conducted to evaluate the impact of the inventory control of the reflux drum. A control scheme that uses two direct composition control loops is selected in the current study and Fig. 5 provides the detailed control structure [7]. The pressure is assumed to be perfectly controlled at the nominal steady-state by the heat removal of condenser. The level of the reflux drum is controlled by reflux flow rate, and a P-only controller is used. The level of rebolier is controlled by the bottom product flow rate, and a P-only controller is chosen. The composition of the bottom product is controlled by the heat duty of reboiler, and a PI controller is adopted. According to the steady-state composition profile in Fig. 2, the composition of stage 3 is selected to be controlled by the feed flow rate of reactant B because it changes rapidly there. This composition loop serves to regulate the stoichiometric balance between the two reactants, and a Ponly controller is adopted. The feed flow rate of reactant A is the production rate handle and flow controlled. The composition controllers are tuned with Ziegler–Nichols tuning method and all the controller parameters are listed in Table 2. A first-order lag with a time constant of 5-min is inserted in all composition control loops.
x C (mole fraction)
a
0.98 5
Stoichiometric balance
Bottom C
11.13 –
0.9 0.42
The composition transmitter spans are set to be 0.1 and all the control valves are designed to be half open at the nominal steady state. The servo responses of the totally refluxed ideal reactive distillation column are illustrated in Fig. 6, when the controller of the bottom product purity is subjected to a ±0.3 mol% step change in its set-point, respectively. It is noted that strong effect has been generated by the variation in the controller gain of the reflux drum for both the positive and negative changes in the set-point of the bottom control loop. With increasingly tight inventory control of the reflux drum, the totally refluxed ideal reactive distillation column displays a consistent improvement in dynamic responses in the case of positive change in the set-point of the bottom control loop. In the case of the negative change in the set-point of the bottom control loop, sustainable oscillations occur when the controller gain of the reflux drum has been set at 0.2 and 1.0 s−1 respectively. After it has been increased to 2.5 s−1 , the sustainable oscillation is vanished. Fig. 7 presents the regulatory responses of the totally refluxed ideal reactive distillation column, when a ±2% step change is introduced to the production rate, respectively, in terms of a corresponding variation in the feed flow rate of reactant A. Quite similar to the circumstances of the step changes in product purity for the bottom control loop, the totally refluxed ideal reactive distillation column exhibits improved responses with the increasingly tight inventory control of the reflux drum. More specifically, whereas the low controller gain (i.e., kcon = 0.2 s−1 ) of the reflux drum results
0.97 5
PC 0.96 5
LC kcon = 0.2
kcon = 1.0
B
kcon = 2.5
2
0.95 5 0
8
16
24
32
3
40
Time (h)
x C (mole fraction)
b
CC
0.99 kcon = 0.2
kcon = 1.0
kcon = 2.5
0.96
A
10
0.93
FC
15 CC
NT=16
0.9 0
8
16
24
32
40
Time (h)
Steam
LC
C Fig. 4. Open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±2% step change in the feed flow rate of reactant B, respectively (Example I). (a) Positive responses and (b) negative responses.
Fig. 5. Control scheme of Example I.
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
0.98 5
b x C (mole fra ction)
x C (mole fra ction)
a
0.98 3
0.98 1 kcon = 0.2
kcon = 1.0
0.99 kcon = 0.2
kcon = 1.0
kcon = 2.5
0.98
0.97
kcon = 2.5
0.97 9
0.96 0
2
4
6 Time (h)
8
10
12
0
37
40
36
37.5 V[16] (mol·s–1 )
V[16] (mol·s–1 )
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35
34
2
4
6 Time (h)
8
10
12
35
32.5 kcon = 0.2
kcon = 1.0
kcon = 2.5
33
kcon = 0.2
30 0
2
4
6 Time (h)
8
10
0
12
37
2
4
6 Time (h)
kcon = 1.0
8
kcon = 2.5
10
12
37
Reflux (mol·s–1 )
Reflux (mol·s–1 )
36 36
35
35 34 33
kcon = 0.2
kcon = 1.0
kcon = 0.2
kcon = 2.5
34
kcon = 1.0
kcon = 2.5
32 0
2
4
6 Time (h)
8
10
12
0
2
4
6 Time (h)
8
10
12
Fig. 6. Servo responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±0.3 mol% step change in the set-point of the bottom composition control loop, respectively (Example I). (a) Positive responses and (b) negative responses.
in sustainable and yet magnified oscillations, the high ones (i.e., kcon = 1.0 or 2.5 s−1 ) achieve a stable operation. 4. Example II: A totally refluxed ideal reactive distillation column with one main and one side reactions
This hypothetical ideal reactive distillation system is developed from Example I with the inclusion of an additional reversible side reaction
(8.2)
In terms of relative volatilities, these four components can be arranged in the following order: A>B>C>D
4.1. Process description
B+C↔D
rs i,j = s i Hj (Kf s,j xB,j xC,j − Kb s,j xD,j )
(9)
which means the component D is also withdrawn with component C from the bottom of the totally refluxed ideal reactive distillation column. Detailed physicochemical properties and operating conditions are tabulated in Table 3. The steady-state profiles of temperature, vapor and liquid flow rates, liquid composition, and net reaction rates are calculated and shown in Fig. 9.
(7) 4.2. Evaluation of open-loop process dynamics
As shown in Fig. 8, the process configuration is exactly similar to that of Example I (c.f., Fig. 1). The main and side reaction rates for component i on reactive stage j are given by rm i,j = m i Hj (Kf m,j xA,j xB,j − Kb m,j xC,j )
(8.1)
In Fig. 10, the open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon (i.e., kcon = 0.2, 1.0, and 2.5 s−1 ) are depicted, when there is a ±5% step change in the reboil flow rate, respectively.
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0.98 2 kcon = 0.2
kcon = 1.0
b
kcon = 2.5
x C (mole fraction)
x C (mole fraction)
a
0.98
0.97 8
0.98 2 kcon = 0.2
kcon = 1.0
kcon = 2.5
0.98
0.97 8 0
2
4
6 Time (h)
8
10
12
0
36
2
4
6 Time (h)
12
kcon = 1.0
kcon = 2.5
34.5 V[16] (mol·s–1 )
35.5 V[16] (mol·s–1 )
10
35 kcon = 0.2
35
34.5
34
33.5 kcon = 0.2
kcon = 1.0
kcon = 2.5
34
33 0
2
4
6 Time (h)
8
10
0
12
35.5
2
4
6 Time (h)
8
10
12
35
Reflux (mol·s–1 )
kcon = 0.2
Reflux (mol·s–1 )
8
35
34.5
kcon = 0.2
kcon = 1.0
kcon = 1.0
kcon = 2.5
34
kcon = 2.5
34
33 0
2
4
6 Time (h)
8
10
12
0
2
4
6 Time (h)
8
10
12
Fig. 7. Regulatory responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±2% step change in the production rate, respectively (Example I). (a) Positive responses and (b) negative responses.
Under-dampness is presented in both the positive and negative responses with its degree inversely proportional to the controller gain of the reflux drum. Similar to Example I, there exists a high degree of asymmetry (i.e., process nonlinearity) between the positive and negative responses. In particular, a much larger variation is caused in the negative response in case kcon is equal to 0.2 s−1 than in case kcon is equal to 1.0 and 2.5 s−1 . It is apparently related to the slow dynamics of the reflux drum. Although not shown here, the difference in steady states can finally vanish after a long time period. Non-minimum phase behavior (i.e., initial inverse response) is observed, which is evidently attributed to the existence of a side reaction. It is noted here that the severity of process nonlinearity and non-minimum phase behavior is also inversely proportional to the controller gain of the reflux drum.
Fig. 11 gives the open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon , when it is in the face of a ±2% step change in the feed flow rate of reactant A, respectively. Extremely similar to Example I, the impact from the variation of the controller gain of the reflux drum is hardly observed in this situation. 4.3. Evaluation of closed-loop control performance Fig. 12 illustrates the detailed control scheme for the totally refluxed ideal reactive distillation column with a main and a side reactions. In the light of the steady-state composition profile given in Fig. 9, stage 11 is chosen and controlled with the feed flow rate of reactant A to regulate the stoichiometric balance between the
Table 4 Controller parameters for Example II. Parameter
k Ti (min)
Control loop Reflux drum level
Base level
Stoichiometric balance
Bottom C
0.2/1.0/2.5 –
0.2 –
4.58 –
2.7 16.67
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Table 3 Physicochemical properties and operating conditions of Example II. Parameter
Value
Column pressure (bar) Number of stages
Liquid hodlup (mol) Main reaction activation energy (J mol−1 ) Side reaction activation energy (J mol−1 ) Specific main reaction rate at 366 K (mol s−1 mol−1 ) Specific side reaction rate at 366 K (mol s−1 mol−1 ) Feed flow rate (mol s−1 ) Feed location Feed thermal condition
Rectifying section Reactive section Stripping section Condenser Column tray Reboiler Forward Backward Forward Backward Forward Backward Forward Backward A B A B A B
Relative volatility (A:B:C:D) Heat of reaction (J mol−1 ) Latent heat of vaporization (J mol−1 ) Bottom product specification (C, mol%) Vapor pressure constants
A (Avp /Bvp ) B (Avp /Bvp ) C (Avp /Bvp ) D (Avp /Bvp )
reactants, A and B. A P-only controller is again used here. The feed flow rate of reactant B is the production rate handle and flow controlled. The controller parameters are tabulated in Table 4. The servo responses of the totally refluxed ideal reactive distillation column are illustrated in Fig. 13, when there is a ±0.3 mol% step change in the set-point of bottom control loop, respectively. In case kcon is set to be 0.2 s−1 , while the totally refluxed ideal
8.0 0 9 5 20,000.0 1000.0 20,000.0 125,520 125,520 125,520 125,520 0.008 0.0004 0.005 0.0002 12.63 12.82 10 2 1.0 1.0 8:4:2:1 0 29,053.7 98 12.3463/3862 11.6531/3862 10.9600/3862 10.2669/3862
reactive distillation column settles down with a large deviation in the face of the positive change in the set-point of bottom control loop, a magnified oscillation occurs in the face of the negative change in the set-point of bottom control loop. In case kcon is set to be 1.0 and 2.5 s−1 , fairly smooth operation can be achieved for both the positive and negative changes in the set-point of bottom control loop. The responses with kcon as 2.5 s−1 outperform those with kcon as 1.0 s−1 . Fig. 14 presents the regulatory responses of the totally refluxed ideal reactive distillation column in the face of a ±2% step change in the feed flow rate of reactant B, respectively. When kcon is set to be 0.2 s−1 , rather oscillating responses are obtained for both the positive and negative changes in the feed flow rate of reactant B. With relatively tight inventory control of the reflux drum (i.e., 1.0 and 2.5 s−1 ), sharp improvement in control performance is achieved, giving much smaller overshoots and shorter settling times. A counterintuitive phenomenon is observed here, i.e., the heat duty of reboiler changes inversely proportional to the production rate. It is considered to be caused by the existence of input multiplicities in the totally refluxed ideal reactive distillation column. The transition between the inherent multiple steady states distorts actually the relationship between the production rate and the heat duty of reboiler. 5. Example III: A high-purity ethylene glycol reactive distillation column 5.1. Process description Ethylene glycol (C2 H6 O2 ) is produced by hydration of ethylene oxide (C2 H4 O), i.e., the main reaction: C2 H4 O(EO) + H2 O → C2 H6 O2 (EG), H R = –80.0 × 103 J mol−1 (10.1)
Fig. 8. Scheme of a totally refluxed ideal reactive distillation column with one main and one side reactions (Example II).
rm = 3.15 × 1015 exp
−9547 xEO xW T
(10.2)
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W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
a
450
140
b L & V Flowrate (mol·s–1)
L
T (K)
430
410
390
105
70
35
370
0
1
4
7
10
13
16
1
4
Stage number 5 A
B
C
7
10
13
16
Stage number
1.0
0.03
D
Reaction Rate (mol·s–1)
0.8 x (mole fraction)
V
0.6 0.4 0.2
0.02
4
0.01
3
0 2
1
6
main reaction
1
11
16
side rea ction
0
0.0 1
4
7
10
13
1
16
Stage number
4
7
10
13
16
Stage number
Fig. 9. Steady-state profiles of Example II. (a) Temperature, (b) vapor and liquid flow rates, (c) liquid composition and (d) net reaction rates.
Table 5 Physicochemical properties and operating conditions of Example III. Parameter
Value
Column pressure (bar) Number of stages Catalyst distribution (m3 ) Liquid hodlup (mol) Reaction rate (mol m−3 s−1 ) Feed flow rate (mol s−1 ) Feed location Feed thermal condition Heat of reaction (J mol−1 )
Rectifying section Reactive section Stripping section Stages 2 ∼ 8 Stages 9 ∼ 14 Condenser Column tray Reboiler Main reaction Side reaction EO W EO W EO W Main reaction Side reaction
Latent heat of vaporization (J mol−1 ) Bottom product specification (EG, mol%) Vapor–liquid equilibrium constants at the atmosphere pressure
EO W EG DEG
1.01325 0 13 2 0.043 0.1 30,000.0 1000.0 30,000.0 3.15 × 1015 exp(–9547/T)xEO xW 6.30 × 1015 exp(–9547/T)xEO xEG 7.65 7.31 8 2 1.0 1.0 −80,000.0 −13,100.0 40,000.0 94.8 71.9 exp[5.72(T–469)/(T–35.9)] 221.2 exp[6.31(T–467)/(T–52.9)] 77.0 exp[9.94(T–645)/(T–71.4)] 47.0 exp[10.42(T–681)/(T–80.6)]
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
a
a
1
1191
1 kcon = 0.2
x C (mole fraction)
x C (mole fraction)
0.8
0.7 kcon = 0.2
kcon = 1.0
0.95
0.92 5
kcon = 2.5
0.9
0.6 0
2
4
6
8
10
12
0
2
4
6
10
8
12
Time (h)
Time (h) 1
b
0.99 kcon = 0.2
kcon = 1.0
kcon = 2.5
0.98 x C (mole fraction)
x C (mole fraction)
kcon = 2.5
0.97 5
0.9
b
kcon = 1.0
0.9
kcon = 0.2
kcon = 1.0
0.97
0.96
kcon = 2.5
0.8
0.95 0
2
4
6
8
10
12
0
2
4
Time (h)
6
10
8
12
Time (h)
Fig. 10. Open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±5% step change in the reboil flow rate, respectively (Example II). (a) Positive responses and (b) negative responses.
Fig. 11. Open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±2% step change in the feed flow rate of reactant A, respectively (Example II). (a) Positive responses and (b) negative responses.
Ethylene oxide can react further with ethylene glycol to form the unwanted by-product diethylene glycol (C4 H10 O3 ), i.e., the side reaction: C2 H4 O + C2 H6 O2 → C4 H10 O3 (DEG), H R = –13.1 × 103 J mol−1 (11.1) rs = 6.3 × 1015 Exp
−9547 xEO xEG T
PC
(11.2)
Both the reactions are irreversible and highly exothermic. Fig. 15 shows a process design with 17 stages including a total condenser at the top and a partial reboiler at the bottom. The process has no rectifying section, either and works in a totally refluxed operation mode with water and ethylene oxide fed onto stages 2 and 8, respectively. Table 5 gives the relevant physicochemical properties and nominal steady state operating conditions. The steady-state profiles of temperature, vapor and liquid flow rates, liquid composition, and net reaction rates are calculated and shown in Fig. 16.
LC B
2
FC
A
10
5.2. Evaluation of open-loop process dynamics In Fig. 17, the open-loop transient responses of the high-purity ethylene glycol reactive distillation column with different kcon (i.e., kcon = 0.5, 2.0, and 4.0 s−1 ) are depicted, when the process is subjected to a ±2% step change in the reboil flow rate, respectively. In addition to the severe non-minimum phase behavior noticed, a high degree of under-dampness is presented in both the positive and negative responses, both showing an inversely proportional relationship with the controller gain of the reflux drum. Fig. 18 presents the open-loop transient responses of the highpurity ethylene glycol reactive distillation column with different kcon in the face of a ±2% step change in the feed flow rate of water, respectively. The impact from the variations of the controller gain
11 CC
15 CC
NT=16 LC
Steam C
Fig. 12. Control scheme of Example II.
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W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
a
0.98 5 kcon = 0.2
kcon = 1.0
b
kcon = 2.5
x C (mole fra ction)
x C (mole fraction)
0.98 3
0.98 1
0.97 9
0.98 5 kcon = 0.2
0.98 1
0.97 7
0.96 9 0
4
8
12
16
20
0
4
8
Time (h) 110
110
107
105
104
kcon = 0.2
12
16
20
Time (h)
V[16] (mol·s–1 )
V[16] (mol·s–1 )
kcon = 2.5
0.97 3
0.97 7
kcon = 1.0
100
kcon = 0.2
kcon = 2.5
kcon = 1.0
kcon = 2.5
95
101 0
4
8
12
16
0
20
4
8
110
107
105
Reflux (mol·s–1 )
110
104
kcon = 0.2
12
16
20
Time (h)
Time (h)
Reflux (mol·s–1 )
kcon = 1.0
kcon = 1.0
100
kcon = 2.5
kcon = 0.2
101
kcon = 1.0
kcon = 2.5
95 0
4
8
12
16
20
0
4
Time (h)
8
12
16
20
Time (h)
Fig. 13. Servo responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±0.3 mol% step change in the set-point of the bottom composition control loop, respectively (Example II). (a) Positive responses and (b) negative responses.
of the reflux drum is not found, either. While non-minimum phase behavior is noticed in both the positive and negative responses, under-dampness cannot be observed.
5.3. Evaluation of closed-loop control performance In Fig. 19, the detailed control structure for the high-purity ethylene glycol reactive distillation column is given. In accordance with the steady-state composition profile shown in Fig. 16, stage 15 is selected to be controlled by the feed flow rate of water, and a P-only controller is adopted to regulate the stoichiometric balance between the two reactants. The feed flow rate of EO is the
production rate handle and flow controlled. The controller parameters are listed in Table 6. The servo responses of the high-purity ethylene glycol reactive distillation column are shown in Fig. 20, when the bottom product purity controller has been subjected to a ±0.1 mol% step change in its set-point, respectively. Remarkable improvements can be seen with the tighter inventory control of the reflux drum. When the controller gain of the reflux drum level is small (i.e., kcon = 0.5 s−1 ), the process is dynamically unstable in the face of both the positive and negative step changes in the set-point of bottom control loop. With the increase of the controller gain (i.e., kcon = 2.0 or 4.0 s−1 ), more stable responses can be gained. Although initial inverse response caused by the non-minimum phase
Table 6 Controller parameters for Example III. Parameter
k Ti (min)
Control loop Reflux drum level
Base level
Stoichiometric balance
Bottom EG
0.5/2.0/4.0 –
0.5 –
3.6 –
4.0 1.8
x C (mole fraction)
a
0.99
b
0.99
0.98
x C (mole fraction)
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
0.98
0.97
kcon = 0.2
kcon = 1.0
1193
0.97
kcon = 2.5
kcon = 0.2
0.96
kcon = 1.0
kcon = 2.5
0.96 0
4
8
12
16
0
20
4
8
Time (h)
12
16
20
Time (h)
120
119 kcon = 0.2
kcon = 1.0
kcon = 0.2
kcon = 2.5
kcon = 1.0
kcon = 2.5
V[16] (mol·s–1 )
V[16] (mol·s–1 )
115
110
113
107
105
100
101 0
4
8
12
16
20
0
4
8
Time (h) 120
16
20
120 kcon = 0.2
kcon = 1.0
kcon = 2.5
kcon = 0.2
kcon = 1.0
kcon = 2.5
115 Reflux (mol·s–1 )
115 Reflux (mol·s–1 )
12 Time (h)
110
105
110
105
100
100 0
4
8
12
16
20
Time (h)
0
4
8
12
16
20
Time (h)
Fig. 14. Regulatory responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±2% step change in the production rate, respectively (Example II). (a) Positive responses and (b) negative responses.
behavior can still be seen, larger controller gains significantly relieve its impact. Fig. 21 presents the regulatory responses of the high-purity ethylene glycol reactive distillation column, when a ±5% step change has been introduced to the feed flow rate of EO, respectively. Non-minimum phase behavior also has a great effect on the dynamic response and the reflux drum level controller with a small kcon (i.e., 0.5 s−1 ) fails to stabilize the high-purity ethylene glycol reactive distillation column and the closed-loop simulation ends up around 10 and 5 h, respectively. When the controller gain of the reflux drum level is assigned to be 2.0 or 4.0 s−1 , stable responses can be gained. 6. Discussions Through the three example systems studied, it has been demonstrated that the totally refluxed reactive distillation columns present generally open-loop under-damped responses and this should be attributed to the sharp difference in process dynamics between the reaction operation and the separation operation
involved. Whether the reaction operation involved has thermal effect (i.e., as in Example III) or not (i.e., as in Examples I and II), it does not alter the presence of the under-dampness in dynamic responses of the system. In spite of the fact that it is extremely difficult to prove the above interpretation in terms of strict theoretical analysis, it is considered that this reflects the general characteristics of the totally refluxed reactive distillation columns. To offer additional insights into this issue, we study further the dynamics and control of several ideal reactive distillation columns working in partially refluxed operation mode (i.e., a ternary and quaternary reaction systems with top and bottom products) or totally reboiled operation mode (i.e., a ternary reaction system with only top product) and the results are contained in the web-published Supplement. It is demonstrated that process dynamics and controllability becomes the severest when the ideal reactive distillation columns have been operated in the totally refluxed operation mode. In the partially refluxed operation mode, the effect of the inventory control of the reflux drum appears to be rather small as compared with the totally refluxed operation mode. For the totally reboiled reactive distillation columns, they can have quite similar dynamic
1194
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
2 Reflux = 181.14 mol·s
FW = 7.31 mol·s–1
x EG (mole fraction)
a
–1
1
0.95
0.9
kcon = 0.5
kcon = 2.0
kcon = 4.0
0.85
FEO = 7.65 mol·s–1
0
5
10
8
15
20
25
Time (h)
b
14
16
x EG (mole fraction)
0.95
V[17] = 166.45 mol·s–1 NT=17
xEO = 0.0000 xW = 0.0026 xEG = 0.9480 xDEG = 0.0494
Bot = 7.31 mol·s–1
0.85
EG
kcon = 0.5
kcon = 2.0
kcon = 4.0
0.75 0
Fig. 15. Scheme of a high-purity ethylene glycol reactive distillation column (Example III).
5
10
15
20
25
Time (h) Fig. 17. Open-loop transient responses of the high-purity ethylene glycol reactive distillation column with different kcon in the face of a ±2% step change in the reboil flow rate, respectively (Example III). (a) Positive responses and (b) negative responses.
500
b L & V Flowrate (mol·s–1)
a
T (K)
450
400
220
165
110
55 L
350
0 1
5
9
13
17
1
5
Stage nu mber
c
1.0
d Reaction Rate (mol·s–1)
0.6 EO
EG
Water
9
13
17
Stage nu mber
0.8 x (mole fraction)
V
DEG
0.4 0.2
2.8 main reaction
side reaction
2.1
1.4
0.7
0
0.0 1
5
9 Stage nu mber
13
17
1
5
9
13
Stage nu mber
Fig. 16. Steady-state profiles of Example III. (a) Temperature, (b) vapor and liquid flow rates, (c) liquid composition and (d) net reaction rates.
17
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
a
0.95 5 kcon = 0.5
x EG (mole fraction)
1195
kcon = 2.0
kcon = 4.0
PC
0.94 5
LC W
2
0.93 5
CC
0.92 5 0
5
10
15
20
25
Time (h)
b
0.95
x EG (mole fraction)
EO
0.94 8
8
14
FC
15 16
0.94 6
CC
NT=17 0.94 4
LC kcon = 0.5
kcon = 2.0
Steam EG
kcon = 4.0
0.94 2 0
5
10
15
20
25
Time (h)
Fig. 19. Control scheme of Example III.
Fig. 18. Open-loop transient responses of the high-purity ethylene glycol reactive distillation column with different kcon in the face of a ±2% step change in the feed flow rate of water, respectively (Example III). (a) Positive responses and (b) negative responses.
Fig. 20. Servo responses of the high-purity ethylene glycol reactive distillation column with different kcon in the face of a ±0.1 mol% step change in the set-point of the bottom composition control loop, respectively (Example III). (a) Positive responses and (b) negative responses.
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W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
0.98 kcon = 0.5
kcon = 2.0
b
kcon = 4.0
x EG (mole fraction)
x EG (mole fra ction)
a
0.96
0.94
0.92 5
10
15 Time (h)
20
25
kcon = 2.0
kcon = 4.0
0.96
0.94
0.92
30
0
171
180
169
175
V[17] (mol·s–1 )
V[17] (mol·s–1 )
kcon = 0.5
0.9 0
167
kcon = 0.5
kcon = 2.0
5
10
15 Time (h)
20
25
30
170
kcon = 0.5
kcon = 4.0
165
kcon = 2.0
kcon = 4.0
165 0
5
10
15 Time (h)
20
25
30
0
5
10
15 Time (h)
20
25
30
186
Reflux (mol·s–1 )
186
Reflux (mol·s–1 )
0.98
184
182
kcon = 0.5
kcon = 2.0
184
182
kcon = 0.5
kcon = 4.0
180
kcon = 2.0
kcon = 4.0
180 0
5
10
15 Time (h)
20
25
30
0
5
10
15 Time (h)
20
25
30
Fig. 21. Regulatory responses of the high-purity ethylene glycol reactive distillation column with different kcon kcon in the face of a ±5% step change in the production rate, respectively (Example III). (a) Positive responses and (b) negative responses.
behaviors as the totally refluxed reactive distillation columns and this fact reminds us again that the special operation modes should be responsible for the unique dynamic behaviors. Understanding the unique behavior of the totally refluxed reactive distillation columns can be especially beneficial to the synthesis and design of their control systems. Contrary to the common practices in the operation of conventional distillation columns and reactive distillation columns with distillate and bottom withdrawal, the inventory control of the reflux drum should be tightly tuned and this helps to reduce the difference in process dynamics between the reaction operation and the separation operation involved. All the simulation studies obtained so far give a good indication that the tight inventory control of the reflux drum can lead to substantial improvement in process dynamics and controllability. Although the current work employs exclusively the bottom composition as the controlled variable to study the dynamics and operation of the totally refluxed reactive distillation columns, it is considered that the gained insights are also useful for other complicated control schemes, including, for example, the temperature inferential control philosophy and model predictive control algorithm.
Al-Arfaj and Luyben also noticed that the heat duty of reboiler ends up at a lower value after more reactants have been added to the ethylene glycol reactive distillation column, and they owned this counterintuitive phenomenon to the release of more heat of reaction [11,15]. It is noted that similar results have also been obtained in Examples II and III of the current work. Since the reaction operation involved is of no thermal effect at all in Example II, the heat duty of reboiler, however, still settles down to a lower value in the face of a +2% step change in the feed flow rate of reactant B. This outcome makes us to think that the thermal heat of reaction should not be responsible for such a counterintuitive phenomenon. Instead, the existence of input multiplicities in Examples II and III should be considered as a substitute. Since Example I displays no such a counterintuitive phenomenon, the involvement of a side reaction into Examples II and III might be one of the primary reasons that trigger the occurrence of input multiplicities. 7. Conclusion Although the totally refluxed reactive distillation columns appear to be relatively simple in process configuration, they
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
present quite unique process dynamics that roots essentially from the totally refluxed operation mode. Owing to the direct connection between the reactive section and reflux drum, the large holdup in the latter retards considerably the dynamics of the reaction operation involved. Because the separation operation involved usually has rather fast process dynamics, the great difference in process dynamics between these two operations leads frequently to the occurrence of under-damped responses. Moreover, the degree of under-dampness is closely related to the inventory control of the reflux drum. With the tight inventory control of the reflux drum, the degree of under-dampness can be reduced and so can the inherent process nonlinearity and non-minimum phase behavior (e.g., as in Examples II and III). The unique dynamics of the totally refluxed reactive distillation columns gives profound implications on the synthesis and design of their control systems. In sharp contrast to conventional distillation columns as well as reactive distillation columns with distillate and bottom withdrawal, the inventory of the reflux drum should be tightly controlled. In terms of two hypothetical ideal reactive distillation columns with and without a side reaction, respectively, and a high-purity ethylene glycol reactive distillation column, the unique process dynamics of the totally refluxed reactive distillation columns has been observed. All the results obtained confirm the fact that noticeable improvement in process dynamics and controllability has been acquired when tight inventory control has been implemented in the reflux drum. Acknowledgements The project is financially supported by the National Science Foundation of China (Grant Number: 21176015) and the Doctoral Programs Foundation of Ministry of Education of China (Grant Number: 20100010110008), and thereby is acknowledged. The authors are also in debt to the anonymous reviewers for their valuable comments and suggestions.
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Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j. jprocont.2012.05.007. References [1] R. Taylor, R. Krishna, Modelling reactive distillation, Chemical Engineering Science 55 (2000) 5183–5229. [2] K. Sundmacher, A. Kienle, Reactive Distillation: Status and Future Directions, Wiley-VCH Verlag, 2003. [3] W.L. Luyben, C.C. Yu, Reactive Distillation Design and Control, John Wiley & Sons, 2008. [4] S.B. Hung, M.J. Lee, Y.T. Tang, Y.W. Chen, I.K. Lai, W.J. Hung, H.P. Huang, C.C. Yu, Control of different reactive distillation configurations, AIChE Journal 52 (2006) 1423–1440. [5] D.B. Kaymak, W.L. Luyben, Evaluation of a two-temperature control structure for a two-reactant/two-product type of reactive distillation column, Chemical Engineering Science 61 (2006) 4432–4450. [6] K. Huang, M. Nakaiwa, A. Tsutsumi, Towards further internal heat integration in design of reactive distillation columns—part II: the process dynamics and operation, Chemical Engineering Science 61 (2006) 5377–5392. [7] W.L. Luyben, Control of ternary reactive distillation columns with and without chemically inert components, Industrial and Engineering Chemistry Research 46 (2007) 5576–5590. [8] D.B. Kaymak, D. Yilmaz, A.Z. Gürer, Inferential temperature control structures for different types of two-reactant reactive distillation systems, Industrial and Engineering Chemistry Research 50 (2011) 6777–6793. [9] D.B. Kaymak, D. Yilmaz, A.Z. Gürer, Effect of relative volatilities on inferential temperature control of reactive distillation columns, Industrial and Engineering Chemistry Research 50 (2011) 8138–8152. [10] A. Kumar, P. Daoutidis, Modeling, analysis and control of ethylene glycol reactive distillation column, AIChE Journal 45 (1999) 51–68. [11] M.A. Al-Arfaj, W.L. Luyben, Control of ethylene glycol reactive distillation column, AIChE Journal 48 (2002) 905–908. [12] F. Zhu, K. Huang, S. Wang, L. Shan, Q. Zhu, Towards further internal heat integration in design of reactive distillation columns—part IV: application to a high-purity ethylene glycol reactive distillation column, Chemical Engineering Science 64 (2009) 3498–3509. [13] K. Huang, F. Zhu, W. Ding, S.J. Wang, Control of high-purity ethylene glycol reactive distillation column with insights of process dynamics, AIChE Journal 55 (2009) 2106–2121. [14] J. Sun, K. Huang, S. Wang, Deepening internal mass integration in design of reactive distillation columns, 1: principle and procedure, Industrial and Engineering Chemistry Research 48 (2009) 2034–2048. [15] W.L. Luyben, Planwide Dynamic Simulators in Chemical Processing and Control, Marcel Dekker, 2002.
Contents lists available at SciVerse ScienceDirect
Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
Dynamics and control of totally refluxed reactive distillation columns Wei Liu a , Kejin Huang a,∗ , Liang Zhang a , Haisheng Chen a , San-Jang Wang b a b
College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China Department of Chemical and Material Engineering, Ta Hwa Institute of Technology, Chiunglin, Hsinchu 307, Taiwan
a r t i c l e
i n f o
Article history: Received 3 August 2011 Received in revised form 3 May 2012 Accepted 10 May 2012 Available online 15 June 2012 Keywords: Reactive distillation column Totally refluxed operation Process dynamics Inventory control Process operation
a b s t r a c t According to the mechanism of the reaction operation involved, reactive distillation columns are often designed to work in a totally refluxed operation mode. The totally refluxed operation mode makes the reflux drum interact solely with the reaction operation involved and retards considerably the dynamics of the latter. The resultant great difference in process dynamics between the reaction operation and the separation operation involved leads frequently to under-damped responses with the degree of underdampness closely dependent on the inventory control of the reflux drum. With the tight inventory control of the reflux drum, the degree of under-dampness can be suppressed and this presents a favorable effect to process dynamics and controllability of the totally refluxed reactive distillation columns. Two hypothetical ideal reactive distillation columns with and without a side reaction, respectively, and a high-purity ethylene glycol reactive distillation column are employed to examine the unique dynamics and controllability of the totally refluxed reactive distillation columns. The results obtained are in good accordance with the above interpretation. The current work reveals the general behaviors of the totally refluxed reactive distillation columns and can be particularly useful in control system synthesis and design. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction According to the mechanism of the reaction operation involved, reactive distillation columns are often designed to work in a totally refluxed operation mode. For example, in case of carrying out a reaction, A + B ↔ C, in a reactive distillation column, the desired product C should be withdrawn from the bottom of the process because it is the heaviest component. At the top of the reactive distillation column, the reactants, A and B, are usually accumulated and they should be recycled back to the reactive section through a totally refluxed operation mode in reflux drum. Since there is no rectifying section in the reactive distillation column, the reflux drum is directly connected to the reactive section and this leads to an intimate interaction between them. Such interaction affects considerably the reaction operation involved and can eventually present a strong impact to the dynamics and controllability of the totally refluxed reactive distillation columns. These should represent the unique behaviors of the totally refluxed reactive distillation columns and are worthwhile to be studied carefully. Recent years have seen increasingly more studies conducted on the dynamics and operation of reactive distillation columns [1–6]. For the totally refluxed reactive distillation columns, there have also appeared a number of papers dealing with their operation
∗ Corresponding author. Tel.: +86 10 64434801; fax: +86 10 64437805. E-mail address: [email protected] (K. Huang). 0959-1524/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jprocont.2012.05.007
issues. Luyben once compared the operation of two ternary ideal reactive distillation columns with and without chemically inert components, respectively [7]. He found that the presence of a chemically inert component could alter the process from a totally refluxed operation mode into a partially refluxed operation mode. In particular, a direct measurement of stage composition is required for the effective process control. Recently, Kaymak et al. further explored the inferential temperature control of the ternary ideal reactive distillation columns with and without chemically inert components [8,9]. They insisted that the direct measurement of stage composition might not be necessary provided that the control structure and control tray locations were properly chosen. Ethylene glycol reactive distillation column represents a typical example for the totally refluxed reactive distillation column and has received considerable attention in the aspects of process dynamics and controllability. Although Kumar and Daoutidis claimed that nonlinear controller was superior to linear PI controller in its operation [10], Al-Arfaj and Luyben showed that the process could be effectively controlled by a simple PI control scheme plus a feed-forward compensator [11]. Huang et al. addressed internal heat integration to a high-purity ethylene glycol reactive distillation column [12,13]. They found that seeking further internal heat integration between the reaction operation and the separation operation involved helped to improve the process dynamics and lessened the difficulties in process operation. Furthermore, the tight inventory control of the reflux drum could function to suppress the inherent process nonlinearity and
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
Nomenclature A Avp B Bot Bvp C CC D DEG E EG EO F FC H HR HV k kcon L LC nc NT PC R r T Ti V W x z
hypothetical component vapor pressure constant (Pa) hypothetical component bottom product flow rate (mol s−1 ) vapor pressure constant (Pa K) hypothetical component composition controller hypothetical component diethylene glycol activation energy of reaction (J mol−1 ) ethylene glycol ethylene oxide feed flow rate (mol s−1 ) flow controller stage holdup (mol) heat of reaction (J mol−1 ) latent heat of vaporization (J mol−1 ) controller gain controller gain of reflux drum level (s−1 ) liquid flow rate (mol s−1 ) level controller total number of components total number of stages pressure controller ideal gas law constant (J mol−1 K−1 ) reaction rate (mol mol−1 s−1 or mol m−3 s−1 ) temperature (K) reset time (s) vapor flow rate (mol s−1 ) water liquid composition feed composition
Greek symbols ˛ pre-exponential factor stoichiometric coefficient of a reaction Subscripts b backward reaction f forward reaction i component index stage index j m main reaction s side reaction
non-minimum phase behavior, favoring therefore process operation significantly. Although this conclusion was acquired from the high-purity ethylene glycol reactive distillation column, it invoked us to anticipate that the interesting phenomenon might be closely related to the particular structure of the totally refluxed reactive distillation columns. This consideration stimulated us to acquire the insights of the phenomenon. To the best of our knowledge, until now no pertinent work has been conducted on the relationship between process configuration and process dynamics and controllability for the totally refluxed reactive distillation columns. The primary objective of the current work is to investigate the dynamics and control of the totally refluxed reactive distillation columns. Two hypothetical ideal reactive distillation columns with and without a side reaction, respectively, and a high-purity ethylene glycol reactive distillation column are chosen as representative examples. Both open-loop process dynamics
1183
and closed-loop operation are examined with special attention focused on the effect of inventory control of reflux drum. The unique behaviors of the totally refluxed reactive distillation columns are analyzed and the acquired insights are then summarized.
2. Unique process dynamics and its implications on the control of the totally refluxed reactive distillation columns Unlike the conventional reactive distillation columns with distillate and bottom withdrawal, the totally refluxed reactive distillation columns contain a reactive section with a direct connection to the reflux drum (because there is no rectifying section). In an extreme point of view, it is even reasonable to treat them as an equivalent reactor with heat exchange in the condenser. Because of the large liquid holdup in the reflux drum, once the light and/or heavy reactants enter the reflux drum, they will generally need a long time to be driven back to the reactive section and this retards considerably the dynamics of the reaction operation involved. In the totally refluxed reactive distillation columns, the separation operation involved is solely carried out in the stripping section. Since the heat duty of reboiler (or the reboil flow rate) serves as the manipulated variable to control the purity of bottom product, it should be taken as the main variable to represent the dynamics of the separation operation involved. Because the bottom reboiler has much faster dynamics than the liquid hydraulics in the reflux drum and in the reactive section compounded further with the reaction kinetics, sharp difference exists in process dynamics between the reaction operation and the separation operation involved. This causes generally under-dampness in the open-loop dynamic responses of the bottom product. The greater the difference is, the severer the degree of the under-dampness becomes. Moreover, the sharp difference in process dynamics between the reaction operation and the separation operation involved can also induce a certain degree of degradation in system properties, e.g., in the aspects of process nonlinearity and non-minimum phase behavior. This inevitably gives rise to additional difficulties in process operation. In terms of the above analysis, one can now understand that the sharp difference in process dynamics between the reaction operation and the separation operation involved is the main reason for the occurrence of the under-damped behaviors in the totally refluxed ideal reactive distillation columns. To suppress the degree of the under-dampness, one must therefore rely on the acceleration of reaction kinetics and liquid hydraulics in the reactive section and in the reflux drum. Although operating pressure can be increased to accelerate reaction kinetics, it is generally determined by the match of operating conditions between the reaction operation and the separation operation involved. For the liquid hydraulics in the reactive section and in the reflux drum, it is decided by process capacity and is usually operationally infeasible to be enhanced through the reduction of their sizes. The only way left is to accelerate the dynamics of the reflux drum through tight control of liquid level. Remember, however, that this differs totally from the conventional design practice, i.e., the inventory of the reflux drum should be loosely controlled in order to avoid strong interaction between different control loops. In the remainder of this paper, we will employ three totally refluxed reactive distillation columns, including two hypothetical ideal reactive distillation columns with and without a side reaction, respectively, and a high-purity ethylene glycol reactive distillation column, to examine their unique dynamics and the effect of inventory control of reflux drum on process operation.
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Table 1 Physicochemical properties and operating conditions of Example I. Parameter
Value 8.0 0 9 5 20,000.0 1000.0 20,000.0 125,520 125,520 0.008 0.0004 12.63 12.82 10 2 1.0 1.0 4:2:1 0 29,053.7 98 12.3463/3862 11.6531/3862 10.9600/3862
Column pressure (bar) Rectifying section Reactive section Stripping section Condenser Column tray Reboiler Forward Backward Forward Backward A B A B A B
Number of stages
Liquid hodlup (mol) Reaction activation energy (J mol−1 ) Specific reaction rate at 366 K (mol s−1 mol−1 ) Feed flow rate (mol s−1 ) Feed location Feed thermal condition Relative volatility (A:B:C) Heat of reaction (J mol−1 ) Latent heat of vaporization (J mol−1 ) Bottom product specification (C, mol%)
A (Avp /Bvp ) B (Avp /Bvp ) C (Avp /Bvp )
Vapor pressure constants
3. Example I: A totally refluxed ideal reactive distillation system involving one reaction with two light reactants and one heavy product 3.1. Process description This hypothetical ideal reactive distillation system is taken from Sun et al. [14]. As is shown in Fig. 1, this reactive distillation column consists of a reactive section above the stripping section with 9 and 5 stages, respectively. The reactive distillation column is operated in a totally refluxed operation mode, and two pure reactant feeds, FA and FB , are fed to stages 10 and 2 (numbered from the top condenser
down to the bottom reboiler, and similarly hereinafter), respectively, with the latter in slight excess. Detailed physicochemical properties and operating conditions are listed in Table 1, and other relevant information can be found in the corresponding reference. In the reactive section of this totally refluxed ideal reactive distillation column, reactants A and B undergo a reversible liquid-phase reaction to form the desired product C. The detailed reaction mechanism is A+B↔C
(1)
The volatilities are such that the product C is the heavy key, A is the light key, and B is the intermediate key. The net reaction rate for component i on reactive stage j is given by ri,j = i Hj (Kf,j xA,j xB,j − Kb,j xC,j )
(2)
where Kf,j and Kb,j are the forward and backward specific reaction rate constants, respectively, described by the Arrhenius law as
2 FB = 12.82 mol·s–1
Kf,j = ˛f exp
Reflux = 34.41 mol·s–1
Kb,j = ˛b exp
−Ef RTj
−Eb RTj
(3.1)
(3.2)
Equimolar overflow is assumed in the nonreactive section. As the reaction reduces the molecular number, i.e., nc
FA = 12.63 mol·s–1
10
i = −1
(4)
i=1
the mass balance equations in the reactive section become
15
Vj = Vj+1 −
V[16] = 34.41 mol·s–1 NT=16
xA = 0.0026 xB = 0.0174 xC = 0.9800
Bot = 12.85 mol·s–1
Lj = Lj−1 +
rj HR
nc i=1
C
Fig. 1. Scheme of a totally refluxed ideal reactive distillation column with one reaction (Example I).
(5)
HV ri,j +
rj HR HV
(6)
Ideal vapor and liquid phase behaviors are assumed in the vapor–liquid equilibrium calculation. The steady-state profiles of temperature, vapor and liquid flow rates, liquid composition, and net reaction rates are calculated and shown in Fig. 2.
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
a
b
450
1185
60
L & V Flowrate (mol·s–1)
L
T (K)
430
410
390
V
40
20
370
0
1
4
7
10
13
16
1
4
7
Stage nu mber
1.0 A
B
C
x (mole fraction)
0.8 0.6 0.4 0.2
d
5
Reaction Rate (mol·s–1)
c
10
13
16
13
16
Stage nu mber
4 3 2 1 0
0.0 1
4
7
10
13
1
16
4
7
10 Stage nu mber
Stage number
Fig. 2. Steady-state profiles of Example I. (a) Temperature, (b) vapor and liquid flow rates, (c) liquid composition and (d) net reaction rates.
3.2. Evaluation of open-loop process dynamics
a
0.98 8 kcon = 0.2
kcon = 1.0
kcon = 2.5
x C (mole fraction)
0.98 6 0.98 4 0.98 2 0.98 0.97 8 0
2
4
6
8
10
12
Time (h)
b
0.99 kcon = 0.2
x C (mole fraction)
In Fig. 3, the open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon (i.e., kcon = 0.2, 1.0, and 2.5 s−1 ) are depicted, when it is subjected to a ± 5% step change in the reboil flow rate, respectively. Sharp differences can be found between the positive and negative responses, irrespective of the detailed parameters assigned to the inventory controller of the reflux drum. In the case of the positive change in the reboil flow rate, the totally refluxed ideal reactive distillation column exhibits under-damped responses with the degree of under-dampness inversely proportional to the controller gain of the reflux drum. In the case of the negative change in the reboil flow rate, the totally refluxed ideal reactive distillation column still displays under-damped responses with the degree of under-dampness inversely proportional to the controller gain of the reflux drum and this can clearly be observed before the time instant of 4 h. After that time instant, transition between the inherent multiple steady states occurs and this moves the totally refluxed ideal reactive distillation column to an unexpected steady state. This is why a high degree of asymmetry has been noticed between the positive and negative responses in this situation. The open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon are shown in Fig. 4, when it is confronted with a ±2% step change in the feed flow rate of reactant B, respectively. No under-damped responses are observed and the variation in the controller gain of the reflux drum shows almost no impact on the system responses. Regardless of the positive and negative changes in the feed flow rate of reactant B, the composition of the bottom product settles down to two values lower than the initial steady state, implying the possible transition between the inherent multiple steady states.
kcon = 1.0
kcon = 2.5
0.97
0.95
0.93 0
8
16
24
32
40
Time (h) Fig. 3. Open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±5% step change in the reboil flow rate, respectively (Example I). (a) Positive responses and (b) negative responses.
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Table 2 Controller parameters for Example I. Parameter
Control loop
k Ti (min)
Reflux drum level
Base level
0.2/1.0/2.5 –
0.2 –
3.3. Evaluation of closed-loop control performance Closed-loop control of the totally refluxed ideal reactive distillation column is conducted to evaluate the impact of the inventory control of the reflux drum. A control scheme that uses two direct composition control loops is selected in the current study and Fig. 5 provides the detailed control structure [7]. The pressure is assumed to be perfectly controlled at the nominal steady-state by the heat removal of condenser. The level of the reflux drum is controlled by reflux flow rate, and a P-only controller is used. The level of rebolier is controlled by the bottom product flow rate, and a P-only controller is chosen. The composition of the bottom product is controlled by the heat duty of reboiler, and a PI controller is adopted. According to the steady-state composition profile in Fig. 2, the composition of stage 3 is selected to be controlled by the feed flow rate of reactant B because it changes rapidly there. This composition loop serves to regulate the stoichiometric balance between the two reactants, and a Ponly controller is adopted. The feed flow rate of reactant A is the production rate handle and flow controlled. The composition controllers are tuned with Ziegler–Nichols tuning method and all the controller parameters are listed in Table 2. A first-order lag with a time constant of 5-min is inserted in all composition control loops.
x C (mole fraction)
a
0.98 5
Stoichiometric balance
Bottom C
11.13 –
0.9 0.42
The composition transmitter spans are set to be 0.1 and all the control valves are designed to be half open at the nominal steady state. The servo responses of the totally refluxed ideal reactive distillation column are illustrated in Fig. 6, when the controller of the bottom product purity is subjected to a ±0.3 mol% step change in its set-point, respectively. It is noted that strong effect has been generated by the variation in the controller gain of the reflux drum for both the positive and negative changes in the set-point of the bottom control loop. With increasingly tight inventory control of the reflux drum, the totally refluxed ideal reactive distillation column displays a consistent improvement in dynamic responses in the case of positive change in the set-point of the bottom control loop. In the case of the negative change in the set-point of the bottom control loop, sustainable oscillations occur when the controller gain of the reflux drum has been set at 0.2 and 1.0 s−1 respectively. After it has been increased to 2.5 s−1 , the sustainable oscillation is vanished. Fig. 7 presents the regulatory responses of the totally refluxed ideal reactive distillation column, when a ±2% step change is introduced to the production rate, respectively, in terms of a corresponding variation in the feed flow rate of reactant A. Quite similar to the circumstances of the step changes in product purity for the bottom control loop, the totally refluxed ideal reactive distillation column exhibits improved responses with the increasingly tight inventory control of the reflux drum. More specifically, whereas the low controller gain (i.e., kcon = 0.2 s−1 ) of the reflux drum results
0.97 5
PC 0.96 5
LC kcon = 0.2
kcon = 1.0
B
kcon = 2.5
2
0.95 5 0
8
16
24
32
3
40
Time (h)
x C (mole fraction)
b
CC
0.99 kcon = 0.2
kcon = 1.0
kcon = 2.5
0.96
A
10
0.93
FC
15 CC
NT=16
0.9 0
8
16
24
32
40
Time (h)
Steam
LC
C Fig. 4. Open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±2% step change in the feed flow rate of reactant B, respectively (Example I). (a) Positive responses and (b) negative responses.
Fig. 5. Control scheme of Example I.
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
0.98 5
b x C (mole fra ction)
x C (mole fra ction)
a
0.98 3
0.98 1 kcon = 0.2
kcon = 1.0
0.99 kcon = 0.2
kcon = 1.0
kcon = 2.5
0.98
0.97
kcon = 2.5
0.97 9
0.96 0
2
4
6 Time (h)
8
10
12
0
37
40
36
37.5 V[16] (mol·s–1 )
V[16] (mol·s–1 )
1187
35
34
2
4
6 Time (h)
8
10
12
35
32.5 kcon = 0.2
kcon = 1.0
kcon = 2.5
33
kcon = 0.2
30 0
2
4
6 Time (h)
8
10
0
12
37
2
4
6 Time (h)
kcon = 1.0
8
kcon = 2.5
10
12
37
Reflux (mol·s–1 )
Reflux (mol·s–1 )
36 36
35
35 34 33
kcon = 0.2
kcon = 1.0
kcon = 0.2
kcon = 2.5
34
kcon = 1.0
kcon = 2.5
32 0
2
4
6 Time (h)
8
10
12
0
2
4
6 Time (h)
8
10
12
Fig. 6. Servo responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±0.3 mol% step change in the set-point of the bottom composition control loop, respectively (Example I). (a) Positive responses and (b) negative responses.
in sustainable and yet magnified oscillations, the high ones (i.e., kcon = 1.0 or 2.5 s−1 ) achieve a stable operation. 4. Example II: A totally refluxed ideal reactive distillation column with one main and one side reactions
This hypothetical ideal reactive distillation system is developed from Example I with the inclusion of an additional reversible side reaction
(8.2)
In terms of relative volatilities, these four components can be arranged in the following order: A>B>C>D
4.1. Process description
B+C↔D
rs i,j = s i Hj (Kf s,j xB,j xC,j − Kb s,j xD,j )
(9)
which means the component D is also withdrawn with component C from the bottom of the totally refluxed ideal reactive distillation column. Detailed physicochemical properties and operating conditions are tabulated in Table 3. The steady-state profiles of temperature, vapor and liquid flow rates, liquid composition, and net reaction rates are calculated and shown in Fig. 9.
(7) 4.2. Evaluation of open-loop process dynamics
As shown in Fig. 8, the process configuration is exactly similar to that of Example I (c.f., Fig. 1). The main and side reaction rates for component i on reactive stage j are given by rm i,j = m i Hj (Kf m,j xA,j xB,j − Kb m,j xC,j )
(8.1)
In Fig. 10, the open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon (i.e., kcon = 0.2, 1.0, and 2.5 s−1 ) are depicted, when there is a ±5% step change in the reboil flow rate, respectively.
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W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
0.98 2 kcon = 0.2
kcon = 1.0
b
kcon = 2.5
x C (mole fraction)
x C (mole fraction)
a
0.98
0.97 8
0.98 2 kcon = 0.2
kcon = 1.0
kcon = 2.5
0.98
0.97 8 0
2
4
6 Time (h)
8
10
12
0
36
2
4
6 Time (h)
12
kcon = 1.0
kcon = 2.5
34.5 V[16] (mol·s–1 )
35.5 V[16] (mol·s–1 )
10
35 kcon = 0.2
35
34.5
34
33.5 kcon = 0.2
kcon = 1.0
kcon = 2.5
34
33 0
2
4
6 Time (h)
8
10
0
12
35.5
2
4
6 Time (h)
8
10
12
35
Reflux (mol·s–1 )
kcon = 0.2
Reflux (mol·s–1 )
8
35
34.5
kcon = 0.2
kcon = 1.0
kcon = 1.0
kcon = 2.5
34
kcon = 2.5
34
33 0
2
4
6 Time (h)
8
10
12
0
2
4
6 Time (h)
8
10
12
Fig. 7. Regulatory responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±2% step change in the production rate, respectively (Example I). (a) Positive responses and (b) negative responses.
Under-dampness is presented in both the positive and negative responses with its degree inversely proportional to the controller gain of the reflux drum. Similar to Example I, there exists a high degree of asymmetry (i.e., process nonlinearity) between the positive and negative responses. In particular, a much larger variation is caused in the negative response in case kcon is equal to 0.2 s−1 than in case kcon is equal to 1.0 and 2.5 s−1 . It is apparently related to the slow dynamics of the reflux drum. Although not shown here, the difference in steady states can finally vanish after a long time period. Non-minimum phase behavior (i.e., initial inverse response) is observed, which is evidently attributed to the existence of a side reaction. It is noted here that the severity of process nonlinearity and non-minimum phase behavior is also inversely proportional to the controller gain of the reflux drum.
Fig. 11 gives the open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon , when it is in the face of a ±2% step change in the feed flow rate of reactant A, respectively. Extremely similar to Example I, the impact from the variation of the controller gain of the reflux drum is hardly observed in this situation. 4.3. Evaluation of closed-loop control performance Fig. 12 illustrates the detailed control scheme for the totally refluxed ideal reactive distillation column with a main and a side reactions. In the light of the steady-state composition profile given in Fig. 9, stage 11 is chosen and controlled with the feed flow rate of reactant A to regulate the stoichiometric balance between the
Table 4 Controller parameters for Example II. Parameter
k Ti (min)
Control loop Reflux drum level
Base level
Stoichiometric balance
Bottom C
0.2/1.0/2.5 –
0.2 –
4.58 –
2.7 16.67
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
1189
Table 3 Physicochemical properties and operating conditions of Example II. Parameter
Value
Column pressure (bar) Number of stages
Liquid hodlup (mol) Main reaction activation energy (J mol−1 ) Side reaction activation energy (J mol−1 ) Specific main reaction rate at 366 K (mol s−1 mol−1 ) Specific side reaction rate at 366 K (mol s−1 mol−1 ) Feed flow rate (mol s−1 ) Feed location Feed thermal condition
Rectifying section Reactive section Stripping section Condenser Column tray Reboiler Forward Backward Forward Backward Forward Backward Forward Backward A B A B A B
Relative volatility (A:B:C:D) Heat of reaction (J mol−1 ) Latent heat of vaporization (J mol−1 ) Bottom product specification (C, mol%) Vapor pressure constants
A (Avp /Bvp ) B (Avp /Bvp ) C (Avp /Bvp ) D (Avp /Bvp )
reactants, A and B. A P-only controller is again used here. The feed flow rate of reactant B is the production rate handle and flow controlled. The controller parameters are tabulated in Table 4. The servo responses of the totally refluxed ideal reactive distillation column are illustrated in Fig. 13, when there is a ±0.3 mol% step change in the set-point of bottom control loop, respectively. In case kcon is set to be 0.2 s−1 , while the totally refluxed ideal
8.0 0 9 5 20,000.0 1000.0 20,000.0 125,520 125,520 125,520 125,520 0.008 0.0004 0.005 0.0002 12.63 12.82 10 2 1.0 1.0 8:4:2:1 0 29,053.7 98 12.3463/3862 11.6531/3862 10.9600/3862 10.2669/3862
reactive distillation column settles down with a large deviation in the face of the positive change in the set-point of bottom control loop, a magnified oscillation occurs in the face of the negative change in the set-point of bottom control loop. In case kcon is set to be 1.0 and 2.5 s−1 , fairly smooth operation can be achieved for both the positive and negative changes in the set-point of bottom control loop. The responses with kcon as 2.5 s−1 outperform those with kcon as 1.0 s−1 . Fig. 14 presents the regulatory responses of the totally refluxed ideal reactive distillation column in the face of a ±2% step change in the feed flow rate of reactant B, respectively. When kcon is set to be 0.2 s−1 , rather oscillating responses are obtained for both the positive and negative changes in the feed flow rate of reactant B. With relatively tight inventory control of the reflux drum (i.e., 1.0 and 2.5 s−1 ), sharp improvement in control performance is achieved, giving much smaller overshoots and shorter settling times. A counterintuitive phenomenon is observed here, i.e., the heat duty of reboiler changes inversely proportional to the production rate. It is considered to be caused by the existence of input multiplicities in the totally refluxed ideal reactive distillation column. The transition between the inherent multiple steady states distorts actually the relationship between the production rate and the heat duty of reboiler. 5. Example III: A high-purity ethylene glycol reactive distillation column 5.1. Process description Ethylene glycol (C2 H6 O2 ) is produced by hydration of ethylene oxide (C2 H4 O), i.e., the main reaction: C2 H4 O(EO) + H2 O → C2 H6 O2 (EG), H R = –80.0 × 103 J mol−1 (10.1)
Fig. 8. Scheme of a totally refluxed ideal reactive distillation column with one main and one side reactions (Example II).
rm = 3.15 × 1015 exp
−9547 xEO xW T
(10.2)
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W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
a
450
140
b L & V Flowrate (mol·s–1)
L
T (K)
430
410
390
105
70
35
370
0
1
4
7
10
13
16
1
4
Stage number 5 A
B
C
7
10
13
16
Stage number
1.0
0.03
D
Reaction Rate (mol·s–1)
0.8 x (mole fraction)
V
0.6 0.4 0.2
0.02
4
0.01
3
0 2
1
6
main reaction
1
11
16
side rea ction
0
0.0 1
4
7
10
13
1
16
Stage number
4
7
10
13
16
Stage number
Fig. 9. Steady-state profiles of Example II. (a) Temperature, (b) vapor and liquid flow rates, (c) liquid composition and (d) net reaction rates.
Table 5 Physicochemical properties and operating conditions of Example III. Parameter
Value
Column pressure (bar) Number of stages Catalyst distribution (m3 ) Liquid hodlup (mol) Reaction rate (mol m−3 s−1 ) Feed flow rate (mol s−1 ) Feed location Feed thermal condition Heat of reaction (J mol−1 )
Rectifying section Reactive section Stripping section Stages 2 ∼ 8 Stages 9 ∼ 14 Condenser Column tray Reboiler Main reaction Side reaction EO W EO W EO W Main reaction Side reaction
Latent heat of vaporization (J mol−1 ) Bottom product specification (EG, mol%) Vapor–liquid equilibrium constants at the atmosphere pressure
EO W EG DEG
1.01325 0 13 2 0.043 0.1 30,000.0 1000.0 30,000.0 3.15 × 1015 exp(–9547/T)xEO xW 6.30 × 1015 exp(–9547/T)xEO xEG 7.65 7.31 8 2 1.0 1.0 −80,000.0 −13,100.0 40,000.0 94.8 71.9 exp[5.72(T–469)/(T–35.9)] 221.2 exp[6.31(T–467)/(T–52.9)] 77.0 exp[9.94(T–645)/(T–71.4)] 47.0 exp[10.42(T–681)/(T–80.6)]
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
a
a
1
1191
1 kcon = 0.2
x C (mole fraction)
x C (mole fraction)
0.8
0.7 kcon = 0.2
kcon = 1.0
0.95
0.92 5
kcon = 2.5
0.9
0.6 0
2
4
6
8
10
12
0
2
4
6
10
8
12
Time (h)
Time (h) 1
b
0.99 kcon = 0.2
kcon = 1.0
kcon = 2.5
0.98 x C (mole fraction)
x C (mole fraction)
kcon = 2.5
0.97 5
0.9
b
kcon = 1.0
0.9
kcon = 0.2
kcon = 1.0
0.97
0.96
kcon = 2.5
0.8
0.95 0
2
4
6
8
10
12
0
2
4
Time (h)
6
10
8
12
Time (h)
Fig. 10. Open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±5% step change in the reboil flow rate, respectively (Example II). (a) Positive responses and (b) negative responses.
Fig. 11. Open-loop transient responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±2% step change in the feed flow rate of reactant A, respectively (Example II). (a) Positive responses and (b) negative responses.
Ethylene oxide can react further with ethylene glycol to form the unwanted by-product diethylene glycol (C4 H10 O3 ), i.e., the side reaction: C2 H4 O + C2 H6 O2 → C4 H10 O3 (DEG), H R = –13.1 × 103 J mol−1 (11.1) rs = 6.3 × 1015 Exp
−9547 xEO xEG T
PC
(11.2)
Both the reactions are irreversible and highly exothermic. Fig. 15 shows a process design with 17 stages including a total condenser at the top and a partial reboiler at the bottom. The process has no rectifying section, either and works in a totally refluxed operation mode with water and ethylene oxide fed onto stages 2 and 8, respectively. Table 5 gives the relevant physicochemical properties and nominal steady state operating conditions. The steady-state profiles of temperature, vapor and liquid flow rates, liquid composition, and net reaction rates are calculated and shown in Fig. 16.
LC B
2
FC
A
10
5.2. Evaluation of open-loop process dynamics In Fig. 17, the open-loop transient responses of the high-purity ethylene glycol reactive distillation column with different kcon (i.e., kcon = 0.5, 2.0, and 4.0 s−1 ) are depicted, when the process is subjected to a ±2% step change in the reboil flow rate, respectively. In addition to the severe non-minimum phase behavior noticed, a high degree of under-dampness is presented in both the positive and negative responses, both showing an inversely proportional relationship with the controller gain of the reflux drum. Fig. 18 presents the open-loop transient responses of the highpurity ethylene glycol reactive distillation column with different kcon in the face of a ±2% step change in the feed flow rate of water, respectively. The impact from the variations of the controller gain
11 CC
15 CC
NT=16 LC
Steam C
Fig. 12. Control scheme of Example II.
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W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
a
0.98 5 kcon = 0.2
kcon = 1.0
b
kcon = 2.5
x C (mole fra ction)
x C (mole fraction)
0.98 3
0.98 1
0.97 9
0.98 5 kcon = 0.2
0.98 1
0.97 7
0.96 9 0
4
8
12
16
20
0
4
8
Time (h) 110
110
107
105
104
kcon = 0.2
12
16
20
Time (h)
V[16] (mol·s–1 )
V[16] (mol·s–1 )
kcon = 2.5
0.97 3
0.97 7
kcon = 1.0
100
kcon = 0.2
kcon = 2.5
kcon = 1.0
kcon = 2.5
95
101 0
4
8
12
16
0
20
4
8
110
107
105
Reflux (mol·s–1 )
110
104
kcon = 0.2
12
16
20
Time (h)
Time (h)
Reflux (mol·s–1 )
kcon = 1.0
kcon = 1.0
100
kcon = 2.5
kcon = 0.2
101
kcon = 1.0
kcon = 2.5
95 0
4
8
12
16
20
0
4
Time (h)
8
12
16
20
Time (h)
Fig. 13. Servo responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±0.3 mol% step change in the set-point of the bottom composition control loop, respectively (Example II). (a) Positive responses and (b) negative responses.
of the reflux drum is not found, either. While non-minimum phase behavior is noticed in both the positive and negative responses, under-dampness cannot be observed.
5.3. Evaluation of closed-loop control performance In Fig. 19, the detailed control structure for the high-purity ethylene glycol reactive distillation column is given. In accordance with the steady-state composition profile shown in Fig. 16, stage 15 is selected to be controlled by the feed flow rate of water, and a P-only controller is adopted to regulate the stoichiometric balance between the two reactants. The feed flow rate of EO is the
production rate handle and flow controlled. The controller parameters are listed in Table 6. The servo responses of the high-purity ethylene glycol reactive distillation column are shown in Fig. 20, when the bottom product purity controller has been subjected to a ±0.1 mol% step change in its set-point, respectively. Remarkable improvements can be seen with the tighter inventory control of the reflux drum. When the controller gain of the reflux drum level is small (i.e., kcon = 0.5 s−1 ), the process is dynamically unstable in the face of both the positive and negative step changes in the set-point of bottom control loop. With the increase of the controller gain (i.e., kcon = 2.0 or 4.0 s−1 ), more stable responses can be gained. Although initial inverse response caused by the non-minimum phase
Table 6 Controller parameters for Example III. Parameter
k Ti (min)
Control loop Reflux drum level
Base level
Stoichiometric balance
Bottom EG
0.5/2.0/4.0 –
0.5 –
3.6 –
4.0 1.8
x C (mole fraction)
a
0.99
b
0.99
0.98
x C (mole fraction)
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
0.98
0.97
kcon = 0.2
kcon = 1.0
1193
0.97
kcon = 2.5
kcon = 0.2
0.96
kcon = 1.0
kcon = 2.5
0.96 0
4
8
12
16
0
20
4
8
Time (h)
12
16
20
Time (h)
120
119 kcon = 0.2
kcon = 1.0
kcon = 0.2
kcon = 2.5
kcon = 1.0
kcon = 2.5
V[16] (mol·s–1 )
V[16] (mol·s–1 )
115
110
113
107
105
100
101 0
4
8
12
16
20
0
4
8
Time (h) 120
16
20
120 kcon = 0.2
kcon = 1.0
kcon = 2.5
kcon = 0.2
kcon = 1.0
kcon = 2.5
115 Reflux (mol·s–1 )
115 Reflux (mol·s–1 )
12 Time (h)
110
105
110
105
100
100 0
4
8
12
16
20
Time (h)
0
4
8
12
16
20
Time (h)
Fig. 14. Regulatory responses of the totally refluxed ideal reactive distillation column with different kcon in the face of a ±2% step change in the production rate, respectively (Example II). (a) Positive responses and (b) negative responses.
behavior can still be seen, larger controller gains significantly relieve its impact. Fig. 21 presents the regulatory responses of the high-purity ethylene glycol reactive distillation column, when a ±5% step change has been introduced to the feed flow rate of EO, respectively. Non-minimum phase behavior also has a great effect on the dynamic response and the reflux drum level controller with a small kcon (i.e., 0.5 s−1 ) fails to stabilize the high-purity ethylene glycol reactive distillation column and the closed-loop simulation ends up around 10 and 5 h, respectively. When the controller gain of the reflux drum level is assigned to be 2.0 or 4.0 s−1 , stable responses can be gained. 6. Discussions Through the three example systems studied, it has been demonstrated that the totally refluxed reactive distillation columns present generally open-loop under-damped responses and this should be attributed to the sharp difference in process dynamics between the reaction operation and the separation operation
involved. Whether the reaction operation involved has thermal effect (i.e., as in Example III) or not (i.e., as in Examples I and II), it does not alter the presence of the under-dampness in dynamic responses of the system. In spite of the fact that it is extremely difficult to prove the above interpretation in terms of strict theoretical analysis, it is considered that this reflects the general characteristics of the totally refluxed reactive distillation columns. To offer additional insights into this issue, we study further the dynamics and control of several ideal reactive distillation columns working in partially refluxed operation mode (i.e., a ternary and quaternary reaction systems with top and bottom products) or totally reboiled operation mode (i.e., a ternary reaction system with only top product) and the results are contained in the web-published Supplement. It is demonstrated that process dynamics and controllability becomes the severest when the ideal reactive distillation columns have been operated in the totally refluxed operation mode. In the partially refluxed operation mode, the effect of the inventory control of the reflux drum appears to be rather small as compared with the totally refluxed operation mode. For the totally reboiled reactive distillation columns, they can have quite similar dynamic
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W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
2 Reflux = 181.14 mol·s
FW = 7.31 mol·s–1
x EG (mole fraction)
a
–1
1
0.95
0.9
kcon = 0.5
kcon = 2.0
kcon = 4.0
0.85
FEO = 7.65 mol·s–1
0
5
10
8
15
20
25
Time (h)
b
14
16
x EG (mole fraction)
0.95
V[17] = 166.45 mol·s–1 NT=17
xEO = 0.0000 xW = 0.0026 xEG = 0.9480 xDEG = 0.0494
Bot = 7.31 mol·s–1
0.85
EG
kcon = 0.5
kcon = 2.0
kcon = 4.0
0.75 0
Fig. 15. Scheme of a high-purity ethylene glycol reactive distillation column (Example III).
5
10
15
20
25
Time (h) Fig. 17. Open-loop transient responses of the high-purity ethylene glycol reactive distillation column with different kcon in the face of a ±2% step change in the reboil flow rate, respectively (Example III). (a) Positive responses and (b) negative responses.
500
b L & V Flowrate (mol·s–1)
a
T (K)
450
400
220
165
110
55 L
350
0 1
5
9
13
17
1
5
Stage nu mber
c
1.0
d Reaction Rate (mol·s–1)
0.6 EO
EG
Water
9
13
17
Stage nu mber
0.8 x (mole fraction)
V
DEG
0.4 0.2
2.8 main reaction
side reaction
2.1
1.4
0.7
0
0.0 1
5
9 Stage nu mber
13
17
1
5
9
13
Stage nu mber
Fig. 16. Steady-state profiles of Example III. (a) Temperature, (b) vapor and liquid flow rates, (c) liquid composition and (d) net reaction rates.
17
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
a
0.95 5 kcon = 0.5
x EG (mole fraction)
1195
kcon = 2.0
kcon = 4.0
PC
0.94 5
LC W
2
0.93 5
CC
0.92 5 0
5
10
15
20
25
Time (h)
b
0.95
x EG (mole fraction)
EO
0.94 8
8
14
FC
15 16
0.94 6
CC
NT=17 0.94 4
LC kcon = 0.5
kcon = 2.0
Steam EG
kcon = 4.0
0.94 2 0
5
10
15
20
25
Time (h)
Fig. 19. Control scheme of Example III.
Fig. 18. Open-loop transient responses of the high-purity ethylene glycol reactive distillation column with different kcon in the face of a ±2% step change in the feed flow rate of water, respectively (Example III). (a) Positive responses and (b) negative responses.
Fig. 20. Servo responses of the high-purity ethylene glycol reactive distillation column with different kcon in the face of a ±0.1 mol% step change in the set-point of the bottom composition control loop, respectively (Example III). (a) Positive responses and (b) negative responses.
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W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
0.98 kcon = 0.5
kcon = 2.0
b
kcon = 4.0
x EG (mole fraction)
x EG (mole fra ction)
a
0.96
0.94
0.92 5
10
15 Time (h)
20
25
kcon = 2.0
kcon = 4.0
0.96
0.94
0.92
30
0
171
180
169
175
V[17] (mol·s–1 )
V[17] (mol·s–1 )
kcon = 0.5
0.9 0
167
kcon = 0.5
kcon = 2.0
5
10
15 Time (h)
20
25
30
170
kcon = 0.5
kcon = 4.0
165
kcon = 2.0
kcon = 4.0
165 0
5
10
15 Time (h)
20
25
30
0
5
10
15 Time (h)
20
25
30
186
Reflux (mol·s–1 )
186
Reflux (mol·s–1 )
0.98
184
182
kcon = 0.5
kcon = 2.0
184
182
kcon = 0.5
kcon = 4.0
180
kcon = 2.0
kcon = 4.0
180 0
5
10
15 Time (h)
20
25
30
0
5
10
15 Time (h)
20
25
30
Fig. 21. Regulatory responses of the high-purity ethylene glycol reactive distillation column with different kcon kcon in the face of a ±5% step change in the production rate, respectively (Example III). (a) Positive responses and (b) negative responses.
behaviors as the totally refluxed reactive distillation columns and this fact reminds us again that the special operation modes should be responsible for the unique dynamic behaviors. Understanding the unique behavior of the totally refluxed reactive distillation columns can be especially beneficial to the synthesis and design of their control systems. Contrary to the common practices in the operation of conventional distillation columns and reactive distillation columns with distillate and bottom withdrawal, the inventory control of the reflux drum should be tightly tuned and this helps to reduce the difference in process dynamics between the reaction operation and the separation operation involved. All the simulation studies obtained so far give a good indication that the tight inventory control of the reflux drum can lead to substantial improvement in process dynamics and controllability. Although the current work employs exclusively the bottom composition as the controlled variable to study the dynamics and operation of the totally refluxed reactive distillation columns, it is considered that the gained insights are also useful for other complicated control schemes, including, for example, the temperature inferential control philosophy and model predictive control algorithm.
Al-Arfaj and Luyben also noticed that the heat duty of reboiler ends up at a lower value after more reactants have been added to the ethylene glycol reactive distillation column, and they owned this counterintuitive phenomenon to the release of more heat of reaction [11,15]. It is noted that similar results have also been obtained in Examples II and III of the current work. Since the reaction operation involved is of no thermal effect at all in Example II, the heat duty of reboiler, however, still settles down to a lower value in the face of a +2% step change in the feed flow rate of reactant B. This outcome makes us to think that the thermal heat of reaction should not be responsible for such a counterintuitive phenomenon. Instead, the existence of input multiplicities in Examples II and III should be considered as a substitute. Since Example I displays no such a counterintuitive phenomenon, the involvement of a side reaction into Examples II and III might be one of the primary reasons that trigger the occurrence of input multiplicities. 7. Conclusion Although the totally refluxed reactive distillation columns appear to be relatively simple in process configuration, they
W. Liu et al. / Journal of Process Control 22 (2012) 1182–1197
present quite unique process dynamics that roots essentially from the totally refluxed operation mode. Owing to the direct connection between the reactive section and reflux drum, the large holdup in the latter retards considerably the dynamics of the reaction operation involved. Because the separation operation involved usually has rather fast process dynamics, the great difference in process dynamics between these two operations leads frequently to the occurrence of under-damped responses. Moreover, the degree of under-dampness is closely related to the inventory control of the reflux drum. With the tight inventory control of the reflux drum, the degree of under-dampness can be reduced and so can the inherent process nonlinearity and non-minimum phase behavior (e.g., as in Examples II and III). The unique dynamics of the totally refluxed reactive distillation columns gives profound implications on the synthesis and design of their control systems. In sharp contrast to conventional distillation columns as well as reactive distillation columns with distillate and bottom withdrawal, the inventory of the reflux drum should be tightly controlled. In terms of two hypothetical ideal reactive distillation columns with and without a side reaction, respectively, and a high-purity ethylene glycol reactive distillation column, the unique process dynamics of the totally refluxed reactive distillation columns has been observed. All the results obtained confirm the fact that noticeable improvement in process dynamics and controllability has been acquired when tight inventory control has been implemented in the reflux drum. Acknowledgements The project is financially supported by the National Science Foundation of China (Grant Number: 21176015) and the Doctoral Programs Foundation of Ministry of Education of China (Grant Number: 20100010110008), and thereby is acknowledged. The authors are also in debt to the anonymous reviewers for their valuable comments and suggestions.
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