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Design and operation of dividing-wall distillation columns. 2. Process dynamics and operation
Chemical Engineering and Processing 91 (2015) 89–103
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Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep
Design and operation of dividing-wall distillation columns. 2. Process dynamics and operation Xin Zong, Kejin Huang * , Yang Yuan, Haisheng Chen, Jieping Yu College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, PR China
A R T I C L E I N F O
A B S T R A C T
Article history: Received 7 February 2015 Accepted 5 March 2015 Available online 10 March 2015
In the first paper of this series, it was demonstrated that the black-hole problem of dividing-wall distillation columns (DWDCs) could effectively be diminished through over-design in terms of careful adjustment of the number of stages. Because the over-design serves to coordinate the relationship between the prefractionator and main distillation column involved, it yields frequently favorable effect to process dynamics and controllability and this represents essentially an additional advantage of such process modifications to diminish the black-hole problem. The three example systems studied in the first paper of this series are again employed to ascertain the anticipation. The DWDCs with and without the over-design are strictly compared in terms of open- and closed-loop controllability studies. It is found that the over-design helps to coordinate the interaction between the prefractionator and main distillation column involved and leads consequently to improved process dynamics and capabilities in disturbance rejection and set-point tracking. The outcome confirms again the necessity of diminishing the black-hole problem in the synthesis and design of the DWDCs. Though the interpretation is gained from the three example systems studied, it should be considered to be of general significance to the design and operation of the DWDCs. ã 2015 Elsevier B.V. All rights reserved.
Keywords: DWDC Over-design Process dynamics Process controllability Dynamic simulation
1. Introduction For a dividing-wall distillation column (DWDC) separating a ternary mixture, it is frequently found infeasible to greatly boost the purity of its intermediate product from its nominal steadystate value even under the extreme operating condition of an infinite boil-up rate or an infinite reflux ratio [1–5]. This represents essentially a serious drawback of the DWDC and can confine considerably process applicability and flexibility. The strong coupling between the prefractionator and main distillation column involved is responsible for such an unfavorable behavior despite the fact that it also represents the main thrust for the enhancement of thermodynamic efficiency of the DWDC. Recently, we referred to the issue as the black-hole problem of the DWDC and pointed out that it could be diminished through over-design in terms of careful adjustment of the number of stages in the prefractionator and main distillation column involved [6]. Because the adjustment of the number of stages affects the flow rates and compositions of the interlinking flows between the prefractionator and main distillation column involved, it can function as an effective method to
* Corresponding author. Tel.: +86 10 64437805; fax: +86 10 64437805. E-mail address: [email protected] (K. Huang). http://dx.doi.org/10.1016/j.cep.2015.03.005 0255-2701/ ã 2015 Elsevier B.V. All rights reserved.
compromise the interaction between these two highly integrated units and serve to diminish the black-hole problem of the DWDC (Fig. 1). Although the over-design philosophy was demonstrated to be effective in enhancing the applicability and flexibility of the DWDC, its impact to process dynamics and controllability has remained yet to be studied so far. Since it is also an important issue that influences the applicability and flexibility of the DWDC, it should be studied in great detail. Since 1990, considerable effort has been dedicated to the studies of the dynamics and control of the DWDCs [7–10]. In particular, open-loop controllability assessment was frequently used to analyze the mass and thermal coupling between the prefractionator and main distillation column involved [11–13]. Hernández and Jiménez conducted detailed comparison between the DWDC and distillation columns with a side rectifier or stripper [14]. They pointed out that although the DWDC needed the lowest utility requirement it exhibited degraded process dynamics and controllability as compared with the distillation columns with a side rectifier or stripper. Serra et al. employed the Morari resiliency index, condition number, relative gain array, and closed-loop disturbance gain as performance indexes to examine the relationship between process synthesis and design and process dynamics and controllability [15,16]. They indicated that the deliberate addition of stages to the DWDC could improve process
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Nomenclature
Symbols Hypothetical component A Avp Vapor pressure constant, kPa B Hypothetical component, benzene, butanol, or bottom product flow rate, mol/s Bvp Vapor pressure constant, kPa C Hypothetical component CC Composition controller CN Condition number d Input signal D Distillate flow rate, mol/s DRGA Dynamic relative gain array DWDC Dividing-wall distillation column E Ethanol F Feed flow rate, mol/s FC Flow rate controller G Process transfer function koff Open-loop gain when the rest of loops are open kon Open-loop gain when the rest of loops are in automatic FPD Final process design IPD Initial process design LC Level controller MASS Maximally achievable steady state MIMO Multiple-input and multiple-output MRI Morari resiliency index n Dimensionality NT Number of stages P Propanol or pressure, Pa PC Pressure controller Q Reboiler heat duty, KW R Reflux flow rate, mol/s SVD Singular value decomposition I Intermediate product flow rate, mol/s T Toluene or temperature, K U Matrix of output singular vectors V Matrix of input singular vectors y Process output signal X o-Xylene Greek Letters Relative volatility Minimum singular value Maximum singular value Matrix of singular values Element of RGA Frequency
a s1 sn S l v
Subscripts A Component index B Component index or bottom product C Component index D Distillate product I Intermediate product E Component index P Component index T Component index X Component index Superscript s Saturation
N1
A
N2
N5 A/B/C
B
N6
N3
N4 C
(a)
A
N1
A/B/C
N 5 N2
B
N6 N3 N4 C
(b) Fig. 1. Sectional stage number and interlinking flows in the Petlyuk distillation column and DWDC (a) Petlyuk distillation column, (b) DWDC.
dynamics and controllability. Regarding the operation of the DWDC, there usually exist two kinds of control configurations, i.e., the three-point and four-point control schemes [1]. In the case of the three-point control scheme, the liquid and vapor split ratios are Table 1 Physical properties, operating conditions, and product specifications of example I. Parameter
Value
Condenser pressure (bar) Stage pressure drop (bar) Feed compositions (mol%)
3.0 0.0068901 33.3 33.3 33.4 27.8 1.0 4:2:1 29,053.7 13.04/3862 12.34/3862 11.65/3862 99 99 99
Feed flow rate (mol/s) Feed thermal condition Relative volatility A:B:C Latent heat of vaporization (kJ/kmol) Vapor pressure constants
Product specifications (mol%)
A B C
A (Avp/Bvp) B (Avp/Bvp) C (Avp/Bvp) A B C
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Table 2 Transfer function matrix of example I.
IPD
XI,
PC 2
R
D
11 1
FC
I
42
CC1
55
LC2
B
(a) PC LC1 2
R
D
14 1 F
26
21 38
FC
CC3
15
A
B
XB,
C
XD,
A
XI, XB,
CC2
41
XD, XI,
FPD (MASS2)
23
37
FPD (MASS1)
CC3
12
19
A
B
XB,C
LC1
F
XD,
B C
D
I
Q
44:250e0:03s 3:51sþ1 51:519e0:09s 8:56sþ1 93:350e0:1s 2:57sþ1
0:283e0:13s 8:69sþ1 8:849e0:01s 0:61sþ1 95:732e0:03s 2:38sþ1
0:131e0:03s 3:74sþ1 0:238e0:03s 0:97sþ1 0:103e0:01s 0:15sþ1
62:890e0:04s 3:97sþ1 31:610e0:07s 20:79sþ1 91:876e0:13s 2:57sþ1
0:290e0:13s 1:49sþ1 10:815e0:01s 0:62sþ1 93:740e0:06s 2:38sþ1
0:175e0:02s 3:49sþ1 0:298e0:03s 0:78sþ1 0:118e0:01s 0:12sþ1
67:750e0:04s 3:34sþ1 25:768e0:05s 29:58sþ1 90:925e0:09s 2:56sþ1
0:310e0:08s 1:6sþ1 12:590e0:01s 0:58sþ1 92:037e0:04s 2:29sþ1
0:175e0:02s 3:09sþ1 0:307e0:03s 0:75sþ1 0:126e0:01s 0:12sþ1
fixed and the top, intermediate, and bottom products are controlled, respectively, with the reflux flow rate (or distillate flow rate and this depends heavily on the detailed reflux ratio), intermediate product flow rate, and reboiler heat duty. A lot of studies confirmed that the control scheme could maintain stable and effective operation of the DWDC [17–19]. In the case of the four-point control scheme, in addition to those controlled and manipulated variables mentioned above, the composition of the heaviest component at the top of the prefractionator and the liquid split ratio are also comprehended and help to tighten the operation of the DWDC. In terms of a DWDC separating a ternary separation of benzene, toluene, and o-xylene, Ling and Luyben demonstrated that the four-point control scheme could achieve self-optimizing control of the DWDC featuring the minimum reboiler heat duty in dynamic operations [20]. Later, various temperature inferential control schemes were reported to avoid the employment of
I CC2
44 45
CC1
58
LC2
B
(b) PC LC1 2
R
D
16 1 F
28
23 40
FC
CC3
17
45 46
I CC2 CC1
59
LC2
B
(c) Fig. 2. Three process designs along with their decentralized control structures (a) IPD, (b) FPD (MASS1), and (c) FPD (MASS2) (example I).
Fig. 3. MRI and CN of the IPD, FPD (MASS1), and FPD (MASS2) (example I).
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X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103 Table 3 DRGA for the IPD, FPD (MASS1), and FPD (MASS2) at 0.0001 and 10000 rad/h (example I).
v = 0.0001 rad/h
v = 10000 rad/h
D
I
Q
D
I
Q
IPD
XD, A XI, B XB, C
0.5831 0.3602 0.0567
0.0027 0.0826 0.9201
0.4196 0.5572 0.0233
0.9032 0.0311 0.0657
0.0006 0.5203 0.4803
0.0974 0.4485 0.4541
FPD (MASS1)
XD, A XI, B XB, C
0.7281 0.2061 0.0658
0.0027 0.1009 0.9018
0.2747 0.6930 0.0324
0.9412 0.0061 0.0527
0.0044 0.5602 0.4442
0.0631 0.4338 0.5031
FPD (MASS2)
XD, A XI, B XB, C
0.7689 0.1582 0.0728
0.0029 0.1169 0.8861
0.2340 0.7249 0.0411
0.9487 0.0026 0.0487
0.0031 0.6022 0.4010
0.0545 0.3953 0.5503
composition measurements and meanwhile still maintain tight control of product qualities [21–24]. Recently, a different fourpoint control scheme aiming to control the ratio between the two impurities in the intermediate product (rather than the composition of the heaviest component at the top of the prefractionator) was also proposed and studied [1,25–27]. In order to cope with the complicated dynamic behaviors of the DWDC, some researcher attempted advanced control algorithms to gain strict control of its product qualities [28,29]. The major purpose of the current work is to elucidate the impact of the over-design through careful adjustment of the number of stages on the dynamics and controllability of the DWDC. Analogous to the first paper of this series, the separations of three ternary mixtures of hypothetical components, A,B, and C, benzene, toluene, and o-xylene, and ethanol, propanol, and butanol, are employed as illustrative examples to show the intricate mechanism. In virtue of open- and closed-loop controllability assessments, strict comparison is made between the DWDCs with and without the over-design through the careful adjustment of the number of stages. Last but not the least, the significance of the over-design is analyzed and some concluding remarks are given in the last section of this article. 2. Methods to evaluate the impact of over-design on the dynamics and controllability of the DWDC
Fig. 4. Diagonal elements of the DRGA of the IPD, FPD (MASS1), and FPD (MASS2) (example I).
Open- and closed-loop controllability assessments are to be used to characterize the effect of the over-design on the dynamics and controllability of the DWDC. In the open-loop controllability assessment, three predictive indexes in the frequency domain, including the Morari resiliency index (MRI), condition number (CN), and dynamic relative gain array (DRGA) are employed.
Table 4 Controller parameters of examples I–III. Controller
Manipulated variable
Controlled variable
KC ()
TI (min)
Example I
CC1 CC2 CC3
Q I D
XB, C XI, B XD, A
3.01 43.68 57.56
66.00 51.48 102.92
Example II
CC1 CC2 CC3
Q I R
XB, X XI, T XD, B
3.22 14.21 6.25
93.72 58.08 105.60
Example III
CC1 CC2 CC3
Q I D
XB, B XI, P XD, E
3.56 54.80 80.68
69.96 50.16 105.60
X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103
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Fig. 5. Regulatory responses of example I for a 5% step change in the feed compositions of component B (a) composition of component A in the top product, (b) flow rate of the top product, (c) composition of component B in the intermediate product, (d) flow rate of the intermediate product, (e) composition of component C in the bottom product, and (f) heat duty of reboiler.
For a square multiple-input and multiple-output (MIMO) process with a dimensionality of n n, the gain between an output and an input can be determined via the following equation [30]. gain ¼
k yðjwÞ k2 k GðjwÞdðjwÞ k2 ¼ k dðjwÞ k2 k dðjwÞ k2
(1)
where y is the process output, d, the process input, and G, the process transfer function. Note that the process gain depends essentially on the direction of the input signal. To analyze the performance of the process, all the process gains needed be determined along with their direction (i.e., the determination of gain directionality). The singular value decomposition (SVD) of G yields G ¼ U SV T
(2)
where V = (v1, v2, . . . , vn) is the unitary matrix of input singular
vectors, S = diag (s 1, s 2, . . . , s n), the diagonal matrix of singular values, and U = (u1, u2, . . . , un), the unitary matrix of output singular vectors. Eqs. (3)–(5) can then be derived and they indicate that the largest gain for any input direction is equal to the maximum
singular value and the smallest gain the minimum singular value. Gvi ¼ s i ui
1in
s i ¼ k Gvi k2 1 i n sn
k Gv k2 ¼ k Gv k2 s 1 k v k2
(3)
(4)
(5)
The MRI is the minimum singular value s n and could be a measurement of influences by potential problems in a dynamic system. The control system should prefer to have a large MRI over the frequency range of interest. The CN is the ratio of the maximum to minimum singular values (c.f., Eq. (6)). It can reflect control system sensitivity under process parameter uncertainties and model/plant mismatches. The control system should prefer to have a small CN over the frequency range of interest. CN ¼
s1 sn
(6)
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Fig. 6. Servo responses of example I for a 0.0012 step change in the purities of the top, intermediate, and bottom products (a) composition of component A in the top product, (b) flow rate of the top product, (c) composition of component B in the intermediate product, (d) flow rate of the intermediate product, (e) composition of component C in the bottom product, and (f) heat duty of reboiler.
The primary use of the RGA is to evaluate interactions of control loop pairings in multi-loop control systems. Its element li, j is dimensionless and is the ratio between two kinds of process gains (c. f., Eq. (7)). If li, j is equal to one it means that the gain from input j to output i is unaffected by closing the rest control loops. In addition, processes with large RGA elements indicate fundamental control problems because of their sensitivity to input uncertainty. The control system should prefer to have a RGA that is as close to one as possible. The DRGA is a kind of improvement of the RGA with the replacement of steady-state models by transfer function ones.
li;j ¼
koff kon
1 i n; and 1 j n
(7)
where koff is the open-loop gain between the output i and the input j when the rest loops are open, kon, the open-loop gain between the output i and the input j when the rest loops are in automatic. In the closed-loop controllability assessment, disturbance rejection and set-point tracking capabilities are used to inspect process behaviors. While the disturbance rejection ability is the behaviors of a control scheme in the face of internal and external disturbances, the set-point tracking capability the behaviors of a control scheme in the
face of variations in set-points. In terms of these two performance indexes, fast convergence and small fluctuations are certainly two desirable properties of a control system. In what follows, three DWDCs fractionating, respectively, three ternary mixtures of hypothetical components, A, B, and C, benzene, Table 5 Physical properties, operating conditions, and product specifications of example II. Parameter
Value
Condenser pressure (atm) Stage pressure drop (atm) Feed compositions (mol%)
0.37 0.0068 30 30 40 1000 1.0 353 385 419 99 99 99
Feed flow rate (mol/s) Feed thermal condition Normal boiling points (K)
Product specifications (mol%)
Benzene (B) Toluene (T) O-xylene (X)
Benzene (B) Toluene (T) O-xylene (X) Benzene (B) Toluene (T) O-xylene (X)
X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103 Table 6 Transfer function matrix of example II.
PC LC1 2
R
1
XD, XI,
B
T X
CC3
9
23
FC
D
FPD (MASS1)
17
14
IPD
XB,
8
F
CC2
30
XD, XI,
I
31
FPD (MASS2)
XB,
X
XD,
B
XI,
CC1
B
T
XB,
T X
43
LC2
I
Q
0:101e0:14s 7:48sþ1 1:060e0:01s 1:26sþ1 1:580e0:08s 1:94sþ1
0:043e0:04s 1:70sþ1 0:009e0:02s 2:17sþ1 0:038e0:01s 0:85sþ1
1:510e0:02s 1:27sþ1 0:274e0:04s 9:9sþ1 0:933e0:11s 1:66sþ1
0:045e0:12s 5:77sþ1 1:481e0:01s 1:11sþ1 1:311e0:08s 1:66sþ1
0:045e0:03s 1:40sþ1 0:002e0:02s 1:67sþ1 0:036e0:01s 0:78sþ1
1:524e0:02s 1:36sþ1 0:265e0:04s 10:52sþ1 0:951e0:12s 1:65sþ1
0:055e0:17s 6:85sþ1 1:404e0:01s 1:19sþ1 1:360e0:08s 1:71sþ1
0:046e0:03s 1:41sþ1 0:003e0:02s 1:78sþ1 0:036e0:01s 0:79sþ1
toluene, and o-xylene, and ethanol, propanol, and butanol are studied to comprehensively assess the impact of the over-design through the adjustment of stage number on process dynamics and operation.
PC LC1 2
R
9 1
D
19
25
I CC2
32
3. Example I: separation of a ternary mixture of hypothetical components, A, B, and C 3.1. Process description
CC3
10
16
FC
R 1:382e0:02s 1:24sþ1 0:050e0:05s 5:65sþ1 1:074e0:12s 1:61sþ1
B
(a)
F
95
33
The physical properties, operating conditions, and product specifications are listed in Table 1. An equi-molar mixture of the hypothetical components A, B, and C is separated into three products with purities of 99 mol%, respectively. Ideal vapor and
CC1
44
LC2
B
(b) PC LC1 2
R
9 1 F
10
CC3
22
17 26
FC
D
34 35
I CC2 CC1
46
LC2
B
(c) Fig. 7. Three process designs along with their decentralized control structures (a) IPD, (b) FPD (MASS1), and (c) FPD (MASS2) (example II).
Fig. 8. MRI and CN of the IPD, FPD (MASS1), and FPD (MASS2) (example II).
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liquid phase behaviors are assumed and the vapor–liquid equilibrium relationship can be expressed by Pj ¼ xA;j PsA þ xB;j PsB þ xC;j PsC
yi;j ¼
xi;j Psi;j Pj
1 j NT
(8)
i ¼ A; B; C; and 1 j NT
(9)
The vapor saturation pressure is calculated via lnPsi;j ¼
Avp;i Bvp;i Tj
i ¼ A; B; C; and 1 j NT
(10)
Fig. 2 reproduces the three designs of the DWDCs obtained in the first paper of this series. While the initial process design (IPD) refers to the one obtained directly from the minimization of total annual cost, namely, without the consideration of the over-design through the adjustment of the number of stages (Fig. 2a), the two final process designs (FPDs) the ones with the further consideration of the over-design through the adjustment of the number of stages (Fig. 2b and c). As a result, the two FPDs have higher maximally achievable steady state (MASS) than the IPD (whose value is only 99.25 mol%). The MASS is a performance index that reflects process flexibility and operability and can be estimated through assuming an equal composition of the primary components in the top, intermediate, and bottom products. The two FPDs differ in process configurations that lead to different MASS. While the FPD (MASS1) shown in Fig. 2b is able to reach the purities of 99.5 mol% in the top, intermediate, and bottom products simultaneously, the FPD (MASS2) shown in Fig. 2c is able to reach the purities of 99.6 mol% in these three products. 3.2. Open-loop controllability assessment
Fig. 9. Diagonal elements of the DRGA of the IPD, FPD (MASS1), and FPD (MASS2) (example II).
In terms of step tests, the transfer function matrixes of the three process designs are obtained and tabulated in Table 2. While the top, intermediate, and bottom product qualities, XD, A, XI, B, and XB, C, are taken as the controlled variable, the distillate flow rate D (since the three reflux ratios are 3.269, 3.207, and 3.242, respectively, and all are greater than 3.0), the intermediate product flow rate I, and the reboiler heat duty Q as the manipulated variables. Fig. 3 shows the variation tendencies of the MRI and CN along with the changes of frequencies for the three process designs after appropriate scaling of their input and output variables. Here, unless otherwise stated, the dotted lines represent the behaviors of the IPD, the dashed lines the behaviors of the FPD (MASS1), and the bold lines the behaviors of the FPD (MASS2). While the black
Table 7 DRGA for the IPD, FPD (MASS1), and FPD (MASS2) at 0.0001 and 10000 rad/h (example II).
v = 0.0001 rad/h
v = 10000 rad/h
R
I
Q
R
I
Q
IPD
XD, B XI, T XB, X
3.1955 0.1534 2.0421
0.0313 0.3184 0.6502
2.2268 0.8349 2.3919
1.4521 0.0059 0.4462
0.0009 0.8919 0.1071
0.4531 0.1140 1.3391
FPD (MASS1)
XD, B XI, T XB, X
2.1659 0.4264 1.5924
0.0140 0.4464 0.5395
1.1800 0.1272 2.0528
1.4642 0.0142 0.4784
0.0003 0.9584 0.0413
0.4646 0.0275 1.4371
FPD (MASS2)
XD, B XI, T XB, X
2.1224 0.4350 1.5574
0.0171 0.4233 0.5596
1.1395 0.1417 1.9978
1.5265 0.0162 0.5427
0.0004 0.9515 0.0481
0.5269 0.0323 1.4945
X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103
curves denote the positive responses, the gray curves the negative ones. As can readily be seen, both the FPD (MASS1) and FPD (MASS2) have relatively larger MRI than the IPD, implying that the over-design through the adjustment of the number of stages helps to abate the occurrences of potential problems in dynamic operations (Fig. 3a). In the case of the CN, both the FPD (MASS1) and FPD (MASS2) give smaller values than the IPD, implying that the over-design through the adjustment of number of stages may reduce the process sensitivity to parameter uncertainties (Fig. 3b). As far as the DRGA is concerned, a more complicated situation happens as shown in Fig. 4. While in the low-frequency region, the DRGA suggests a diagonal pairing for the top product and offdiagonal pairings for the intermediate and bottom products, in the intermediate and high frequency region, it suggests diagonal pairings for the top, intermediate, and bottom products. Table 3 compares two DRGAs at frequencies of 0.0001 and 10,000 rad/h, respectively and shows clearly the contradiction. Since the DRGA in the intermediate and high frequency region takes into account loop interactions in dynamic state, it is considered to be more reasonable than the one by the DRGA in the low-frequency region and should be followed in the synthesis and design of decentralized control systems. The over-design through the adjustment of
97
the number of stages makes the l1,1 closer to one in the low and high frequency regions in the FPD (MASS1) and FPD (MASS2) than in the IPD. With regard to the l2,2 and l3,3, they all locate somewhere around 0.5 in the high frequency region and indicate potentially strong interactions between the intermediate and bottom products. The FPD (MASS1) and FPD (MASS2) still present values slightly closer to one than the IPD and the tendencies can clearly be confirmed in Table 3. These outcomes imply again that the FPD (MASS1) and FPD (MASS2) have alleviated interactions between the top, intermediate, and bottom products than the IPD. 3.3. Closed-loop controllability assessment According to the above DRGA analysis, the decentralized control systems are determined and also shown in Fig. 2. For all the three process designs, the qualities of the top, intermediate, and bottom products are controlled, respectively, with the top product flow rate, the intermediate product flow rate, and the heat duty of reboiler. A five minutes dead-time is assumed in each composition sensor. With reference to the IPD, all the composition controllers are tuned in a sequential iteration manner with the embedded Tyreus–Luyben rule and the convergent controller parameters are
Fig. 10. Regulatory responses of example II for a 5% step change in the feed compositions of toluene (a) composition of benzene in the top product, (b) reflux flow rate, (c) composition of toluene in the intermediate product, (d) flow rate of the intermediate product, (e) composition of o-xylene in the bottom product, and (f) heat duty of reboiler.
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listed in Table 4. For the FPD (MASS1) and FPD (MASS2), they are made to share the same controller parameters with the IPD and this serves to facilitate the comparison of closed-loop controllability between the three process designs. Fig. 5 displays the responses of the controlled and manipulated variables of the three process designs in the face of a 5% step change in the feed compositions of component B at the time instant of 0.5 h. The ratio between the feed compositions of components A and C is kept the same as in the nominal operating conditions. All the three process designs are found to be able to return to their designated steady states. The IPD appears to be inferior to the FPD (MASS1) and FPD (MASS2) because of the relatively long settling times and enlarged peak deviations in the intermediate and bottom products. Fig. 6 depicts the responses of the controlled and manipulated variables of the three process designs in the face of a 0.0012 step change in the purities of the top, intermediate, and bottom products. In the case of the IPD, although the top and intermediate products exhibit relatively small overshoots, they give much strong interaction to the bottom product and worsen considerably its dynamic performance
Table 8 Physical properties, operating conditions, and product specifications of example III. Parameter
Value
Condenser pressure (atm) Stage pressure drop (atm) Feed compositions (mol%)
1 0.0068 33.3 33.3 33.4 83.33 1.0 352 370 392 99 99 99
Feed flow rate (mol/s) Feed thermal condition Normal boiling points (K)
Product specifications (mol%)
Ethanol (E) Propanol (P) Butanol (B)
Ethanol (E) Propanol (P) Butanol (B) Ethanol (E) Propanol (P) Butanol (B)
especially in the positive changes of the three set-points. On the contrary, the FPD (MASS1) and FPD (MASS2) can still effectively track the new steady states with an acceptable degree of oscillations in the controlled and manipulated variables.
Fig. 11. Servo responses of example II for a 0.002 step change in the purities of the top, intermediate, and bottom products (a) composition of benzene in the top product, (b) reflux flow rate, (c) composition of toluene in the intermediate product, (d) flow rate of the intermediate product, (e) composition of o-xylene in the bottom product, and (f) heat duty of reboiler.
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99
Table 9 Transfer function matrix of example III.
IPD
XI,
PC 2
R
D
10 1
38
FC
FPD (MASS2)
23
I
42
CC1
XD,
E
XI,
P B
I
Q
0:113e0:03s 10:36sþ1 4:136e0:01s 1:04sþ1 30:860e0:03s 3:15sþ1
0:028e0:03s 5:71sþ1 0:064e0:03s 0:76sþ1 0:036e0:01s 0:16sþ1
25:849e0:04s 6:92sþ1 2:699e0:07s 33:46sþ1 28:756e0:13s 4:12sþ1
0:109e0:08s 7:03sþ1 5:458e0:01s 1:08sþ1 29:586e0:02s 3:15sþ1
0:041e0:08s 4:42sþ1 0:086e0:02s 0:75sþ1 0:043e0:01s 0:19sþ1
28:282e0:06s 7:38sþ1 0:867e0:07s 73:98sþ1 28:167e0:13s 4:26sþ1
0:149e0:17s 6:83sþ1 6:579e0:01s 1:12sþ1 28:541e0:07s 3:15sþ1
0:051e0:03s 3:9sþ1 0:101e0:02s 0:69sþ1 0:048e0:01s 0:18sþ1
4.1. Process description
LC2
B
(a) PC LC1 2
R
D
12 1
CC3
13 25
20 39
FC
B
D 16:885e0:06s 4:94sþ1 13:744e0:1s 11:2sþ1 29:888e0:13s 4:12sþ1
4. Example II: separation of a ternary mixture of benzene, toluene, and o-xylene
56
F
E
P
XB,
XB,
CC2
41
B
XD, XI,
CC3
11
19
FPD (MASS1)
E
P
XB,
LC1
F
XD,
Table 5 summarizes the physical properties, operating conditions, and product specifications. A ternary mixture of benzene, toluene, and o-xylene with a composition of 0.3/0.3/0.4 is separated into three products with purities of 99 mol%, respectively. Fig. 7 reproduces the three designs of the DWDCs obtained in the first paper of this series. The IPD has a MASS of only 99.29 mol%. With the over-design through the adjustment of the number of stages, while the FPD (MASS1) shown in Fig. 7b is able to reach the purities of 99.6 mol% in the top, intermediate, and bottom products, the FPD (MASS2) shown in Fig. 7c is able to reach the purities of 99.8 mol% in these three products.
I CC2
43 44
CC1
58
LC2
B
(b) PC LC1 2
R
D
13 1 F
26
24 43
FC
CC3
14
44 45
I CC2 CC1
59
LC2
B
(c) Fig. 12. Three process designs along with their decentralized control structures (a) IPD, (b) FPD (MASS1), and (c) FPD (MASS2) (example III).
Fig. 13. MRI and CN of the IPD, FPD (MASS1), and FPD (MASS2) (example III).
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4.2. Open-loop controllability assessment While the reflux flow rate R (since the reflux ratios are now 2.635, 2.598, and 2.596, respectively, and all are smaller than 3.0), the intermediate product flow rate I, and the reboiler heat duty Q are chosen as the manipulated variables, the top, intermediate, and bottom product compositions, XD, B, XI, T, and XB, X, as the controlled variables. Table 6 lists the transfer function matrixes of the three process designs. The open-loop controllability assessment is then conducted and the changing tendencies of the MRI and CN in the frequency domain are shown in Fig. 8. With the overdesign through the adjustment of the number of stages in the IPD, while the MRI is enhanced, the CN is reduced in the resultant FPD (MASS1) and FPD (MASS2), implying a certain extent of improvement in process dynamics and controllability. The variation tendencies of the diagonal elements of the DRGA in the frequency domain are depicted in Fig. 9 and two detailed DRGAs at frequencies of 0.0001 and 10,000 rad/h are compared in Table 7. Note that the diagonal control structure is recommended for all the three process designs studied. With the over-design through the adjustment of the number of stages in the IPD, while the l2,2 is increased in the full range of frequency domain and made quite close to one in the high frequency region, the l1,1 and l3,3 are decreased in the low frequency region in the resultant FPD (MASS1) and FPD (MASS2) (i.e., made closer to one). This may forebode that the interaction between the top, intermediate, and bottom products has become weaker in the FPD (MASS1) and FPD (MASS2) than in the IPD. Moreover, the interaction affecting the intermediate product is abated more greatly than those affecting the top and bottom products. 4.3. Closed-loop controllability assessment In accordance with the above DRGA analysis, the decentralized control systems are determined for the three process designs studied and also shown in Fig. 7. The controller parameters are listed in Table 4. Fig. 10 displays the responses of the controlled and manipulated variables of the three process designs in the face of a 5% step change in the feed compositions of toluene. The ratio between the feed compositions of benzene and o-xylene is again kept the same as in the nominal operating conditions. It can be noted that the FPD (MASS1) and FPD (MASS2) present slightly better responses than the IPD in the aspect of settling times. Fig. 11 depicts the responses of the controlled and manipulated variables of the three process designs in the face of a 0.002 step change in the purities of the top, intermediate, and bottom products. The FPD (MASS) and FPD (MASS2) show slightly more rapid and smoother transitions to the new steady states than the IPD. Fig. 14. Diagonal elements of the DRGA of the IPD, FPD (MASS1), and FPD (MASS2) (example III).
Table 10 DRGA for the IPD, FPD (MASS1), and FPD (MASS2) at 0.0001 and 10000 rad/h (example III).
v = 0.0001 rad/h D
v = 10000 rad/h I
Q
I
Q
IPD
XD, E XI, P XB, B
0.7071 0.2304 0.0626
0.0032 0.1157 0.8875
0.2961 0.6539 0.0499
D 0.96807 0.0101 0.0219
0.0006 0.5230 0.4776
0.0325 0.4650 0.5005
FPD (MASS1)
XD, E XI, P XB, B
0.8834 0.0405 0.0760
0.0032 0.1540 0.8492
0.1198 0.8055 0.0748
0.9625 0.0008 0.0366
0.0014 0.5338 0.4676
0.0389 0.4654 0.4957
FPD (MASS2)
XD, E XI, P XB, B
0.9213 0.0129 0.0916
0.0044 0.1873 0.8171
0.0831 0.8256 0.0913
0.9571 0.0001 0.0430
0.0019 0.5515 0.4504
0.0447 0.4486 0.5067
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5. Example III: separation of a ternary mixture of ethanol, propanol, and butanol 5.1. Process description Table 8 tabulates the physical properties, operating conditions, and product specifications. An equi-molar ternary mixture of ethanol, propanol, and butanol is separated into three products with purities of 99 mol%, respectively. Fig. 12 reproduces the three designs of the DWDCs obtained in the first paper of this series. The IPD has a MASS of only 99.24 mol%. With the over-design through the adjustment of the number of stages, while the FPD (MASS1) shown in Fig. 12b is able to reach the purities of 99.4 mol% in the top, intermediate, and bottom products, the FPD (MASS2) shown in Fig. 12c is able to reach the purities of 99.6 mol% in these three products. 5.2. Open-loop controllability assessment While the distillate flow rate D (since the reflux ratios are now 3.361, 3.337, and 3.325, and all are greater than 3.0), the intermediate product flow rate I, and the reboiler heat duty Q
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are selected as the manipulated variables, the top, intermediate, and bottom product qualities, XD, E, XI, P, and XB, B, as the controlled variables. Table 9 lists the transfer function matrixes of the three process designs studied. Open-loop controllability assessment is performed again in terms of the MRI, CN, and DRGA in the frequency domain and their variation tendencies are illustrated in Figs. 13 and 14. In particular, the comparison of two detailed DRGAs at frequencies of 0.0001 and 10,000 rad/h is made in Table 10. While the MRI is enhanced the CN reduced in the full range of frequency domain by the over-design through the adjustment of the number of stages, implying a certain degree of improvement in process dynamics and controllability from the IPD to the FPD (MASS1) and FPD (MASS2). In the case of the DRGA, although only fairly small changes observed in the l3,3, the over-design through the adjustment of the number of stages in the IPD increases the l2,2 in the full range of frequency domain (i.e., becomes closer to one) and makes the l1,1 approach to one in the low frequency region. The same tendency can also be identified in Table 10. These outcomes indicate possibly the alleviation of the interaction between the top, intermediate, and bottom products from the IPD to the FPD (MASS1) and FPD (MASS2).
Fig. 15. Regulatory responses of example III for a 5% step change in the feed compositions of propanol (a) composition of ethanol in the top product, (b) flow rate of the top product, (c) composition of propanol in the intermediate product, (d) flow rate of the intermediate product, (e) composition of butanol in the bottom product, and (f) heat duty of reboiler.
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5.3. Closed-loop controllability assessment In accordance with the above DRGA analysis, the decentralized control systems are determined and also shown in Fig. 12. The controller parameters are comprehended in Table 4. Fig. 15 displays the responses of the controlled and manipulated variables of the three process designs in the face of a 5% step change in the feed compositions of propanol. The ratio between the feed compositions of ethanol and butanol is again kept the same as in the nominal operating conditions. Although the FPD (MASS1) and FPD (MASS2) exhibit relatively extended settling times in the top product in comparison with the IPD, they leads to suppressed oscillations and reduced settling times in the intermediate and bottom products. Fig. 16 depicts the responses of the controlled and manipulated variables of the three process designs in the face of a 0.0015 step change in the purities of the top, intermediate, and bottom products. Although slightly better performance (in the aspect of overshoots) are achieved in the top and intermediate products in the IPD than in the FPD (MASS1) and FPD (MASS2), the reverse is true in the bottom product due to the severe interaction encountered and the
dramatically extended settling times especially in face of the positive changes in the set-points of the top, intermediate, and bottom products. 6. Discussion In terms of the three example systems studied in the current work, it has clearly been demonstrated that the over-design through the careful adjustment of the number of stages can frequently yield a favorable influence to the dynamics and controllability of the DWDC. Despite the fact that it is extremely difficult to present a detailed theoretical analysis about their intricate interplay, the compromise of the relationship between the prefractionator and main distillation column involved should be taken here as the primary reason. Since the over-design through the careful adjustment of the number of stages also represent an effective method for the diminution of the black-hole problem as shown in the first paper of this series, it is not difficult to understand the great importance of considering the over-design of the DWDC during process synthesis and design. Under some circumstances, it can even serve as an effective strategy to trade-off
Fig. 16. Servo responses of example III for a 0.0015 step change in the purities of the top, intermediate, and bottom products (a) composition of ethanol in the top product, (b) flow rate of the top product, (c) composition of propanol in the intermediate product, (d) flow rate of the intermediate product, (e) composition of butanol in the bottom product, and (f) heat duty of reboiler.
X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103
process synthesis and design and process dynamics and controllability during the development of the DWDC. It is worthwhile to mention here the fact that the over-design through the careful adjustment of the number of stages leads to rather obvious improvement in the dynamic behaviors of the intermediate product. Both open-loop and closed-loop controllability assessments have corroborated definitely such tendency (c.f., Figs. 4–6 in example I, Figs. 9–11 in example II, and Figs. 14–16 in example III). Since the over-design is originally aimed to enhance the flexibility and operability of the DWDC through the careful adjustment of the number of stages, its direct effect lies definitely on the extension of the rangeability of the intermediate product (as clearly indicated in the first paper of this series), however, a good consistency has been observed between process design and process operation. This represents actually a potential advantage of the over-design philosophy proposed in the current work. The extended rangeability allows the FPD (MASS1) and FPD (MASS2) to have dynamic improved performance in disturbance rejection and set-point tracking as compared with the IPD. In face of great load disturbances and set-point changes, whereas the IPD is quite likely to give degraded performance or even become instable, the FPD (MASS1) and FPD (MASS2), due to their enhanced redundancy in process synthesis and designs, can still do a good job and settle back to their expected steady states. This is why the over-design through the adjustment of the number of stages is interpreted to lead to alleviated interaction between the top, intermediate, and bottom products. 7. Conclusions In the current work, the impact of the over-design through the adjustment of the number of stages of the DWDC has been examined to process dynamics and controllability. With reference to the three example systems studied in the first paper of this series, the DWDCs with and without the over-design have been compared via open- and closed-loop controllability assessments. It has been found that such over-design is quite likely to present beneficial effect to the dynamics and controllability of the DWDC and the reason is attributed to the refined relationship between the prefractionator and main distillation column involved. These outcomes endorse further the necessity of diminishing the blackhole problem in process synthesis and design. The over-design through the adjustment of the number of stages can also serve as an effective strategy to compromise process synthesis and design and process dynamics and controllability. Although the above insight has been acquired from the three example systems studied, it should be viewed as a general interpretation to the design and operation of the DWDC. Acknowledgements The current work is financially supported by The National Natural Science Foundation of China under the grant numbers of 21076015 and 21376018 and The Doctoral Programs Foundation of Ministry of Education of China under the grant number of 20100010110008.
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References [1] E.A. Wolff, S. Skogestad, Operation of integrated three-product (Petlyuk) distillation columns, Ind. Eng. Chem. Res. 34 (1995) 2094–2103. [2] I.J. Halvorsen, S. Skogestad, Optimal operation of Petlyuk distillation: steadystate behavior, J. Process Control 9 (1999) 407–424. [3] G. Dünnebier, C.C. Pantelides, Optimal design of thermally coupled distillation columns, Ind. Eng. Chem. Res. 38 (1999) 162–176. [4] I.J. Halvorsen, S. Skogestad, Shortcut analysis of optimal operation of Petlyuk distillation, Ind. Eng. Chem. Res. 43 (2004) 3994–3999. [5] S. Wang, D.S.H. Wong, Controllability and energy efficiency of high purity divided wall column, Chem. Eng. Sci. 62 (2007) 1010–1025. [6] W. Chen, K. Huang, H. Chen, Design and operation of dividing-wall distillation columns. 1. Diminishing the black-hole problem through over-design, Chem. Eng. Process. 75 (2014) 90–109. [7] N. Asprion, G. Kaibel, Dividing wall columns: fundamentals and recent advances, Chem. Eng. Process. 49 (2010) 139–146. [8] I. Dejanovi c, L. Matijaševi c, Ž. Oluji c, Dividing wall column—a breakthrough towards sustainable distilling, Chem. Eng. Process. 49 (2010) 559–580. [9] Ö. Yildirim, A.A. Kiss, E.Y. Kenig, Dividing wall columns in chemical process industry: a review on current activities, Sep. Purif. Technol. 80 (2011) 403–417. [10] Y.H. Kim, Structural design and operation of a fully thermally coupled distillation column, Chem. Eng. J. 85 (2002) 289–301. [11] A. Jimenez, S. Hernández, F.A. Montoy, Analysis of control properties of conventional and nonconventional distillation sequences, Ind. Eng. Chem. Res. 40 (2001) 3757–3761. [12] F.I. Go’mez-Castro, J.G. Segovia-Hernández, S. Hernández, Dividing wall distillation columns: optimization and control properties, Chem. Eng. Technol. 31 (2008) 1246–1260. [13] F.I. Go’mez-Castro, M.A. Rodríguez-A’ngeles, J.G. Segovia-Hernández, Analysis of dynamic performance for multiple dividing wall distillation columns, Ind. Eng. Chem. Res. 52 (2013) 9922–9929. [14] S. Hernández, A. Jiménez, Controllability analysis of thermally coupled distillation systems, Ind. Eng. Chem. Res. 38 (1999) 3957–3963. [15] M. Serra, A. Espuna, L. Puigjaner, Control and optimization of the divided wall column, Chem. Eng. Process. 38 (1999) 549–562. [16] M. Serra, M. Perrier, A. Espuna, L. Puigjaner, Study of the divided wall column controllability: influence of design and operation, Comput. Chem. Eng. 24 (2000) 901–907. [17] A.A. Kiss, C.S. Bildea, A control perspective on process intensification in dividing-wall columns, Chem. Eng. Process. 50 (2011) 281–292. [18] R.C. Diggelen, A.A. Kiss, A.W. Heemink, Comparison of control strategies for dividing-wall columns, Ind. Eng. Chem. Res. 49 (2010) 288–307. [19] M.I. Abdul Mutalib, R. Smith, Operation and control of dividing wall columns. Part 1: degree of freedom and dynamic simulation, Chem. Eng. Res. Des. 76 (1998) 308–318. [20] H. Ling, W.L. Luyben, New control structure for divided-wall columns, Ind. Eng. Chem. Res. 48 (2009) 6034–6049. [21] H. Ling, W.L. Luyben, Temperature control of the BTX divided-wall column, Ind. Eng. Chem. Res. 49 (2010) 189–203. [22] S. Luan, K. Huang, N. Wu, Operation of dividing-wall distillation columns. 1. A simplified temperature difference control scheme, Ind. Eng. Chem. Res. 52 (2013) 2642–2660. [23] N. Wu, K. Huang, S. Luan, Operation of dividing-wall distillation columns. 2. A double temperature difference control scheme, Ind. Eng. Chem. Res. 52 (2013) 5365–5383. [24] Y. Yuan, K. Huang, Operation of dividing-wall distillation columns. 3. A simplified double temperature difference control scheme, Ind. Eng. Chem. Res. 53 (2014) 15969–15979. [25] Y. Wang, K. Huang, S. Luan, W. Chen, S.J. Wang, D.S.H. Wong, Circumventing the black-hole problem in design and control of dividing-wall distillation columns, Ind. Eng. Chem. Res. 51 (2012) 14771–14792. [26] J. Gao, K. Huang, S. Luan, Y. Jiao, S. Li, Avoiding the black-hole problem by the arrangements of multiple intermediate products to dividing-wall distillation columns, Ind. Eng. Chem. Res. 52 (2013) 4178–4201. [27] Y. Yuan, H. Chen, J. Yu, K. Huang, S. Li, W. Chen, Suppressing the black-hole problem through feed splitting in dividing-wall distillation columns, Ind.Eng. Chem. Res. (Under review). [28] R.R. Rewagad, A.A. Kiss, Dynamic optimization of a dividing-wall column using model predictive control, Chem. Eng. Sci. 68 (2012) 132–142. [29] S.T. Avila, A.J. Gutie’rrez, Control of dividing-wall columns via fuzzy logic, Ind. Eng. Chem. Res. 52 (2013) 7492–7503. [30] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control Analysis and Design, second ed., Wiley, 2001.
Contents lists available at ScienceDirect
Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep
Design and operation of dividing-wall distillation columns. 2. Process dynamics and operation Xin Zong, Kejin Huang * , Yang Yuan, Haisheng Chen, Jieping Yu College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, PR China
A R T I C L E I N F O
A B S T R A C T
Article history: Received 7 February 2015 Accepted 5 March 2015 Available online 10 March 2015
In the first paper of this series, it was demonstrated that the black-hole problem of dividing-wall distillation columns (DWDCs) could effectively be diminished through over-design in terms of careful adjustment of the number of stages. Because the over-design serves to coordinate the relationship between the prefractionator and main distillation column involved, it yields frequently favorable effect to process dynamics and controllability and this represents essentially an additional advantage of such process modifications to diminish the black-hole problem. The three example systems studied in the first paper of this series are again employed to ascertain the anticipation. The DWDCs with and without the over-design are strictly compared in terms of open- and closed-loop controllability studies. It is found that the over-design helps to coordinate the interaction between the prefractionator and main distillation column involved and leads consequently to improved process dynamics and capabilities in disturbance rejection and set-point tracking. The outcome confirms again the necessity of diminishing the black-hole problem in the synthesis and design of the DWDCs. Though the interpretation is gained from the three example systems studied, it should be considered to be of general significance to the design and operation of the DWDCs. ã 2015 Elsevier B.V. All rights reserved.
Keywords: DWDC Over-design Process dynamics Process controllability Dynamic simulation
1. Introduction For a dividing-wall distillation column (DWDC) separating a ternary mixture, it is frequently found infeasible to greatly boost the purity of its intermediate product from its nominal steadystate value even under the extreme operating condition of an infinite boil-up rate or an infinite reflux ratio [1–5]. This represents essentially a serious drawback of the DWDC and can confine considerably process applicability and flexibility. The strong coupling between the prefractionator and main distillation column involved is responsible for such an unfavorable behavior despite the fact that it also represents the main thrust for the enhancement of thermodynamic efficiency of the DWDC. Recently, we referred to the issue as the black-hole problem of the DWDC and pointed out that it could be diminished through over-design in terms of careful adjustment of the number of stages in the prefractionator and main distillation column involved [6]. Because the adjustment of the number of stages affects the flow rates and compositions of the interlinking flows between the prefractionator and main distillation column involved, it can function as an effective method to
* Corresponding author. Tel.: +86 10 64437805; fax: +86 10 64437805. E-mail address: [email protected] (K. Huang). http://dx.doi.org/10.1016/j.cep.2015.03.005 0255-2701/ ã 2015 Elsevier B.V. All rights reserved.
compromise the interaction between these two highly integrated units and serve to diminish the black-hole problem of the DWDC (Fig. 1). Although the over-design philosophy was demonstrated to be effective in enhancing the applicability and flexibility of the DWDC, its impact to process dynamics and controllability has remained yet to be studied so far. Since it is also an important issue that influences the applicability and flexibility of the DWDC, it should be studied in great detail. Since 1990, considerable effort has been dedicated to the studies of the dynamics and control of the DWDCs [7–10]. In particular, open-loop controllability assessment was frequently used to analyze the mass and thermal coupling between the prefractionator and main distillation column involved [11–13]. Hernández and Jiménez conducted detailed comparison between the DWDC and distillation columns with a side rectifier or stripper [14]. They pointed out that although the DWDC needed the lowest utility requirement it exhibited degraded process dynamics and controllability as compared with the distillation columns with a side rectifier or stripper. Serra et al. employed the Morari resiliency index, condition number, relative gain array, and closed-loop disturbance gain as performance indexes to examine the relationship between process synthesis and design and process dynamics and controllability [15,16]. They indicated that the deliberate addition of stages to the DWDC could improve process
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Nomenclature
Symbols Hypothetical component A Avp Vapor pressure constant, kPa B Hypothetical component, benzene, butanol, or bottom product flow rate, mol/s Bvp Vapor pressure constant, kPa C Hypothetical component CC Composition controller CN Condition number d Input signal D Distillate flow rate, mol/s DRGA Dynamic relative gain array DWDC Dividing-wall distillation column E Ethanol F Feed flow rate, mol/s FC Flow rate controller G Process transfer function koff Open-loop gain when the rest of loops are open kon Open-loop gain when the rest of loops are in automatic FPD Final process design IPD Initial process design LC Level controller MASS Maximally achievable steady state MIMO Multiple-input and multiple-output MRI Morari resiliency index n Dimensionality NT Number of stages P Propanol or pressure, Pa PC Pressure controller Q Reboiler heat duty, KW R Reflux flow rate, mol/s SVD Singular value decomposition I Intermediate product flow rate, mol/s T Toluene or temperature, K U Matrix of output singular vectors V Matrix of input singular vectors y Process output signal X o-Xylene Greek Letters Relative volatility Minimum singular value Maximum singular value Matrix of singular values Element of RGA Frequency
a s1 sn S l v
Subscripts A Component index B Component index or bottom product C Component index D Distillate product I Intermediate product E Component index P Component index T Component index X Component index Superscript s Saturation
N1
A
N2
N5 A/B/C
B
N6
N3
N4 C
(a)
A
N1
A/B/C
N 5 N2
B
N6 N3 N4 C
(b) Fig. 1. Sectional stage number and interlinking flows in the Petlyuk distillation column and DWDC (a) Petlyuk distillation column, (b) DWDC.
dynamics and controllability. Regarding the operation of the DWDC, there usually exist two kinds of control configurations, i.e., the three-point and four-point control schemes [1]. In the case of the three-point control scheme, the liquid and vapor split ratios are Table 1 Physical properties, operating conditions, and product specifications of example I. Parameter
Value
Condenser pressure (bar) Stage pressure drop (bar) Feed compositions (mol%)
3.0 0.0068901 33.3 33.3 33.4 27.8 1.0 4:2:1 29,053.7 13.04/3862 12.34/3862 11.65/3862 99 99 99
Feed flow rate (mol/s) Feed thermal condition Relative volatility A:B:C Latent heat of vaporization (kJ/kmol) Vapor pressure constants
Product specifications (mol%)
A B C
A (Avp/Bvp) B (Avp/Bvp) C (Avp/Bvp) A B C
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Table 2 Transfer function matrix of example I.
IPD
XI,
PC 2
R
D
11 1
FC
I
42
CC1
55
LC2
B
(a) PC LC1 2
R
D
14 1 F
26
21 38
FC
CC3
15
A
B
XB,
C
XD,
A
XI, XB,
CC2
41
XD, XI,
FPD (MASS2)
23
37
FPD (MASS1)
CC3
12
19
A
B
XB,C
LC1
F
XD,
B C
D
I
Q
44:250e0:03s 3:51sþ1 51:519e0:09s 8:56sþ1 93:350e0:1s 2:57sþ1
0:283e0:13s 8:69sþ1 8:849e0:01s 0:61sþ1 95:732e0:03s 2:38sþ1
0:131e0:03s 3:74sþ1 0:238e0:03s 0:97sþ1 0:103e0:01s 0:15sþ1
62:890e0:04s 3:97sþ1 31:610e0:07s 20:79sþ1 91:876e0:13s 2:57sþ1
0:290e0:13s 1:49sþ1 10:815e0:01s 0:62sþ1 93:740e0:06s 2:38sþ1
0:175e0:02s 3:49sþ1 0:298e0:03s 0:78sþ1 0:118e0:01s 0:12sþ1
67:750e0:04s 3:34sþ1 25:768e0:05s 29:58sþ1 90:925e0:09s 2:56sþ1
0:310e0:08s 1:6sþ1 12:590e0:01s 0:58sþ1 92:037e0:04s 2:29sþ1
0:175e0:02s 3:09sþ1 0:307e0:03s 0:75sþ1 0:126e0:01s 0:12sþ1
fixed and the top, intermediate, and bottom products are controlled, respectively, with the reflux flow rate (or distillate flow rate and this depends heavily on the detailed reflux ratio), intermediate product flow rate, and reboiler heat duty. A lot of studies confirmed that the control scheme could maintain stable and effective operation of the DWDC [17–19]. In the case of the four-point control scheme, in addition to those controlled and manipulated variables mentioned above, the composition of the heaviest component at the top of the prefractionator and the liquid split ratio are also comprehended and help to tighten the operation of the DWDC. In terms of a DWDC separating a ternary separation of benzene, toluene, and o-xylene, Ling and Luyben demonstrated that the four-point control scheme could achieve self-optimizing control of the DWDC featuring the minimum reboiler heat duty in dynamic operations [20]. Later, various temperature inferential control schemes were reported to avoid the employment of
I CC2
44 45
CC1
58
LC2
B
(b) PC LC1 2
R
D
16 1 F
28
23 40
FC
CC3
17
45 46
I CC2 CC1
59
LC2
B
(c) Fig. 2. Three process designs along with their decentralized control structures (a) IPD, (b) FPD (MASS1), and (c) FPD (MASS2) (example I).
Fig. 3. MRI and CN of the IPD, FPD (MASS1), and FPD (MASS2) (example I).
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X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103 Table 3 DRGA for the IPD, FPD (MASS1), and FPD (MASS2) at 0.0001 and 10000 rad/h (example I).
v = 0.0001 rad/h
v = 10000 rad/h
D
I
Q
D
I
Q
IPD
XD, A XI, B XB, C
0.5831 0.3602 0.0567
0.0027 0.0826 0.9201
0.4196 0.5572 0.0233
0.9032 0.0311 0.0657
0.0006 0.5203 0.4803
0.0974 0.4485 0.4541
FPD (MASS1)
XD, A XI, B XB, C
0.7281 0.2061 0.0658
0.0027 0.1009 0.9018
0.2747 0.6930 0.0324
0.9412 0.0061 0.0527
0.0044 0.5602 0.4442
0.0631 0.4338 0.5031
FPD (MASS2)
XD, A XI, B XB, C
0.7689 0.1582 0.0728
0.0029 0.1169 0.8861
0.2340 0.7249 0.0411
0.9487 0.0026 0.0487
0.0031 0.6022 0.4010
0.0545 0.3953 0.5503
composition measurements and meanwhile still maintain tight control of product qualities [21–24]. Recently, a different fourpoint control scheme aiming to control the ratio between the two impurities in the intermediate product (rather than the composition of the heaviest component at the top of the prefractionator) was also proposed and studied [1,25–27]. In order to cope with the complicated dynamic behaviors of the DWDC, some researcher attempted advanced control algorithms to gain strict control of its product qualities [28,29]. The major purpose of the current work is to elucidate the impact of the over-design through careful adjustment of the number of stages on the dynamics and controllability of the DWDC. Analogous to the first paper of this series, the separations of three ternary mixtures of hypothetical components, A,B, and C, benzene, toluene, and o-xylene, and ethanol, propanol, and butanol, are employed as illustrative examples to show the intricate mechanism. In virtue of open- and closed-loop controllability assessments, strict comparison is made between the DWDCs with and without the over-design through the careful adjustment of the number of stages. Last but not the least, the significance of the over-design is analyzed and some concluding remarks are given in the last section of this article. 2. Methods to evaluate the impact of over-design on the dynamics and controllability of the DWDC
Fig. 4. Diagonal elements of the DRGA of the IPD, FPD (MASS1), and FPD (MASS2) (example I).
Open- and closed-loop controllability assessments are to be used to characterize the effect of the over-design on the dynamics and controllability of the DWDC. In the open-loop controllability assessment, three predictive indexes in the frequency domain, including the Morari resiliency index (MRI), condition number (CN), and dynamic relative gain array (DRGA) are employed.
Table 4 Controller parameters of examples I–III. Controller
Manipulated variable
Controlled variable
KC ()
TI (min)
Example I
CC1 CC2 CC3
Q I D
XB, C XI, B XD, A
3.01 43.68 57.56
66.00 51.48 102.92
Example II
CC1 CC2 CC3
Q I R
XB, X XI, T XD, B
3.22 14.21 6.25
93.72 58.08 105.60
Example III
CC1 CC2 CC3
Q I D
XB, B XI, P XD, E
3.56 54.80 80.68
69.96 50.16 105.60
X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103
93
Fig. 5. Regulatory responses of example I for a 5% step change in the feed compositions of component B (a) composition of component A in the top product, (b) flow rate of the top product, (c) composition of component B in the intermediate product, (d) flow rate of the intermediate product, (e) composition of component C in the bottom product, and (f) heat duty of reboiler.
For a square multiple-input and multiple-output (MIMO) process with a dimensionality of n n, the gain between an output and an input can be determined via the following equation [30]. gain ¼
k yðjwÞ k2 k GðjwÞdðjwÞ k2 ¼ k dðjwÞ k2 k dðjwÞ k2
(1)
where y is the process output, d, the process input, and G, the process transfer function. Note that the process gain depends essentially on the direction of the input signal. To analyze the performance of the process, all the process gains needed be determined along with their direction (i.e., the determination of gain directionality). The singular value decomposition (SVD) of G yields G ¼ U SV T
(2)
where V = (v1, v2, . . . , vn) is the unitary matrix of input singular
vectors, S = diag (s 1, s 2, . . . , s n), the diagonal matrix of singular values, and U = (u1, u2, . . . , un), the unitary matrix of output singular vectors. Eqs. (3)–(5) can then be derived and they indicate that the largest gain for any input direction is equal to the maximum
singular value and the smallest gain the minimum singular value. Gvi ¼ s i ui
1in
s i ¼ k Gvi k2 1 i n sn
k Gv k2 ¼ k Gv k2 s 1 k v k2
(3)
(4)
(5)
The MRI is the minimum singular value s n and could be a measurement of influences by potential problems in a dynamic system. The control system should prefer to have a large MRI over the frequency range of interest. The CN is the ratio of the maximum to minimum singular values (c.f., Eq. (6)). It can reflect control system sensitivity under process parameter uncertainties and model/plant mismatches. The control system should prefer to have a small CN over the frequency range of interest. CN ¼
s1 sn
(6)
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Fig. 6. Servo responses of example I for a 0.0012 step change in the purities of the top, intermediate, and bottom products (a) composition of component A in the top product, (b) flow rate of the top product, (c) composition of component B in the intermediate product, (d) flow rate of the intermediate product, (e) composition of component C in the bottom product, and (f) heat duty of reboiler.
The primary use of the RGA is to evaluate interactions of control loop pairings in multi-loop control systems. Its element li, j is dimensionless and is the ratio between two kinds of process gains (c. f., Eq. (7)). If li, j is equal to one it means that the gain from input j to output i is unaffected by closing the rest control loops. In addition, processes with large RGA elements indicate fundamental control problems because of their sensitivity to input uncertainty. The control system should prefer to have a RGA that is as close to one as possible. The DRGA is a kind of improvement of the RGA with the replacement of steady-state models by transfer function ones.
li;j ¼
koff kon
1 i n; and 1 j n
(7)
where koff is the open-loop gain between the output i and the input j when the rest loops are open, kon, the open-loop gain between the output i and the input j when the rest loops are in automatic. In the closed-loop controllability assessment, disturbance rejection and set-point tracking capabilities are used to inspect process behaviors. While the disturbance rejection ability is the behaviors of a control scheme in the face of internal and external disturbances, the set-point tracking capability the behaviors of a control scheme in the
face of variations in set-points. In terms of these two performance indexes, fast convergence and small fluctuations are certainly two desirable properties of a control system. In what follows, three DWDCs fractionating, respectively, three ternary mixtures of hypothetical components, A, B, and C, benzene, Table 5 Physical properties, operating conditions, and product specifications of example II. Parameter
Value
Condenser pressure (atm) Stage pressure drop (atm) Feed compositions (mol%)
0.37 0.0068 30 30 40 1000 1.0 353 385 419 99 99 99
Feed flow rate (mol/s) Feed thermal condition Normal boiling points (K)
Product specifications (mol%)
Benzene (B) Toluene (T) O-xylene (X)
Benzene (B) Toluene (T) O-xylene (X) Benzene (B) Toluene (T) O-xylene (X)
X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103 Table 6 Transfer function matrix of example II.
PC LC1 2
R
1
XD, XI,
B
T X
CC3
9
23
FC
D
FPD (MASS1)
17
14
IPD
XB,
8
F
CC2
30
XD, XI,
I
31
FPD (MASS2)
XB,
X
XD,
B
XI,
CC1
B
T
XB,
T X
43
LC2
I
Q
0:101e0:14s 7:48sþ1 1:060e0:01s 1:26sþ1 1:580e0:08s 1:94sþ1
0:043e0:04s 1:70sþ1 0:009e0:02s 2:17sþ1 0:038e0:01s 0:85sþ1
1:510e0:02s 1:27sþ1 0:274e0:04s 9:9sþ1 0:933e0:11s 1:66sþ1
0:045e0:12s 5:77sþ1 1:481e0:01s 1:11sþ1 1:311e0:08s 1:66sþ1
0:045e0:03s 1:40sþ1 0:002e0:02s 1:67sþ1 0:036e0:01s 0:78sþ1
1:524e0:02s 1:36sþ1 0:265e0:04s 10:52sþ1 0:951e0:12s 1:65sþ1
0:055e0:17s 6:85sþ1 1:404e0:01s 1:19sþ1 1:360e0:08s 1:71sþ1
0:046e0:03s 1:41sþ1 0:003e0:02s 1:78sþ1 0:036e0:01s 0:79sþ1
toluene, and o-xylene, and ethanol, propanol, and butanol are studied to comprehensively assess the impact of the over-design through the adjustment of stage number on process dynamics and operation.
PC LC1 2
R
9 1
D
19
25
I CC2
32
3. Example I: separation of a ternary mixture of hypothetical components, A, B, and C 3.1. Process description
CC3
10
16
FC
R 1:382e0:02s 1:24sþ1 0:050e0:05s 5:65sþ1 1:074e0:12s 1:61sþ1
B
(a)
F
95
33
The physical properties, operating conditions, and product specifications are listed in Table 1. An equi-molar mixture of the hypothetical components A, B, and C is separated into three products with purities of 99 mol%, respectively. Ideal vapor and
CC1
44
LC2
B
(b) PC LC1 2
R
9 1 F
10
CC3
22
17 26
FC
D
34 35
I CC2 CC1
46
LC2
B
(c) Fig. 7. Three process designs along with their decentralized control structures (a) IPD, (b) FPD (MASS1), and (c) FPD (MASS2) (example II).
Fig. 8. MRI and CN of the IPD, FPD (MASS1), and FPD (MASS2) (example II).
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X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103
liquid phase behaviors are assumed and the vapor–liquid equilibrium relationship can be expressed by Pj ¼ xA;j PsA þ xB;j PsB þ xC;j PsC
yi;j ¼
xi;j Psi;j Pj
1 j NT
(8)
i ¼ A; B; C; and 1 j NT
(9)
The vapor saturation pressure is calculated via lnPsi;j ¼
Avp;i Bvp;i Tj
i ¼ A; B; C; and 1 j NT
(10)
Fig. 2 reproduces the three designs of the DWDCs obtained in the first paper of this series. While the initial process design (IPD) refers to the one obtained directly from the minimization of total annual cost, namely, without the consideration of the over-design through the adjustment of the number of stages (Fig. 2a), the two final process designs (FPDs) the ones with the further consideration of the over-design through the adjustment of the number of stages (Fig. 2b and c). As a result, the two FPDs have higher maximally achievable steady state (MASS) than the IPD (whose value is only 99.25 mol%). The MASS is a performance index that reflects process flexibility and operability and can be estimated through assuming an equal composition of the primary components in the top, intermediate, and bottom products. The two FPDs differ in process configurations that lead to different MASS. While the FPD (MASS1) shown in Fig. 2b is able to reach the purities of 99.5 mol% in the top, intermediate, and bottom products simultaneously, the FPD (MASS2) shown in Fig. 2c is able to reach the purities of 99.6 mol% in these three products. 3.2. Open-loop controllability assessment
Fig. 9. Diagonal elements of the DRGA of the IPD, FPD (MASS1), and FPD (MASS2) (example II).
In terms of step tests, the transfer function matrixes of the three process designs are obtained and tabulated in Table 2. While the top, intermediate, and bottom product qualities, XD, A, XI, B, and XB, C, are taken as the controlled variable, the distillate flow rate D (since the three reflux ratios are 3.269, 3.207, and 3.242, respectively, and all are greater than 3.0), the intermediate product flow rate I, and the reboiler heat duty Q as the manipulated variables. Fig. 3 shows the variation tendencies of the MRI and CN along with the changes of frequencies for the three process designs after appropriate scaling of their input and output variables. Here, unless otherwise stated, the dotted lines represent the behaviors of the IPD, the dashed lines the behaviors of the FPD (MASS1), and the bold lines the behaviors of the FPD (MASS2). While the black
Table 7 DRGA for the IPD, FPD (MASS1), and FPD (MASS2) at 0.0001 and 10000 rad/h (example II).
v = 0.0001 rad/h
v = 10000 rad/h
R
I
Q
R
I
Q
IPD
XD, B XI, T XB, X
3.1955 0.1534 2.0421
0.0313 0.3184 0.6502
2.2268 0.8349 2.3919
1.4521 0.0059 0.4462
0.0009 0.8919 0.1071
0.4531 0.1140 1.3391
FPD (MASS1)
XD, B XI, T XB, X
2.1659 0.4264 1.5924
0.0140 0.4464 0.5395
1.1800 0.1272 2.0528
1.4642 0.0142 0.4784
0.0003 0.9584 0.0413
0.4646 0.0275 1.4371
FPD (MASS2)
XD, B XI, T XB, X
2.1224 0.4350 1.5574
0.0171 0.4233 0.5596
1.1395 0.1417 1.9978
1.5265 0.0162 0.5427
0.0004 0.9515 0.0481
0.5269 0.0323 1.4945
X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103
curves denote the positive responses, the gray curves the negative ones. As can readily be seen, both the FPD (MASS1) and FPD (MASS2) have relatively larger MRI than the IPD, implying that the over-design through the adjustment of the number of stages helps to abate the occurrences of potential problems in dynamic operations (Fig. 3a). In the case of the CN, both the FPD (MASS1) and FPD (MASS2) give smaller values than the IPD, implying that the over-design through the adjustment of number of stages may reduce the process sensitivity to parameter uncertainties (Fig. 3b). As far as the DRGA is concerned, a more complicated situation happens as shown in Fig. 4. While in the low-frequency region, the DRGA suggests a diagonal pairing for the top product and offdiagonal pairings for the intermediate and bottom products, in the intermediate and high frequency region, it suggests diagonal pairings for the top, intermediate, and bottom products. Table 3 compares two DRGAs at frequencies of 0.0001 and 10,000 rad/h, respectively and shows clearly the contradiction. Since the DRGA in the intermediate and high frequency region takes into account loop interactions in dynamic state, it is considered to be more reasonable than the one by the DRGA in the low-frequency region and should be followed in the synthesis and design of decentralized control systems. The over-design through the adjustment of
97
the number of stages makes the l1,1 closer to one in the low and high frequency regions in the FPD (MASS1) and FPD (MASS2) than in the IPD. With regard to the l2,2 and l3,3, they all locate somewhere around 0.5 in the high frequency region and indicate potentially strong interactions between the intermediate and bottom products. The FPD (MASS1) and FPD (MASS2) still present values slightly closer to one than the IPD and the tendencies can clearly be confirmed in Table 3. These outcomes imply again that the FPD (MASS1) and FPD (MASS2) have alleviated interactions between the top, intermediate, and bottom products than the IPD. 3.3. Closed-loop controllability assessment According to the above DRGA analysis, the decentralized control systems are determined and also shown in Fig. 2. For all the three process designs, the qualities of the top, intermediate, and bottom products are controlled, respectively, with the top product flow rate, the intermediate product flow rate, and the heat duty of reboiler. A five minutes dead-time is assumed in each composition sensor. With reference to the IPD, all the composition controllers are tuned in a sequential iteration manner with the embedded Tyreus–Luyben rule and the convergent controller parameters are
Fig. 10. Regulatory responses of example II for a 5% step change in the feed compositions of toluene (a) composition of benzene in the top product, (b) reflux flow rate, (c) composition of toluene in the intermediate product, (d) flow rate of the intermediate product, (e) composition of o-xylene in the bottom product, and (f) heat duty of reboiler.
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listed in Table 4. For the FPD (MASS1) and FPD (MASS2), they are made to share the same controller parameters with the IPD and this serves to facilitate the comparison of closed-loop controllability between the three process designs. Fig. 5 displays the responses of the controlled and manipulated variables of the three process designs in the face of a 5% step change in the feed compositions of component B at the time instant of 0.5 h. The ratio between the feed compositions of components A and C is kept the same as in the nominal operating conditions. All the three process designs are found to be able to return to their designated steady states. The IPD appears to be inferior to the FPD (MASS1) and FPD (MASS2) because of the relatively long settling times and enlarged peak deviations in the intermediate and bottom products. Fig. 6 depicts the responses of the controlled and manipulated variables of the three process designs in the face of a 0.0012 step change in the purities of the top, intermediate, and bottom products. In the case of the IPD, although the top and intermediate products exhibit relatively small overshoots, they give much strong interaction to the bottom product and worsen considerably its dynamic performance
Table 8 Physical properties, operating conditions, and product specifications of example III. Parameter
Value
Condenser pressure (atm) Stage pressure drop (atm) Feed compositions (mol%)
1 0.0068 33.3 33.3 33.4 83.33 1.0 352 370 392 99 99 99
Feed flow rate (mol/s) Feed thermal condition Normal boiling points (K)
Product specifications (mol%)
Ethanol (E) Propanol (P) Butanol (B)
Ethanol (E) Propanol (P) Butanol (B) Ethanol (E) Propanol (P) Butanol (B)
especially in the positive changes of the three set-points. On the contrary, the FPD (MASS1) and FPD (MASS2) can still effectively track the new steady states with an acceptable degree of oscillations in the controlled and manipulated variables.
Fig. 11. Servo responses of example II for a 0.002 step change in the purities of the top, intermediate, and bottom products (a) composition of benzene in the top product, (b) reflux flow rate, (c) composition of toluene in the intermediate product, (d) flow rate of the intermediate product, (e) composition of o-xylene in the bottom product, and (f) heat duty of reboiler.
X. Zong et al. / Chemical Engineering and Processing 91 (2015) 89–103
99
Table 9 Transfer function matrix of example III.
IPD
XI,
PC 2
R
D
10 1
38
FC
FPD (MASS2)
23
I
42
CC1
XD,
E
XI,
P B
I
Q
0:113e0:03s 10:36sþ1 4:136e0:01s 1:04sþ1 30:860e0:03s 3:15sþ1
0:028e0:03s 5:71sþ1 0:064e0:03s 0:76sþ1 0:036e0:01s 0:16sþ1
25:849e0:04s 6:92sþ1 2:699e0:07s 33:46sþ1 28:756e0:13s 4:12sþ1
0:109e0:08s 7:03sþ1 5:458e0:01s 1:08sþ1 29:586e0:02s 3:15sþ1
0:041e0:08s 4:42sþ1 0:086e0:02s 0:75sþ1 0:043e0:01s 0:19sþ1
28:282e0:06s 7:38sþ1 0:867e0:07s 73:98sþ1 28:167e0:13s 4:26sþ1
0:149e0:17s 6:83sþ1 6:579e0:01s 1:12sþ1 28:541e0:07s 3:15sþ1
0:051e0:03s 3:9sþ1 0:101e0:02s 0:69sþ1 0:048e0:01s 0:18sþ1
4.1. Process description
LC2
B
(a) PC LC1 2
R
D
12 1
CC3
13 25
20 39
FC
B
D 16:885e0:06s 4:94sþ1 13:744e0:1s 11:2sþ1 29:888e0:13s 4:12sþ1
4. Example II: separation of a ternary mixture of benzene, toluene, and o-xylene
56
F
E
P
XB,
XB,
CC2
41
B
XD, XI,
CC3
11
19
FPD (MASS1)
E
P
XB,
LC1
F
XD,
Table 5 summarizes the physical properties, operating conditions, and product specifications. A ternary mixture of benzene, toluene, and o-xylene with a composition of 0.3/0.3/0.4 is separated into three products with purities of 99 mol%, respectively. Fig. 7 reproduces the three designs of the DWDCs obtained in the first paper of this series. The IPD has a MASS of only 99.29 mol%. With the over-design through the adjustment of the number of stages, while the FPD (MASS1) shown in Fig. 7b is able to reach the purities of 99.6 mol% in the top, intermediate, and bottom products, the FPD (MASS2) shown in Fig. 7c is able to reach the purities of 99.8 mol% in these three products.
I CC2
43 44
CC1
58
LC2
B
(b) PC LC1 2
R
D
13 1 F
26
24 43
FC
CC3
14
44 45
I CC2 CC1
59
LC2
B
(c) Fig. 12. Three process designs along with their decentralized control structures (a) IPD, (b) FPD (MASS1), and (c) FPD (MASS2) (example III).
Fig. 13. MRI and CN of the IPD, FPD (MASS1), and FPD (MASS2) (example III).
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4.2. Open-loop controllability assessment While the reflux flow rate R (since the reflux ratios are now 2.635, 2.598, and 2.596, respectively, and all are smaller than 3.0), the intermediate product flow rate I, and the reboiler heat duty Q are chosen as the manipulated variables, the top, intermediate, and bottom product compositions, XD, B, XI, T, and XB, X, as the controlled variables. Table 6 lists the transfer function matrixes of the three process designs. The open-loop controllability assessment is then conducted and the changing tendencies of the MRI and CN in the frequency domain are shown in Fig. 8. With the overdesign through the adjustment of the number of stages in the IPD, while the MRI is enhanced, the CN is reduced in the resultant FPD (MASS1) and FPD (MASS2), implying a certain extent of improvement in process dynamics and controllability. The variation tendencies of the diagonal elements of the DRGA in the frequency domain are depicted in Fig. 9 and two detailed DRGAs at frequencies of 0.0001 and 10,000 rad/h are compared in Table 7. Note that the diagonal control structure is recommended for all the three process designs studied. With the over-design through the adjustment of the number of stages in the IPD, while the l2,2 is increased in the full range of frequency domain and made quite close to one in the high frequency region, the l1,1 and l3,3 are decreased in the low frequency region in the resultant FPD (MASS1) and FPD (MASS2) (i.e., made closer to one). This may forebode that the interaction between the top, intermediate, and bottom products has become weaker in the FPD (MASS1) and FPD (MASS2) than in the IPD. Moreover, the interaction affecting the intermediate product is abated more greatly than those affecting the top and bottom products. 4.3. Closed-loop controllability assessment In accordance with the above DRGA analysis, the decentralized control systems are determined for the three process designs studied and also shown in Fig. 7. The controller parameters are listed in Table 4. Fig. 10 displays the responses of the controlled and manipulated variables of the three process designs in the face of a 5% step change in the feed compositions of toluene. The ratio between the feed compositions of benzene and o-xylene is again kept the same as in the nominal operating conditions. It can be noted that the FPD (MASS1) and FPD (MASS2) present slightly better responses than the IPD in the aspect of settling times. Fig. 11 depicts the responses of the controlled and manipulated variables of the three process designs in the face of a 0.002 step change in the purities of the top, intermediate, and bottom products. The FPD (MASS) and FPD (MASS2) show slightly more rapid and smoother transitions to the new steady states than the IPD. Fig. 14. Diagonal elements of the DRGA of the IPD, FPD (MASS1), and FPD (MASS2) (example III).
Table 10 DRGA for the IPD, FPD (MASS1), and FPD (MASS2) at 0.0001 and 10000 rad/h (example III).
v = 0.0001 rad/h D
v = 10000 rad/h I
Q
I
Q
IPD
XD, E XI, P XB, B
0.7071 0.2304 0.0626
0.0032 0.1157 0.8875
0.2961 0.6539 0.0499
D 0.96807 0.0101 0.0219
0.0006 0.5230 0.4776
0.0325 0.4650 0.5005
FPD (MASS1)
XD, E XI, P XB, B
0.8834 0.0405 0.0760
0.0032 0.1540 0.8492
0.1198 0.8055 0.0748
0.9625 0.0008 0.0366
0.0014 0.5338 0.4676
0.0389 0.4654 0.4957
FPD (MASS2)
XD, E XI, P XB, B
0.9213 0.0129 0.0916
0.0044 0.1873 0.8171
0.0831 0.8256 0.0913
0.9571 0.0001 0.0430
0.0019 0.5515 0.4504
0.0447 0.4486 0.5067
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5. Example III: separation of a ternary mixture of ethanol, propanol, and butanol 5.1. Process description Table 8 tabulates the physical properties, operating conditions, and product specifications. An equi-molar ternary mixture of ethanol, propanol, and butanol is separated into three products with purities of 99 mol%, respectively. Fig. 12 reproduces the three designs of the DWDCs obtained in the first paper of this series. The IPD has a MASS of only 99.24 mol%. With the over-design through the adjustment of the number of stages, while the FPD (MASS1) shown in Fig. 12b is able to reach the purities of 99.4 mol% in the top, intermediate, and bottom products, the FPD (MASS2) shown in Fig. 12c is able to reach the purities of 99.6 mol% in these three products. 5.2. Open-loop controllability assessment While the distillate flow rate D (since the reflux ratios are now 3.361, 3.337, and 3.325, and all are greater than 3.0), the intermediate product flow rate I, and the reboiler heat duty Q
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are selected as the manipulated variables, the top, intermediate, and bottom product qualities, XD, E, XI, P, and XB, B, as the controlled variables. Table 9 lists the transfer function matrixes of the three process designs studied. Open-loop controllability assessment is performed again in terms of the MRI, CN, and DRGA in the frequency domain and their variation tendencies are illustrated in Figs. 13 and 14. In particular, the comparison of two detailed DRGAs at frequencies of 0.0001 and 10,000 rad/h is made in Table 10. While the MRI is enhanced the CN reduced in the full range of frequency domain by the over-design through the adjustment of the number of stages, implying a certain degree of improvement in process dynamics and controllability from the IPD to the FPD (MASS1) and FPD (MASS2). In the case of the DRGA, although only fairly small changes observed in the l3,3, the over-design through the adjustment of the number of stages in the IPD increases the l2,2 in the full range of frequency domain (i.e., becomes closer to one) and makes the l1,1 approach to one in the low frequency region. The same tendency can also be identified in Table 10. These outcomes indicate possibly the alleviation of the interaction between the top, intermediate, and bottom products from the IPD to the FPD (MASS1) and FPD (MASS2).
Fig. 15. Regulatory responses of example III for a 5% step change in the feed compositions of propanol (a) composition of ethanol in the top product, (b) flow rate of the top product, (c) composition of propanol in the intermediate product, (d) flow rate of the intermediate product, (e) composition of butanol in the bottom product, and (f) heat duty of reboiler.
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5.3. Closed-loop controllability assessment In accordance with the above DRGA analysis, the decentralized control systems are determined and also shown in Fig. 12. The controller parameters are comprehended in Table 4. Fig. 15 displays the responses of the controlled and manipulated variables of the three process designs in the face of a 5% step change in the feed compositions of propanol. The ratio between the feed compositions of ethanol and butanol is again kept the same as in the nominal operating conditions. Although the FPD (MASS1) and FPD (MASS2) exhibit relatively extended settling times in the top product in comparison with the IPD, they leads to suppressed oscillations and reduced settling times in the intermediate and bottom products. Fig. 16 depicts the responses of the controlled and manipulated variables of the three process designs in the face of a 0.0015 step change in the purities of the top, intermediate, and bottom products. Although slightly better performance (in the aspect of overshoots) are achieved in the top and intermediate products in the IPD than in the FPD (MASS1) and FPD (MASS2), the reverse is true in the bottom product due to the severe interaction encountered and the
dramatically extended settling times especially in face of the positive changes in the set-points of the top, intermediate, and bottom products. 6. Discussion In terms of the three example systems studied in the current work, it has clearly been demonstrated that the over-design through the careful adjustment of the number of stages can frequently yield a favorable influence to the dynamics and controllability of the DWDC. Despite the fact that it is extremely difficult to present a detailed theoretical analysis about their intricate interplay, the compromise of the relationship between the prefractionator and main distillation column involved should be taken here as the primary reason. Since the over-design through the careful adjustment of the number of stages also represent an effective method for the diminution of the black-hole problem as shown in the first paper of this series, it is not difficult to understand the great importance of considering the over-design of the DWDC during process synthesis and design. Under some circumstances, it can even serve as an effective strategy to trade-off
Fig. 16. Servo responses of example III for a 0.0015 step change in the purities of the top, intermediate, and bottom products (a) composition of ethanol in the top product, (b) flow rate of the top product, (c) composition of propanol in the intermediate product, (d) flow rate of the intermediate product, (e) composition of butanol in the bottom product, and (f) heat duty of reboiler.
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process synthesis and design and process dynamics and controllability during the development of the DWDC. It is worthwhile to mention here the fact that the over-design through the careful adjustment of the number of stages leads to rather obvious improvement in the dynamic behaviors of the intermediate product. Both open-loop and closed-loop controllability assessments have corroborated definitely such tendency (c.f., Figs. 4–6 in example I, Figs. 9–11 in example II, and Figs. 14–16 in example III). Since the over-design is originally aimed to enhance the flexibility and operability of the DWDC through the careful adjustment of the number of stages, its direct effect lies definitely on the extension of the rangeability of the intermediate product (as clearly indicated in the first paper of this series), however, a good consistency has been observed between process design and process operation. This represents actually a potential advantage of the over-design philosophy proposed in the current work. The extended rangeability allows the FPD (MASS1) and FPD (MASS2) to have dynamic improved performance in disturbance rejection and set-point tracking as compared with the IPD. In face of great load disturbances and set-point changes, whereas the IPD is quite likely to give degraded performance or even become instable, the FPD (MASS1) and FPD (MASS2), due to their enhanced redundancy in process synthesis and designs, can still do a good job and settle back to their expected steady states. This is why the over-design through the adjustment of the number of stages is interpreted to lead to alleviated interaction between the top, intermediate, and bottom products. 7. Conclusions In the current work, the impact of the over-design through the adjustment of the number of stages of the DWDC has been examined to process dynamics and controllability. With reference to the three example systems studied in the first paper of this series, the DWDCs with and without the over-design have been compared via open- and closed-loop controllability assessments. It has been found that such over-design is quite likely to present beneficial effect to the dynamics and controllability of the DWDC and the reason is attributed to the refined relationship between the prefractionator and main distillation column involved. These outcomes endorse further the necessity of diminishing the blackhole problem in process synthesis and design. The over-design through the adjustment of the number of stages can also serve as an effective strategy to compromise process synthesis and design and process dynamics and controllability. Although the above insight has been acquired from the three example systems studied, it should be viewed as a general interpretation to the design and operation of the DWDC. Acknowledgements The current work is financially supported by The National Natural Science Foundation of China under the grant numbers of 21076015 and 21376018 and The Doctoral Programs Foundation of Ministry of Education of China under the grant number of 20100010110008.
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References [1] E.A. Wolff, S. Skogestad, Operation of integrated three-product (Petlyuk) distillation columns, Ind. Eng. Chem. Res. 34 (1995) 2094–2103. [2] I.J. Halvorsen, S. Skogestad, Optimal operation of Petlyuk distillation: steadystate behavior, J. Process Control 9 (1999) 407–424. [3] G. Dünnebier, C.C. Pantelides, Optimal design of thermally coupled distillation columns, Ind. Eng. Chem. Res. 38 (1999) 162–176. [4] I.J. Halvorsen, S. Skogestad, Shortcut analysis of optimal operation of Petlyuk distillation, Ind. Eng. Chem. Res. 43 (2004) 3994–3999. [5] S. Wang, D.S.H. Wong, Controllability and energy efficiency of high purity divided wall column, Chem. Eng. Sci. 62 (2007) 1010–1025. [6] W. Chen, K. Huang, H. Chen, Design and operation of dividing-wall distillation columns. 1. Diminishing the black-hole problem through over-design, Chem. Eng. Process. 75 (2014) 90–109. [7] N. Asprion, G. Kaibel, Dividing wall columns: fundamentals and recent advances, Chem. Eng. Process. 49 (2010) 139–146. [8] I. Dejanovi c, L. Matijaševi c, Ž. Oluji c, Dividing wall column—a breakthrough towards sustainable distilling, Chem. Eng. Process. 49 (2010) 559–580. [9] Ö. Yildirim, A.A. Kiss, E.Y. Kenig, Dividing wall columns in chemical process industry: a review on current activities, Sep. Purif. Technol. 80 (2011) 403–417. [10] Y.H. Kim, Structural design and operation of a fully thermally coupled distillation column, Chem. Eng. J. 85 (2002) 289–301. [11] A. Jimenez, S. Hernández, F.A. Montoy, Analysis of control properties of conventional and nonconventional distillation sequences, Ind. Eng. Chem. Res. 40 (2001) 3757–3761. [12] F.I. Go’mez-Castro, J.G. Segovia-Hernández, S. Hernández, Dividing wall distillation columns: optimization and control properties, Chem. Eng. Technol. 31 (2008) 1246–1260. [13] F.I. Go’mez-Castro, M.A. Rodríguez-A’ngeles, J.G. Segovia-Hernández, Analysis of dynamic performance for multiple dividing wall distillation columns, Ind. Eng. Chem. Res. 52 (2013) 9922–9929. [14] S. Hernández, A. Jiménez, Controllability analysis of thermally coupled distillation systems, Ind. Eng. Chem. Res. 38 (1999) 3957–3963. [15] M. Serra, A. Espuna, L. Puigjaner, Control and optimization of the divided wall column, Chem. Eng. Process. 38 (1999) 549–562. [16] M. Serra, M. Perrier, A. Espuna, L. Puigjaner, Study of the divided wall column controllability: influence of design and operation, Comput. Chem. Eng. 24 (2000) 901–907. [17] A.A. Kiss, C.S. Bildea, A control perspective on process intensification in dividing-wall columns, Chem. Eng. Process. 50 (2011) 281–292. [18] R.C. Diggelen, A.A. Kiss, A.W. Heemink, Comparison of control strategies for dividing-wall columns, Ind. Eng. Chem. Res. 49 (2010) 288–307. [19] M.I. Abdul Mutalib, R. Smith, Operation and control of dividing wall columns. Part 1: degree of freedom and dynamic simulation, Chem. Eng. Res. Des. 76 (1998) 308–318. [20] H. Ling, W.L. Luyben, New control structure for divided-wall columns, Ind. Eng. Chem. Res. 48 (2009) 6034–6049. [21] H. Ling, W.L. Luyben, Temperature control of the BTX divided-wall column, Ind. Eng. Chem. Res. 49 (2010) 189–203. [22] S. Luan, K. Huang, N. Wu, Operation of dividing-wall distillation columns. 1. A simplified temperature difference control scheme, Ind. Eng. Chem. Res. 52 (2013) 2642–2660. [23] N. Wu, K. Huang, S. Luan, Operation of dividing-wall distillation columns. 2. A double temperature difference control scheme, Ind. Eng. Chem. Res. 52 (2013) 5365–5383. [24] Y. Yuan, K. Huang, Operation of dividing-wall distillation columns. 3. A simplified double temperature difference control scheme, Ind. Eng. Chem. Res. 53 (2014) 15969–15979. [25] Y. Wang, K. Huang, S. Luan, W. Chen, S.J. Wang, D.S.H. Wong, Circumventing the black-hole problem in design and control of dividing-wall distillation columns, Ind. Eng. Chem. Res. 51 (2012) 14771–14792. [26] J. Gao, K. Huang, S. Luan, Y. Jiao, S. Li, Avoiding the black-hole problem by the arrangements of multiple intermediate products to dividing-wall distillation columns, Ind. Eng. Chem. Res. 52 (2013) 4178–4201. [27] Y. Yuan, H. Chen, J. Yu, K. Huang, S. Li, W. Chen, Suppressing the black-hole problem through feed splitting in dividing-wall distillation columns, Ind.Eng. Chem. Res. (Under review). [28] R.R. Rewagad, A.A. Kiss, Dynamic optimization of a dividing-wall column using model predictive control, Chem. Eng. Sci. 68 (2012) 132–142. [29] S.T. Avila, A.J. Gutie’rrez, Control of dividing-wall columns via fuzzy logic, Ind. Eng. Chem. Res. 52 (2013) 7492–7503. [30] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control Analysis and Design, second ed., Wiley, 2001.