Operation of Dividing-Wall Distillation Columns. 2. A Double

Article pubs.acs.org/IECR

Operation of Dividing-Wall Distillation Columns. 2. A Double Temperature Difference Control Scheme Ning Wu, Kejin Huang,* and Shujun Luan College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China ABSTRACT: In the first paper of this series, it has been demonstrated that, although the temperature difference control (TDC) and simple temperature difference control (STDC) schemes render stricter control of the three product qualities of the dividingwall distillation column (DWDC) than the temperature control (TC) scheme, they still exhibit somewhat great steady-state offsets, which deteriorate system performance. To deal with this drawback, we devise a double temperature difference control (DTDC) scheme to facilitate the operation of the DWDC in the current work. The proposed DTDC scheme consists of four double temperature difference control loops, with one in the prefractionator, whose objective is to maintain a certain degree of separation of the fed mixture, and the remainder in the main distillation column, whose objective is to maintain the purities of top, intermediate, and bottom products. In particular, the three temperature measurements in the control loops for the prefractionator and intermediate product are required to be in the each side of the dividing wall, and this makes the DTDC scheme behave in good accordance with the working principle of the DWDC. Separation of two ternary mixtures of hypothetical components A, B, and C and benzene, toluene, and o-xylene are chosen as illustrative examples to evaluate the performance of the DTDC scheme, and thorough comparison is conducted with the currently available TDC scheme. The obtained results indicate that the proposed DTDC scheme is much superior to the TDC scheme in the operation of the DWDC with substantially reduced steady-state discrepancies. This permits the DTDC scheme to be capable of handling severe disturbances as great as 30% in feed compositions with acceptable steady-state deviations in the three product qualities. The great advantages of the DTDC scheme justify the employment of four additional temperature measurements in the operation of the DWDC, compared with the currently available TDC scheme. respectively.18,19 Four temperatures/temperature differences were controlled in the TC/TDC scheme with the reflux flow rate, intermediate product flow rate, reboiler heat duty, and liquid split ratio as the corresponding manipulated variables. Although the TC scheme could well handle feed flow rate disturbances (reinforced with feedforward compensations), it failed to suppress feed composition disturbances greater than 10% in magnitude. Although the TDC scheme could suppress the feed composition disturbances up to 20%, a certain extent of steadystate deviations were found in the three product purities. With regard to the simplified temperature difference control (STDC) scheme devised in the first paper of this series, it was demonstrated to suffer from the same drawback in the maintenance of the three product purities.20 These example studies exemplified the common drawbacks of the temperature inferential control of the DWDC and indicated the great incentives of devising effective strategies to tackle them. It is reasonable to employ more temperature measurements than those in the TDC scheme for the operation of the DWDC, and this leads to the creation of more-complicated temperature inferential control schemes. One of such control configurations is the double temperature difference control (DTDC) scheme. It was used in the operation of conventional distillation columns

1. INTRODUCTION Although temperature inferential control is preferred to direct composition control in the operation of various distillation columns, it is usually not an easy matter to maintain strict control of their product qualities.1−6 Frequently, some specific strategies, e.g., over reflux operation regime and soft sensor techniques, must be employed to guarantee the product qualities and this inevitably gives rise to additional utility consumption in process operation and more complexities in the synthesis and design of the temperature inferential control systems.7−10 In the case of the dividing-wall distillation column (DWDC), it is not exceptional.11−16 Wang and Wong studied the temperature control of a DWDC separating a ternary mixture of ethanol, n-propanol, and n-butanol into three products with purities up to 99.9 mol %, respectively.17 A temperature in the prefractionator and two temperatures in the main distillation column were selected as controlled variables and the corresponding manipulated variables were reflux flow rate, intermediate product flow rate, and reboiler heat duty. Although stable operation was achieved and the three products could return to their desired purity levels in the face of feed flow rate disturbances, large steady-state deviations in the product purities occurred in the face of feed composition disturbances. They ultimately relied on a temperature plus composition cascade control mechanism to deal with this deficiency. Recently, Ling and Luyben compared the performance of the temperature control (TC) and temperature difference control (TDC) schemes in the operation of a DWDC separating a ternary mixture of benzene, toluene, and o-xylene into three products with purities of 99 mol %, © 2013 American Chemical Society

Received: Revised: Accepted: Published: 5365

December 10, 2012 March 4, 2013 March 13, 2013 March 13, 2013 dx.doi.org/10.1021/ie303395d | Ind. Eng. Chem. Res. 2013, 52, 5365−5383

Industrial & Engineering Chemistry Research

Article

and found to present favorable effects to the reduction of steadystate deviations in their top and bottom product qualities.21 In the case of the DWDC, there have been no relevant studies reported yet. The current paper attempts to devise a DTDC scheme for the operation of the DWDC, and the major purpose is to achieve tight control of the three product qualities in the face of changes from feed flow rates and/or compositions. The principle for the synthesis and design of the DTDC scheme is introduced, and two DWDC systems involving the separation of two ternary mixtures of hypothetical components, A, B, and C and benzene, toluene, and o-xylene are employed as illustrative examples to evaluate the performance of the proposed DTDC scheme. Thorough comparison is also conducted with the currently available TDC scheme in the face of feed composition disturbances. The features of the proposed DTDC scheme are further analyzed and some concluding remarks are finally summarized in the last section of the article.

Figure 2. Optimum design of the DWDC and its DTDC scheme (Example I): (a) the DWDC scheme and (b) the DTDC scheme. Figure 1. Double temperature difference control (DTDC) scheme proposed for the operation of the dividing-wall distillation column (DWDC).

2. A DTDC SCHEME PROPOSED FOR THE OPERATION OF THE DWDC For the DWDC, its control objectives are to maintain stable and yet energy efficient on-specification operation in the face of disturbances from feed flow rates and/or compositions. It is a greatly challenging work to devise an effective control structure to fulfill these tasks simultaneously. The DTDC scheme proposed for the operation of the DWDC is sketched in Figure 1. Four double temperature difference control loops are included (i.e., DTDC1 to DTDC4), with one in the rectifying section, one in the stripping section, and two in both sides of the dividing wall. They are in charge of the control of the top, intermediate, bottom product qualities of the main distillation column and the top and/or bottom product qualities of the prefractionator with the distillate flow rate D (or the reflux flow rate R, and this depends closely on the steady-state value of reflux ratio RR), intermediate product flow rate I, reboiler heat duty QR, and liquid split ratio RL as manipulated variables, respectively. Although the DTDC scheme is configured here, in terms of the principle that controlled variables should be paired, respectively, with the manipulated variables that are near them in location, it should finally be determined through RGA analysis and/or detailed closed-loop operation studies. Since 4 double

Table 1. Physical Properties and Design Specifications for the DWDC (Example I) parameter condenser pressure (bar) stage pressure drop (bar) feed compositions (mol %) A B C feed flow rate (kmol/s) feed thermal condition (liquid fraction) relative volatility, A:B:C vapor pressure constants A(Avp/Bvp) B(Avp/Bvp) C(Avp/Bvp) product specifications (mol %) A B C

value 3 6.8901 × 10−3 33.33 33.33 33.34 1.0 1.0 4:2:1 13.04/3862 12.34/3862 11.65/3862 99 99 99 5366

dx.doi.org/10.1021/ie303395d | Ind. Eng. Chem. Res. 2013, 52, 5365−5383


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Figure 3. Sensitivity analysis of the DWDC (Example I): (a) prefractionator, (b) prefractionator, (c) main distillation column, and (d) main distillation column.

Figure 4. Analysis by singular value decomposition (Example I): (a) rectifying section of the main distillation column, (b) intermediate section of the main distillation column, (c) stripping section of the main distillation column, and (d) prefractionator.

temperature differences should be controlled, at least, 12 temperature sensors should be employed. More specifically, 1 sensitive stage temperature and 2 reference stage temperatures should be chosen and measured in each DTDC loop. The sensitive stage temperature works to detect the changes in operation conditions and the two reference stage temperatures serve to remove the influences of the coexistent components as well as the variations in pressures and pressure drops. With reference to the top product, the double temperature difference (Δ2TR) is expressed by eqs 1−3, which involves essentially two temperature differences, ΔTR1 and ΔTR2.

ΔTR1 = (TR − TR r1)

(1)

ΔTR2 = (TR r2 − TR )

(2)

Δ2 TR = ΔTR2 − ΔTR1 = (TR r2 − TR ) − (TR − TR r1) = (TR r1 + TR r2) − 2TR

(3)

While sensitivity analysis is usually performed for selecting the four sensitive stage temperatures (i.e., TR, TI, TS, and TP), singular value decomposition (SVD) technique is conducted for 5367



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Figure 5. Temperature difference and double temperature difference variations in the face of changes in the feed compositions of component A (Example I): (a) rectifying section of the main distillation column, (b) intermediate section of the main distillation column, (c) stripping section of the main distillation column, and (d) prefractionator.

Figure 6. Aspen Dynamics implementation of the DTDC (Example I).

TDC schemes in the number of reference stage temperature measurements. Since the reference stage temperature measurements help to extract more-accurate information about the three product qualities, as well as the operation of the prefractionator, the DTDC scheme can be expected to give better system performance than the TC and TDC schemes in the

selecting the eight reference stage temperatures (i.e., TRr1, TRr2, TIr1, TIr2, TSr1, TSr2, TPr1, and TPr2). If the sensitive stage temperatures and reference stage temperatures are chosen, respectively, in terms of these two methods for the four DTDC loops, the resultant DTDC scheme differs only from the TC and 5368



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the mixture fed to the prefractionator, it also facilitates the rejection of the disturbances from the top and bottom of the prefractionator to the intermediate product. The capability of fulfilling these multiple demands also reflects the merit of using the DTDC scheme to operate the DWDC. For the Δ2TR and Δ 2TS control loops, the arrangement of the three temperature measurements can be relatively flexible. Although they can be located exclusively in the rectifying section and stripping section, respectively, the arrangement of one reference stage temperature measurement (i.e., the bottom and top ones in the Δ2TR and Δ2TS control loops) to the prefractionator sometimes benefits process operation, because it quickly senses changes in the feed flow rate and/or compositions. In what follows, two DWDC systems separating, respectively, two ternary mixtures of hypothetical components A, B, and C and benzene, toluene, and o-xylene are employed as illustrative examples to evaluate the DTDC scheme proposed for the operation of the DWDC. Both the steady-state analysis

Table 2. Controller Parameters for the TDC and DTDC Schemes (Example I) controller

manipulated variable

controlled variable

TC

TI (min)

TDC1 TDC2 TDC3 TDC4 DTDC1 DTDC2 DTDC3 DTDC4

D I QR RL D I QR RL

ΔT10 ΔT35 ΔT46 ΔTP22 Δ2T10 Δ2T35 Δ2T46 Δ2TP22

5.12 0.91 0.31 2.13 0.14 0.15 0.05 0.57

34.32 18.84 10.56 19.8 14.52 17.16 14.52 21.12

face of disturbances from feed flow rate and/or compositions.22,23 For the DTDC scheme to behave in good accordance with the working principle of the DWDC, the three temperature measurements should be located exclusively in each side of the dividing wall in the Δ2TP and Δ2TI control loops, respectively.20 While the strategy helps to keep a certain extent of separation of

Figure 7. Regulatory responses of the DTDC in the face of a ±10% step change in the feed compositions of component A (Example I). 5369



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Figure 8. Comparison between the regulatory responses of the TDC and DTDC schemes for a ±10% step change in the feed compositions of components A, B, and C (Example I): (a) component A, (b) component B, and (c) component C.

Table 4. Integral Absolute Error for a ±10% Step Change in the Feed Compositions (Example I)

Table 3. Relative Errors of the Three Product Qualities (Example I) Relative Error (%)

a

scenario

product

TDCa

DTDCb

+10% ZA

A B C

0.073 −0.211 −0.030

−0.067 −0.049 0.016

−10% ZA

A B C

0.056 0.152 0.028

0.084 0.075 −0.013

+10% ZB

A B C

−0.134 −0.190 −0.045

−10% ZB

A B C

+10% ZC

−10% ZC

Integral Absolute Error, IAE scenario +10% ZA

product A B C

TDC 0.0073 0.0209 0.0025

DTDC 0.0058 0.0050 0.0014

−10% ZA

A B C

0.0072 0.0137 0.0025

0.0061 0.0071 0.0012

0.058 0.017 0.034

+10% ZB

A B C

0.0120 0.0141 0.0039

0.0051 0.0027 0.0031

0.133 0.133 0.058

−0.049 −0.002 −0.035

−10% ZB

A B C

0.0119 0.0104 0.0052

0.0042 0.0012 0.0029

A B C

0.053 0.266 0.092

0.024 0.055 −0.047

+10% ZC

A B C

0.0048 0.0222 0.0079

0.0021 0.0045 0.0037

A B C

−0.100 −0.467 −0.069

−0.022 −0.055 0.051

−10% ZC

A B C

0.0083 0.0381 0.0057

0.0018 0.0041 0.0043

Temperature difference control. control.

b

and dynamic studies are conducted in addition to the thorough comparison with the currently available TDC scheme.

Double temperature difference 5370



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Figure 9. Regulatory responses of the DTDC scheme for a ±30% step change in the feed compositions of component A (Example I).

3. EXAMPLE I: OPERATION OF A DWDC SEPARATING A TERNARY MIXTURE OF HYPOTHETICAL COMPONENTS, A, B, AND C 3.1. Process Design. Ideal vapor and liquid phase behaviors are assumed for the hypothetical ternary mixture separated, and the vapor−liquid equilibrium relationship can be expressed by Pj = xA, jPAs + x B, jPBs + xC, jPCs yi , j =

xi , jPis, j Pj

(i = A, B, C

(1 ≤ j ≤ N ) and

The physical properties and design specifications are listed in Table 1. A DWDC is employed to separate an equimolar mixture of the hypothetical components A, B, and C into three products with purities of 99 mol %, respectively. The DWDC is simulated using the commercial software Aspen Plus, in which it is represented by a stripping column (with only a reboiler), two absorber columns in parallel (with neither reboiler nor condenser), and a rectifying column (with only a condenser).18−20 The number of stages in these four columns, the locations of the feed processed and the intermediate product withdrawn, and the liquid and vapor split ratios are used as decision variables in process synthesis and design. The minimization of total annual cost (TAC), which includes utility cost and discounted capital investment, is taken as an objective function, and the optimum design of the DWDC is generated in terms of a simple search methodology proposed in our early work.24 The sizing relationships and economic basis are also taken from the same work. The resultant DWDC is shown in Figure 2a, which accommodates totally 53 stages, with the

(4)

1 ≤ j ≤ N) (5)

The vapor saturation pressure is calculated via the following equation: ln Pis, j = A vp, i − 1 ≤ j ≤ N)

Bvp, i Tj

(i = A, B, C

and (6) 5371



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Figure 10. Regulatory responses of the TDC scheme for a ±30% step change in the feed compositions of component A (Example I).

and the distillate flow rate D are delineated in Figure 3c, and those between stage temperatures and the intermediate product flow rate I and the liquid split ratio RL are shown in Figure 3d. It is readily understood that the sensitive stages should be stages 10, 35, and 46, since great changes also occur there. 3.3. Selection of the Reference Stages. In terms of the obtained steady-state gain matrixes in the above sensitivity analysis, the matrixes of the gain differences between the chosen sensitive stages and the remaining stages are calculated, and this can easily be accomplished through simple matrix operation. In contrast with the selection of sensitive stages, the reference stages should be chosen to yield as small a magnitude of change as possible over the variations of feed compositions. The gain difference matrixes are then factorized using the SVD technique and the left singular vectors are yielded, from which the stages with the largest absolute values should be chosen as reference stages for the DTDC scheme. Figure 4 delineates the outcomes of the SVD analysis of the DWDC. With reference to Figure 4a, the reference stages should be chosen as stage 16 in the prefractionator and stage 2 in the main distillation column,

rectifying section running from stage 2 to stage 12 and the stripping section from stage 40 to stage 52. The dividing-wall runs from stage 13 down to stage 39. The feed processed is fed onto stage 27 in the prefractionator and the intermediate product is withdrawn from stage 23 on the other side of the dividing wall. The reflux ratio is given as RR = 3.56, and the reboiler heat duty is given as QR = 44.75 MW. The liquid split ratio (RL) is 0.385, and the vapor split ratio (RV) is 0.61. 3.2. Selection of the Sensitive Stages. As mentioned in section 2, sensitivity analysis is used to select sensitive stages, whose temperatures should hold close relationships with the three product qualities.25 Figure 3 presents the results of sensitivity analysis for the DWDC. For the prefractionator, the steady-state gains between stage temperatures and the reboiler heat duty QR and the distillate flow rate D are given in Figure 3a and those between stage temperatures and the intermediate product flow rate I and the liquid split ratio RL are depicted in Figure 3b. The sensitive stage is found to be stage 22, because a great change occurs there. For the main distillation column, the steady-state gains between stage temperatures and the reboiler heat duty QR 5372



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Figure 11. Comparison between the regulatory responses of the TDC and DTDC schemes for a ±30% step change in the feed compositions of components A, B, and C (Example I): (a) component A, (b) component B, and (c) component C.

Table 5. Integral Absolute Error for a ±30% Step Change in the Feed Compositions (Example I)

column as one of the reference stages, the dynamic consideration precludes this alternative, because it is physically far from the sensitive stage 10 in the rectifying section, compared to stage 16 in the prefractionator. Thus, a double temperature difference of

Integral Absolute Error, IAE scenario

product

TDC

DTDC

+30% ZA

A B C

0.0209 2.4798 0.2588

0.0035 0.0298 0.0091

A B C

0.0207 0.03738 0.0100

0.00114 0.02497 0.0079

A B C

0.0386 0.0051 0.0295

0.0197 0.0450 0.0235

A B C

0.0365 0.0521 0.0348

0.0070 0.0156 0.0326

+30% ZC

A B C

0.0096 0.0490 0.0310

0.0085 0.0154 0.0275

−30% ZC

A B C

0.3538 1.3179 0.0231

0.0057 0.0123 0.0117

−30% ZA

+30% ZB

−30% ZB

Δ 2T10 = (TP18 − T10) − (T10 − T2)

is configured for the top product. Similarly, with reference to Figures 4b−4d, the other three double temperature differences can be constructed as follows: for the intermediate product: Δ 2T35 = (T39 − T35) − (T35 − T21)

for the bottom product: Δ 2T46 = (T52 − T46) − (T46 − T40)

for the prefractionator: Δ 2TP22 = (TP30 − TP22) − (TP22 − TP14)

Figure 2b gives the sketch of the resultant DTDC scheme of the DWDC. Note that the three temperature measurements of the Δ2TP22 and Δ2T35 are in each side of the dividing wall, thereby facilitating operation of the DWDC. 3.4. Steady-State Analysis. Under the condition of strictly keeping the three products on their specifications, respectively, the variations of the four temperature differences and four double temperature differences are compared in Figure 5, in the face of changes in the feed compositions of component A. Here, the four temperature differences include ΔT10 = (TP18 − T10) for the top product, ΔT35 = (T35 − T26) for the intermediate product,

because they both display very large absolute values. Although it also appears reasonable to choose stage 22 in the main distillation 5373



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ΔT46 = (T46 − T40) for the bottom product, and ΔTP22 = (TP22 − TP14) for the prefractionator. It is found that the double temperature differences are exclusively much smaller in magnitude than the temperature differences. For example, as is indicated in Figure 5a, while the ΔTR2 between stage 10 of the main distillation column and stage 16 of the prefractionator and the ΔTR1 between stages 2 and 10 of the main distillation column decreases, respectively, by ∼2 and 3 K as the feed compositions of component A increase from 26.67 mol % to 39.96 mol %, the Δ2T10 between ΔTR1 and ΔTR2 reduces only by ∼0.8 K, which is much smaller than either ΔTR1 or ΔTR2. With reference to Figures 5b−5d, similar tendencies can also be identified for the remaining corresponding temperature differences and double temperature differences. These outcomes imply that the DTDC scheme can be expected to give tighter control of the DWDC than the TDC scheme. 3.5. Closed-Loop Evaluations. Since the TDC scheme was already found to be more effective than the TC and STDC schemes in the operation of the DWDC, the comparison between the TDC and DTDC schemes are to be mainly examined in the current work.18−20 Dynamic simulations of the DWDC, controlled, respectively, with the TDC and DTDC schemes are carried out with the commercial software Aspen Dynamics, and the representation of the DWDC, along with the DTDC scheme, is shown in Figure 6. In consideration of the dynamic relationships between the double temperature differences and the candidate manipulated variables, ΔT10/Δ2T10 is paired with the distillate flow rate D (since RR = 3.56, which is greater than 3.0), ΔT35/Δ2T35 is paired with the intermediate product flow rate I, ΔT46/Δ2T46 is paired with the reboiler heat duty QR, and ΔTP22/Δ2TP22 is paired with the liquid split ratio RL in the TDC/DTDC scheme. All temperature sensors are assumed to have a 1-min dead-time element. Both of the TDC and DTDC schemes are tuned using a sequential method proposed by Ling and Luyben.19 More specifically, the bottom product control loop is tuned first, because it affects all of the other controlled variables. Then, the top product control loop is

Figure 12. Optimum design of the DWDC and its DTDC scheme (Example II): (a) DWDC and (b) DTDC scheme.

Figure 13. Sensitivity analysis of the DWDC (Example II): (a) prefractionator, (b) prefractionator, (c) main distillation column, and (d) main distillation column. 5374



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Figure 14. Analysis by singular value decomposition (Example II): (a) rectifying section of the main distillation column, (b) intermediate section of the main distillation column, (c) stripping section of the main distillation column, and (d) prefractionator.

Figure 15. Aspen Dynamics implementation of DTDC (Example II).

tuned in terms of the built-in Tyreus−Luyben rule; the outcomes are shown in Table 2. Figure 7 gives the closed-loop responses of the DWDC, controlled with the DTDC scheme, in the face of a ±10% step change in the feed compositions of component A, respectively. The ratio between the feed compositions of components B and C

tuned with the bottom product control loop on automatic. Third, the intermediate product control loop is tuned with both the bottom and top product control loops on automatic. Finally, the control loop of the prefractionator is tuned with the other three control loops on automatic. Proportional-plus-integral (PI) controllers are employed in these four TDC/DTDC loops and 5375



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Figure 16. Temperature difference and double temperature difference variations in the face of changes in the feed compositions of benzene (Example II): (a) rectifying section of the main distillation column, (b) intermediate section of the main distillation column, (c) stripping section of the main distillation column, and (d) prefractionator.

Table 6. Physical Properties and Design Specifications for the DWDC (Example II) parameter

Table 8. Relative Errors of the Three Product Qualities (Example II)

value

condenser pressure (atm) stage pressure drop (atm) feed compositions (mol %) benzene toluene o-xylene feed flow rate (kmol/s) feed thermal condition relative volatility, B:T:X normal boiling-points (K) benzene toluene o-xylene product specifications (mol %) benzene toluene o-xylene

Relative Error (%)

0.37 6.90 × 10−3 30 30 40 1.0 1.0 7.1: 2.2:1 353 385 419 99 99 99

Table 7. Controller Parameters of the TDC and DTDC Schemes (Example II) controller

manipulated variable

controlled variable

TC

TI (min)

TDC1 TDC2 TDC3 TDC4 DTDC1 DTDC2 DTDC3 DTDC4

R I QR RL R I QR RL

ΔT7 ΔT27 ΔT39 ΔTP15 Δ2T7 Δ2T27 Δ2T39 Δ2TP15

0.18 0.40 2.15 0.76 0.15 0.40 0.10 0.27

10.56 10.56 11.88 22.44 14.52 17.16 10.56 22.44

scenario

product

TDC

DTDC

+20% ZB

B T X

0.183 0.003 −0.055

−0.048 0.044 0.026

−20% ZB

B T X

−0.183 0.009 0.033

0.057 0.039 −0.019

+20% ZT

B T X

−0.189 0.117 −0.144

0.041 −0.093 0.059

−20% ZT

B T X

0.194 −0.145 0.221

−0.027 0.095 −0.034

+20% ZX

B T X

−0.019 −0.183 0.463

0.027 0.022 −0.038

−20% ZX

B T X

−0.048 0.137 −0.186

−0.011 0.037 0.117

deviations in the four controlled variables. Figure 8 presents the comparisons between the DWDC, controlled, respectively, with the TDC and DTDC schemes, in the face of a ±10% step change in the feed compositions of the three components A, B, and C. Throughout this work, the solid lines are assumed to represent the responses of the DTDC scheme and the dashed lines the responses of the TDC scheme. Although somewhat similar settling times were displayed, the DTDC scheme leads to smaller

has been kept the same as in the nominal operating conditions. Fairly rapid and smooth regulations of the four double temperature differences are achieved, leaving rather small peak 5376



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Figure 17. Regulatory responses of the DTDC scheme in the face of a ±20% step change in the feed compositions of benzene (Example II).

steady-state deviations in the three product qualities than the TDC scheme. Table 3 summarizes the relative steady-state errors of the three product purities. It can readily be noted that, although the DTDC scheme keeps the three product compositions within the range of 99% ± 0.467%, the TDC scheme remains within the range of 99% ± 0.084%. In Table 4, integral absolute error (IAE) is tabulated and the DTDC scheme is found to give a much-reduced IAE, compared to the TDC scheme. Figure 9 gives the closed-loop responses of the DWDC, controlled with the DTDC scheme, in the face of a ±30% step change in the feed compositions of component A, respectively. The ratio between the feed compositions of components B and C has still been kept the same as in the nominal operating conditions. Fairly rapid and smooth regulations of the double temperature differences are achieved again, leaving small peak deviations in the four controlled variables. Figure 10 gives the closed-loop responses of the DWDC, controlled with the TDC scheme, in the face of a ±30% step change in the feed

compositions of component A. The ratio between the feed compositions of components B and C has been kept the same as in the nominal operating conditions. The temperature difference ΔT35 cannot return to its setpoint, because of the strong fluctuations in the controlled and manipulated variables in the case of the positive disturbance in the feed compositions of component A. Figure 11 details the comparisons between the DWDC, controlled, respectively, with the TDC and DTDC schemes, in the face of a ±30% step change in the feed compositions of components A, B, and C. While the DTDC scheme maintains stable operation of the DWDC, the TDC scheme fails to suppress the disturbances in the cases of positive change in the feed compositions of component A and negative change in the feed compositions of component C, presenting great decreases, respectively, in the intermediate and bottom product compositions in the former situation and in the top and intermediate product compositions in the latter situation. Under the other circumstances, the DTDC scheme still exhibits smaller 5377



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Figure 18. Comparison between the regulatory responses of the TDC and DTDC schemes for a ±20% step change in the feed compositions of benzene, toluene, and o-xylene (Example II): (a) benzene, (b) toluene, and (c) o-xylene.

Table 9. IAE for a ±20% Step Change in the Feed Compositions (Example II)

Table 10. Integral Absolute Error for a ±30% Step Change in the Feed Compositions (Example II)

Integral Absolute Error, IAE

Integral Absolute Error, IAE

scenario

product

TDC

DTDC

scenario

product

TDC

DTDC

+20% ZB

B T X

0.0144 0.0053 0.0035

0.0037 0.0046 0.0019

+30% ZB

B T X

0.0181 0.0126 0.0033

0.0047 0.0114 0.0026

−20% ZB

B T X

0.0144 0.0043 0.0021

0.0043 0.0045 0.0015

−30% ZB

B T X

0.0205 0.0066 0.0031

0.0059 0.0092 0.0019

+20% ZT

B T X

0.0142 0.0119 0.0100

0.0036 0.0084 0.0038

+30% ZT

B T X

0.0329 0.0756 0.0177

0.0084 0.0572 0.0064

−20% ZT

B T X

0.0138 0.0141 0.0148

0.0025 0.0076 0.0023

−30% ZT

B T X

0.0180 0.0432 0.0252

0.0030 0.0069 0.0024

+20% ZX

B T X

0.0021 0.0156 0.0297

0.0020 0.0038 0.0022

+30% ZX

B T X

2.2251 4.0456 0.0532

0.0033 0.0090 0.0010

−20% ZX

B T X

0.0038 0.0119 0.0126

0.0012 0.0056 0.0074

−30% ZX

B T X

0.0158 0.0277 0.0153

0.0024 0.0166 0.0106

steady-state deviations in the three product qualities than the TDC scheme. Table 5 summarizes the IAE for these

circumstances, and the DTDC scheme is still found to lead to a much smaller IAE than the TDC scheme. 5378



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Figure 19. Regulatory responses of the DTDC scheme for a ±30% step change in the feed compositions of o-xylene (Example II).

4. EXAMPLE II: OPERATION OF A DWDC SEPARATING A TERNARY MIXTURE OF BENZENE, TOLUENE, AND o-XYLENE 4.1. Process Design. A ternary mixture of benzene (B), toluene (T), and o-xylene (X) is to be separated with a DWDC; the physical properties and design specifications are listed in Table 4. The sizing relationships and economic basis are taken from Wang et al.24 The steady-state simulation of the DWDC is still conducted with the commercial software Aspen Plus, in which the Chao-Seader physical model is used to describe the thermodynamic properties of the ternary mixture separated. The minimization of the TAC is chosen as an objective function and the optimum design of the DWDC is generated with a grid-search method, which is depicted in Figure 12a. The DWDC contains a total of 46 stages, with its rectifying section from stage 2 to stage 9 and the stripping section from stage 34 to stage 45. The dividing-wall runs between stage 10 and stage 33 with a liquid split ratio of RL = 0.353 and a vapor split ratio of RV = 0.627. The reflux ratio is

given as RR = 2.84 and the reboiler heat duty is given as QR = 35.69 MW. 4.2. Selection of the Sensitive and Reference Stages. The sensitive and reference stage temperatures are selected in a similar manner as done in Example I. In terms of the outcomes of sensitivity analysis shown in Figure 13, the sensitive stages are chosen to be stages 7, 27, and 39 in the main distillation column and stage 15 in the prefractionator. Figure 14 gives the results of the SVD analysis and this leads to the selection of four double temperature differences: Δ 2T7 = (T7 − T2) − (TP12 − T7) Δ 2T27 = (T27 − T19) − (T32 − T27) Δ 2T39 = (T45 − T39) − (T39 − T34) Δ 2TP15 = (TP15 − TP11) − (TP24 − TP15) 5379



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Figure 20. Regulatory responses of the TDC scheme for a ±30% step change in the feed compositions of o-xylene (Example II).

Note again that the three temperature measurements of the Δ2TP15 and Δ2T27 are located in the each side of the dividing wall, thereby facilitating the operation of the DWDC. With the consideration of the dynamic relationships between the double temperature differences and the candidate manipulated variables, Δ2T7 is paired with the reflux flow rate R (since RR = 2.84 is less than 3.0), Δ2T27 is paired with the intermediate product flow rate I, Δ2T39 is paired with the reboiler heat duty QR, and Δ2TP15 is paired with the liquid split ratio RL. In terms of the commercial software Aspen Dynamics, the implementation of the DWDC, along with the DTDC scheme, is shown in Figure 15. 4.3. Steady-State Analysis. With strictly keeping the three products on their specifications, respectively, the variations of the temperature differences and double temperature differences in the face of changes in the feed compositions of benzene are illustrated in Figure 16. Here, the four temperature differences are defined as

ΔT7 = (TP12 − T7) ΔT27 = (T32 − T27) ΔT39 = (T39 − T34) ΔTP15 = (TP15 − TP11) The double temperature differences show again much smaller variations than the corresponding temperature differences, and this implies that the DTDC scheme is likely to provide tighter control of the DWDC than the TDC scheme. 4.4. Closed-Loop Evaluations. All temperature sensors are assumed to have a 1-min dead-time element. The tuning of the TDC and DTDC schemes is based on the same method adopted in Example I, and the resultant outcomes are listed in Table 5. Figure 17 displays the closed-loop responses of the DWDC, controlled with the DTDC scheme, in the face of a ±20% step change in the feed compositions of benzene, respectively. (Physical properties and design specifications for the DWDC for 5380



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Figure 21. Comparison between the regulatory responses of the TDC and DTDC schemes for a ±30% step change in the feed compositions of o-xylene (Example II): (a) benzene, (b) toluene, and (c) o-xylene.

note that the temperature differences ΔT27, ΔT39, and ΔTP15 can no longer return to their setpoints in the case of the positive disturbance in the feed compositions of o-xylene, indicating that the TDC scheme fails to do a good job in this situation. Figure 21 details the comparisons of the closed-loop responses of the DWDC, controlled with the TDC and DTDC schemes, respectively, in the face of a ±30% step change in the feed compositions of benzene, toluene, and o-xylene. The outcomes are in good accordance with the observations made from Figure 20, that a drift in the three product qualities occurs in the case of the positive step change in the feed compositions of o-xylene. Under other circumstances, the TDC scheme does not appear to be as effective in the product quality control as the DTDC scheme and leaves much-larger peak and steady-state deviations. In Table 10, the IAE is also calculated and listed for these circumstances, which shows a good accordance with the above observations.

Example II are given in Table 6.) The ratio between the feed compositions of toluene and o-xylene has been kept the same as in the nominal operating conditions. Rapid and smooth regulations of the double temperature differences are achieved, leaving rather small peak deviations in the four controlled variables. In Figure 18, the closed-loop responses of the DWDC, controlled with the TDC and DTDC schemes, respectively, are compared in the face of a ±20% step change in the feed compositions of benzene, toluene, and o-xylene. (Controller parameters of the TDC and DTDC schemes for Example II are given in Table 7.) It can readily be found that the DTDC scheme presents tighter control of the three product qualities than the TDC scheme. The relative steady-state errors of the three product purities are summarized in Table 8. While the DTDC scheme keeps the three product compositions within the range of 99% ± 0.463%, the TDC scheme within the range of 99% ± 0.117%. The IAE is calculated and listed in Table 9 for these circumstances, which justifies the above observations. Figure 19 displays the closed-loop responses of the DWDC, controlled with the DTDC scheme, in the face of a ±30% step change in the feed compositions of o-xylene, respectively. The ratio between the feed compositions of benzene and toluene has been kept the same as in the nominal operating conditions. It can be found that fairly rapid and smooth regulations of the double temperature differences are achieved, leaving small peak deviations in the four controlled variables. Figure 20 gives the regulatory responses of the DWDC, controlled with the TDC scheme, in the face of a ±30% step change in the feed compositions of o-xylene, respectively. The ratio between the feed compositions of benzene and toluene has been kept the same as in the nominal operating conditions. One can readily

5. DISCUSSION In terms of the two example systems studied in the current work, the DTDC scheme has been demonstrated to be generally advantageous over the currently available TDC scheme in the operation of the DWDC. Not only have the steady-state deviations in the three product qualities been reduced, but the capabilities of suppressing the feed composition disturbances also have been enhanced. The reason undoubtedly lies in the more-accurate inferences of the changes in the three product qualities, as well as in the operation of the prefractionator with the additional four reference temperature measurements added, compared to the TDC scheme. Although the DTDC scheme 5381



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paper. Therefore, one of our future research tasks will be centered on the studies of the SDTDC in the operation of the DWDC.

incurs relatively more instrumentation cost and needs more engineering efforts than the TDC scheme in control system synthesis and design, the great improvement acquired in the product quality control justifies its application in the operation of the DWDC. The great improvement in operation robustness acquired by the application of the DTDC scheme to the DWDC is noteworthy here. Because of the availability of three temperature measurements in each side of the dividing wall, the requirement on the locations of these temperature measurements is, to a certain extent, relaxed, which permits the DTDC scheme to be more robust than the TC, DTC, and STDC schemes in the face of the changes in operating conditions, as well as changes in the thermodynamic properties of the ternary mixtures separated. The capability of the DTDC scheme to handle ±30% feed composition disturbances in the two example systems studied in the current work is a good demonstration to this interpretation. Although not shown here, the DTDC scheme is also found to yield better performance than the TC, DTC, and STDC schemes in the separation of ternary mixtures with widely different ease of separation indexes and the outcomes serve again to be a concrete corroboration to this interpretation. The relatively high robustness of the DTDC scheme, on the other hand, somewhat diminishes the complexities involved in control system development.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +86 10 64434801. Fax: +86 10 64437805. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The current work is financially supported by The National Science Foundation of China (under Grant No. 21076015) and The Doctoral Programs Foundation of Ministry of Education of China (under Grant No. 20100010110008).

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6. CONCLUSION In this work, a double temperature difference control (DTDC) scheme has been proposed for the operation of the dividing-wall distillation column (DWDC). It involves four DTDC loops, with one being in the prefractionator, for the purpose of maintaining a certain degree of separation of the fed mixture, and the remainder being in the main distillation column, for the purpose of maintaining the purities of the top, intermediate, and bottom products. The three temperature measurements in the control loops for the prefractionator and intermediate product should be in each side of the dividing wall, and this ensures that the DTDC scheme behaves in good accordance with the working principle of the DWDC. In terms of two DWDC systems separating, respectively, two ternary mixtures of hypothetical components, A, B, and C and benzene, toluene, and o-xylene, it has been demonstrated that the DTDC scheme behaves much better than the TDC scheme in guaranteeing process stability and keeping the purities of the three products close to their desired specifications. The former is also characterized by its greater capability to handle feed composition disturbances than the latter. These advantages are considered to stem from the improved predictions on the top, intermediate, and bottom product qualities of the main distillation column, as well as the composition profiles of the prefractionator. They justify the employment of four additional temperature sensors in the DTDC scheme, compared to the TDC scheme. Since great improvements in system performance can be secured with the application of the DTDC strategy in the operation of the DWDC, it reminds us of employing the same strategy to facilitate the STDC scheme proposed in the first paper of this series.17 With the replacement of the two temperature difference control (TDC) loops by the two DTDC loops in both sides of the dividing wall, a simplified double temperature difference control (SDTDC) scheme is thus generated. The SDTDC scheme is expected to give better system performance than the STDC and TDC schemes and behavior comparable to the DTDC scheme addressed in the current

NOTATION A, B, C = hypothetical components Avp = vapor pressure constant, Pa Bvp = vapor pressure constant, Pa K COMP = compressor D = distillate flow rate, kmol/s DTDC = double temperature difference control DWDC = dividing-wall distillation column F = feed flow rate, kmol/s FC = flow rate controller i = component index I = intermediate product flow rate, kmol/s j = stage index KC = proportional gain L = liquid flow rate, kmol/s LC = level controller N = number of stages P = pressure, bar PC = pressure controller QC = heat duty of condenser, MW QR = heat duty of reboiler, MW q = feed thermal condition R = reflux flow rate, kmol/s RC = ratio controller RL = liquid split ratio RV = vapor split ratio RR = reflux ratio SVD = singular value decomposition T = temperature, K TC = temperature control TDC = temperature difference control ΔT = temperature difference, K Δ2T = double temperature difference, K TAC = total annual cost, $ TI = integral time, s x = liquid composition y = vapor composition Z = feed composition

Subscripts

A = component index B = component index or bottom product C = component index D = distillate product I = intermediate product or intermediate sections P = prefractionator or stage index of the prefractionator 5382



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(22) Halvorsen, I. J.; Skogestad, S. Optimal Operation of Petlyuk Distillation: Steady-State Behavior. J. Process Control 1999, 9, 407. (23) Halvorsen, I. J.; Skogestad, S. Shortcut Analysis of Optimal Operation of Petlyuk Distillation. Ind. Eng. Chem. Res. 2004, 43, 3994. (24) Wang, P.; Chen, H.; Wang, Y.; Zhang, L.; Huang, K.; Wang, S. J. A Simple Algorithm for the Design of Fully Thermally Coupled Distillation Columns (FTCDC). Chem. Eng. Commun. 2011, 199, 608. (25) Luyben, W. L. Evaluation of Criteria for Selecting Temperature Control Trays in Distillation Columns. J. Process Control 2006, 16, 115.

r = reference stage R = rectifying section S = stripping section i = stage index of the main distillation column Superscripts

s = saturation

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Operation of Dividing-Wall Distillation Columns. 2. A Double

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