Distillate Cascade Composition Control Using a Two-Temperature

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Ind. Eng. Chem. Res. 2006, 45, 6828-6841

RESEARCH NOTES Distillate Cascade Composition Control Using a Two-Temperature Measurement Secondary Component Rafael Urrea, Eduardo Castellanos-Sahagun, Jesus Alvarez, and Jose Alvarez-Ramirez* DiVision de Ciencias Basicas e Ingenieria, UniVersidad Autonoma Metropolitana-Iztapalapa, Mexico D.F., 09340 Mexico

The intent of this note is to show that the incorporation of a temperature measurement in the stripping section improves the performance of distillate cascade controller. The proposed controller regulates the distillate composition by manipulating the reflux flow rate, based on a primary component, which is driven by the distillate composition measurement, and a secondary component, which is driven by two temperature measurements located at the most sensitive trays of the stripping and rectifying sections. Compared to the standard single-temperature cascade control scheme, the proposed two-temperature cascade controller has better behavior, because of the improvement of the feed-forward-like disturbance rejection capability of the secondary control component. Numerical simulations are used to illustrate the performance of the control scheme in the face of feed flow and composition disturbances. 1. Introduction

Table 1. Flow Rates and Compositions for the Characterized Distillation Column

Chemical and petrochemical industries make extensive use of distillation processes that require the regulation of product (distillate and/or bottom) compositions for efficient operation (i.e., minimum energy consumption).1,2 The composition control problem has been extensively studied with linear3,4 and nonlinear control schemes.5 In high-purity columns with poor input-tocomposition output sensitivity and measurement delays, these control schemes may exhibit sluggish responses, because the control actions happen after the entire composition profile has been upset by disturbances. To overcome this problem, cascade composition-to-temperature control schemes have been applied,6 according to the rationale that the fast temperature secondary controller performs most of the disturbance rejection task, and the slow primary composition controller is dedicated to regulate the product compositions. The cascade control scheme can be reasonably handled with conventional linear control designs7 (e.g., internal model control (IMC)) and temperature sensor locations.8 In the case of distillate composition control, a primary (delayed) composition loop is combined with a secondary temperature controller, which is driven by a temperature measurement located at a sufficiently sensitive tray of the rectifying section. Given that composition measurements generally are delayed, because of constraints in the response of measurement devices and internal liquid transport, a feedback loop based only on composition measurements can suffer from poor performance and even instabilities. To alleviate this situation, compositiontemperature cascade control schemes have been proposed. The basic idea is that feed flow rate or composition disturbance effects can be detected faster by the temperature loop, so that anticipated corrective actions can be taken to counteract its adverse effects. Criteria for the selection of the temperature measurement location are well-established,8 relying on maximizing the loop sensitivity by taking the temperature at the stage with maximum steady-state gradient. However, intrinsic delays that are due to liquid transport through the distillation column * To whom correspondence should be addressed. Tel.: 52 55 5724 4600 or 4648. Fax: 52 55 5804 4900. E-mail address: [email protected].

parameter

value

liquid feed flow rate vapor feed flow rate feed temperature number of components feed composition, xF distillate composition, xD reflux flow rate distillate flow rate reboiler heat input

800.0 lb-mol/h 200.0 lb-mol/h 120.0 °F 5 0.6 mol fraction 0.9738 mol fraction 400.0 lb-mol/h 200.0 lb-mol/h 5.0 × 106 kcal/h

can limit the performance of the cascade control scheme. For instance, feed flow rate disturbances in columns with liquidphase feed travel first through the stripping section, to affect the distillate composition via vapor transport eventually. Even if a temperature loop is present, the liquid flow rate disturbance is detected only after a certain (possibly time-varying) delay, because of internal liquid transport along the tray sequence. These considerations suggest the possibility of improving the cascade control behavior by incorporating a temperature measurement in the stripping section, or equivalently, by improving the feed-forward-like cascade control scheme disturbance rejection capability of the fast secondary control component. In other words, a secondary controller that is driven by two temperature measurements instead of one should enable better detection and compensation of feed disturbances in the secondary control loop. The idea of using two temperatures in a single-composition control scheme is consistent with the distillation estimation and control approaches, based on nonlinear wave models,9 and old average temperature-based control schemes.10 The two-temperature single-composition control problem has not been addressed with the wave model approach, and the corresponding pursuit should lead to a complex, nonlinear, and highly model-dependent control scheme that would increase reliability and implementation cost concerns among practitioners. On the other hand, there are few reported studies of the application-oriented average-temperature single-composition control schemes, and there is a lack of systematic procedures to design these control schemes. In this note, our contribution consists of presenting an improved conventional-like distillation column control scheme

10.1021/ie060421a CCC: $33.50 © 2006 American Chemical Society Published on Web 08/29/2006

Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006 6829

Figure 1. Responses of product mole fraction and temperature to disturbances in the feed flow rate.

that exploits the (inexpensive) information provided by an additional temperature sensor located in the stripping section. The adjustable-weight average of two temperatures, one per section, is regarded as an output to be regulated by a singleinput-single-output (SISO) secondary loop, with an average temperature set point generated by a primary composition controller. Thus, the consideration of an average output temperature yields a standard composition-to-temperature SISO cascade control design with the average temperature weight parameter as

an additional tuning parameter. The result is a two-temperaturedriven conventional cascade control scheme whose control signal amounts to the adjustable weighted linear combination of two individually designed standard cascade controllers. The proposed technique is tested through numerical simulations, showing control behavior benefits from the application of a cascade scheme with a two-temperature secondary component.

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Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006

Figure 2. Responses of product mole fraction and temperature to disturbances in the feed composition.

2. Distillation Column Dynamics In this section, the response of the column temperature to feed flow rate disturbances is characterized. To this end, let us consider a five-component distillation column,11 with 15 trays (plus reboiler and condenser), with feed at the fifth tray, trays numbered from bottom to top, and 4 (or 10) trays in the stripping (or rectifying) section. (See Table 1.) Equimolar flow cannot be assumed; therefore, the model includes an energy balance per tray. A bubble-point subroutine is provided with known

liquid compositions and pressure; a linear bottom-to-top pressure drop is assumed, coolant and steam dynamics are assumed to be negligible in the condenser and reboiler, and liquid hydraulics are described via the Francis weir formula. The reflux flow rate (R) is the manipulated variable used to regulate the product distillate composition. Following Tolliver and McCune’s guidelines,8 temperature measurements were selected at trays with the largest tray-to tray gradient, yielding the third and thirteenth trays for the stripping

Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006 6831 Table 2. Input/Output (I/O) Linear Model Parameters parameter

steady-state gain

time constant

composition temperature rectifying section stripping section

3.2714 × 10-4 mol fraction h/lb-mol

0.282 h

-1.39 × 10-1 °F h/lb-mol -2.7 × 10-2 °F h/lb-mol

0.2549 h 0.0055 h

and rectifying sections, respectively. In the sequel, Ts (or Tr) will denote the stripping (or rectifying) temperature in the third (or thirteenth) tray. Figure 1 shows the responses of the distillate composition (xD) and the temperature (Ts and Tr) to a (5% step disturbance in the feed flow rate. As observed in Figure 1, (i) the product composition xD has a response time constant of ∼1 h, reflecting the fact that the distillate composition is formed through a slow rectifying process that is induced by internal reflux, and (ii) the stripping (or rectifying) temperature response time constant is ∼0.075 h (or 0.2 h), meaning that the stripping temperature quickly reflects the effect of disturbances. Temperature dynamics detect the effects of the disturbance faster. Thus, the stripping temperature reacts ∼3 times faster than the enriching temperature, because the feed flow rate disturbance is transported first through the stripping section to eventually be transported through the rectifying section via vapor flow. Similar results are observed for a (2.5% disturbance in feed composition (see Figure 2). In other words, high-frequency (or low-frequency) column responses are better reflected by the stripping (or enriching) column temperature. In this work, this dynamical feature is exploited to improve the behavior of a standard distillate composition-temperature cascade control scheme by including a temperature measurement in the stripping column section also. 3. Cascade Control Design In a conventional distillate composition-temperature cascade control design, a fast secondary rectifying-section temperature loop is used to achieve safe composition regulation in the face of measurement and transport delays. Indeed, safe operation is guaranteed by the fast action of the (nondelayed) temperature feedback loop. Given that, in the presence of feed disturbances, the stripping-section temperature response (Ts) is faster than that of the rectifying-section temperature (Tr), the appropriateincorporation of the measurement Ts into the cascade control loop should accelerate the response of the temperature loop, and this, in turn, should improve the behavior of the product composition primary loop. For the control design purpose at hand, let us consider firstorder plus delay I/O models computed from a +1.0% disturbance reflux step response:

xD(s) R(s)

) GRx(s) ) Tr(s) R(s) Ts(s) R(s)

KRx exp(-θRxs) τRxs + 1

) GRTr(s) ) ) GRTs(s) )

KRTr τRTrs + 1 KRTs τRTss + 1

where θ > 0 is the delay due to measurements and internal transport. Steady-state gains, time constants, and delay values are reported in Table 2.

3.1. Conventional Cascade Control Design. The standard distillate composition-temperature cascade controller that is driven by a temperature measurement in the rectifying section is designed according to the following procedure. 3.1.1. Composition Loop. The primary loop is driven by the distillate composition measurement, xD, and calculates the required rectifying section temperature Tr, which is regarded as a “virtual controller”. Because GRx(s) ) (xD(s)/R(s)) and GRTr(s) ) (Tr(s)/R(s)), one has

GTrx(s) ) GRx(s)/GRTr(s) ) KTrx

(

)

τRTrs + 1 τRxs + 1

exp(-θs)

where KTrx ) KRx/KRTr. For conditions close to steady state, (τRTrs + 1)/(τRxs + 1) ≈ 1, and the delay dynamics exp(-θs) dominates the overall I/O dynamics. This implies that GTrx(s) ≈ KTrx exp(-θs) and, consequently, that the regulation of the composition loop can be performed with a simple integral feedback, i.e.,

CxTr(s) )

KI,x s

KI,x )

1 KTrxτI,x

with the integral gain KI,x being set according to internal model control (IMC) tuning rules:12 τI,x is ∼1.5-2 times greater than θ. 3.1.2. Temperature Loop. This control component manipulates the reflux flow, R (see Figure 3), to track the time-varying set point (T*r), which is calculated using the primary composition controller. Hence, the temperature loop design is based on the model (Tr(s)/R(s)) ) GRTr(s) ) (KRTr/τRTrs + 1). If eTr(s) ) Tr,ref(s) - Tr(s) is the temperature tracking error, and R(s) is the control output (i.e., eTs(s): CTr(s) f R(s)), the corresponding PI controller is given by

(

CTr(s) ) Kc,Tr 1 +

)

1 τI,Tr

Kc,Tr )

1 τRTr τI,Tr ) τRTr KRTr τc Tr

with the gain Kc,Tr and integral time τI,Tr chosen according to the IMC tuning rule:12 τTc r is ∼0.75-1.5 times greater than τRTr. The resulting feedback scheme is composed by a fast temperature loop that performs most of the disturbance rejection task, and a slow composition loop that is dedicated to ensure the distillate composition regulation. 3.2. Proposed Cascade Control Design. To exploit of the fast response of the stripping section temperature Ts, let us redesign the preceding cascade control scheme according to the following rationale: (i) Assume that the reflux flow control input R has two weighted components; one (Rr) is associated with the slowresponse of the rectifying-section temperature (Tr), and the other is (Rs) associated with the fast response of the stripping-section temperature (Ts):

R ) RRs + (1 - R)Rr

(0 e R e 1)

where the weighting factor R is regarded as a tuning parameter. (ii) Design two conventional cascade controllers, one for each temperature measurement (Tr for the rectifying section and Ts for the stripping section) and denote them CxTr(s) and CTr(s). Accordingly, assume that the output of the model that underlies

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Figure 3. Distillation column and the cascade control scheme.

the secondary control component is the convex combination ofthe two measured temperatures (Tr and Ts), i.e.,

Tv ) RTr + (1 - R)Ts

(R∈ [0, 1])

CxTv(s) ) RCxTr(s) + (1 - R)CxTs(s)

(0 e R e 1)

CTv(s) ) RCTr(s) + (1 - R)CTs(s) where CxTv(s) is the primary composition regulator, and CTv(s) is the secondary temperature regulator. If R ) 1, the standard composition-temperature control scheme is recovered. As the tuning parameter R is decreased, the stripping section temperature has a greater effect on the cascade control dynamics. This two-temperature cascade controller is along the idea of the wave model-based nonlinear control approach,13 in the sense that relevant information column is contained in its two wave propagation fronts, one per section. Consequently, two temperature sensors enable the monitoring of the wave generated by feed disturbances. In principle, in comparison with the standard single-temperature cascade control scheme, the proposed two-measurement scheme should have a

faster temperature disturbance rejection capability, or equivalently, better closed-loop behavior. 3.3. Simulation Results. Some simulation examples were used to illustrate the performance of the proposed control scheme. In all cases, a 10-min composition loop delay was considered. 3.3.1. Example 1. Consider the five-component distillation column used to illustrate the distillation column dynamics in Section 2. After the column was stabilized about the prescribed composition (x ) 0.974), the column was subjected to the following sequence of feed disturbances: a +5% step change in feed flow rate at t ) 7.5 h, a -5% step change in the feed flow rate at t ) 10.5 h, +2.5% step change in feed composition at t ) 14.5 h, and a -2.5% step change in feed composition at t ) 16.5 h. The square composition regulation error,

E2 )

∫0t (xref - x(t′))2 dt

was used as an index to quantify the performance of the control scheme. The results of the numerical simulations are shown in Figure 4 for three different values of the parameter R. The conventional cascade controller corresponds to R ) 1, which


Figure 4. Performance of the cascade control scheme: Example 1 (multicomponent distillation) for a sequence of feed disturbances, with temperature measurements at the third and thirteenth trays.

is outperformed when R < 1. We have observed that better control performance, with respect to the conventional cascade controller, is obtained for R ∈ (0.75, 1.0). For values R < 0.75, the control loop displays severe oscillations that increase the integral error E2 significantly. Because of changes in the operating conditions or uncertainties in the optimal temperature sensor location, a conventional cascade control scheme can undergo unstable behavior. For a conventional cascade control scheme (R ) 1) with the temper-

ature sensor located at the second tray, Figure 5 shows that the lack of a tight tuning can lead to unstable behavior, represented, in this case, by oscillations with excessively large settling times. The incorporation of a stripping section temperature measurement into the feedback control loop can stabilize these oscillations. In fact, Figure 5 shows that oscillations are drastically reduced when the sensitivity to feed disturbances are shared by temperature measurements in both column sections. This results show that the stripping section temperature also enhances the

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Figure 5. Performance of the cascade control scheme: Example 1 (multicomponent distillation) for a sequence of feed disturbances, with temperature measurements at the second and thirteenth trays.

robustness margin of the composition-temperature cascade control scheme. We have tested the performance of the cascade control scheme for changes in the feed tray location. A change in the feed tray location can induce significant changes in the sensitivity of the tray temperatures. In particular, the former optimal temperature measurements (according to Tolliver and McCune’s guidelines8) may be not optimal for a different feed

tray location. The cascade control scheme should be robust, to some extent, to changes in the distillation column configuration. That is, the cascade control scheme with two temperature measurements should lead to smaller regulation errors than the scheme with a single temperature measurement. Numerical simulations for the distillation column with the feed at the sixth tray (former feed was located at the fifth tray) are shown in Figure 6. Notice that, despite the fact that no control recon-


Figure 6. Performance of the cascade control scheme for a sequence of feed disturbances: Example 1 (multicomponent distillation), after changing the feed tray location from the fifth tray to the sixth tray. Note that the cascade control scheme was not reconfigured.

figuration was made (i.e., temperature measurements are not changed), the cascade control scheme retains the same disturbance rejection capabilities. That is, the stripping temperature measurement acts as a feed-forward element that allows fast rejection of the high-frequency effects of the feed disturbance.

The addition of trays increases the capacity of the column, which has the effect of increasing the damping effects. That is, larger column capacities increase the column time constants (e.g., the R/xD time constant). Although a large column capacity gives slower column dynamics, the wash-out (i.e., low-pass)

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Figure 7. Performance of the cascade control scheme: Example 1 (multicomponent distillation), when the column is added with a tray in the stripping section.

filtering effects against feed disturbances are enhanced. As commented previously, the cascade control scheme with multiple temperature measurements can be considered as a feed-forward configuration that improves the disturbance rejection capabilities.

Figure 7 shows numerical simulations for the distillation column with an additional tray in the stripping section. Note that, because of the increased column capacity, the improvement introduced by the additional temperature measurement is not


Figure 8. Performance of the cascade control scheme: Example 2 (binary benzene-toluene distillation), for a sequence of feed disturbances, with temperature measurements at the seventh and fourteenth trays.

as significant as the cases in Figures 4-6. This is because the increased capacity in the stripping section has reduced the feed

disturbance effects. However, such reduction is obtained at the expense of a larger (maybe not optimal) column with slower

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Figure 9. Performance of the cascade control scheme: Example 2 (binary benzene-toluene distillation), for a sequence of feed disturbances, with temperature measurements at the sixth and fourteenth trays.

dynamics. In this form, the proposed cascade control scheme has its more important performance improvements in distillation columns with relatively low capacity.

3.3.2. Example 2. Now consider an 18-tray benzene-toluene column, as described in Table 1 of Castellanos-Sahagun et al.14 The column has 18 trays, not including the reboiler and


Figure 10. Performance of the cascade control scheme: Example 3 (binary methanol-water distillation with saturated feed), for a sequence of feed disturbances.

condenser drum. The feed is located at the tenth tray, counting from below. Liquid hydraulics are described via the Francis weir formula. Following Tolliver and McCune’s guidelines,8 temperature measurements were selected at trays with the largest tray-to tray gradient, yielding the seventh and fourteenth trays. The corresponding time constants are 0.415 h for composition, 0.2428 h for the 14th tray temperature (in the rectifying section), and 0.0356 h for the seventh tray (in the stripping section). After the column was stabilized about the prescribed composition x

) 0.983, the column was subjected to the following sequence of feed disturbances: a +5% step change in feed flow rate at t ) 7.5 h, and a -10% step change in the feed flow rate at t ) 9.5 h. As in the multicomponent column case, the numerical simulations in Figure 8 show that the secondary (i.e., stripping section) temperature measurement induces a fast disturbance rejection action. Note that, for the sake of detailing the secondary measurement effects, only feed flow rate disturbance effects are shown, and that feed composition effects (not shown) display

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Figure 11. Performance of the cascade control scheme: Example 3 (binary methanol-water distillation with subcooled feed), for a sequence of feed disturbances.

behavior similar to that observed in the former example. Interestingly, an important control performance improvement (∼40% reduction of the square integral error) is obtained with only 5%-10% weighting of the secondary measurement. Numerical simulations show that the cascade control scheme is robust against changes in the feed tray location, column

capacity, etc. Figure 9 illustrates the point for a change in the location of the stripping temperature measurement (from the seventh tray to the sixth tray). Interestingly, note that no significant improvement is obtained for R j 0.93. This value can be seen as a bound for the cascade control improvement via secondary (stripping section) measurements. In fact, although


over-weighting the secondary temperature measurement can yield fast disturbance rejection for short times, it can also leads to slowing of the control loop response for large time scales. 3.3.3. Example 3. As a further robustness test, the proposed methodology was applied to a nonideal 12-tray methanol-water column, which does not satisfy the equimolecular flow assumption, meaning that stage-by-stage energy balances must be taken into account. The model included vapor-liquid equilibrium with activity coefficients (Wilson’s equation) and enthalpy calculations. The characteristics of the column are summarized in Table 1 of our previous work.14 The feed tray is located at the third location, counting from below, and the temperature sensors are located at the second and tenth trays,8 for the stripping and the rectifying sections, respectively. For a series of feed flow-rate and composition disturbances (at t ) 7 h the feed flow rate changes from 1.0 mol/s to 1.1 mol/s; at t ) 12 h, the feed flow rate changes from 1.1 mol/s to 0.9 mol/s; at t ) 15 h, the feed composition changes from 0.5 mole fraction to 0.55 mole fraction; and at t ) 21 h, the feed composition chnages from 0.55 mole fraction to 0.45 mole fraction), Figures 10 and 11 show the results for saturated feed and subcooled feed, respectively. For saturated feed, significant integral error decrements are obtained as the controller weight R is decreased. However, for the subcooled feed, the behavior of the integral error, with respect to R, is not monotonic. In fact, feed composition disturbances acting at t ) 7 h and t ) 12 h are poorly rejected when the controller weights (i.e., R < 1) the stripping section measurement. This behavior can be explained by the combination of subcooled feed and small-size stripping section (two trays), which yields a weak sensitivity of the stripping section temperatures to feed composition disturbances.14 However, the controller shows a good disturbance rejection in the face of feed flow-rate disturbances, yielding smaller integral errors when the stripping temperature measurement is taken into consideration in the feedback loop (i.e., R < 1). In summary, the proposed two-temperature measurement (one per section) cascade control design methodology is capable of regulating the effluent compositions for both ideal and nonideal mixtures and handles measurement lags and delays properly. 4. Conclusions This note has shown that the incorporation of a stripping section temperature measurement into a cascade control design improves the behavior of the control scheme by enhancing the

disturbance rejection capability of the secondary control component. The behavior improvement can be explained in terms of the wave theory for the propagation of disturbances into the column: (i) the stripping section temperature induces a feedforward-type error compensation path, which allows a faster rejection of feed disturbances, and (ii) the rectifying section temperature is more devoted to inducing a faster composition regulation. An important feature is that the control design is quite simple in the sense that it amounts to the adjustable-weight combination of two conventional composition-temperature cascade control designs. Literature Cited (1) Humphrey, J. L.; Seibert, A. F.; Koort, R. A. Separation Technologiess Advances and Priorities. OE Contract AC07-901D12920, February 1991. (2) Luyben, W. L. Steady-state energy conservation aspects of distillation column control design. Ind. Eng. Chem. Fundam. 1975, 14 (4), 321. (3) Castellanos-Sahagu´n, E.; Alvarez, J. Synthesis of two-point linear controllers for binary distillation columns. Chem. Eng. Commun. 2006, 193, 206. (4) Skogestad, S. Dynamics and control of distillation columns: A critical survey. Model. Identif. Control 1997, 18 (3), 177. (5) Castro, R.; Alvarez, Ja.; Alvarez, Jo. Nonlinear disturbance decoupling control of a binary distillation column. Automatica 1990, 26 (3), 567. (6) Fuentes, C.; Luyben, W. L. Control of high-purity distillation columns. Ind. Eng. Chem. Process Des. DeV., 1983, 22, 361. (7) Wolff, E. A.; Skogestad S. Temperature cascade control of distillation columns. Ind. Eng. Chem. Res. 1996, 35, 475. (8) Tolliver, T. L.; McCune, L. C. Finding the optimum temperature control trays for distillation columns. InTech 1980, 27 (9), 75. (9) Han, M.; Park, S. Control of high-purity distillation column using a nonlinear wave theory. AIChE J. 2004, 39 (5), 787. (10) Luyben, W. L. Feedback control of distillation columns by double differential temperature control. Ind. Eng. Chem. Fundam. 1969, 8 (4), 739. (11) Luyben, W. L. Process Modeling, Simulation and Control for Chemical Engineers; McGraw-Hill: New York, 1989. (12) Morari, M.; Zafiriou, E. Robust Process Control; Prentice-Hall: Englewood Cliffs, NJ, 1989. (13) Shin, J.; Seo, H.; Han, M.; Park, S. A nonlinear profile observer using tray temperatures for high-purity binary distillation columns. Chem. Eng. Sci. 2000, 55, 807. (14) Castellanos-Sahagun, E.; Alvarez-Ramirez, J.; Alvarez, J. Twopoint temperature control structure and algorithm design for binary distillation columns. Ind. Eng. Chem. Res. 2005, 44, 142.

ReceiVed for reView April 4, 2006 ReVised manuscript receiVed June 8, 2006 Accepted August 10, 2006 IE060421A

Distillate Cascade Composition Control Using a Two-Temperature

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