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Ind. Eng. Chem. Res. 2000, 39, 1840-1849
Distillate-Bottoms Control of Middle-Vessel Distillation Columns James R. Phimister and Warren D. Seider* Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6393
The performance of a middle-vessel distillation column subject to distillate-bottoms (DB) control configurations is analyzed. DB-controlled columns, where the distillate flow rate is manipulated to control the distillate composition and the bottoms flow rate is manipulated to control the bottoms composition, have been labeled as inoperable for continuous distillation. In this paper, two DB-control configurations for a middle-vessel column (MVC) are shown to overcome the deficiencies commonly associated with DB control. In the first configuration, the feed flow rate from the middle vessel is fixed and the flow rate of the heating medium controls the sump level. In the second “on-demand” configuration, the feed flow rate from the middle vessel is manipulated to control the sump level. To demonstrate the effectiveness of these configurations, the distillation of a near-ideal binary mixture is examined. Introduction Papina1
Recently, Barolo and added a middle vessel to improve the dual-composition control of a continuous distillation column. They examined various control configurations, illustrating the improved performance facilitated by the middle vessel as compared to a conventional column (without a middle vessel). However, they did not consider the uncommon DB-control configuration, a configuration that, due to operating difficulties, has been rarely implemented in continuous distillation columns. Distillate-bottoms control, or DB control, in which the distillate flow rate is manipulated to control the composition in the reflux drum and the bottoms flow rate is adjusted to control the composition of the bottoms product, is widely labeled as inoperable. This configuration, for a “feed-driven” column, to which the feed flow rate is flow-controlled, is shown in Figure 1. In this paper, a middle vessel is added and two DB-control configurations are presented to overcome these operating difficulties, with their performance analyzed for the separation of a near-ideal binary pair. In an analysis of several dual-composition control configurations, Skogestad and Morari2 and Skogestad et al.3 emphasize the importance of reducing interactions between the controllers. They conclude that the (L/D)(V/B)- or LV-control configurations are generally the most favorable. Furthermore, they observe that, because of controller interactions, most industrial distillation columns have just a single-composition controller or have no composition control loops, that is, operate with reflux and boilup flow rates fixed, and the distillate and bottoms flow rates used for inventory control. When controller interactions are reduced, dualcomposition control configurations are more attractive because they substantially reduce the energy costs. Furthermore, in the semicontinuous, pressure-swing process introduced by Phimister and Seider,4 dualcomposition control is essential because failure to control the concentrations of both product streams significantly deteriorates the throughput. Note that when controller interactions are significant, and meeting the specifications for one product is more important * To whom correspondence should be addressed.
than for the other (e.g., when the other product is recycled), single-point composition control is favored. Given small interactions, however, normally a dualcomposition control configuration is favored. The DB-control configuration is simple and requires few sensors. Each controller increases the product flow rate when its product concentration is above specification and decreases its flow rate when its concentration is below specification. When a product concentration is far off specification, its control valve can be shut, to prevent off-specification product from leaving the column. However, the DB-control configuration is flawed when implemented on common continuous columns, that is, columns without a middle vessel. For these columns, the actions of the distillate and bottoms controllers are not independent of each other, but are linked by the mass-balance constraint:
F) D + B
(1)
This constraint removes a degree of freedom, preventing the independent manipulation of both variables. Consequently, when the flow rate of one product stream is reduced, if the flow rate of the other product does not increase, the feed accumulates in the column, the sump level rises above the bottom trays, and the tower becomes inoperable. This arises because the decentralized control strategy does not account for eq 1, and hence, when sidedraws are not included, operating problems arise. Despite this limitation, Finco et al.5 show by simulation and experiment, for an industrial C3 splitter, that when the DB-control configuration is applied for towers with a large number of trays, it performs well compared with a (L/D)(V/B)-control configuration. Because of difficulties with valve saturation, they label this configuration as “fragile” and include override controllers. Skogestad et al.6 confirm that successful operation is possible when the column can accumulate mass temporarily, that is, through increased holdup in the sump and reflux drum in response to changes in the feed composition. They show satisfactory performance for towers having >100 stages. In both studies, a large liquid holdup in the column is necessary to decouple the composition controllers.
10.1021/ie990890v CCC: $19.00 © 2000 American Chemical Society Published on Web 05/06/2000
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1841
Figure 1. DB-control configuration.
Even in these select cases, dual-composition control fails when one of the four controllers (either for composition or level control) becomes saturated. Additionally, if the flow rate of either the bottoms product or the distillate is temporarily fixed by its controller, the composition of the alternate stream cannot be controlled because of mass-balance constraints. There are added difficulties during start-up because both controllers must be overridden until a steady state with onspecification products is achieved. For these reasons, the DB-control configuration is rarely used in industry. The two DB-control configurations in Figures 2 and 3, proposed for a continuous tower with a middle vessel, provide dual-composition control without succumbing to the difficulties above. Furthermore, a large number of stages is not required to decouple the composition controllers. Figure 2 shows a configuration with the feed rate flow controlled, in which the flow rate of the heating medium is adjusted to control the sump level. In this configuration, an external middle vessel feeds the column and is fed by a full-liquid sidedraw from the column. The second “on-demand” configuration, shown in Figure 3, uses the column feed rate to maintain the holdup in the sump, with the reboiler duty separately controlled. Herein, on-demand refers to the use of a downstream measurement (sump level) to adjust the feed flow rate. More commonly, the flow rate leaving the process is measured. Middle-Vessel Columns (MVCs). During the past decade, there has been increased interest in the performance of alternative distillation configurations, including the middle-vessel column,7 multivessel columns, series of heat-integrated, batch-distillation columns,8 and semicontinuous distillation configurations.9 Although the MVC was initially suggested by Robinson and Gilliland10 (p 388), few papers included middle
vessels until the 1990s. Two of the first papers, by Hasebe and co-workers,11,12 reintroduced the MVC, presenting a campaign policy with simulation results. More recently, MVC control configurations have been discussed by Barolo et al.,13 Farschman and Diwekar,14 Phimister and Seider,15 and Barolo and Papini.1 All emphasize the importance of the middle vessel in decoupling the composition controllers. Note that Phimister and Seider15 were the first to apply the DBcontrol configuration to the MVC. They illustrate its performance for the semicontinuous, cyclic distillation of ternary mixtures. On-Demand Control. In the on-demand configuration, the feed flow rate from the middle vessel is manipulated to control the sump level. Whereas in alternative configurations, more commonly used in continuous processing, the flow rates of one or both of the product streams are manipulated to maintain the levels in the reflux accumulator and sump. Luyben16 discusses on-demand control configurations for distillation processes and for reactor-separatorrecycle structures. While it is recognized that these configurations can provide effective control, problems associated with dynamic lags (e.g., through the stripping section), controller interactions, and disturbances traveling both upstream and downstream, generally render on-demand configurations less favorable than “material driven” operations (e.g., processes with flow controllers on the feed streams). For the semicontinuous distillation of near-ideal ternary mixtures, however, Phimister and Seider15 show that an on-demand configuration performs well. In contrast to the continuous process, the on-demand DBcontrol configuration is more effective when few trays are present in the stripping section. This is because a large liquid lag between the feed tray and the sump,
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Figure 2. DB-control configuration with feed flow control.
with a high-order transfer function between the feed flow rate and the liquid level in the sump, degrades the response of the control system. Control Analysis The inherent problem with traditional DB-control configurations, in which the feed stream is flowcontrolled, is that the level controllers are rendered ineffective when the product concentrations in the distillate and bottoms streams are below their specifications. To provide effective DB control, the interactions between the level controllers must be reduced. When this is accomplished, the interactions between the composition controllers are reduced as well. The interactions between the level controllers are complicated by the mass-balance constraint. Consider, for example, a binary column. When the concentration of the light species in the distillate is below its specification, the distillate flow rate is reduced. As a consequence, the reflux flow rate is increased, more of the light species concentrates in the sump, the concentration of the heavy species in the bottoms product is driven below its specification, the bottoms flow rate is reduced, and the vapor flow rate is increased. This undesirable spiral drives both products off specification, causing the valves for the distillate and bottoms product to shut down gradually. Furthermore, as the inventory of the column builds, the liquid levels in the sump and reflux drum rise rapidly and the column becomes inoperable. In contrast, with a middle vessel, the proposed control configurations perform well when the flow rates of the
distillate and bottoms products are decreased, or even when their valves are shut. For both control configurations, as the column approaches total reflux, the flow rate of the sidedraw to the middle vessel approaches the flow rate from the middle vessel to the column. Note that the full-liquid sidedraw permits the level controllers to continue functioning properly. Were a fraction of the liquid withdrawn, the difficulties using a DBcontrol configuration without a middle vessel would be encountered. To illustrate the performance of the two control configurations, a column to separate n-hexane from n-heptane is considered. A dynamic model involving the MESH equations4 is used to compute gain matrices and an auto-tuning method by A° stro¨m and Ha¨gglund17 is used to determine the ultimate gain and frequency of a proportional controller. Then, the Ziegler-Nichols tuning rules are used to compute gains and the integral time constants for PI controllers. Note that more integral action is used, with τI’s chosen such that, for a 10% step change in the mole fraction of n-hexane in the feed stream and middle vessel, the product mole fractions are returned to within 0.001 of their set points within 10 h. Column Design and Model. The feed to the process (middle vessel) is a saturated liquid comprised of 50 mol % n-hexane and 50 mol % n-heptane at 22.38 kmol/h and 1 atm. It is assumed to have a constant relative volatility of 3.0. The column has 20 stages, including a condenser and a reboiler, with 1-m diameter, a weir length-to-diameter ratio of 0.77, an active area of 0.6 m2, a weir height of 5 cm, and a tray residence time of
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Figure 3. On-demand DB-control configuration.
21 s. The feed to the column, from the middle vessel, is to tray 10 from the top of the column. It is at 5 m3/h when under flow control, with 5 m3/h being the midpoint (or bias) for the on-demand control configuration. A fullliquid sidedraw from tray 9 is fed to the middle vessel. The sump and reflux drum have capacities of 10 m3 with set points at 5 m3, and the middle vessel has a capacity of 10 m3 with a set point at 5 m3. The set points for the compositions of the distillate and bottoms products are 98 mol % n-hexane and 98 mol % n-heptane, respectively. The corresponding reflux and reboil ratios are 0.94 and 2.13, and the condenser and reboiler are sized to provide sufficient area for heat transfer under these conditions. The tray hydraulics are modeled using the Francis weir formula which relates the liquid flow rate from a tray, L (kg/s), to the crest height over the weir, how (mm):
[ ]
L ) FLlw
how 750
1.5
(2)
where lw (m) is the weir length. During each time step, the overall mass balances are used to estimate the change in the molar liquid holdup on each tray. Assuming a uniform density on each tray (FL), the volume of the holdup is computed, from which the crest height above the weir is computed by difference. Then, the liquid flow rate from each tray, L, is computed using eq 2 and the vapor flow rates from each tray are updated using the overall energy balances. The tri-diagonal species balances are integrated using an implicit-Euler integrator with a step-size adjustment algorithm. Then, the tray temperatures are estimated using the bubble-
Table 1. Steady-State Performance feed flow control (kmol/h) Fc D B L0 V S
33.44 11.19 11.19 10.57 23.85 11.06
point equations and the enthalpies and densities are updated to complete a time step. In summary, the Francis weir formula and overall mass balance model the hydraulics on each tray. When combined in series, the hydraulic models for the trays in the stripping section permit a representation of the lag of the liquid level in the sump in response to a change in the feed flow rate. This model is the key to the analysis of the level controller in the on-demand DB-control configuration. When sizing the valves, it is important to provide sufficient capacity to allow for normal operating swings. To initiate start-up, the column is loaded with the feed mixture, and the sump, reflux accumulator, and middle vessel contain 5 m3 of the feed mixture. After the reboiler and condenser are placed in operation, total reflux is achieved, at which the recirculation rates throughout the tower and middle vessel are at a maximum. Then, gradually, the flows of the distillate and bottoms product and the flow of the feed stream to the middle vessel are increased from zero. At steady state, the principal flow rates are shown in Table 1. Note that they are identical for the two DB-control configurations. Upper and lower bounds (zU and zL) and midpoints (z*) for the manipulated variables and upper and lower bounds (xU and xL) and set points (xsp) for the controlled
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Table 2. Controller SpecificationssDB Configuration with Flow Control of Feed Rate xsp xU xL z* zU zL Kc τI
C-101
C-102
C-103
C-104
C-105
5 m3/h
0.98c 0.999c 0.96c 1.18 m3/h 2.37 m3/h 0 m3/h 22.2 0.5 h
5 m3 10 m3 2 m3 5 m3/h 7.5 m3/ha 1.25 m3/ha 0.78 5h
0.98b 0.999b 0.96+ 1.31 m3/h 2.63 m3/h 0 m3/h 24.7 0.5 h
5 m3 10 m3 2 m3 5 m3/ha 7.5 m3/ha 2.5 m3/ha 0.63 5h
5 m3/h
a Refers to boilupsmust convert to bounds on flow rate of heating medium. b Mole fraction of the heavy species. c Mole fraction of the light species.
Table 3. Controller SpecificationssOn-Demand DB Configuration C-101
C-102
C-103
C-104
C-105
xsp 5 m3 0.98c 5 m3 0.98b xU 10 m3 0.999c 10 m3 0.999b xL 2 m 3 0.96c 2 m3 0.96b z* 5 m3/h 1.18 m3/h 5 m3/h 1.31 m3/h 23.85 kmol/ha zU 7.5 m3/h 2.37 m3/h 7.5 m3/h 2.63 m3/h zL 2.5 m3/h 0 m3/h 1.25 m3/h 0 m3/h Kc 0.63 22.2 0.78 24.7 τI 5h 0.5 h 5h 0.5 h a Refers to boilupsmust convert to bounds on flow rate of heating medium. b Mole fraction of the heavy species. c Mole fraction of the light species.
variables are shown in Table 2, for the DB-control configuration with a flow controller for the feed stream, and Table 3, for the on-demand DB-control configuration. For both configurations, at steady state, the flow rate of the feed stream to the column is 5 m3/h. Beginning with Table 2, the flow controller, C-101, is assumed to hold the manipulated variable at its set point, 5 m3/h. The concentration controller, C-102, has an upper bound on the flow rate equal to the flow rate of the light species in the feed stream to the process, with a lower bound at zero and a midpoint that bisects the two. Note that the measured output has a set point at 0.98 mole fraction of the light species. When its mole fraction exceeds 0.999, the valve is opened entirely, and when its mole fraction falls below 0.96, the valve is shut. The concentration controller, C-104, operates similarly. Its upper bound on the flow rate equals the flow rate of the heavy species in the feed stream. For the level controller in the reflux accumulator, C-103, the upper bound on the flow rate of the reflux stream is set 50% higher than the maximum feed flow rate to provide sufficient capacity at total reflux. Its lower bound is set to maintain sufficient liquid traffic on the trays, and its midpoint corresponds to steady operation at total reflux. The upper bound for the controlled variable, the volumetric holdup, is the capacity of the tank. A reasonable lower bound is at 20% full, with a midpoint at 50% full. The bounds are set similarly for the flow rates associated with the level controller in the sump, C-105. Note, however, that the simulations were carried out with the flow rates of the boilup adjusted, rather than the flow rates of the heating medium, steam. Consequently, Table 2 shows the bounds on the flow rates of the boilup. Corresponding bounds on the flow rate of the heating medium are obtained by energy balance in the reboiler. The settings in Table 3, for the on-demand DB-control configuration, are identical to those in Table 2, except that controller C-101 is assigned the bounds for the level
controller in the sump and controller C-105 is assumed to operate at its set point under flow control. In summary, the controllers are designed such that when an upper or lower bound for a controlled variable is violated, its manipulated variable is set to its corresponding bound. The upper bounds on the distillate and bottoms products are constrained by the volumetric flows corresponding to xL,FFc and xH,FFc, respectively, where xL,F and xH,F are the mole fractions of the light and heavy species in the process feed. The lower bounds for the product flow rates are zero, with the midpoints corresponding to the volumetric flows of 0.5xL,FFc and 0.5xL,FFc. When the upper bounds and the product flow rates are too small, high reflux and reboil ratios result and the composition controllers are ineffective as both product compositions exceed their specifications. The lower bounds on the boilup and reflux flow rate are constrained to maintain sufficient boilup and reflux for proper tray operation, with liquid seals and without weeping. The upper and lower bounds on the heating and cooling duties must be adjusted accordingly. Note that while less important, the upper bound should be above the flow rate of the feed to the column because, at total reflux, the boilup and reflux flow rates are approximately equal to the feed flow rate on a molar basis, assuming constant molal overflow. Each PI controller adjusts its manipulated variable, z, by
[
z ) z* + Kc e +
1 τI
∫e dt
]
(3)
where e is the error in the measured variable, z* is the midpoint of the manipulated variable (or bias at e ) 0), Kc is the controller gain, and τI is the integral time constant. The gains on the composition controllers are set using the ATV tuning method17 and the Ziegler-Nichols tuning rules for PI controllers. For both configurations, the distillate and bottoms composition controllers have gains of 22.2 and 24.7, respectively. The ATV tuning method gives an ultimate period of 2 h and a reset time of 1.67 h, using Ziegler-Nichols tuning rules. Because a sluggish response results, the reset times are reduced to 10 min, providing more integral action. The gains for the level controllers provide percent-to-percent changes; that is, a deviation of 1% in the level, x, over its range (xU - xL), results in a 1% change in the corresponding manipulated variable, z, over its range (zU - zL). Column Performance. To compare the performance of the two DB-control configurations, step changes in the feed mole fractions and flow rate are imposed. Note that to provide a clearer response of the column and its controllers, the step changes in the mole fractions are imposed jointly on the feed stream to the column, the feed to the process, and the large middle vessel. Also, for the on-demand configuration, because the feed flow rate is the manipulated variable, the set point on the boilup rate is adjusted to provide an equivalent change in Fc at steady state. When the feed is flow-controlled, the responses to a 10% change in the mole fraction of the light species from 0.5 to 0.45 and a 10% increase in Fc are shown in Figures 4 and 5, respectively. In response to the mole fraction change, Figure 4A shows that the flow rates of the product streams adjust quickly to their new values at steady state. Furthermore, the composition trajecto-
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Figure 4. Responses to a 10% change in the mole fraction of n-hexane in the feed stream and middle vessel, from 0.5 to 0.45, when the feed stream is flow-controlled. (A) Distillate (solid line) and bottoms (dashed line) flow rates. (B) Mole fraction of light species in distillate (solid line) and heavy species in bottoms product (dashed line). (C) Reflux flow rate (solid line) and boilup rate (dashed line). (D) Sump (dashed line) and reflux drum (solid line) volumetric holdup.
Figure 6. Responses to a 10% change in the mole fraction of n-hexane in the feed stream and middle vessel, from 0.5 to 0.45, for the on-demand control configuration. (A) Distillate (solid line) and bottoms (dashed line) flow rates. (B) Mole fraction of light species in distillate (solid line) and heavy species in bottoms product (dashed line). (C) Feed flow rate (dark line), reflux flow rate (dashed line), and boilup rate (solid line). (D) Reflux drum (solid line) and sump (dashed line) volumetric holdup.
Figure 5. Responses to a 10% increase in the flow rate of the feed stream when the feed stream is flow-controlled. (A) Distillate (solid line) and bottoms (dashed line) flow rate. (B) Mole fraction of light species in distillate (solid line) and heavy species in bottoms (dashed line). (C) Reflux flow rate (solid line) and boilup rate (dashed line). (D) Sump (dashed line) and reflux drum (solid line) volumetric holdup.
Figure 7. Responses to a 10% increase in the flow rate of the feed for the on-demand control configuration. (A) Distillate (solid line) and bottoms (dashed line) flow rate. (B) Mole fraction of light species in distillate (solid line) and heavy species in bottoms (dashed line). (C) Feed flow rate (dark line), reflux flow rate (solid line), and boilup rate (dashed line). (D) Reflux drum (solid line) volumetric holdup and sump (dashed line).
ries in Figure 4B show that the mole fraction of n-hexane in the distillate does not fall below 0.977 and returns to 0.980 ( 0.001 within 7 h. Because the feed is a saturated liquid, a larger variation in the mole fraction of n-heptane in the bottoms product is observed. Similarly, however, the mole fraction of n-heptane in
the bottoms product returns to 0.980 ( 0.001 within 7 h. Figure 4C shows that the reflux and boilup rates remain nearly constant, and Figure 4D shows that the holdups in the reflux drum and sump remain nearly constant. These illustrate a desirable feature of the MVC, with its DB-control configuration; that is, despite
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a large upset in the feed composition, small changes in the internal flow rates are observed. The responses to a 10% increase in the flow rate of the feed stream are shown in Figure 5. Because the feed stream is a saturated liquid, the liquid flow rates in the stripping section increase. Initially, as shown in Figures 5A, an inverse response is observed. Although the flow rate of the bottoms product is increased at the new steady state, initially the concentration of n-hexane increases in the stripping section and the flow rate of the bottoms product decreases. As expected, the boilup and reflux rates increase, as shown in Figure 5C, and the holdups in the sump and reflux drum return to their set points, as shown in Figure 5D. For the on-demand control configuration, the responses to a 10% change in the mole fraction of the light species from 0.5 to 0.45 and a 10% increase in Fc are shown in Figures 6 and 7. Figure 6A,B shows that the responses to composition changes are similar to those when the feed is flow controlled, in that the mole fractions in the distillate and bottoms product remain near their set points, with the bottoms mole fraction taking longer to recover. Note that the feed flow rate increases slightly, as shown in Figure 6C, which does not occur when the feed rate is flow-controlled. When subjected to an increase in the boilup rate (which increases Fc), the on-demand control configuration provides different responses. Initially, the increased boilup reduces the mole fraction of the light species in the sump, as shown in Figure 7B, and hence, the bottoms flow rate increases, as shown in Figure 7A. This causes the sump level to decrease and the feed rate to be increased to return the sump level to its set point, as shown in Figure 7C,D. In summary, both control configurations perform well in rejecting these disturbances. Similar responses to feed concentration changes are observed, while the responses to increases in the boilup rate differ considerably. Gain Matrix. The success of the DB-control configurations is anticipated when examining the steady-state gain matrix that relates the changes in the mole fractions of the light species in the distillate and bottoms product to changes in the flow rates of the distillate and bottoms product:
[ ][
][ ]
dxD g11 g12 dD dxB ) g21 g22 dB
(4)
|
FxF ) BxB0 + DxD0
(5)
where xF, xB, and xD are the mole fractions of the light species in the process feed, the bottoms product, and the distillate, and the superscript “0” denotes the mole fractions prior to the change. After the change, the overall mass balance is N (F + ∆D)xF ) BxN B + (D + ∆D)xD
(6)
where the superscript “N” denotes the modified mole fractions. Note that ∆F ) ∆D. When eq 6 is subtracted from eq 5, then rearranged, and the limit taken as ∆D approaches zero, the relationship between the elements of the gain array is well-defined:
dxB dD
|
) B
(
|
dxD 1 x -D B F dD
)
- xD
B
(7)
Similarly, when a small change is imposed in the flow rate of the bottoms product, ∆B, the relationship becomes
|
dxD dB
)
D
(
|
dxB 1 xF - B D dB
D
- xB
)
(8)
Small perturbations in the flow rates of the distillate and bottoms product ∆D ) ∆B ) 0.01 kmol/h, from the steady state in Table 1, are used to compute by finite difference the gain matrices for the two DB-control configurations. First, when the feed rate, Fc, is flowcontrolled, it is
[ ][
][ ]
dxD -0.024 -0.008 dD dxB ) -0.019 0.051 dB
(9)
which is quite comparable to that for the on-demand configuration:
where
|
In contrast, when a middle vessel is included, the gain matrices are determinate. For both control configurations, an increase in D results in a decrease in the side draw flow rate and an increase in the flow rate of the process feed to the middle vessel, F, to maintain the level in the middle vessel. To derive the relationship between the elements in the gain matrix, for the ondemand control configuration, a small change in the distillate flow rate, ∆D, is imposed with the flow rate of the bottoms product held constant. The overall mass balance prior to the change is
|
dxD dxD dxB g11 ) , g12 ) , g21 ) , and dD B dB D dD B g22 )
[ ][
][ ]
dxD -0.026 0.005 dD dxB ) -0.018 0.037 dB
|
dxB dB
D
First, it is noted that the gain matrix is indeterminate for a conventional column, without a middle vessel, under DB control, as explained by Papastathopoulou and Luyben.18 Consider the element, g11, which represents the change in xD with D, holding B constant, with all of the other controllers assumed perfect, that is, F constant. Clearly, a change in D cannot occur at steady state when B and F are constant. Stated differently, such a change violates the overall mass balance, and consequently, the gain matrix is indeterminate.
(10)
The gain matrices in eqs 9 and 10 are used to compute the relative gain array (RGA), where the relative gain for the pairing D-xD is
λ11 )
g11 g11 - g12(g21/g22)
(11)
Values of 0.890 and 1.103 are computed, respectively. As anticipated, the relative gains are close to unity because small interactions are observed between the control loops in both DB-control configurations. The signs and relative values of elements in the gain matrix can be determined by examination of the reflux
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1847
and reboil ratios before and after the perturbations. Consider an increase in the distillate flow rate, ∆D, with B held constant and all other controllers performing normally. For both configurations, the reflux ratios are reduced from L0/D to (L0 - ∆D)/(D + ∆D). The reduction in the reflux ratio causes an increase in the mole fraction of the heavy species in the distillate, and hence, g11 is negative. Because the reflux ratio at the new steady state is the same for both configurations, as anticipated, the values of g11 are comparable for both configurations. Note that g21 is linked to g11 by eq 7, which applies for both configurations. In contrast, when the flow rate of the bottoms product is increased by ∆B, the reboil ratios at the new steady state differ for the two configurations. When the feed rate is flow-controlled, the reboil ratio is reduced from V/B to (V - ∆B)/(B + ∆B), whereas for the on-demand configuration, it is reduced to V/(B + ∆B). Because the mole fraction of the light species in the bottoms product is increased, g22 is positive for both configurations. Furthermore, because the resulting reboil ratio is smaller for the on-demand configuration, it has a smaller gain, g22, as observed. Note also that g12 is obtained from eq 8. Transfer Functions. Insights into the dynamic performance of the DB-control configuration are obtained by examination of its transfer functions. In this section, the transfer functions are compared for a MVC with the feed rate under flow control to that for a tower without a middle vessel. Papastathopoulou and Luyben18 show that the transfer function matrix for a distillation column without a middle vessel, under the DB-control configuration, is obtained by transforming the gain matrix associated with the more common DV-control configuration. Herein, the transformation is shown for the MVC. Beginning with the transfer function matrix for a column without a middle vessel under the DV-control configuration,
[ ][
][ ]
dxD G11{s} G12{s} dD dxB ) G21{s} G22{s} dV
(13)
applies, where LN is the liquid flow rate from the bottom tray; that is, small changes in the distillate flow rate are transmitted from tray-to-tray throughout the column. With a middle vessel, having a full-liquid side draw, and under flow control of the feed rate, the liquid flow rates in the stripping section are nearly independent of the reflux flow rate, and dLN = 0. Consequently,
dV ) -dB
(14)
and
[ ] [
][ ]
dD 1 0 dD ) dV 0 -1 dB
(15)
Substituting in eq 12,
[ ][
][ ][ ]
dxD G11{s} G12{s} 1 0 dD dxB ) G21{s} G22{s} 0 -1 dB
and
][ ]
dxD G11{s} -G12{s} dD dxB ) G21{s} -G22{s} dB
(17)
The transfer function matrices in eqs 12 and 17 are identical, except for the directionality difference in response to changes in the bottoms flow rate. This confirms that the dynamic performance of the DBcontrol configuration, with the feed rate under flow control, is similar to that for a column without a middle vessel under the commonly used DV-control configuration. In contrast, for a column without a middle vessel under the DB-control configuration, Papastathopoulou and Luyben18 derive the transfer function matrix,
[ ]
dxD dxB )
[
G11{s} -
G12{s}GL{s}
-G12{s}
]
[ ]
1 - GL{s} 1 - GL{s} dD G22{s}GL{s} G22{s} dB G21{s} 1 - GL{s} 1 - GL{s} (18)
where
GL{s} )
1 (1 + τLs)NT
τL is the hydraulic time constant, and NT is the number of trays in the column. Comparing eqs 17 and 18, the dynamic performance of the MVC, with a DB-control configuration and the feed rate under flow control, is closer to that for a column without a middle vessel under the commonly used DV-control configuration than under the DB-control configuration. The transfer function, GL{s}, represents a series of first-order lags, each of which relates the liquid flow rates to and from a tray, for the NT trays in the column. Discussion
(12)
the overall mass balance about the reboiler
dV ) dLN - dB
[ ][
(16)
Both DB-control configurations provide decentralized, dual-composition control, avoiding the valve-saturation problems encountered when DB control is implemented without a middle vessel. When the flow rate of the bottoms product or distillate is restrained by a controller, the flow rate of the alternate stream can be manipulated to maintain its composition. Furthermore, when B and D are both set to zero, operation is at total reflux with the flow rate of the sidedraw equal to that of the column feed; that is, S ) Fc. Simulations show that the performance of the MVC, with the DB-control configuration, is not improved when larger middle vessels (as large as 100 m3) are employed. This suggests that the middle vessel is not simply a surge tank, in place to “absorb” upstream composition disturbances because if this were the case, larger middle vessels would show improved performance. Rather, the middle vessel acts as an inventory regulator, allowing the MVC with the DB-control configuration to recover from the aforementioned inventory control problems. Because there is no advantage to using a large middle vessel, smaller and less expensive vessels should be used. To avoid the propagation of changes in the flow rate of the process feed, a feedforward controller can be used. Rather than adjust the flow rate of the process feed to maintain the level in the middle vessel, the feedforward controller would either manipulate Fc or V.
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Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000
When an intermediate impurity is present, using the DB-control configuration, it builds in the middle vessel. Eventually, the product concentrations in the distillate and bottoms fall below their specifications, and operation at total reflux results. When an intermediate impurity is present, it is recommended that the semicontinuous, middle-vessel column (SCMVC) introduced by Phimister and Seider15 be used to separate the three species. Configuration Comparison. The DB-control configuration with the feed rate under flow control appears preferable as the on-demand configuration is expected to transmit disturbances upstream. Note, however, that both configurations are on-demand because the flow rate of the process feed is adjusted to maintain the inventory in the middle vessel. Furthermore, the on-demand configuration does not upset the flow rate of the process feed appreciably more, and consequently, both configurations are comparable with respect to this limitation. Both DB-control configurations experience similar dynamic responses, but there are important distinctions. Because the boilup rate is controlled separately in the on-demand configuration, the flow rates in the rectification section are less variable. Furthermore, when the flow rate of the feed stream is increased (through an increase in the boilup rate), inverse response does not occur, whereas inverse response is experienced when the feed rate is flow-controlled. Despite these advantages, the on-demand configuration may be less effective when there are numerous trays in the stripping section. As the number of trays is increased, the ultimate gain of the PI controller is decreased and the nonlinearities in the relationship between the feed flow rate and the sump holdup are increased. Furthermore, the on-demand configuration is less effective when the concentration of the heavy species decreases in the process feed, and as the quality of the feed decreases from unity (saturated liquid.)
Nomenclature B ) bottoms flow rate (kmol/h) D ) distillate flow rate (kmol/h) e ) error (xsp - x) F ) flow rate of process feed (kmol/h) Fc ) flow rate of column feed (kmol/h) gij ) elements of the gain matrix Gij ) elements of the transfer function matrix how ) crest height over weir (mm) Kc ) controller gain lw ) weir length (m) L ) liquid flow rate over weir (kg/s) Li ) liquid flow rate from tray i (kmol/h) LN ) liquid flow rate from the bottom tray (kmol/h) MB ) molar holdup in sump (kmol) NT, NS ) number of stages in column and in stripping section (including reboiler) s ) Laplace transform variable S ) sidedraw flow rate (kmol/h) t ) time (h) V ) boilup rate (kmol/h) xsp, xL, xU ) set point and lower and upper bounds for the controlled variable xD, xB, xF ) mole fraction of light species in distillate, bottoms product, and process feed xH,F ) mole fraction of heavy species in process feed xL,F ) mole fraction of light species in process feed z*, zL, zU ) midpoint and lower and upper bounds for manipulated variable Greek Letters λij ) relative gain for the i-j pairing FL ) liquid density (kg/m3) τI ) integral time constant (h) τL ) hydraulic time constant for a tray (s) Superscripts N ) final 0 ) initial
Conclusions It is concluded that 1. While inoperable for most distillation columns without a middle vessel, the DB-control configuration is effective for distillation columns with a middle vessel. 2. Two DB-control configurations are attractive for distillation columns with a middle vessel. One involves flow control of the feed rate with the sump level controlled by the boilup rate. The other “on-demand configuration” adjusts the feed flow rate to control the sump level. Both are shown to provide fine control. Their advantages and disadvantages are discussed herein. 3. For each controller configuration, one element of the relative gain array is near unity, suggesting the existence of little interaction between the control loops. This is confirmed using dynamic simulations. 4. The middle vessel with a full-liquid sidedraw renders the DB-control configuration effective. This is consistent with the observation that the structure of the transfer function matrix for a MVC under the DBcontrol configuration, with the feed rate under flow control, is similar to that of the transfer function matrix for the DV-control configuration without a middle vessel. Acknowledgment Partial support of this research was provided by NSF Grant CTS96-32992 and is gratefully acknowledged.
Literature Cited (1) Barolo, M.; Papini, C. A. Improving Dual Composition Control in Continuous Distillation Through a Novel Column Design. AIChE J. 2000, 46, 146-159. (2) Skogestad, S.; Morari, M. Control Configuration Selection for Distillation Columns. AIChE J. 1987, 33, 1620-1635. (3) Skogestad, S.; Jacobsen, E. W.; Morari, M. Inadequacy of Steady-State Analysis for Feedback Control: Distillate-Bottom Control of Distillation Columns. Ind. Eng. Chem. Res. 1990, 29, 2339-2346. (4) Phimister, J. R.; Seider, W. D. Semi-continuous, PressureSwing Distillation. Ind. Eng. Chem. Res. 2000, 39, 122-130. (5) Finco, M. V.; Luyben, W. L.; Polleck, R. E. Control of Distillation Columns with Low Relative Volatilities. Ind. Eng. Chem. Res. 1989, 28, 75-83. (6) Skogestad, S.; Lundstrom, R.; Jacobsen, E. W. Selecting the Best Distillation Control Configuration. AIChE J. 1990, 36, 753764. (7) Skogestad, S.; Wittgens, B.; Litto, R.; Sorensen, E. Multivessel Batch Distillation. AIChE J. 1997, 43, 971-978. (8) Hasebe, S.; Noda, M.; Hashimoto, I. Optimal Operation Policy for Total Reflux and Multi-effect Batch Distillation Systems. Comput. Chem. Eng. 1999, 23, 523-532. (9) Phimister, J. R.; Seider, W. D. Semi-continuous Operation of a Middle-Vessel Distillation Column. In Foundations of ComputerAided Process Design (FOCAPD’99 Conference); AIChE: New York, 2000; in press. (10) Robinson, C. S.; Gilliland, E. R. Elements of Fractional Distillation; McGraw-Hill: New York, 1950. (11) Haesbe, S.; Abdul Aziz, B. B.; Hashimoto, I.; Watanabe, T. Optimal Design and Operation of Complex Batch Distillation
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1849 Column. In IFAC Workshop on the Interaction of Process Design and Process Control; London, 1992; pp 177-182. (12) Hasebe, S.; Kurooka, T.; Aziz, B. B. A.; Hashimoto, I.; Watanabe, T. Simultaneous Separation of Light and Heavy Impurities by a Complex Batch Distillation Column. J. Chem. Eng. Jpn. 1996, 29, 1000-1006. (13) Barolo, M.; Guarise, G. B.; Ribon, N.; Rienzi, S. A.; Trotta, A.; Macchietto, S. Some Issues in the Design and Operation of a Batch Distillation Column with a Middle Vessel. Comput. Chem. Eng. 1996, 20, S37-S42. (14) Farschman, C. A.; Diwekar, U. Dual Composition Control in a Novel Batch Distillation Column. Ind. Eng. Chem. Res. 1998, 37, 89-96. (15) Phimister, J. R.; Seider, W. D. Semi-continuous, MiddleVessel Distillation of Ternary Mixtures. AIChE J. 2000, in press.
(16) Luyben, W. L. Inherent Dynamic Problems with Ondemand Control Structures. Ind. Eng. Chem. Res. 1999, 38, 23152329. (17) A° stro¨m, K. J.; Ha¨gglund, T. Automatic Tuning of Simple Regulators and Specifications on Phase and Amplitude Margins. Automatica 1984, 20, 645-651. (18) Papastathopoulou, H. S.; Luyben W. L. Tuning Controllers on Distillation Columns with the Distillate-Bottoms Structure. Ind. Eng. Chem. Res. 1990, 29, 1859-1868.
Received for review December 8, 1999 Revised manuscript received March 16, 2000 Accepted March 21, 2000 IE990890V
