Energy Flow Patterns and Control Implications for Integrated

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Ind. Eng. Chem. Res. 2010, 49, 8048–8061

Energy Flow Patterns and Control Implications for Integrated Distillation Networks Sujit S. Jogwar and Prodromos Daoutidis* Department of Chemical Engineering and Materials Science, UniVersity of Minnesota, Minneapolis, Minnesota 55455

This paper focuses on the dynamics and control of energy-integrated distillation networks. Typical examples of networks of distillation columns, with significant energy integration, are analyzed with the help of energy flow diagrams. Common patterns (e.g., recycle loops, throughput paths) are identified, which influence the dynamic characteristics of these networks. An example of double effect distillation is considered for detailed dynamic analysis and control. A simulation case study is presented to illustrate the application of the proposed framework. 1. Introduction Distillation is one of the most energy consuming separation processes. In a typical chemical plant, it accounts for ∼40% of the total energy consumption.1 Motivated by this, the synthesis of energy integrated distillation columns has been an area of research for quite some time, resulting in configurations such as vapor recompression distillation, multieffect distillation, thermally coupled distillation (divided wall columns), etc.2,3 Multicomponent separations typically result in a network of distillation columns. The synthesis of such networks has also been an area of rich research activity.4,5 In a single distillation column, energy is input in the reboiler (energy sink) and is taken out in the condenser (energy source). Since the reboiler temperature is higher than the condenser temperature, energy integration through the coupling of reboiler and condenser in a column is not straightforward. However, in a network of distillation columns, the presence of reboilers and condensers operating at different temperatures allows for the coupling of some of these energy sources and sinks through the use of combined reboiler-condensers. It has been welldocumented that such integrated configurations lead to significant energy and, thus, cost savings.6 However, these benefits come at the expense of operating and control challenges. Integration usually leads to stronger dynamic interactions between different units in a system. The tighter the integration, the more severe these interactions. These interactions give rise to slowly evolving network dynamics, in addition to the dynamics of the individual units. The coupling of heating and cooling requirements also results in the reduction of available degrees of freedom. For example, in the case of a combined reboiler-condenser, the condenser and reboiler duties cannot be manipulated independently. Such a coupling also results in feedback interactions, which can further complicate the control problem. Yet, not much effort has been put toward analyzing the dynamics of such networks and its impact on control. Also, the majority of the existing contributions are system specific (see, for example, Table 1). In a different vein, the control of (more general) process networks has recently attracted increased attention in the literature.7-11 In this paper, we establish, through the use of “energy flow diagrams”, that most of these integrated distillation column networks possess energy flow patterns that are combinations * To whom correspondence should be addressed. E-mail: daoutidi@ cems.umn.edu.

of those in two fundamental classes of integrated process networks: networks with large energy recycle12 and networks with large energy throughput.13 These have been analyzed previously in a different context and were shown to give rise to hierarchies of time scales in their dynamic response, which strongly affect the design of the control system. We also focus on a particular configuration (double effect distillation (DED) networks) and develop a generic platform for the dynamic analysis, model reduction, and hierarchical control for such networks, exploiting the underlying energy flow structure. The rest of the paper is organized as follows. In section 2, a brief review of existing literature on the synthesis, dynamics, and control of energy-integrated distillation column networks is presented. In section 3, various examples of energy-integrated distillation configurations are discussed, pointing at common energy flow patterns and their connection to those in generic energy-integrated networks. In section 4, a particular example of a double effect distillation column configuration is considered to illustrate, in detail, the use of the proposed analysis and control framework. Table 1. Control Studies on Energy Integrated Distillation Columns contribution

configuration studieda

system studied

Tyreus and Luyben (1976)34 Lenhoff and Morari (1982)39 Frey et al. (1984)40 Abu-Eishah and Luyben (1985)41 Elaahi and Luyben (1985)42 Alatiqi and Luyben (1986)35 Levien and Morari (1987)43 Chiang and Luyben (1988)36 Al-Elg and Palazoglu (1989)44 Ding and Luyben (1990)45 Pohlmeier and Rix (1996)46 Han and Park (1996)47 Gross et al. (1998)48 Mizsey et al. (1998)37 Engelien et al. (2003)49 Bansal et al. (2000)50 Wang and Lee (2002)51 Hernandez et al. (2004)52 Zhu and Liu (2005)38 Hernandez et al. (2007)53

DED with FS DED with FS and MI DED with FS and MI with FEHE with FEHE Petlyuk Petlyuk DED with MI DED with FS DED with MI DED with FS DED with FS and MI DED with FS DED with MI, Petlyuk DED with MI DED with DED with MI Petlyuk ITCDIC Petlyuk

C3/C3), methanol/water methanol/water A/B (artificial) THF/water HC (quaternary) benzene/toluene/o-xylene methanol/ethanol/water methanol/water methanol/water benzene/toluene/m-xylene methanol/water methanol/water A/B/C (artificial) HC (ternary) methanol/water methanol/water methanol/water HC (ternary) benzene/toluene HC (ternary)

a

Legend for abbreviations: DED, double effect distillation; FS, feed split; MI, material integration; FEHE, feed effluent heat exchanger; ITCDIC, internal thermally coupled distillation column; C3/C3):, propane/propylene; and HC, hydrocarbon mixture.

10.1021/ie101006v  2010 American Chemical Society Published on Web 07/19/2010

Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010

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2. Literature Review 2.1. Distillation Networks with Energy Integration. The synthesis of energy-integrated distillation column networks has attracted significant attention in the past 30 years. Earlier works14,15 focused largely on the synthesis of distillation networks based on heuristics; a review article16 has reported some of these approaches. The concepts of pinch analysis17,18 and bounding technique19-21 (application of bounds on utility usage) have also been used for the synthesis of energy-integrated distillation systems. A two-step synthesis method, addressing separation sequencing, followed by heat-exchanger network design, has been proposed22 for rectification systems. Comparisonbased synthesis, wherein various plausible energy-integrated configurations are rigorously compared, have been proposed for specific cases.23-27 Rigorous mathematical formulations such as mixed integer linear/nonlinear programming28-31 and genetic algorithms32,33 have also been used to synthesize an optimal heat-integrated distillation network. Energy integration typically results in dynamic interactions between various units in the network. Yet, studies analyzing the effect of energy integration on the dynamic performance of such systems are relatively scarce. One of the earliest contributions34 analyzed the dynamics of a double effect distillation column and demonstrated the presence of oscillations stemming from severe interactions between the two columns. A comparison of the dynamic performance of a complex sidestream column/stripper configuration with a conventional nonintegrated column configuration35 was performed for a ternary system. A similar comparative study was also performed36 to select the best (stable and acceptable) configuration from three heatintegrated distillation configurations. The presence of nonlinearity, oscillations (due to internal feedback), inverse response, and sluggish response has also been documented for two energyintegrated distillation configurations.37 Similar dynamic analysis of high-purity thermally coupled columns38 showed the presence of strong asymmetric nonlinearity, inverse response, and high sensitivity to disturbances. One of the first control studies on energy-integrated distillation systems34 proposed the use of auxiliary equipment (condenser/reboiler) to reduce interactions between different columns. Various control studies performed subsequently on different configurations of energy-integrated distillation columns are tabulated in Table 1. The control systems considered include SISO linear control,35,36 internal model control,38,43 MIMO nonlinear control,47 nonlinear decoupling and full linearizationbased control,51 and self-optimizing control.49 Various controllability studies48,54-56 have also been reported for energyintegrated distillation systems, comparing their performance with nonintegrated systems with the help of numerous controllability indices. Most of these dynamics and control studies involved comparison with conventional distillation for a particular separation task and gave little perspective on the application of the proposed solutions to other configurations of distillation networks. 2.2. Simple Prototype Networks with Energy Integration. 2.2.1. Networks with Large Energy Recycle.12 Energy integration involves the recovery and recycle of energy from a source to a sink. Tight energy integration thus results in the recycle of a large amount of energy. A simple prototype network with a large energy recycle is shown in Figure 1; it captures the structural properties of numerous such process networks. For example, in the case of heat-integrated reactors, a large

Figure 1. Prototype network with a large energy recycle.

Figure 2. Prototype network with a large energy throughput.

amount of energy associated with the outlet of an exothermic reactor is recycled back to the cold inlet via a feed effluent heat exchanger. Such networks with a large energy recycle have been shown to exhibit a two-time scale dynamics for the energy balance variables, with fast evolution of the individual unit enthalpy and slow evolution of the total network enthalpy. This time scale multiplicity naturally leads to the following hierarchical control strategy: • Control objectives related to the energy balance of the indiVidual units (e.g., temperature regulation of a particular unit) of the network should be addressed in the fast time scale using the large internal energy flows. • Control and, more importantly, optimization of the energy utilization at the network leVel should be undertaken in the slow time scale, using the small external energy flows of the network as manipulated inputs. 2.2.2. Networks with Large Energy Throughput.13 Figure 2 shows a prototype network with large energy throughput. Qin and Qout represent a large (external) energy source and sink, respectively, leading to a large throughput of energy from Qin to Qout. This prototype network captures structural properties of numerous process networks. For example, in a multieffect evaporator system, a large amount of energy enters the system in the effect operating at the highest pressure and is subsequently rejected from the effect operating at the lowest pressure.58 Such networks with a large energy throughput have been shown to exhibit a two-time scale dynamic evolution, with the entire energy balance dynamics evolving in the fast time scale and the material balance dynamics evolving in the slow time scale. This time scale multiplicity leads to the following hierarchical control strategy: • Control objectives pertaining to the energy balance (e.g., temperature control) should be addressed in the fast time scale, using the large energy flows (Qin and Qout) and, potentially, the flow rates of a subset of the internal material streams as manipulated inputs. • Control objectives related to the material balance, (e.g., the material holdups of the individual units, the total holdup of the network, and the purity of the product(s)) should be addressed over a longer time horizon, employing the remaining flow rates as manipulated inputs. 3. Examples of Energy-Integrated Distillation Networks Let us now consider typical example networks of distillation columns with energy integration. To investigate the underlying

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Figure 3. Energy flow diagram for a process-to-process heat exchanger.

Figure 6. Schematic of the sidestream rectifier.

Figure 4. Double effect distillation (DED) configuration with reverse energy integration.

energy flow structure of these distillation column networks, we will use energy flow diagrams. These diagrams are schematic representations used to visualize the energy flow structure (which, unlike material flows, is not readily apparent) in a network. For example, Figure 3 shows the energy flow diagram for a process-to-process heat exchanger, wherein the hot and the cold sides of the exchanger are represented as two separate blocks connected via an energy transfer stream. 3.1. Double Effect Distillation. Double effect distillation (DED), as shown in Figure 4, represents one of the simplest networks of energy-integrated distillation columns. It consists of two distillation columns operating at different pressures. The heating load of the reboiler of the low-pressure (LP) column is coupled with the cooling load of the condenser of the highpressure (HP) column through a combined reboiler-condenser (RC) (a process-to-process heat exchanger). The remaining reboiler and condenser duties (QCL and QBH) thus form external utilities. To visualize the underlying energy flow structure, we construct an energy flow diagram, as shown in Figure 5. The reboiler and the condenser sides of the RC are shown as two separate blocks (RB-LP and CD-HP) connected by an energy transfer stream. The contribution of latent heat to the enthalpy is typically much larger than the contribution of sensible heat. Therefore, energy flow corresponding to a vapor flow will be larger, compared to a liquid flow. Similarly, the energy transfer in reboilers and condensers is dominated by latent heat effects and

Figure 7. Energy flow diagram for the sidestream rectifier in Figure 6.

is thus comparable in magnitude to the other large energy flows in the network. Representing these large flows with thick lines, we can note that this network acts as a conduit for large energy flow between the external source and the sink. A simpler generic network with a large energy throughput (as shown in Figure 2 and reviewed briefly in section 2.2) shares this energy flow structure. Therefore, this prototype network should capture the structural and dynamic (particularly, the energy dynamics) properties of this DED network. A detailed analysis of this network is pursued in section 4. 3.2. Sidestream Rectifiers/Strippers. Sidestream rectifiers/ strippers are typically used in the petroleum industry as well as in cryogenic air separations (wherein the side stripper is used for the recovery of argon). Figure 6 shows a sidestream rectifier. Note that no separate reboiler is needed for the side column (which results in significant energy savings). The corresponding energy flow diagram is shown in Figure 7. Recognizing again the discrepancies in the energy flows arising due to the presence of vapor and liquid flows as well as the heat transfer dominated by latent heat effects, the energy

Figure 5. Energy flow diagram for the DED configuration with reverse energy integration (DC, column; RB, reboiler; CD, condenser).


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Figure 10. Schematic of the Petlyuk column. Figure 8. Schematic describing heat-pump-assisted distillation.

Figure 9. Energy flow diagram for the heat-pump-assisted distillation column in Figure 8 (in this figure, C denotes compressor).

flow structure for this column points to a forklike structure of large energy flows. This forklike structure can also be viewed as a fused combination of two large energy throughputs: one from the reboiler of the parent column to the condenser of the parent column and the other from the same reboiler to the condenser of the side column. 3.3. Heat-Pump-Assisted Distillation Columns. As noted earlier, in a distillation column, the energy source is at a lower temperature than the energy sink. To facilitate thermal coupling of the source and the sink, the principle of heat pumping is used. Figure 8 shows one such network. Note that the network in Figure 8 represents a more-integrated case of the network in Figure 6, wherein all the energy sources and sinks are thermally coupled. This network uses two RC units to capture the latent heat of condensation of vapor streams and, in turn, to vaporize the liquid stream from column 1. Because the temperature of the condensing vapor is lower than the vaporizing liquid, the boiling temperature in RC1 is reduced using an expansion valve and the condensation temperature in RC2 is elevated through vapor compression. In the corresponding energy flow diagram (see Figure 9), the two RCs are represented as separate blocks. Recognizing the discrepancies in the energy flows due to latent heat effects, the various energy flows in the network span two different orders of magnitude (distinguished by different thickness). We can note

the presence of two large energy recycle loops (DC1-CD1RB1-C1-DC1 and DC1-DC2-C2-CD2-RB2-DC1), one each corresponding to two RCs. A large energy recycle loop forms the core of a class of energy-integrated networks with a large energy recycle12 (as shown in Figure 1 and reviewed briefly in section 2.2). This distillation column network represents an extension of the prototype network with a large energy recycle in Figure 1, with two interconnected loops. 3.4. Petlyuk Columns. Petlyuk columns, as shown in Figure 10, have been the focus of research for more than five decades. It has been well-established that they lead to significant cost savings over conventional sequences for multicomponent separations. The Petlyuk column in Figure 10 separates a threecomponent mixture fed to the prefractionator. The corresponding energy flow diagram is shown in Figure 11. The main column is represented by three blocks: DC-top (with connections to the top of the prefractionator), DC-bot (with connections to the bottom of the prefractionator), and DC-mid (the middle part). The large energy flows are shown with thick lines. Note that this network also acts as a conduit for large energy flow between the energy source and the sink, but with two parallel paths. This can also be viewed as a fused combination of two large energy throughputs (similar, while not identical, to the case of sidestream rectifiers). 3.5. Ideal Heat-Integrated Distillation Columns (HIDiC). The ideal heat-integrated distillation column configuration involves thermal coupling of rectifying and stripping sections.57 Figure 12 shows a typical configuration of HIDiC, wherein the rectifying and the stripping sections are separated into two columns with several internal heat exchangers allowing for heat transfer between the two sections. Because the temperatures in the rectifying section are typically lower than the stripping section, the former is operated at a higher pressure (and, hence, higher temperature) than the latter. The energy flow diagram for this configuration is shown in Figure 13. There is a recycle loop (RS-SS-C-RS) corresponding to large energy flows, showing structural similarity to large energy recycle networks. Note that this configuration operates on a principle similar to double effect distillation; however, the underlying energy flow structures (see Figures 5 and 13) are quite different from each other. 3.6. Complex Networks of Distillation Columns. Lastly, let us consider the network shown in Figure 14. This configuration was proposed in Andrecovich and Westerberg29 as a solution to a five-component separation problem considered therein. It consists of four distillation columns with condensers

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Figure 11. Energy flow diagram for the Petlyuk column in Figure 10 (in this figure, PF denotes the prefractionator).

Figure 12. Schematic of an ideal heat-integrated distillation column.

Figure 14. Multicomponent separation with five species.

Figure 13. Energy flow diagram for the ideal heat-integrated distillation column in Figure 12 (RS, rectifying section; SS, stripping section; TV, throttling valve).

of the first and the third column providing energy to the reboiler of the fourth column through two RCs. The corresponding energy flow diagram is shown in Figure 15. The two RCs are represented by separate reboilers (RB4a and RB4b) and condensers (CD1 and CD3), connected via energy transfer streams. The various energy flows span three different orders of magnitude. The additional segregation of energy flows is due to the discrepancies in material flows (in the case of large reflux ratios), which act as energy carriers. Implementation of large reflux ratios is common in the case of separations involving species with close boiling points.

In this system, we can identify a large energy throughput path between QB2 and QC2. There are two more large throughput paths (QB1fQC4 and QB3fQC4) leading to a forklike structure similar to side rectifiers. There are also six large energy recycle loops (DC4-RB4a-RB4b-DC4, DC4-CD4-DC4, DC2-CD2DC2, DC2-RB2-DC2, DC3-CD3-DC3, and DC3-RB3-DC3) with intermediate magnitude energy flows (note that the block CD represents a condensed form for a column section consisting of N trays). Structurally, this system is thus a combination of the two fundamental networks: networks with a large energy recycle and networks with a large energy throughput. The examples of distillation networks discussed above show that their energy flow structures are combinations of those in the prototype networks with large recycle/throughput. However, the connection between the time scale properties of such networks and those of the prototype networks is not clear a priori. In what follows, we focus on the double effect distillation configuration and correlate the underlying dynamic properties with the established results for the prototype configurations.


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Figure 15. Energy flow diagram for the network of distillation columns in Figure 14 (DC, distillation column; RB, reboiler; CD, condenser).

4. Dynamics and Control of a Double Effect Distillation (DED) Column Network

Column section: For 1 e i < NF, j:

Let us consider a DED configuration with reverse energy integration (reverse with respect to material integration) for separation of a ternary mixture, shown in Figure 4. The feed with components A, B, and C enters the NF, L tray of a lowpressure (LP) column, with NL trays, at a flow rate FL. The bottom stream from the LP column is fed to the high-pressure (HP) column at the NF, H tray. The vapor from the HP column condenses in the reboiler of the LP column. QC, L and QB, H are the condenser and reboiler heat duties for the LP and HP columns, respectively.

dx1,i,j 1 ) [V (y - y1,i,j) + Rj(x1,i-1,j - x1,i,j)] dt Mi,j j 1,i+1,j dx2,i,j 1 ) [V (y - y2,i,j) + Rj(x2,i-1,j - x2,i,j)] dt Mi,j j 2,i+1,j dTi,j 1 ) [V (h˜ (T ) - h˜V(Ti,j)) + dt Mi,jcpl j V i+1,j

For simplicity, let us assume constant tray holdups, constant heat capacities, and constant relative volatility in each column. Under these assumptions, the dynamic material and energy balance equations of this network can be formulated as follows:

Fj(x1,F,j - x1,NF,j) + Rj(x1,NF-1,j - x1,NF,j)] dx2,NF,j 1 ) [V (y - y2,NF,j) + dt MNF,j j 2,NF+1,j

Condenser section: dMC,j ) V j - Rj - D j dt dx1,C,j 1 ) [V (y - x1,C,j)] dt MC,j j 1,1,j dx2,C,j 1 ) [V (y - x2,C,j)] dt MC,j j 2,1,j dTC,j 1 ) [V (h˜ (T ) - h˜l(TC,j)) - QC,j] dt MC,jcpl j V 1,j

Rj(h˜l(Ti-1,j) - h˜l(Ti,j))] dx1,NF,j 1 ) [V (y - y1,NF,j) + dt MNF,j j 1,NF+1,j

(1)

Fj(x2,F,j - x2,NF,j) + Rj(x2,NF-1,j - x2,NF,j)] dTNF,j 1 ) [V (h˜ (T ) - h˜V(TNF,j)) + dt MNF,jcpl j V NF+1,j Fj(h˜l(TF) - h˜l(TNF,j))+Rj(h˜l(TNF-1,j) - h˜l(TNF,j))] For NF, j < i e Nj: dx1,i,j 1 [V (y - y1,i,j) + (Rj + Fj)(x1,i-1,j - x1,i,j)] ) dt Mi,j j 1,i+1,j dx2,i,j 1 [V (y - y2,i,j) + (Rj + Fj)(x2,i-1,j - x2,i,j)] ) dt Mi,j j 2,i+1,j dTi,j 1 [V (h˜ (T ) - h˜V(Ti,j)) + (Rj + Fj)(h˜l(Ti-1,j) - h˜l(Ti,j))] ) dt Mi,jcpl j V i+1,j

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Reboiler section:

dMB,j ) R j + F j - Vj - Bj dt dx1,B,j 1 ) [(R + Fj)(x1,N,j - x1,B,j) - Vj(y1,B,j - x1,B,j)] dt MB,j j dx2,B,j 1 ) [(R + Fj)(x2,N,j - x2,B,j) - Vj(y2,B,j - x2,B,j)] dt MB,j j

dTB,j 1 ) [(R + Fj)(h˜l(TN,j) - h˜l(TB,j)) - Vj(h˜V(TB,j) - h˜l(TB,j)) + QB,j] dt MB,jcpl j where the index j represents the column (L for LP and H for HP); h˜ represents the molar enthalpy of a stream, with the subscripts V and l distinguishing between the vapor and liquid streams. The integrated structure leads to the following equality relations: BL ) FH QB,L ) QC,H As noted earlier, the contribution of latent heat to the enthalpy is much larger than the contribution of sensible heat. Therefore, we can define a small parameter so that h˜l(Ti,s) ) l h˜V(Ti,s) ki and FL,sh˜l(TF,L,s) ) QC,L,s/B,H,s kC,L/B,H where k are O(1) constants. Let us now proceed with the rigorous time scale analysis of this system. Defining the scaled material (u) and energy (w) flows as the ratio of the actual to the steady-state value, we can cast the dynamics in eq 1 in vector form: dx ) f + gu dt

(2)

dζ 1 ) F + gsws + glwl dt with u ) [uV,L uR,L uD,L uB,L uV,H uR,H uD,H uB,H ]T l l l l uV,L · · · wN,L uV,L w1,H uV,H · · · wN,H uV,H wB,H ]T wl ) [wC,L w1,L

[ ]

s s s s s s uV,L wC,L uR,L w1,L uR,L · · · wN,H uR,H wB,H uR,L wB,H uV,H ]T ws ) [wC,L

f ) FL,s

g ) FL,s

[

0 l 0 uL(x1,F,L - x1,NF,L)/MNF,L uL(x2,F,L - x2,NF,L)/MNF,L uL(x1,NF,L - x1,NF+1,L)/MNF+1,L uL(x2,NF,L - x2,NF+1,L)/MNF+1,L l kFuF(x1,N,H - x1,B,H)/MB,H kFuF(x2,N,H - x2,B,H)/MB,H

kV,L -kR,L kV,L(y1,1,L - x1,C,L) 0 MC,L kV,L(y2,1,L - x2,C,L) 0 MC,L kV,L(y1,2,L - y1,1,L) kR,L(x1,C,L - x1,1,L) M1,L M1,L 0

·

0

· · ·

··

-kD,L

0

· · · 0

l 0 -kV,H(y2,B,H - x2,B,H) kR,H(x2,N,H - x2,B,H) MB,H MB,H

0 0

]

F)

[ ]

FL,sh˜l(TF,L,s) × cpl

0 l 0

s s uF,L(wF,L - kNF,LwNF,L )/MNF,L

s s uF,L(kNF,LwNF,L - kNF+1,LwNF+1,L )MNF+1,L l s - ksB,LwsB,L)MB,L uF,L(kN,LwN,L 0 l 0 s s kF,HuF,H(wF,H - kNF,HwNF,H )MNF,H

s s kF,HuF,H(kNF,HwNF,H - kNF+1,HwNF+1,H )MNF+1,H l s s s kF,HuF,H(kN,HwN,H - kB,H wB,H )MB,H

[

FL,sh˜l(TF,L,s) × gs ) cpl -kV,LksC,L MC,L

gl )


0

· · ·

0

0

l

-kR,LksC,L kR,Lk1,L M1,L M1,L

0

·

l

· · ·

0

[

0

·· 0 0 s s kR,HkN,H kR,HkB,H kV,HkB,H MB,H MB,H MB,H

FL,sh˜l(TF,L,s) × cpl -kC,L kV,Lkl1,Lk1,L MC,L MC,L 0

kV,Lkl1,Lk1,L kV,Lkl2,Lk2,L M1,L M1,L ·

l

0

0

· · ·

· · ·

0

0

l

··

0

0 l kB,H kB,H kV,HkB,H MB,H MB,H

] ]

Here, u represents the vector of scaled material flows and the vectors ws and wl represent, respectively, the scaled energy flows corresponding to small and large energy flows. The presence of the small parameter makes the dynamic equations given in eq 2 stiff, showing a potential for two-time scale dynamics. We use singular perturbations to investigate these time scale properties. A fast time scale τ is defined by stretching the original time scale t as τ ) t/. Substituting τ in eq 2, we get dx ) [f + gu] dτ dζ ) [F + gsws] + glwl dτ The description of the dynamics in the fast time scale can be obtained by taking the limit f 0, to get

dx )0 dτ dζ ) glwl dτ

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(3)

We can note that only the energy dynamics (tray temperatures) evolves in this fast time scale. The fast dynamics is influenced only by the scaled inputs corresponding to the large energy flows (wl). These flows serve as potential manipulated inputs to address control objectives in this fast time scale. The control objectives such as temperature regulation at a particular tray should be addressed in this time scale, using one of these scaled large internal energy flows or scaled external large energy flows. This fast dynamics converges to a quasi-steady state captured by the following constraints: 0 ) glwl

(4)

It can be verified that the constraints 4 are linearly independent. Since the quasi-steady-state constraints are linearly independent, we can solve for the quasi-steady-state values for the various temperatures in the network, T* ) φ(x, uˆ) (where uˆ are the material flows not set by fast controllers). Thus, the energy dynamics entirely evolves over a fast time scale and the material dynamics corresponds to the slow dynamics of the network. The slow time scale material dynamics can now be given as dx ) f(x, T*) + g˜(x, T*)uˆ dt

(5)

Equation 5 can now be used to address control objectives in the slow time scale, such as the regulation of condenser/reboiler holdups, the control of exit concentrations, etc. uˆ represents the subset of the scaled material flows, which are not set by fast controllers and, hence, are available for manipulation in the slow time scale. The dynamic properties of this network have a striking similarity with the properties of the prototype network with high energy throughput. These two networks share the same core structure with respect to energy flows. Therefore, it can be reasonably argued that this core structure is at the origin of this time scale multiplicity in dynamics and that the hierarchical control strategy discussed in section 2.2 also applies to other Table 2. Nominal Values of Parameters for the Double Effect Distillation (DED) Configuration parameter

nominal value

parameter

nominal value

x1, C, L x2, C, L x2, B, L x3, B, L x1, F, L x2, F, L DL BL FL RL VL TC, L TB, L TF, L QC, L QB, L MC, L MB, L PC, L NL NF, L

0.9126 kg/kg 0.0874 kg/kg 0.6169 kg/kg 0.3696 kg/kg 0.2119 kg/kg 0.5000 kg/kg 1.69 kg/s 5.98 kg/s 7.68 kg/s 2.30 kg/s 7.98 kg/s 309.96 K 397.00 K 288.50 K 1.71 MW 2.98 MW 65.32 kg 905.12 kg 0.2 bar 37 16

x1, C, H x2, C, H x2, B, H x3, B, H x1, F, H x2, F, H DH BH FH RH VH TC, H TB, H TF, H QC, H QB, H MC, H MB, H PC, H NH NF, H

0.0202 kg/kg 0.9089 kg/kg 0.0363 kg/kg 0.9637 kg/kg 0.0135 kg/kg 0.6169 kg/kg 4.00 kg/s 1.98 kg/s 5.98 kg/s 4.08 kg/s 9.20 kg/s 387.08 K 442.25 K 397.00 K 2.98 MW 3.09 MW 64.68 kg 903.52 kg 1.06 bar 44 29

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Figure 16. Open-loop response of the DED system for a step change in QB,H from 3.09 MW to 3.15 MW.

Figure 17. Open-loop response of the DED system for a step change in FL from 7.68 kg/s to 8.00 kg/s.

networks, which are structurally similar (i.e., sharing the same core structure) to these networks. Let us now consider a simulation case study to further illustrate these points. We considered the separation of a benzene-toluene-m-xylene mixture in a DED configuration, as shown in Figure 4. Primary control objectives considered were the control of concentrations of the three exit streams: DL, DH, and BH. The nominal values of the process parameters are given in Table 2.

The system was modeled with gPROMS,59 which provides an equation-oriented modeling system for building firstprinciples models within a flow-sheeting framework. A comprehensive library containing typical standard unit operations (such as heat exchangers, pumps and compressors, kinetic reactors, and distillation columns) with open models (possible inspection and change of the model equations) allows for the implementation of complex integration loops. Integration with various thermodynamics packages facilitates simulating real

Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010 Table 3. Controller Summary for the Double Effect Distillation (DED) Case Study control variable

manipulated input

LC, L LC, H LB, L LB, H PC, L PC, H TB, H xC, L

DL DH BL BH QC, L QC, H QB, H RL

xC, H

RH

xB, H

TB, H, set

controller details proportional, K ) -20 proportional, K ) -20 proportional, K ) -750 proportional, K ) -750 PI, K ) 10, τI ) 30s PI, K ) 10, τI ) 30s PI, K ) -6.44 × 104, τI ) 60s input/output linearizing controller with output feedback β2 ) 9 min2, β1 ) 6 min input/output linearizing controller with output feedback β2 ) 9 min2, β1 ) 6 min PI, K ) 25, τI ) 600 s

component systems. In this case study, the distillation columns and the heat exchangers were modeled using gPROMS Process Model Library v3.1. The built-in IPPFO (ideal physical proper-

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ties foreign object) package was used for property estimation. In gPROMS, the condenser, reboiler, and column sections are built separately. A constraint equating the duties of the HP condenser and the LP reboiler formulated the RC. For the demonstration of the time scale properties, we implemented stabilizing hold-up controllers for the reboilers and the condensers and pressure controllers for the condensers. No product-purity controller was implemented. First, we analyzed the effect of a step change (3.09 MW to 3.15 MW) in the heat input of the reboiler of the HP column (QB, H), and Figure 16 shows the evolution of two temperatures: one from each column and the corresponding mass fractions. With QB, H being a fast disturbance to the system, the temperatures showed a fast transient response (with a subsequent slow approach to a new steady state). The mass fractions only showed a slow approach to a new steady state, confirming their evolution in the slow time scale. Next, we considered a step change (7.67 kg/s to 8.00 kg/s) in the feed flow to the LP column (FL). This is a

Figure 18. Closed-loop response of the DED system for a set-point change in x1, C,L.

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Figure 19. Closed-loop response of the DED system for a set-point change in x2, C,H and unmeasured disturbance in feed composition.

slow disturbance and, hence, affects only the slow dynamics of the system. This was confirmed from the response of the system, as shown in Figure 17. Let us now proceed to the design of a control system for this network. Level controllers (equivalent to hold-up control) for condensers and reboilers were installed using the distillate (D) and bottoms (B) flow rates as the manipulated inputs respectively. Pressures in the LP and HP condensers were controlled using the condenser duty of the LP column (QC,L) and the heattransfer rate across the RC (QC,H ≡ QB,L), respectively. In the fast time scale, the temperature in the reboiler of the HP column (TB,H) was controlled using the reboiler duty QB,H (large energy flow). The internal material flows (VL and VH) depend, through constitutive relations, on the corresponding reboiler temperatures (TB,L and TB,H). Since the duties of the two reboilers were already fixed (through energy integration and pressure/temperature control), these internal material flows were not available for manipulation in the fast time scale.

In the slow time scale, control of exit concentrations (x1, C,L, x2, C,H, and x3, B,H) was addressed. The material flows RL and RH were the available manipulated inputs. There was a shortage of one manipulated input, so we chose the set point for the temperature controller (TB,H, set) as the third manipulated input, resulting in a cascaded configuration. Such a cascaded configuration was possible because the temperature controller was faster, compared to the concentration controller. Table 3 tabulates the details of all these controllers. Since the level is not required to be controlled exactly at the set point, simple proportional controllers were used for level control. Linear proportional integral (PI) controllers were used for pressure control to guarantee tight regulation of pressure, which is essential for the operation of the system. We considered nonlinear controllers for the control of exit concentrations, since they can be subjected to frequent transitions. Specifically, we implemented output feedback input/output linearizing control.60 The slow model (eq 5) was used for controller derivation and


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Figure 20. Closed-loop response of the DED system for an increase in the feed flow rate (FL).

also as an open loop observer. The relative degree for each output is 2, so we requested second-order responses of the form β2

d2y dy + β1 + y ) V 2 dt dt

y: ) xC,L, xC,H

To get offset-free response, we added external error feedback action of the form

(

V ) yset + Ks (yset - y) +

1 τs

∫ (y t

0

set

)

- y) dtˆ

with Ks ) 2/β1 and τs ) β1.60 The values of the controller parameters βi are chosen to result in a critically damped response with a time constant on the order of the process slow time constant. For the first simulation run, we considered a set-point change (0.9126 to 0.95) in x1, C, L, and the corresponding response is shown in Figure 18. We can note that the set-

point change in the top composition of the LP column leads to activation of the control loops in the HP column, demonstrating the dynamic coupling in the system. The hierarchical control strategy yields a smooth operating point transition. In the next simulation run, we considered a set-point change (0.9089 to 0.92) in x2, C, H, and the corresponding response in the presence of an unmeasured disturbance in the feed composition is shown in Figure 19. We can note that the controllers are quite robust to unmeasured disturbances and modeling errors (resulting from to the mismatch between the process model and the slow model (eq 5) used for the controller design). Finally, we considered a 5% increase in the feed rate, and the corresponding response is shown in Figure 20. The proposed control strategy also yields a very good performance in production rate change.

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5. Conclusions and Outlook for Complex Networks In this paper, we studied the dynamics and control of networks of distillation columns with significant energy integration. Various configurations of energy-integrated distillation columns were analyzed with the help of energy flow diagrams. The underlying energy flow structure showed connections with two fundamental classes of energy integrated networkssnetworks with a large energy recycle and networks with a large energy throughputsfor which the time scale properties and the subsequent use of hierarchical control strategy has already been established. We performed a rigorous dynamic analysis for a ternary separation in a double effect distillation (DED) configuration and correlated its dynamic properties with the underlying energy flow structure. We confirmed the predicted time scale properties through open-loop simulations. A hierarchical control strategy was proposed, and the closed-loop system was tested for various operational scenarios. The proposed control strategy demonstrated very good performance for various operating scenarios. The example of DED emphasizes the importance of the energy flow structure on the dynamics and control of networks of distillation columns. For the case of simple networks (with a single integration loop), the correlation between the underlying energy flow structure and the dynamic properties is an easier task. However, note that many of the configurations presented in section 3 are complex (with multiple integration loops). The underlying energy flow structure represents combinations of one or both the prototype networks, with fused or independent loops, and with energy flows spanning multiple orders of magnitude. A rational approach for the analysis of such complex networks is to systematically decompose the corresponding energy flow structure into its constituents (the core structures) via structure reduction algorithms based on graph theory. This and the detailed analysis of such networks is part of our ongoing research. Nomenclature B ) bottoms molar flow [mol/s] cp ) molar heat capacity at constant pressure [J/mol/K] D ) distillate molar flow [mol/s] F ) feed molar flow [mol/s] Fi ) material flows [mol/s] f ) vector function F ) vector function g ) matrix function h ) enthalpy flow [W] Hi ) molar enthalpy [J/mol] h˜ ) molar enthalpy [J/mol] k ) O(1) constant K ) proportional gain of a controller L ) exit liquid molar flow [mol/s] M ) molar holdup [mol] N ) number of trays in distillation column NF ) feed tray in distillation column P ) pressure [bar] Q ) energy flow [W] R ) reflux molar flow [mol/s] T ) temperature of a stream [K] T* ) quasi-steady-state temperature [K] u ) scaled material flows u ) vector of scaled material flows uˆ ) flows set by slow controller V ) vapor molar flow [mol/s]

V ) controller input w ) scaled energy flow w ) vector of scaled energy flow x ) material balance variable x ) mole fraction in liquid stream y ) mole fraction in vapor stream Greek Symbols β ) nonlinear controller tuning parameter ) singular perturbation parameter τ ) stretched time scale [s] τI ) integral time constant [s] ζ ) energy balance variables Subscripts i, j, k ) indexes (e.g., ith component, jth tray, kth column) s ) steady-state value V ) vapor phase l ) liquid phase f ) feed in ) inlet stream out ) outlet stream set ) set point value

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ReceiVed for reView May 2, 2010 ReVised manuscript receiVed June 21, 2010 Accepted June 23, 2010 IE101006V

Energy Flow Patterns and Control Implications for Integrated

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