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Ind. Eng. Chem. Res. 2010, 49, 8670–8684
Rigorous Separation Design. 2. Network Design Solutions for Mixtures with Various Volatility Differences and Feed Compositions Seon B. Kim and Andreas A. Linninger* Laboratory for Product and Process Design, UniVersity of Illinois at Chicago, Chicago, Illinois 60607
This article presents progress in the computer-aided synthesis of energy-efficient complex column networks based on the automatic temperature collocation algorithm. Three prefractionating complex networks are employed to split several quaternary mixtures into four almost-pure end products. Network solutions for two different mixtures with different difficulties of separation as well as various feed compositions are presented. Twelve network designs, all satisfying stringent product quality requirements, but with different energy demands and capital costs, are laid out in detail. The design computations with the temperature collocation methodology are demonstrated to agree closely with rigorous mass, equilibrium, summation, and heat computations verified with the industrial-standard flow sheet simulator AspenPlus. In fact, detailed design specifications were so accurate that their use to initialize AspenPlus flow sheet simulations leads to convergence in a few iterations to essentially the same end products. The success of the method demonstrates that rigorous solutions to separation problems can be obtained in a fraction of valuable engineering design time by computer methods. The automatic and rigorous flow sheet synthesis is apt to systematically address process design problems such as the synthesis of energy-efficient separation networks, the layout of biorefineries with novel feedstocks, or a sustainable process for reduction of greenhouse gases emissions. 1. Introduction Recently there has been a renewed interest from academic and industrial researchers in the systematic synthesis of separation processes to meet modern energy efficiency and environmental sustainability requirements.1 The prevalence of distillation as the most eminent unit operation in the chemical industry makes it an ideal target for industry-wide energy and emission reduction efforts.2 We have discovered new algorithmic approaches to computer-aided synthesis of separation networks to achieve desired purity targets with optimal performance. An earlier collocation method reduces the number of dimensions drastically by converting the tray number into a continuous variable.6,7 The mathematical foundation of the novel temperature collocation approach was derived in an earlier paper.3 A fully automatic synthesis procedure for the generation of simple column networks was presented elsewhere.4 The first part of this series5 demonstrated an extension of temperature collocation for the design of complex distillation networks. Despite search space reduction, the temperature collocation solutions will be shown to be equivalent to column profiles obtained with rigorous mass, equilibrium, summation, and heat (MESH) equations. Complex columns are distillation units with more than one feed stream or multiple product streams in addition to a single bottoms and a single distillate leaving conventional columns. Complex separation networks are significant, because there appears to be consensus that they offer the potential of meeting high purity specifications with much less energy than conventional configurations.5,8–16 The synthesis of a complex separation network is an interesting challenge, because several units such as prefractionators do not produce final product cuts, but merely split the mixture into such intermediate streams that the final purity targets can be met in a subsequent multifeed complex column. In effect, the design of these special columns like prefractionators is not covered by traditional design strategies such as the need for sharp splits. * To whom correspondence should be addressed, E-mail:
[email protected].
These special integrated configurations need to be optimized in their complete ensemblesthe entire separation networksas opposed to the traditional piecemeal unit-by-unit optimization. Especially for separating nonideal mixtures, there are currently no rigorous computer algorithms to guide design engineers toward optimal network solutions. Part 1 of the series5 demonstrated that the temperature collocation methodology succeeds in synthesizing entire separation flow sheets with complex or traditional configurations, or the combination of simple or complex assemblies. Network solutions were generated for the separation for mixtures, whose phase equilibrium may be described by a nonideal vapor-liquid equilibrium models including azeotropes, ideal solution behavior, or constant relative volatility models. We generated a feasible separation flow sheet for multicomponent mixtures with numerous species. A 10-compound separation task was solved without resorting to any limiting assumption such as lumping of species, arbitrary selection of key or nonkey compounds, admission of shortcut equations, or simplified phase equilibrium models. The second part of this paper series will illustrate potential benefits in terms of energy and capital cost that can be achieved by rigorous separation synthesis. We will arrive at the important conclusion that the synthesis of optimal networkssespecially heat-integrated complex networkssunfortunately cannot be performed with shortcut methods, performed with rule-of-thumb or combinatorial separation heuristics, or addressed by intuition. For a complex separation network solution to be optimal, the separation problem needs to be solved rigorously for the specific feed compositions and the appropriate phase equilibrium model. This paper extends the temperature collocation methodology5,28 with minimum bubble point distance feasibility criterion3 for the automatic and rigorous design of entire flow sheets. Numerous approximate techniques, such as the rectification body method17 and the Underwood method,18,19 have been proposed to facilitate the design of distillation systems. There are also other separation synthesis techniques such as the zero volume
10.1021/ie100355c 2010 American Chemical Society Published on Web 08/09/2010
20
21
method, eigenvalue methods, and minimum vapor diagrams.22 However, previous works mainly address merely simple column configurations. Topological methods often do not apply to general nonideal vapor-liquid phase equilibrium models. We will show how to separate multicomponent mixtures with prefractionating and complex column configurations for different feed specifications and mixtures with various degrees of separability due to the vapor pressure differences of the chemical compounds to be separated. The paper is organized as follows: The Methodology section describes a practical implementation of our inverse method to synthesize entire separation networks. The Results and Applications section presents specific separation network solutions to split two distinct quaternary mixtures into almost pure products. In addition, the impact of two different feed compositions on the structure and operating conditions of heat-integrated complex column networks will be discussed. All network solutions will be analyzed in terms of their energy consumption. We will also show in detail how the differences in the ease of separations determines the structure and operating parameters of the resulting network solutions. To demonstrate that the proposed design solutions are realistic and industrially realizable, every network configuration for all case studies is validated with rigorous AspenPlus23 simulations. Finally, the paper closes with Discussion and Conclusions. 2. Methodology For the comprehensive derivation of the mathematical foundations of temperature collocation, bubble point distance, and the global feasibility criterion used to design complex column networks, we refer the interested reader to the first part5 and earlier papers.3,4,24,25 The practical implementation of these concepts for the applications in this paper is summarized in the next subsections. We will show a three-step procedure: (i) a method to enforce the purity targets for all desired product cuts, (ii) how to compute rigorous composition profiles traversing each equilibrium stage, and (iii) the mathematical relations to identify column specification that solve the separation task with columns that can be built in the real world. This last task will require a rigorous column and network feasibility criterion. 2.1. Step 1. How To Set Up Separation Networks for Desired Product Purities. Before tackling the detailed column design, it is necessary to find the total product flow rates, Pm, as a function of given purity targets. We propose to specify desired fractional recoveries, fj,Pm, or molar product compositions, xj,Pm. Equation 1 is an input-output species balance relating total molar product flow rates to desired fractional recoveries as well as total molar flow rates of each species j entering a column nj. In multifeed complex columns, nj is the sum of molar fluxes of K component j in all feed streams, nj ) ∑k)1 Fkzj,k, where Fk is the total feed flow rate of feed stream k, and zj,k is its component j molar composition. The product flow rates, Pi, i ) 1, ..., m, can be calculated as functions of product purities in terms of molar composition as in eq 2.
[][
f1,P1 P1 f1,P2 P2 ) l l PM f1,PM
f2,P1 f2,P2 l f2,PM
··· ··· · ·· ···
fJ,P1 fJ,P2 l fJ,PM
][ ] n1 n2 l nJ
(1)
[][
][
Ind. Eng. Chem. Res., Vol. 49, No. 18, 2010
x1,P1 P1 x2,P1 P2 ) l l PM xJ,P1
x1,P2 · · · x2,P2 · · · · ·· l xJ,P2 · · ·
x1,PM x2,PM l xJ,PM
-1
z1,F1 z1,F2 · · · z2,F1 z2,F2 · · · · ·· l l zJ,F1 zJ,F2 · · ·
z1,FK z2,FK l zJ,FK F1 F2 l FK
[]
]
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(2)
Knowledge of the product streams in addition to selecting operating conditions such as specifying the column’s reboil or reflux ratios, s or r, allows the determination of the internal vapor flows, Vi, and liquid flows, Li, in each column section i of the network. A total mass balance for a section yields the internal flow rates in each column section, a result needed to derive the identities 3-6 valid for section S1. Figure 1 also depicts the labels of all internal and external species balances, feed stages, and product stages, and the relationships between the internal flow rates and compositions. x∆j,1 ) xj,P1
(3)
y∆j,1 ) xj,P1
(4)
L1 ) rP1
(5)
V1 ) P1(r + 1)
(6)
The relationships for the other sections can readily be derived leading to the generalized procedure for equivalent rectifying and stripping sections. The equations for the equiValent rectifying sections are given in eqs 7-10. Li ) Li-1 - P(i+1)/2
(7)
Figure 1. Schematic diagram of complex column equipment consisting of K feed streams, F1, ..., FK, and M product flows, P1, ..., PM. This unit is composed of I sections where eq 19 is employed. Odd stage indices correspond to equivalent rectifying sections and even stage indices apply to equivalent stripping sections.
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Vi ) Vi-1
(8)
x∆j,i ) xj,P(i+1)/2
(9)
y∆j,i ) y∆j,i-1
(10)
The relations of internal flow rates and key section parameters for equiValent stripping sections are given in eqs 11-14. Li ) Li-1 + qi/2Fi/2 Vi ) Vi-1 - (1 - qi/2)Fi/2
dxj,i ) dT -
( ∑ {[(
) )
1+
1 1 (x - yj,i) + (X - xj,i) R∆i j,i R∆i ∆j,i
1+
1 1 (x - yj,i) + (X - xj,i) Kj,i R∆i j,i R∆i ∆j,i
c
j)1
] }
c
y∆j,i )
(13)
j)1
∑ {[(1 + R1 )(x - y ) + R1 (X c
Finally, for the bottom section, an equilibrium relation is needed for the reboiler as in eqs 15-17. Here, x∆j and y∆j are the liquid and vapor compositions of species j leaving the reboiler. Kj is the equilibrium constant between vapor and liquid streams of compound j leaving an equilibrium stage.
x∆j,I )
1 (V y + PMxj,PM) LI I ∆j,I s)
VI PM
j)1
∆
j
∆j
]}
- xj) Kj
∆
∆i ) Vi - Li
(20) (14)
y∆j,I ) Kj(TPM, xPM)xj,PM
dKj
with R∆i ) Li /∆i, X∆j,i ) (Viy∆j,i - Lix∆j,i)/∆i,
1 (L x + Vi-1y∆j,i - Li-1x∆j,i - Fi/2zj,i/2) Vi i ∆j,i
(19)
j
dn )dT
j
x∆j,i ) xj,Pi/2+1
j)1
dKj,i x dT j,i
∑ dT x
(11) (12)
c
∑
(15) (16)
(17)
In each section, the bubble point temperature corresponding to the product composition is also equal to the first stage temperature. Equation 18 allows the determination of key parameters R∆i, X∆j,i, and ∆i for each column section i of the entire network. R∆i ) Li /∆i, X∆j,i ) (Viy∆j,i - Lix∆j,i)/∆i, ∆i ) Vi - Li (18) 2.2. Step 2. How To Compute Column Profiles. Our temperature collocation method can be understood as an approximate method to solve the rigorous mass, equilibrium, summation, and heat (MESH) equations. Instead of directly formulating component and equilibrium balances on each tray, we deploy a set of continuous differential equations for the evolution of the liquid composition profiles. The locus of the liquid composition profiles is called a column profile, first proposed by Doherty.26 Recently, Glasser and Hildebrand27 derived generalized column profile equations for any type of column section, so they apply to column sections in complex columns, side strippers, and side rectifiers, but also to conventional stripping and rectifying sections. Column sections in a complex column can be classified as an equivalent stripping section, if its vapor rate is higher than the liquid flow; or an equivalent rectifying section in the opposite case. Our group has further suggested the beneficial transformation from tray numbers, n, to the bubble point temperature, T. Equation 19 is the column profile equation with the bubble point temperature along the column length serving as an independent variable. Furthermore, the transformation relation between stage numbers and temperatures is governed by eq 20.
In the generalized profile equations for each section appear the generalized reflux, R∆i, and the difference point composition for species j, X∆j,i. The generalized reflux for section i, R∆i, is the recirculation ratio with a role similar to the reflux in conventional columns. The difference point compositions, X∆j,i, represent the concentration difference between the vapor and liquid stream leaving or entering the section, analogous to the role of the distillate or bottom composition in conventional stripping or rectifying sections. ∆i is the net flow in the column section i. Positive ∆ constitutes an equivalent rectifying section, while negative net flow represents an equivalent stripping section. More details on the physical interpretation of generalized reflux, difference points, and their role in separation design can be found elsewhere.12 In eqs 19 and 20, xj,i and yj,i are the liquid and vapor compositions in the phase equilibrium relationship leaving an equilibrium stage related by the equilibrium constant Kj,i. x∆j,i and y∆j,i are the liquid and vapor compositions of the first stage used to initialize eq 19, Li and Vi are the internal liquid and vapor flow rates of each column section i. The profile equations are solved for each species j in any column section i, xj,i, by global collocation on orthogonal polynomials on finite elements with temperature as independent variable. All profiles are integrated up to the stable pinch point of each section, xpinch j, Tpinch. Thus eq 19 is solved between the first stage, x∆j, T∆, and the stable pinch, xpinch j, Tpinch. During the composition profile computation with temperature collocation, pinch points play an important role. In an equivalent rectifying section, the temperature of the stable pinch point delineates the furthest point of any column profile, marking also the highest bubble point temperature any column section can reach. We also found that the bubble point temperatures of the saddle pinches coincide with regions of the highest curvature in the composition profiles.5 Even though the exact pinch point location is not a requirement, we usually deploy it for advancing higher precision in the composition profiles by proper element and node placement. Accordingly, we recommend placing finite element boundaries more densely around saddle bubble point temperature regions. As a result, the adaptive node placement using the pinch point temperatures can improve the accuracy of the collocation method.5 Strategies to identify pinch points are described in Appendix A. 2.3. Step 3. Computation of Bubble Point Distance To Ensure a Feasible Column. After finding the liquid composition profiles of all column sections, the minimum bubble point distance criterion is performed to check whether the current
Ind. Eng. Chem. Res., Vol. 49, No. 18, 2010 Table 1. Antoine Parameters of the Alkane Mixture of Pentane (A), Hexane (B), Heptane (C), and Octane (D) Used for the Calculation of the Vapor Pressurea components (xA, xB, xC, xD) Antoine parameter A Antoine parameter B Antoine parameter C a
[pentane, hexane, heptane, octane] [6.85221; 6.8702; 6.8938; 6.9094] [1064.630; 1168.7; 1264.4; 1349.8] [232.000; 224.2100; 216.6360; 209.3850]
Log[P (mmHg)] ) A - B/[(T °C) + C].
choices for the specifications such as product and operating conditions lead to a realizable column. According to the bubble point distance method,3 we calculate the globally minimum distance between each pair of adjacent column sections of all columns of the network; each result is called a minimum bubble point distance, minBPD. If all these scalar minimum distances, minBPD, are below a small tolerance, the column is realizable as in expression 21. A single minBPD for any section in the entire network larger than zero means that the corresponding columnsand thus the connected networksis guaranteed to be not realizable. In practice, a tolerance, minBPD e 10-4, was sufficient. As in all numerical computations, the choice of BPD tolerance is significant. In our case studies, even a sloppy tolerance on the order of minBPD e 10-4 was found to coincide with rigorous MESH results. Nevertheless, we recommend accepting final design solutions only after passing plausibility and visual inspection that a profile intersection has really occurred. It is conceivable to construct pathological cases, in which even closeness does not mean intersection such as two profiles approaching each other, yet on two different sides of a distillation boundary. Such extreme cases are theoretically possible, especially when using digital computers with floating point arithmetic, but we have not encountered such cases in our design experience. Since the tolerance criterion with collocation is equivalent to a rigorous MESH equation, when it fails, also the MESH calculations are likely to fail. The entire network is feasible, if and only if all its simple or complex columns k are feasible as expressed in eq 22. When infeasibility is encountered, some design choices (purities of nonkey species or intermediate streams, refluxes) need to be altered to arrive at feasible specifications. The search for these parameters has been successfully implemented with specialized stochastic algorithms.3 The global search for the minBPD cansthanks to the choice of a collocation method on orthogonal polynomialss be effectively implemented in polynomial arithmetic. The implementation of global optimization in polynomial arithmetic is described in Appendix B. φ(k) )
∑ minBPD(T) < ε
1
(21)
K
Ψ(k) )
∑ φ(k) < ε
2
(22)
k)1
When all sections are feasible, the entire network is feasible and is smoothly connected through the intermediate feed and product streams according to the global species balances of step 1. The following example better illustrates the three steps for the design of a feasible single complex column with three feeds and four product streams. 2.4. Design Example: Multifeed Complex Column with Four Product Streams. A quaternary mixture of pentane (A), hexane (B), heptane (C), and octane (D) with vapor-liquid phase equilibrium data given in Table 1 is to be separated into products with purity of 99% or higher. We assume that some prefractionation within the complex separation network has already occurred upstream so that three different feeds with
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Table 2. Feed and Product Specifications To Design a Four Product Stream Complex Column Shown in Figure 2 To Separate a Quaternary Alkane Mixture of Pentane (A), Hexane (B), Heptane (C), and Octane (D) stream
[xA, xB, xC, xD]
F1 F2 F3 P1 P2 P3 P4
[0.5765; 0.4235; 0.0000; 0.0000] [0.0000; 0.3207; 0.6793; 0.0000] [0.0000; 0.0000; 0.3126; 0.6874] [0.99; 9.9999 × 10-3; 9.99 × 10-8; 1.0 × 10-10] [4.9999 × 10-3; 0.99; 4.9999 × 10-3; 2.0 × 10-7] [0.0000; 0.0041; 0.99; 0.0059] [0.0000; 1.0 × 10-7; 9.9999 × 10-3; 0.99]
flow rate q (kmol/h) 1 1 1 1 1 1 1
43.3880 20.1485 36.4636 25.1399 24.7317 24.9578 25.1705
Table 3. Antoine Parameters for a Mixture of Benzene (A), Toluene (B), Octane (C), and Nonane (D) Used to Compute the Partial Vapor Pressurea components (xA, xB, xC, xD) Antoine parameter A Antoine parameter B Antoine parameter C a
[benzene, toluene, octane, nonane] [6.90565; 6.95464; 6.90940; 6.93440] [1211.033; 1344.800; 1349.820; 1429.460] [220.790; 219.482; 209.385; 201.820]
Log[P (mmHg)] ) A - B/[(T °C) + C].
flows and compositions listed in Table 2 are to be processed. We choose to address the separation task with a single complex column with four product streams including two side products as shown in Figure 2. The final separation should meet stringent product purity requirements given in Table 2. Step 1 consists in finding product streams and internal molar flow rates. For a specific reflux of r ) 3.5, the product flows Pm for the four products, the internal flow rates Vi and Li, parameters such as the characteristic column profile parameters R∆i, X∆j,i, and ∆i, the initial section stage compositions x∆j and y∆j, and the associated bubble point temperatures T∆j for the six column sections were computed using eqs 1-18. The resulting values for each section are summarized in Table 4. In step 2, we also located pinch points as indicated in Appendix A. Then, we solved eq 19 to compute composition profiles for each column section between the bubble point temperatures of the appropriate products and the section stationary pinch point, respectively. Figure 2b depicts the resulting composition profiles. In step 3, polynomial arithmetic allowed the global determination of the minimum bubble point distance, minBPD, between each pair of adjacent equivalent rectifying and stripping sections. The minBPD values between three pairs of adjacent rectifying and stripping equivalent sections are also reported in Table 4. The near-zero minBPD proves that the complex column is feasible, because criterion 21 is satisfied. Again referring to Figure 2b, it depicts these composition profiles for a feasible design whose trajectories connect all product and feed streams without a gap. In contrast, Figure 2c shows the liquid composition profiles, computed in similar fashion through steps 1-3, but for a smaller reflux ratio of r ) 2.5. This specification is unfortunately infeasible, because the globally minimum bubble point distance between section pairs S1-S2, S3-S4, and S5-S6 are clearly larger than zero. These three violations can be easily identified as a gap in the liquid composition trajectories of Figure 2c. In general, a single violation of the minimum bubble distance criterion is sufficient to guarantee infeasibility of the given specification! Figure 2b,c also plots the equivalent stage numbers obtained by converting temperature back to stages with the help of eq 20. The unique and clear distinction between feasible and infeasible specifications is one of the major advantages of the inverse design methodology based in temperature collocation
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Figure 2. Panel A: Case study of a complex column with three feeds and four product streams employed to separate a quaternary mixture of pentane (A), hexane (B), heptane (C), and octane (D) into products of 99% purity. The column was designed by solving eq 19 in all six sections. Panel B: Liquid composition profiles of a quaternary alkane mixture of pentane (A), hexane (B), heptane (C), and octane (D) through a feasible six section complex column on bubble point temperature stage (right axis) and liquid composition versus number of stages. For a reflux of r ) 3.5, the profiles have zero bubble point distance (BPD) between section pairs S1-S2, S3-S4, and S5-S6. Panel C: For a reflux of r ) 2.5, the liquid composition profiles between all three pairs of sections, S1-S2, S3-S4, and S5-S6 have a gap. This nonzero BPD characterizes infeasible specifications. Component A, blue -O-; component B, green -×-; component C, pink -0-, component D, red - + -. Table 4. Operating Conditions for Each Section of a Four Product Complex Column Shown in Figure 1B Separating a Quaternary Alkane Mixture of Pentane (A), Hexane (B), Heptane (C), and Octane (D) into Products with Purity of at least 99%a r ) 3.5 x∆j xA xB xC xD T∆, °C xpinch j xA xB xC xD Tpinch j, °C y∆j xA xB xC xD X∆j,i xA xB xC xD Li, kmol/h Vi, kmol/h R∆i ∆i minBPD minBPT, °C a
S1
S2
S3
S4
S5
S6
0.9900 9.999 × 10-3 9.999 × 10-8 1.000 × 10-10 36.2758
4.999 × 10-3 0.9900 4.999 × 10-3 2.000 × 10-7 68.5700
4.999 × 10-3 0.9900 4.999 × 10-3 2.000 × 10-7 68.5700
1.000 × 10-17 4.100 × 10-3 0.9900 5.900 × 10-3 98.3679
1.000 × 10-17 4.100 × 10-3 0.9900 5.900 × 10-3 98.3679
8.205 × 10-17 3.797 × 10-7 1.872 × 10-2 0.9813 124.9427
2.922 × 10-2 7.732 × 10-4 2.503 × 10-8 9.7000 116.8324
0.7962 0.2038 -2.142 × 10-8 -1.986 × 10-11 40.4954
1.103 × 10-17 1.698 × 10-2 1.056 × 10-3 0.9820 123.5522
1.0547 -5.382 × 10-4 -0.0541 1.831 × 10-8 39.4596
1.157 × 10-17 -6.941 × 10-9 9.849 × 10-2 0.9015 121.9711
-8.572 × 10-3 1.434 × 10-7 1.089 × 10-2 0.9977 42.0960
0.9900 9.999 × 10-2 9.999 × 10-8 1.000 × 10-10
4.713 × 10-3 0.9895 5.807 × 10-3 2.323 × 10-7
4.713 × 10-3 0.9895 5.806 × 10-3 2.323 × 10-7
1.050 × 10-16 3.691 × 10-3 0.9897 6.613 × 10-3
1.050 × 10-16 3.691 × 10-3 0.9897 6.613 × 10-3
9.808 × 10-17 4.420 × 10-7 2.065 × 10-2 0.9793
0.9900 6.776 × 10-3 9.999 × 10-3 0.9932 9.999 × 10-8 -1.376 × 10-7 1.000 × 10-10 -1.378 × 10-10 87.9897 131.3777 113.1297 113.1297 3.5000 -7.1996 25.1399 -18.2480 1.036 × 10-5 42.8742
1.644 × 10-15 -7.766 × 10-16 0.9809 7.489 × 10-3 1.907 × 10-2 0.9925 7.633 × 10-7 -3.622 × 10-7 106.6460 126.7945 113.1297 113.1297 16.4484 -9.2789 6.4837 -13.6648 1.316 × 10-5 96.0757
9.618 × 10-15 1.000 × 10-17 -2.229 × 10-7 9.999 × 10-7 0.9870 9.999 × 10-3 1.303 × 10-2 0.9900 101.8366 138.3002 113.1297 113.1297 9.0177 -5.4945 11.2930 -25.1705 2.526 × 10-5 119.4306
When the column operates at reflux of r ) 3.5, the flow sheet is feasible, approaching the desired product purity.
with minimum BPD. The Results and Applications section will apply the methodology for realistic network separation problems of interest. 3. Results and Applications This section demonstrates the proposed inverse design procedure to synthesize entire networks composed of complex
columns. In all cases, we assume isobaric operation at standard pressure, P ) 1 atm; pressure drops are neglected. 3.1. Design of Prefractionating Networks with Different Feed Compositions. We will consider the separation of a mixture entitled MIX1 containing aromatic and alkane compounds: benzene (A), toluene (B), octane (C), and nonane (D). Two different separation problems corresponding to feed composi-
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Figure 3. Liquid composition profiles over the bubble point temperature domain on each column of the networks NW1, NW2, and NW3. These feasible designs were found by the temperature collocation method. The quaternary mixture MIX1 constituted by benzene (A), toluene (B), octane (C), and nonane (D) is separated in each one of the three networks at two different feed specifications: equimolar feed, FEED1, of [0.25; 0.25; 0.25; 0.25]. The composition profiles on the top correspond to the first column, those on the second row correspond to the second column, and the bottom profiles correspond to the third column for each network. Component A, blue -O-; component B, green -×-; component C, pink -0-, component D, red - + -.
tions equimolar FEED1 and FEED2 ) [0.178; 0.4046; 0.2771; 0.1403] should be addressed. The network, NW1, shown in Figure 3a is a candidate solution for equimolar feed, FEED1. The application of steps 1-3 of the temperature collocation method easily yields the desired operating conditions that make the three columns of NW1 feasible. The first column, C1, is a prefractionator integrated with the complex column, C2. It splits the quaternary mixture into two product streams, removing the light component, benzene (A), as the distillate. In addition, the second lightest species, toluene (B), is separated by a nonsharp split; some amount of B is directed to the distillate and the rest of B leaves the column at the bottom accompanied by the two heavy components. The need to perform nonsharp splits is typical for prefractionator columns. From a single unit point of view, as this is often the focus of attention in traditional design methodologies, such sloppy cuts cannot be addressed methodically. However, when considering the entire structure, the incomplete separation of B is energy-efficient overall. Accordingly, the two product streams leaving column C1 with the sloppy B split are fed to a second complex column, which completely separates A and B as top and side products that satisfy the high product purities. The bottoms containing almost all of octane (C) and nonane (D) is processed to the simple column, C3. It generates two product cuts to complete the
separation task for the equimolar feed, FEED1. The methodology also determines the following detailed key design results: reflux ratios of all columns, location of feed and side-product trays, number of trays, liquid column profiles, and internal liquid and vapor flow rates. The detailed network design parameters are summarized in Table 5. For the same four species with the same vapor-liquid equilibrium model given in Table 3, we showed the separation synthesis problem for a different feed composition, FEED2. Is the network configuration NW1, also capable of delivering the products with the same 99% purity requirement as shown for the equimolar feed? Certainly, the operating conditions, tray number, locations, and compositions of feeds would have to be different. Yet, can the high purity requirements for all cuts be met for this new feed? What would be the energy demand as well as the operating and design conditions such as the column height, tray locations of feed, and side products for this task? We applied the temperature collocation synthesis methodology to determine feasible specifications for NW1 to solve the separation task for FEED2 of mixture MIX1. The main parameter modifications occur in the total trays and refluxes for the first and last columns. The first column runs at a 43% higher reflux. Yet, the third column needs nine fewer stages with the reflux diminished by 28%. The second column can keep similar
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Table 5. Detailed Network Design Specifications and Operating Conditions Obtained by Solving the Inverse Design Problem To Separate Two Quaternary Mixtures of MIX1 Constituted by Benzene (A), Toluene (B), Octane (C), and Nonane (D) and MIX2 Constituted by Pentane (A), Hexane (B), Heptane (C), and Octane (D) Using Two Different Molar Compositions of Equimolar FEED1 of [0.25; 0.25; 0.25; 0.25] and FEED2 of [0.1780; 0.4046; 0.2771; 0.1403] in Three Different Complex Networks NW1, NW2, and NW3 NW1
NW2
MIX1
MIX2
NW3
MIX1
FEED1
FEED2
FEED1
FEED2
FEED1
25 1 42 42 100 38.88 61.12 1.675
26 1 43 43 100 52.54 47.46 2.392
21 1 43 43 100 37.78 62.22 0.500
22 1 40 40 100 38.21 61.79 0.620
17 1 28 28 100 61.25 38.75 1.600
8 45 1 22 58 58 37.88 62.22 25 25.05 49.95 4.100
11 41 1 20 55 55 52.54 47.46 17.82 40.25 41.93 4.200
8 36 1 21 40 40 37.78 62.22 25.00 25.05 49.95 2.000
6 27 1 20 35 35 38.21 61.79 17.94 40.30 41.76 3.5
24 55 1 42 64 64 61.25 37.75 49.78 25.40 24.82 3.700
MIX2
MIX1
MIX2
FEED2
FEED1
FEED2
FEED1
FEED2
FEED1
FEED2
17 1 29 29 100 70.67 29.33 1.550
18 1 21 21 100 52.21 47.79 1.800
21 1 23 23 100 78.84 21.16 0.600
21 1 32 32 100 44 56 0.950
22 1 34 34 100 56.06 43.94 1.500
17 1 21 21 100 50 50 0.700
18 1 20 20 100 56.97 43.03 1.200
26 48 1 37 66 66 70.67 29.33 57.89 28.15 13.96 3.800
12 28 1 15 30 30 52.21 47.79 49.85 24.95 25.20 3.900
13 33 1 16 35 35 78.84 21.16 57.94 27.98 14.08 2.700
7 32 1 14 35 35 44 56 25.21 43.32 31.47 3.000
13 35 1 24 47 47 56.06 43.94 17.97 46.63 35.40 2.900
6 22 1 16 23 23 50 50 24.86 28.35 46.79 4.100
7 33 1 19 36 36 56.97 43.03 17.85 43.08 39.07 2.7
14 59 1 45 63 63 43.32 31.47 25.37 24.29 25.13 3.5
16 51 1 40 61 61 46.63 35.40 40.49 27.61 13.93 2.7
8 29 1 20 39 39 28.35 46.79 25.18 24.96 25.00 3.800
11 29 1 16 33 33 43.08 39.07 40.26 27.74 14.15 1.900
C1 feed stage location product stage location total stages feed flows (kmol/h) product flows (kmol/h) reflux
C2 feed stage location product stage location total stages feed flows (kmol/h) product flows (kmol/h) reflux
C3 feed stage location product stage location total stages feed flows (kmol/h) product flows (kmol/h) reflux
10
17
16
16
14
9
7
24
1
1
1
1
1
1
1
1
34
25
23
22
20
20
24
30
34
25
23
22
20
20
24
30
49.95
41.93
49.95
41.76
49.78
57.89
49.85
57.94
24.82
27.90
24.82
27.73
25.03
17.60
25.05
17.60
25.13
14.03
25.13
14.03
24.75
40.29
24.80
40.34
2.973
2.139
2.460
1.766
1.928
3.308
1.457
2.543
reflux and total stages. Overall, NW1 requires 11% less energy to separate FEED1 than to separate FEED2. The investment cost, expressed roughly by considering tray numbers only, suggests that FEED2 requires less capital cost. A detailed list of the specifications for FEED2 can be found in Table 5. Figures 3a and 4a depict the composition profiles for the two network solutions. Both networks use the same structure, but have marked differences in the intermediate streams as well as in the detailed design specification for each column. For example, the prefractionator purity leaving C1 at the top is much richer in A, but the concentration of B is 50% less than for the first mixture MIX1. Finally, the purity of the terminal products is almost the same, consistent with the high purity requirements (>99%). The final molar flow rates of products B and C are larger for FEED2 as is expected given the different amounts in the inlet molar stream of FEED2. 3.2. Alternative Structures for the Separation of FEED1 and FEED2. In addition to NW1, part 1 of this series showed three more configurations to separate a quaternary mixture like MIX1. Here, we will present two new network configurations to achieve the same product purity. These new networkss previously not shownsare termed NW2 and NW3. For FEED1, the complex configuration of the flow sheet as well as the compositions of all column sections are shown in Figure 3b,c. NW2 deploys a nonsharp split prefractionator C1 followed by a complex column C2. Here, the component C (octane) is split
sloppily. In contrast to NW1, the complex column C2 products C and D (nonane) split, while the lights A (benzene) and B (toluene) are purified in column C3. The purities of the final products produced by NW2 satisfy the specifications as desired. NW3 has two complex columns including several nonsharp splits. It engages column C1 for a sloppy split of two components, B and C. The complex column C2 produces only benzene (A) in high purity. The complex column C3 finally completes the task by purifying simultaneously toluene (B), octane (C), and nonane (D). The detailed design specifications for NW2 and NW3 are listed in Table 5. An important feature is that, for FEED1, NW3 is more energy efficient than NW1. The energy savingsswithout the claim for rigorous optimalitys amount to 10%. From the energy point of view, configuration NW3 is best to separate FEED1. Interestingly, for FEED2, NW1 and NW3 have almost the same energy requirement. We also infer from Figure 10 that NW2 seems to be suboptimal for either feed composition from an energy consumption point of view. The detailed comparisons of the structural alternatives for the feed compositions, FEED1 and FEED2, suggest that different structural network solutions have a strong impact on the oVerall energy consumption required for addressing a separation task. Moreover, our methodology seems suitable for performing these detailed comparisons, as the purities are ensured for realistic columns without dropping rigor in the calculations or the need to employ shortcut methods or the restrictions of sharp splits.
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Figure 4. Liquid composition profiles on each column of the networks NW1, NW2, and NW3. These feasible designs were found by the temperature collocation method. The quaternary mixture MIX1 constituted by benzene (A), toluene (B), octane (C), and nonane (D) is separated in each one of the three networks with feed purity, FEED2, of [0.1780; 0.4046; 0.2771; 0.1403]. The composition profiles on the top correspond to the first column, those on the second row correspond to the second column, and the bottom profiles correspond to the third column for each network. Component A, blue -O-; component B, green -×-; component C, pink -0-, component D, red - + -.
The networks are really comparable as they all solve with high precision identical separation tasks. 3.3. Design of Prefractionating Networks with Different Mixtures. The next subsection documents the solution of the task of purifying a completely different quaternary mixture, MIX2, composed of the alkanes pentane (A), hexane (B), heptane (C), and octane (D). This mixture is treated as ideal, but it has different vapor pressures and relative volatilities as noted in Table 1. The feed composition FEED1 and the network configuration NW1 will be considered again for comparison. Because the thermodynamic phase equilibrium properties are completely different in this mixture, it is expected that network NW1 should be designed with different key parameters compared to the previous examples concerning a mixture of aromatics and alkanes. What are the properties with which NW1 can be used to split MIX2 into pure products? Applying our novel design methodology, we find easily the feasible and reasonable parameters for NW1 which are summarized in Table 5. Again, we also discuss solutions for splitting of a second feed composition FEED2 for the mixture MIX2. We find the new design parameters for NW1 given in Table 5. For this feed, the reflux ratio of the second column is increased from r ) 2.0 to r ) 3.5 and the energy consumption for the last column can be reduced from r ) 2.46 to r ) 1.77. However, the total number
of stages in each column remains similar in the first column with just a small variation of three and two stages in the second and third columns, respectively. It is interesting to note that even though the thermodynamics are completely different, the design methodology can be easily used to obtain in detail all design specifications such as product streams, feed stages, and product stages as before. The composition profiles for MIX2 with different feeds FEED1 and FEED2 are plotted in Figures 5a and 6a, respectively. A comparison of the energy consumption for this MIX2 between the two feed scenarios shows that the vapor duty for both networks is similar, requiring about 200 kmol/h. In addition, we infer that the separation of alkane is easier than MIX1, which had both aromatics and alkanes. This result can be explained by the relative ease of separation: expressed roughly in terms of relative volatilities, the alkane mixture MIX2 has relative volatilities R ) [18.49, 6.79, 2.57, 1] at the feed temperature, while the aromatic-alkane mixture MIX1 has R ) [7.97, 3.37, 2.17, 1]. Thus, mixture MIX1 is more difficult to separate, because of smaller R differences (1-7.97) compared to the range of volatilities of the second mixture (1-18.49). Next, we expand our search for solutions to the separation task with different networks. We will again revisit for better comparison NW2 and NW3 that we have also employed for MIX1. The temperature collocation methodology easily produced
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Figure 5. Liquid composition profiles on each column of the networks NW1, NW2, and NW3. These feasible designs were found by the temperature collocation method. The quaternary mixture MIX2 constituted by pentane (A), hexane (B), heptane (C), and octane (D) is separated in each one of the three networks at two different feed specifications: equimolar feed, FEED1, of [0.25; 0.25; 0.25; 0.25]. The composition profiles on the top correspond to the first column, those on the second row correspond to the second column, and the bottom profiles correspond to the third column for each network. In the composition triangle, the labels, Fi, refer to the feed tray compositions, not the feed compositions.
rigorous designs for all the networks; column profiles of the network solutions can be found in Figure 5b,c and Figure 6b,c, respectively. Even though the intermediate product specifications of each column vary widely, the method systematically creates feasible networks meeting the desired targets for the final cuts. When comparing the energy requirements to separate FEED1 of MIX1 with the effort required for FEED2 of MIX2, NW2 and NW3 are more expensive than NW1. NW1 appears to be the most energy-efficient network solution for the alkane separation. The comparison between the separation of MIX2 and MIX1 confirms the trend that in our example the alkane separation was easier than the split of the aromatic-alkane mixture with a smaller relative volatility difference between the species. The results demonstrate that for any desirable separation structure we can systematically find the detailed network configuration for the separation problem. Note also that the designs presented in this article focus on the creation of feasible designs; there is yet not claim for globally optimized designs. The global optimization of separation flow sheets will be the subject of a follow-up paper. 3.4. Validation of Synthesized Designs with Rigorous Flow Sheet Simulation. The proposed methodology has been shown to yield the detailed design specifications for all column sections and columns of the entire network. It would be desirable
to verify with rigorous tray-by-tray MESH equation algorithms that these designs are sound and realizable. We will show in this section that all the flow sheet design specifications obtained by our synthesis methodology hold the stringent validation test with the commercial flow sheet simulator Aspen. Twelve different networks were designed rigorously by temperature collocation to separate two quaternary mixtures with two different feed purities. The calculated liquid and vapor profiles resulting from the predicted operating conditions such as the reflux and reboil ratios, total tray numbers, and the location of side-product removal and feed trays for all network combinations are collected in Table 5. This detailed information is used as input to validate all three prefractionating networks NW1, NW2, and NW3 in AspenPlus for the 12 cases. After creating AspenPlus flow sheets representing each of the three different networks, all 12 synthesis problems using the design solution data as input converged in a few iterations. Figure 7 offers a direct comparison of the liquid composition profiles for each column of NW1. Each diagram displays the liquid composition trajectories of all components. In this figure, the left column shows the design solution obtained by temperature collocation. The right column represents the converged solution from AspenPlus. The Aspen results approach the product requirements very closely at the product nodes. The
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Figure 6. Liquid composition profiles on each column of the networks NW1, NW2, and NW3. These feasible designs were found by the temperature collocation method. The quaternary mixture MIX2 constituted by pentane (A), hexane (B), heptane (C), and octane (D) is separated in each one of the three networks with feed FEED2 of [0.1780; 0.4046; 0.2771; 0.1403]. The composition profiles on the top correspond to the first column, those on the second row correspond to the second column, and the bottom profiles correspond to the third column for each network. In the composition triangle, the labels, Fi, refer to the feed tray compositions, not the feed compositions.
agreement between the liquid composition profiles in all three columns is satisfactory. The global product streams consisting of two product streams from C2, and the two products from C3 match very closely the final product purities by the two methods with less than 0.1% difference. AspenPlus was used to validate the design of network NW2 separating mixture MIX2 with the second feed composition FEED2. Again, similar liquid profiles satisfying the global product purity were obtained as shown in Figure 8. Finally, the design of network NW3 to separate MIX1 for the first feed specification FEED1 is used to compare the proposed inverse design methodology with the AspenPlus forward simulation. Figure 9 depicts again a very close match between the liquid composition profiles through the column stages for all distillation columns of each complex network for both methods. There are minor differences between the trajectories of the two design approaches, temperature collocation and Aspen, in the middle section of some columns. Fortunately, compositions in the middles of column sections are not very significant. More important is the exact matching in the initial and final product compositions at the product nodes. The product compositions are the most important design outcome. When using commercial flow sheet simulators, the initialization of the complex flow sheets with several columns requires
detailed knowledge of all the feed and product trays, compositions, flows, and number of trays. These detailed specifications are a necessary condition for a rigorous flow sheet simulation to converge. For an entirely new separation problem, the task of identifying the key parameters for such a flow sheet can be a challenge. Therefore, in industrial practice is often a timeconsuming effort requiring expert knowledge to put together flow sheets, especially when complex configurations with heat integrations are to be designed. Therefore, our temperature collocation method could also be considered an invaluable tool to initialize commercial flow sheet simulations. Without rigorous inverse design algorithms, the initialization requires extensive trial-and-error efforts, which are expensive and time-consuming. 4. Discussion It was demonstrated that the temperature collocation algorithm combined with minimum BPD can adjust the design specifications so that the product purity targets are met, if these targets are at all attainable. Using the fractional recoveries as degrees of freedom, we enforced global column species balances. Column species balances would thus be close provided that their respective profiles pass the rigorous feasibility test. Suitable fractional recoveries can be selected by the designer. The search
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Figure 7. Liquid composition profiles in the stage domain found by the temperature collocation model (left) and AspenPlus simulation results (right) for prefractionating complex column network NW1. The quaternary mixture MIX2 constituted by pentane (A), hexane (B), heptane (C), and octane (D) is separated in network NW1 using an equimolar feed, FEED1, of [0.25; 0.25; 0.25; 0.25]. Component A, blue -O-; component B, green -×-; component C, pink -0-, component D, red - + -.
space is usually small because of the specified high product purities. Recoveries would typically range in the region of 0.99 and higher. Alternatively, a black box search can be performed automatically using suitable global search algorithms such as the hybrid algorithm described previously.4 Unfortunately, there are no obvious criteria to identify feasible network recoveries a priori; accordingly, our semiautomatic design feasibility test needs to be deployed repeatedly or a global search can be run as described elsewhere. Figure 10 depicts the total vapor flow rates of each network to meet the high purity separation targets. Network NW1 has among all structures the least total vapor rate for the separation of MIX2. With almost the same amount of energyswith less
than 1% differencesboth feed compositions of FEED1 and FEED2 for MIX2 can be purified. Our synthesis method precisely identified the necessary adjustments in terms of number of trays as well as locations of feed and side products. These detailed results could also permit a detailed capital cost assessment combined with a more explicit operating cost analysis. These types of detailed cost optimization are left to the industrial practitioners, but are ideal follow-up steps intended to be performed after the use of our synthesis methodology. For mixture MIX1, NW3 has the lowest energy consumption. This configuration, NW3, can again split both feed compositions with about 20% of the energy demand. NW1 and NW3 have almost the same energy requirement for splitting FEED2 of
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Figure 8. Liquid composition profiles in the stage domain found by the temperature collocation model (left) and AspenPlus simulation results (right) for prefractionating complex column network NW2. The quaternary mixture MIX2 constituted by pentane (A), hexane (B), heptane (C), and octane (D) is separated in network NW2 using a feed, FEED2, of [0.1780; 0.4046; 0.2771; 0.1403]. Component A, blue -O-; component B, green -×-; component C, pink -0-, component D, red - + -.
mixture MIX1 with only 2.6% difference in terms of total vapor flow. NW2 has the worst energy performance in all cases, because the two feed specifications do not benefit from the pseudoindirect separation occurring in this particular network. It is also the only system in which the equimolar mixture, MIX2-FEED1, consumes more heat than the second feed composition, MIX2-FEED2. Another interesting result is given by the total vapor consumption in NW3 to solve the four different problem specifications. This network needed an average vapor rate of 313.5 kmol/h for all four cases with only 15% deviation from the average vapor consumption in the system MIX1-FEED2. In this example, its performance may be explained by its
intensive degree of heat integration, leading to almost constant heat duty for any separation task in our examples. However, we recommend not drawing general conclusions from specific examples like the ones we show here, but consider the rigorous synthesis and analysis results for each particular case as a function of the different thermodynamic systems with their specific feed compositions and purity requirements. The perfect test for the validity of the synthesis results would be to build a pilot or process plant in the real world. Since this approach would lead to unreasonable cost, we resorted to the next best validation based on commercial flow simulators which solve the rigorous mass, equilibrium, summation, and heat equations (MESH). Even though the temperature collocation
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Figure 9. Liquid composition profiles in the stage domain found by the temperature collocation model (left) and AspenPlus simulation results (right) for prefractionating complex column network NW3. The quaternary mixture MIX1 constituted by benzene (A), toluene (B), octane (C), and nonane (D) is separated in network NW3 using an equimolar feed, FEED1, of [0.25; 0.25; 0.25; 0.25]. Component A, blue -O-; component B, green -×-; component C, pink -0-, component D, red - + -.
approach approximates the discrete MESH equations by the continuous generalized profile eq 19, a remarkable agreement between our synthesis and the flow sheet simulation occurred in all cases we investigated. The achieved purity targets in both methods were practically identical, and the flow sheet simulation converged in a few iterations, in essence to the same profiles as computed by our approach. Small differences in the middle of column section trajectories obtained by temperature collocation and Aspen are due to model differences such as discrete versus continuous profiles as well as possible heat effects not accounted for in our method, while Aspen does. The inverse design methodology based on temperature collocation can also be used to effectively initialize flow sheets for rigorous flow sheet simulation like AspenPlus. The agree-
ment between the converged AspenPlus product specifications and the specified product requirements indicates that the temperature collocation synthesis algorithm is practically equivalent to AspenPlus results. 5. Conclusions The applicability and stringency of the minimum bubble point distance algorithm combined with temperature collocation for the successful design of heat-integrated complex separation networks with various feeds and volatility differences is reported in this article. The minimum BPD criterion was robust and efficient to characterize feasible network specifications guaranteeing con-
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or new sustainable process designs for reduction of greenhouse gas emissions. Acknowledgment Financial support by DOE Grant DE-FG36-06GO16104 is gratefully acknowledged. G. Ruiz is mentioned for his contributions in the early stages of this work. We acknowledge Dr. ChauChyun Chen for his support in providing an Aspen software research license. Figure 10. Total vapor flow rate required in each one of the 12 different synthesis problems to separate two quaternary mixtures of MIX1 constituted by benzene (A), toluene (B), octane (C), and nonane (D) and MIX2 constituted by pentane (A), hexane (B), heptane (C), and octane (D) using two different molar compositions of equimolar, FEED1, of [0.25; 0.25; 0.25; 0.25] and FEED2 of [0.1780; 0.4046; 0.2771; 0.1403] in three different complex networks NW1, NW2, and NW3.
tinuous liquid composition profiles connecting all product and feed stages in each column section of complex flow sheets. Specifications leading to a nonzero bubble point distance are guaranteed to be impossible to realize. The proposed feasibility test is therefore rigorous, subject of course to the limits of floating point operations on a digital computer. Three different heat-integrated networks were designed and discussed in detail to separate two different quaternary mixtures with two different purities by the inverse design methodology. Each one of these 12 synthesis problems was rigorously solved to compute key design parameters. Our results suggest that it is not possible to generalize that a particular network structure could be the best in terms of energy requirements without studying in detail the separation task as a function of feed, the thermodynamic system, and the feed composition of the mixture. Finally, a remarkable agreement between the inverse design solutions and rigorous forward flow sheet simulations was demonstrated. Three particular feasible designs were validated by initialization of AspenPlus simulations which converged in a few iterations in excellent agreement with the solution from the temperature collocation method. The converged product specifications in the Aspen simulations match closely the final product purities computed by inverse design. These results show that temperature collocation can be used to automate the synthesis of complex separation networks. The proposed method has not been used for azeotropic separations. In principle, temperature collocation applies to azeotropic mixtures as shown elsewhere.3 The synthesis of azeotropic separation networks, however, requires a more complex logic, which this article does aim to address. In addition, temperature monotonicity might not hold.17,28–30 A more detailed list of limitations can be found in the conclusions of part 1 of this series.5 We have so far not attempted to solve for optimal designs, while enforcing feasibility. This task would involve a global multilevel optimization with a high-level objective function subject to structural (which flow sheet, which configuration) and parametric (operating conditions, actual product purities) decisions. At level 2, the feasibility criterion would have to be enforced as a side condition; this subproblem is already a global optimization problem. In this paper, we have not discussed the multilevel optimization. This article suggests that the synthesis methodology can be adapted to address modern process design problems such as computer-aided synthesis of energy-efficient separations, detailed design of large-scale biorefineries with novel feedstocks,
Appendix A: Estimation of Pinch Points for Temperature Collocation Robust computation of pinch points in each column section is an important task; rigorous methods for solving the pinch equation (A1) using homotopy continuation methods are discussed elsewhere.14 The pinch point location for ideal and constant volatility mixtures can be simplified to a onedimensional root finding problem for any number of species.3
(
1+
)
1 1 (x - yj,i) + (X - xj,i) ) 0 R∆i j,i R∆i ∆j,i
(A1)
After computing all pinch points, the stable pinch point of each section, xpinch j, Tpinch, is employed as the final stationary condition of the composition profiles. The solution accuracy benefits from node placement in regions with steep gradients.3 In the composition profiles, these regions occur about the saddle point temperatures Tsaddle i. Therefore, the saddle point temperatures in the temperature integration should be used as element boundaries. The pinch temperature values also help to check the existence of an attainable temperature window between any pair of equivalent rectifying and stripping column sections.3 If the temperature window does not intersect, the column sections are guaranteed to be infeasible for the current specifications. Only if the current specifications meet this shortcut feasibility test, the liquid composition profiles for all species in each section are computed. Equation 19 is then solved between the first stage of the section column, x∆j, T∆, and the stable pinch, xpinch j, Tpinch, using orthogonal collocation on a finite element. Appendix B: Polynomial Arithmetic for Computing the Globally Minimum Bubble Point Distance Function This section describes the global search for the minimum bubble point distance function for two adjacent column section profiles with polynomial arithmetic. The solution of eq 19 by finite element on orthogonal collocation generates an approximate analytical solution represented by a set of polynomial temperature functions for each composition on each column section. Equations B1 and B2 represent composition profiles for any pair of adjacent column sections r and s of a multicomponent mixture parametrized in terms of temperature T. Xr(T) ) [gA(T), gB(T), gC(T), gD(T)]T
(B1)
Xs(T) ) [fA(T), fB(T), fC(T), fD(T)]T
(B2)
Then, a scalar function representing the Euclidean difference between any pair of adjacent rectifying and stripping equivalent sections, BPD ) |Xr(T) - Xs(T)|, evaluated at the same temperature value defines the bubble point distance function, BPD. Hence, the BPD is also a piecewise polynomial function in T of order equal to 2 times the highest degree of the two
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polynomial orders. The BPD is a continuous scalar field in terms of temperature as in (B3): BPD(T) ) |Xr(T) - Xs(T)|
(B3)
Using eq B3, it is possible to compute continuous values of the BPD for any temperature over which the pair of adjacent profiles exists simultaneously. This range is determined by the temperature window given by the bubble point temperature of the first stage and the stationary pinch point. Finally, the minimum bubble point distance, BPD, between two adjacent column sections is the solution of a global optimization problem, (B4). β ) min BPD(T) T
(B4)
Now, suppose N nodes are used per element to discretize the composition solution in the temperature domain. In that case, the solution of the nonlinear profile equations are polynomial functions gj(T) and fj(T) of order N - 1 after implementation of eq B3. Then, the BPD function is a piecewise polynomial function of order 2(N - 1) given as BPD(T) ) a1T2(N-1) + a2T2(N-1)-1 + ... + a2(N-1)+1
(B5) To find its global minimum for a fixed design variable set and product requirements, the optimality condition of eq B5 generates another piecewise polynomial of 1 order less as shown in eq B6. d[BPD(T)] ) [2(N - 1)]a1T2(N-1)-1 + [2(N - 1) dT 1]a2T2(N-1)-2 + ... + a2(N-1) ) 0
(B6)
The Newton Horner scheme4 with synthetic division can be used to solve eq B6 for all positive real zeros. The arrangement with the smallest BPD is the desired minBPD. Note that polynomial arithmetic can solve the minimum bubble point distance criterion globally, even if the phase equilibrium models are nonlinear. Literature Cited (1) EPA Pollution Prevention: Laws and Policy. http://www.epa.gov/ p2/pubs/laws.htm. (2) DOE Technical Topic Description. www.doe.gov. (3) Zhang, L.; Linninger, A. A. Temperature collocation algorithm for fast and robust distillation design. Ind. Eng. Chem. Res. 2004, 43 (12), 3163– 3182. (4) Zhang, L.; Linninger, A. A. Towards computer-aided separation synthesis. AIChE J. 2006, 52 (4), 1392–1409. (5) Kim, S. B.; Ruiz, G. J.; Linninger, A. A. Rigorous Separation Synthesis. 1. Multicomponent Mixtures, Nonideal Mixtures, and Prefractionating Column Networks. Ind. Eng. Chem. Res. 2010, 49 (14), 64996513. DOI: 10.1021/ie1000532. Published Online: June 21, 2010. (6) Seferlis, P.; Hrymak, A. N. Adaptive collocation on finite elements models for the optimization of multistage distillation units. Chem. Eng. Sci. 1994, 49 (9), 1369–1382. (7) Seferlis, P. Optimal design and sensitivity analysis of reactive distillation units using collocation models. Ind. Eng. Chem. Res. 2001, 40 (7), 1673–1685.
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ReceiVed for reView February 15, 2010 ReVised manuscript receiVed June 24, 2010 Accepted July 8, 2010 IE100355C
