Rigorous Separation Design. 1. Multicomponent Mixtures, Nonideal

Ind. Eng. Chem. Res. 2010, 49, 6499–6513

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Rigorous Separation Design. 1. Multicomponent Mixtures, Nonideal Mixtures, and Prefractionating Column Networks Seon B. Kim, Gerardo J. Ruiz, and Andreas A. Linninger* Laboratory for Product and Process Design, Department of Chemical and Bio-Engineering, UniVersity of Illinois at Chicago, Chicago, Illinois

Currently, there is a lack of reliable computational methods to automatically synthesize separation networks within specific product targets. Computational methods exploring the combinatorial wealth of different separation configurations, while simultaneously selecting feasible or detecting globally optimal operating conditions, are not available for problems of practical size. In this paper, we extend the minimum bubble point distance algorithm embedded in the temperature collocation methodology to rigorously design complex networks to separate nonideal multicomponent mixtures into products of desired purity using heat-integrated prefractionating columns. Our employed inverse design procedure enables the systematic design of physically realizable separations for mixtures with a large number of species. The computer procedure robustly converges to the desired purity targets, unless the desired purity target is thermodynamically impossible to realize. The algorithm also rapidly identifies infeasible specifications without fail. Finally, synthesized networks were validated with AspenPlus matching exactly the inverse design results with the target purity. The rigorous flowsheet design combined with validation of the networks with commercial flowsheet simulators enables the systematic design of energy-efficient separation networks. The methodology is ready to address currently unresolved design problems such as the computer-aided design of energy-efficient separations, the design of biorefineries, or new process designs for carbon sequestration. 1. Introduction The increasing demand on energy efficiency and environmental sustainability of chemical manufacturing renews the interest in systematic process design with ecological or energy targets.1,2 Distillation is arguably the most significant chemical unit operation in industrial practice, responsible for about 3% of total U.S. energy consumption.3 Accordingly, even a small improvement in the distillative separations would yield huge energy savings. To harness possible energy improvements, the use of thermodynamically integrated complex columns has been suggested.4–12 For example, thermally coupled distillation columns known as Petlyuk configurations13 only require one reboiler and one condenser, independent of the number of components to be separated. Wright14 proposed the diVided wall column, in which the prefractionator is transferred inside the second tower so that the entire configuration can be realized in a simple unit. Both structures are typically more energy efficient than conventional simple columns. Recent work by Luyben’s group15 shows practical and effective control strategies for heat integrated columns including diVided wall columns. Although the final verdict is still out, it appears that controllability of complex heat-integrated columns is not a problem, if methods such as Luyben’s are applied with care. Therefore, current research aims at novel methods for discovering energy-efficient configurations systematically. Until recently, however, there were no rigorous algorithms to synthesize complete separation flowsheets without limiting assumptions of the thermodynamic vapor-liquid equilibrium model or restrictions in its applicability like sharp splits or nondistributing species. Purely combinatorial superstructure approaches have been pioneered by the mathematical programming community.4,16,17 Recently, structural approaches specifically addressing synthesizing complex network structures have * To whom correspondence should be addressed. E-mail: linninge@ uic.edu.

led to the possibility of enumerating all possible configurations systematically.9,18 However, to judge whether a certain configuration can actually achieve the desired purity, a detailed design including realizable column profiles is needed. Flowsheet synthesis poses two main problems. The first design stage asks for configuration and operating conditions of a single column such that a known feed or multiple feed streams can be separated into a set of product streams with desired compositions. Columns with only one feedsa distillate and bottom productssare generally known as simple columns. Columns with more than one feed or more than two products are termed complex columns. The configuration parameters for a column, simple or complex, include the number of trays in each section as well as the location of the feed tray and the number and composition of the product cuts. The operational parameters include the reflux and reboil ratios which are linked by energy balances. For synthesizing entire networks, the product purities must not only satisfy final product purity targets but, more challenging, each intermediate product must reach such compositions so that the subsequent step or multiple steps are also thermodynamically possible. Especially in complex networks, these intermediate cuts do not always fall into the sharp split category, a requirement often demanded by existing design methods. The second design stage seeks for the specifications of the entire network. Classical separation design aimed at sequencing series of simple columns each equipped with a condenser and reboiler.19 Recent research and modern practice such as crude refinery trains suggest that simple energy networks are seldom energy optimal. However, a complex heat-integrated network is overall feasible, only when each of its steps and therefore each and every column section in the network are thermodynamically feasible. Thus, separation design requires a multiple step design strategy in which a single operationssimple or complexsmay make sense only when completing the entire network. Again, the individual columns of the Petlyuk config-

10.1021/ie1000532  2010 American Chemical Society Published on Web 06/21/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 14, 2010

uration render a meaningful and energy efficient flowsheet as an ensemble; the prefractionator could never been found by classical sequential (step-by-step) reasoning. Accordingly, the second design stage requires a creative search for structural alternatives. Also, for an entire network to satisfy minimum energy demands, a single column should not be optimized locally without considering the entire separation train. For example, the Petlyuk column prefractionator produces only a sloppy split; the entire configuration with thermal coupling produces pure products with reduced energy consumption compared to a sequence of simple columns. To avoid computational intractability, we have for the moment incorporated specific flowsheets as generated by Agrawal’s method. This step can be automated as described by Agrawal but has been omitted here to focus on ensuring rigorous profiles. We have successfully implemented orthogonal collocation on finite elements (OCFE). For complex separation network design, this approach substantially reduces the problem size without a significant loss in accuracy compared to the rigorous tray-bytray computations. Although collocation has been used previously in separation design,20–28 this paper demonstrates the temperature collocation29 with a minimum bubble point distance (BPD)30,31 criterion for the rigorous design of entire flowsheets to separate nonideal mixtures, multicomponent mixtures, including prefractionating and complex column configurations. Finally, among all feasible separation networks, we typically are interested in identifying the solution that minimizes a userdefined cost objective. Recently, energy cost and emission reduction have gained significance in the search for new separation solutions. The paper is organized as follows: First, a methodology section addresses both the structural as well as the parametric challenges of the process design problem. Results and applications demonstrate a rigorous design to separate a quaternary nonideal mixture into almost pure products. A column design to separate a 10-component mixture with three different product distributions is addressed in section 3. Furthermore, the design of prefractionating complex column networks to separate a quaternary mixture of alkanes is described. To demonstrate that the proposed designs are realistic and industrially realizable, equivalent network configurations are validated with rigorous tray-by-tray AspenPlus simulation in section 4. Finally, sections 5 and 6 offer a discussion and conclusions. 2. Methodology 2.1. Inverse Design Procedure. Despite the common misconception of distillation as a mature field, complex column configurations for enhanced energy-efficiency and extractive or reactive separations are an open field of research.32–35 In general, the design of distillation columns can be performed by forward performance simulation. The classical design simulation computes product purities based on a given feed and column design specifications (e.g., total column tray, feed tray, reflux ratio, etc). This approach requires trial-and-error adjustments of operating conditions and cannot guarantee desired product purities, because they are results of the forward computation not input design specifications. Specifying product purities before searching for feasible column configurations leads to an inVerse design method. Thus, inverse design seeks optimal column dimensions and operating conditions for desired product purities. Inverse separation network design to achieve desired product quality specifications requires a rigorous convergence criterion, so that the network can be built in practice. Our group advocates the precise intersection of liquid composition profiles

for any pair of equivalent rectifying and stripping sections as a rigorous criterion to ensure thermodynamic feasibility and practical realizability of a separation column. This criterion has been formulated mathematically as the minimum bubble point distance criterion in previous work.29 In addition, a rigorous inverse design methodology for complex column networks has to meet the following requirements: (i) the search for configurations should incorporate all thermodynamically admissible combinations of simple and complex column equipment to achieve any type of product distribution such as in prefractionating, sharp, nonsharp, and sloppy splits; (ii) liquid composition profile equations should admit all phase equilibrium relationships for ideal, nonideal, as well as azeotropic mixtures; and (iii) column profiles must exactly intersect in feasible designs, while infeasible design specifications are characterized by a gap in at least one pair of composition profiles in adjacent column sections. Thermodynamically impossible specifications lead to nonzero bubble point distances.29 These requirements have been incorporated in an inverse design methodology known as temperature collocation with rigorous computation of the minimum bubble point distance (BPD). The methodology has two main elements briefly reviewed in the next subsections: (1) rigorous liquid composition profile computations and (2) a bubble point distance criterion for network feasibility. More details can be found elsewhere.29 2.2. Rigorous Profile Computation. Our method departs for the generalized column profile equations proposed by Hildebrand and co-workers36,37 based on early work by Doherty and co-workers.38,39 Equation 1 introduces two new design quantities, the generalized reflux, R∆, and the difference point composition, X∆j. The difference point composition, X∆j, is equal to the distillate or bottom composition in conventional rectifying or stripping sections. In complex column sections, such as the intermediate section of the Petlyuk configuration, it can be interpreted as the concentration difference between the vapor and liquid stream exiting the section. In this sense, X∆j expresses the operating line conditions in a similar fashion to that of the product purity in simple column sections. The net flow of each section, ∆, is equal to the difference between vapor and liquid flow rates, ∆ ) V - L. Positive net flow means an upward stream corresponding to an equivalent rectifying section. Negative net flow signifies a downward stream as it occurs in equivalent stripping sections. Accordingly, intermediate sections can be categorized into equivalent rectifying sections, in which net flow is upward, and equivalent stripping sections, with a net downward flow. The generalized reflux, R∆, is the recirculation ratio belonging to a column section with the same conceptual function as the reflux in conventional columns.

(

)

dxj 1 1 ) 1+ (x - yj) + (X - xj) dn R∆ j R∆ ∆j with R∆ ) L/∆, X∆j ) (Vy∆j - Lx∆j)/∆

(1)

Where, xj and yj are the liquid and vapor compositions in the phase equilibrium relationship, and x∆j and y∆j are the liquid and vapor compositions of the streams entering a column section. Temperature collocation transforms the generalized column profile equations in eq 1 into continuous composition profiles with the bubble point temperature, T, as a new independent integration variable, as shown in eq 2. This temperature, T, is equal to the tray temperature at each stage but also corresponds to its bubble point, since vapor and liquid streams are in phase equilibrium. Fortunately, the difference point equations also describe the liquid composition of simple column sections, thus

Ind. Eng. Chem. Res., Vol. 49, No. 14, 2010 Table 1. Pseudo-Code Implementation to Compute the Liquid Composition Profiles Using Orthogonal Collocation on Finite Elements for Each Column 1 2 3 4 5 6 7 8 9 10 11 12 13 14

specify the product purities, and design parameters such as reflux or reboil ratio calculate all product flow rates, internal vapor and liquid flow, and stationary points (Fout, x∆j, y∆j, L, V) compute initial integration temperature, T∆ calculate the column profiles parameters, R∆, X∆j, ∆ compute the stable pinch point, xpinch_j,Tpinch in each column section check if there is an attainable temperature window between any pair of equivalent rectifying and stripping column sections if the shortcut criterion is infeasible, then the column specifications are infeasible; stop specify the placements and total number of elements and nodes create the temperature vector where each component of this vector denotes the temperature value find the elements of the derivative matrix A: λm(T) ) d/(dT) lm(T) lm(T) ) ∏n ) 1;n * m N(T - Ten)/(Tme - Ten) find g with equations [A](x) )(g) evaluate F(x) ) [A](x) - (g) repeat steps 10, 11, 12, and 13 while |F(x)| > ε e if |F(xn,c )| e ε, output the feasible column section profiles

rendering a single mathematical expression for all sections in any separation network, whether they are complex or simple. dxj dT

[( ∑ {[(

1+

)-

c

j)1

) )

] ]}

1 1 (x - yj) + (X - xj) R∆ j R∆ ∆j

×

1 1 1+ (x - yj) + (X - xj) Kj R∆ j R∆ ∆j

c

dKj

∑ dT x

j

(2)

j)1

6501

difference point equation does not correspond to a product leaving the section. Yet, the component balances for an equivalent rectifying section closes the component balances with the difference point in the same way the distillate determines the operating line equations. For the computation of a complex column section, we choose an intermediate product, whose composition constitutes a starting point of the composition profiles. Accordingly, the composition profiles described by eq 2 for any two complex column sections emerge from the respective intermediate product node. Similar considerations apply to equivalent stripping sections. 2.3. Bubble Point Distance and Global Feasibility Test. After computing the liquid composition profiles of all column sections, the minimum BPD is defined as the globally minimum distance between each pair of adjacent column sections. In each section, the continuous temperature column profile equations are a set of polynomial functions for each composition in terms of the bubble point temperature as an independent variable. Since all composition profiles, xj(T), are approximated by polynomial functions, the minimum BPD is found easily using polynomial arithmetic. A complex column k is feasible, if and only if the sum of all minimum profile distances of any pair of equivalent rectifying, r, and stripping, s, column sections is within a small ε tolerance of zero, as in expression 3, in which the sum of all its column section bubble point distances is still a scalar. An entire separation network is feasible, if and only if all its simple or complex columns, k, are feasible. The network feasibility is given in expression 4. φ(k) )

∑ min BPD(T) < ε

1

(3)

K

The profile equations in eq 2 are solved for each species composition, xj, by global collocation of orthogonal polynomials on finite elements with temperature as independent variable. A detailed derivation of the temperature transformation is given elsewhere.29 The procedure to compute the composition profiles of all columns in a network based on solving eq 2 for each column section is displayed in Table 1 and Figure 1. For each column, the first step consists in specifing the product purities for a simple or complex column in terms of molar composition, xPj, or fraction recovery, fPj, and design parameters such as reflux ratio. Step 2 calculates all product flow rates by global mass balances in each column of the network; this step is a linear algebraic problem. Step 3 involves the computation of the stationary nodes in each column section by the implementation of consecutive mass balances in each section and bubble point temperature calculation; these stationary nodes are defined by the values of x∆j, y∆j, T∆, L, and V. The calculation of the column profile parameters, R∆, X∆j, and ∆, is given in step 4. The stable pinch point of each section, xpinch_j,Tpinch, is computed in step 5 to be used in the next two actions. Steps 6 and 7 inspect a shortcut feasibility test based on the existence of an attainable temperature window29 between any pair of equivalent rectifying and stripping column sections. If specifications pass the shortcut feasibility test, steps 8 to 14 are executed to compute all liquid composition profiles of all species in each section by solving eq 2 between the stationary node, x∆j,T∆, and stable pinch, xpinch_j,Tpinch, using orthogonal collocation on a finite element in each column section. The detail information about orthogonal collocation on finite elements can be found in a previous publication.29 For regular rectifying sections, the generalized reflux becomes the classical reflux ratio, and the difference point is equal to the distillate composition. In equivalent rectifying sections, the

Ψ(k) )

∑ φ(k) < ε

2

(4)

k)1

The procedure for meeting feasible design specifications for a single column is given in Table 1. The information flow for the entire flowsheet is depicted in Figure 1. Product purities of the final products are fixed in stage 1 of Figure 1. These purities can be specified in terms of product compositions or fractional recoveries. The intermediate product purities and operating conditions are at this point still to be determined so that the network becomes feasible and optimal. In stage 2, initial guesses for these internal degrees of freedom such as intermediate product compositions as well as reboil or reflux ratios of columns are specified. These choices combined with suitable component balances allow the determination of all internal and exiting process streams in the network. With all input and output streams of the network fixed, the BPD criteria are easily computed for all pairs of column sections in the network. To find a feasible network configuration, the minimum BPD algorithm finds the minimum distance between each pair of adjacent column sections of the entire network in stage 3. If all minimum distances are inside suitable tolerances close to zero, the entire network is feasible. However, evaluating the BPD criterion for the initial guesses is not sufficient to conclude a design. Different choices of the internal degrees of freedom may also be feasible with even lower energy demand. Accordingly, the inner design optimization loop (stages 2 to 3) is a global search. In the case of infeasible specifications, these internal degrees of freedom are updated by a stochastic methodology or by the design engineer until a feasible design is approached. More details about the implementation of stochastic optimization for network synthesis can be found elsewhere.19 The inverse design methodology based on temperature collocation is ex-

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Figure 1. Detailed information flow diagram to find feasible design specifications for an entire flowsheet. Stage 1 uses the feed and sets up the final product purities. In stage 2, initial guesses of intermediate products’ purities and operating conditions (reflux or reboil ratios) are specified. Then, the product flow rate, internal vapor and liquid flow rate, stationary points, and difference points parameters of each column section are computed using species balances. Stage 3 performs the BPD criterion with obtained composition profiles by OCFE and polynomial arithmetic to assess feasibility. The schematics on the right side depict the specifications for the computations of each stage.

ecuted consecutively until all intersected column profiles are resolved. Also, a globally optimum energy column can be found, but not without pushing against the BPD criterion constraint. We therefore do not present globally optimal structures here. A quantitative recommendation of how to maintain feasibility, while searching for energy minima, will be the subject of a follow-up paper. It will be shown in section 4 that this process shown in Figure 1 for each column in the network is equivalent to solving the mass, equilibrium, summation, and heat (MESH) equations. 2.4. Nonideal Solution Models. In eq 2, the equilibrium between vapor and liquid streams existing in equilibrium stages is described by the equilibrium constant, Kj. For ideal mixtures, the equilibrium constant for each species is given by the ratio of the pure component vapor pressure over the total pressure, P. Accordingly, the K value changes with temperature along the column profiles. The vapor pressures as a function of temperature are typically modeled by the well-known Antoine equation. Nonideal phase equilibrium expressions including activity coefficients with implicit dependency on composition and temperature are described by eqs 5 and 6. The nonideal vapor-liquid equilibrium model uses liquid phase activity coefficients. The vapor phase fugacities were neglected to avoid the introduction of too many empirical

parameters. The temperature dependent binary parameters can be obtained using a commercial database such as the Aspen properties tool.40 Analytical derivatives of the activities coefficients with respect to composition and temperature are computed by general expressions for several nonideal models.41 To find the liquid composition profile, the temperature continuous model equation in eq 2 is solved for all sections belonging to each column of any complex network following the procedure described in Table 1. Kj )

(

V dKj 1 dPj γ + PVj ) dT P dT j

PVj γj P

[

(5)

c

∂γj dxi

i)1

i

∂γj

∑ ∂x dT + ∂T

])

(6)

Equation 2 can be solved implicitly after substituting eq 6, resulting in a system of ordinary differential equations. Using the temperature collocation for the entire network requires no modification of the minimum BPD and a global feasibility test,29 since the liquid profiles are again polynomial with piecewise elemental support whether the solution is ideal or nonideal. In addition, eq 7 gives the number of trays along the change in bubble point temperature along a column section, which is


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Table 2. NRTL Binary Interaction Parameters of the Nonideal Mixture of Methanol (A), Ethanol (B), n-Propanol (C), and Acetic Acid (D) Used for Modeling the Activity Coefficients component i

component j

aij

aji

bij

bji

R

methanol methanol methanol ethanol ethanol n-propanol

ethanol n-propanol acetic acid n-propanol acetic acid acetic acid

0 0 0 0 0 0

0 0 0 0 0 0

268.389644 259.918335 -392.58623 124.474895 -153.74537 -282.88883

-162.74727 -146.48108 568.71156 -107.23555 169.001873 392.600529

0.3 0.3 0.3 0.3 0.3 0.3

Table 3. Antoine Parameters of the Nonideal Mixture of Methanol (A), Ethanol (B), n-Propanol (C), and Acetic Acid (D) Used for the Calculation of the Partial Vapor Pressure components (xA,xB,xC,xD)

[methanol, ethanol, n-propanol, and acetic acid]

Antoine parameter A Antoine parameter B Antoine parameter C

[8.07240, 8.11220, 7.61920, 7.5596] [1574.990, 1592.864, 1375.140, 1644.05] [238.870, 226.184, 193.010, 233.524]

needed to compute the capital cost. The one-dimensional minimum BPD algorithm31 is executed without difficulty to ascertain the feasibility of each column of the network. c

dn )dT

dKj

∑ dT x

j

j)1

∑ {[( c

j)1

)

]}

(7)

1 1 1+ (x - yj) + (X - xj) Kj R∆ j R∆ ∆j

3. Result and Application The following section will demonstrate the efficacy of the inverse design procedure based on temperature collocation combined with the BPD criterion to synthesize feasible processes for the separation of nonideal mixtures and mixtures with 10 different species and a rigorous design of prefractionating complex networks for splitting a quaternary mixture into almost pure products. 3.1. Case Study 1sRigorous Network Design with Prefractionators for Nonideal Mixture Separation. Consider purifying a quaternary nonideal mixture of equimolar fraction in methanol

(A), ethanol (B), n-propanol (C), and acetic acid (D) into products of 99% purity. The activity coefficients were computed using the NRTL model with binary interaction parameters as listed in Table 2. Pure component vapor pressure values were approximated by the Antoine equation with the coefficients given in Table 3. In a recent article,30 we discussed the detailed designs of nine different networks to separate an ideal quaternary mixture. The most energy efficient network of that previous work is shown in Figure 2; this example is now generalized for a nonideal mixture model. The separation network configuration, NW1, consists of one simple quaternary column where pure A is obtained as the distillate. The bottom product of this column is fed to a second column operating as a prefractionator. Finally, the two product streams of the prefractionator are transferred to a double feed column in which the ternary mixture of B, C, and D is purified. Figure 2 depicts the liquid composition profiles obtained by solving eq 2 in each column section of the entire network. The precise intersection between all pairs of adjacent column sections in each distillation tower ensures that the network is realizable in practice. The global BPD feasibility criterion for the entire network was employed, the purities on intermediate streams and refluxes were adjusted with a stochastic global search to satisfy the column and network feasibility criteria in expressions 3 and 4. 3.2. Case Study 2sColumn Design of a 10-Component Mixture Separation. The next case studies show that our method can find feasible design solutions for all possible separations of a hydrocarbon mixture containing many species. In this case,

Figure 2. Liquid composition profiles of a feasible design for the complex column network NW1 to separate a quaternary nonideal mixture of methanol (A), ethanol (B), n-propanol (C), and acetic acid (D). The behavior of the nonideal mixture was modeled using NRTL.

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Figure 3. Global minimum BPD search in the reflux ratio space for chosen product purity (sloppy split case study of a 10-component mixture shown in Table 5). This figure demonstrates that the BPD algorithm can be extended to multicomponent mixtures without any restriction.

Figure 4. Feed and product specifications of a 10-component hydrocarbon mixture and liquid composition profiles computed by a temperature collocation approach in the temperature domain for direct split (top), indirect split (middle), and sloppy split (bottom). The mixture is constituted of i-pentane (N1), n-pentane (N2), cyclopentane (N3), 2,2-dimethylbutane (N4), 2,3-dimethylbutane (N5), 2-methylpentane (N6), 3-methylpentane (N7), n-hexane (N8), methylcyclopentane (N9), and benzene (N10).

realizable design specifications at a minimum operating cost were found for a conventional distillation tower like the ones used in a refinery to separate a 10-alkane mixture. The global search for minimum BPD in the space of reflux ratios for the fixed product purity of a 10-component mixture was performed. Figure 3 illustrates the topology of the thermo-

dynamic search space in terms of bubble point distance for different reflux specifications of a sloppy split separation column. With given equimolar feed and fixed desired product purities in terms of fractional recoveries, all available column degrees of freedom for the 10-component separation problem are fully determined.


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Table 4. Stream Table of a 10-Component Alkane Mixture Distillation Problem to Meet Direct and Indirect Fractional Recoveries of 99% and Sloppy Splits Recovery (90% for N1 and N2, 40% for N3 and N4 at the Top; 93% for N5, 99% for N6, and 100% for N7 to N10 at the Bottom) Requirements direct

indirect

sloppy

component

feeda

topa

bottoma

topa

bottoma

topa

bottoma

i-pentane, N1 n-pentane, N2 cyclopentane, N3 2,2-dimethylbutane, N4 2,3-dimethylbutane, N5 2-methylpentane, N6 3-methylpentane, N7 n-hexane, N8 methyl-cyclopentane, N9 benzene, N10

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.979999 0.019607 2.1081d-4 2.1031d-4 3.2591d-5 1.9974d-5 9.1875d-6 5.3079d-6 3.8540d-6 1.5274d-6

1.0013d-3 0.10905 0.11123 0.11123 0.11125 0.11125 0.11125 0.11125 0.11125 0.11125

0.11110 0.11110 0.11110 0.11110 0.11109 0.11108 0.11090 0.11040 0.10877 3.3331d-3

3.1000d-13 1.9010d-10 9.9051d-7 1.9710d-6 9.0247d-5 2.1011d-4 1.8810d-3 6.3033d-3 0.021011 0.970501

0.337197 0.327463 0.144720 0.136764 0.036013 0.015816 1.6320d-3 3.0256d-4 8.7790d-5 3.1703d-6

2.8429d-5 4.1308d-3 0.081152 0.084505 0.126968 0.135481 0.141459 0.142020 0.142110 0.142146

a

All compositions values are given in molar fractions.

Table 5. Antoine Parameters of the Alkane Mixture of Pentane (A), Hexane (B), Heptane (C), and Octane (D) Used for the Calculation of the Partial Vapor Pressure components (xA,xB,xC,xD)

[pentane, hexane, heptane, octane]

Antoine parameter A Antoine parameter B Antoine parameter C

[6.85221, 6.8702, 6.8938, 6.9094] [1064.630, 1168.7, 1264.4, 1349.8] [232.000, 224.2100, 216.6360, 209.3850]

Again, the temperature collocation to solve eq 2 with a minimum BPD criterion ensured realistic designs with exact intersection between individual liquid composition profiles between rectifying and stripping column sections. The reflux, r, was uniquely determined to meet the design specifications. The reflux, r ) 2.0, gives the globally minimum BPD at 53.01 °C; this is the desired operating condition for a feasible sloppy split separation target as shown in Figure 4. Some purity targets, especially those with unsuitable nonkey purities, cannot be matched with any reflux. Such a specification has to be rejected as thermodynamically impossible. The saturated liquid stream of an equimolar 10-component mixture is separated to meet different product specifications for the distillate and bottom summarized in Table 4. Three scenarios to achieve desirable product purities are shown. The first case is a direct split separation of the light component, i-pentane, with a recovery of at least 99%. The second case describes an indirect split of benzene with a recovery of 99% or more. Finally, the third case distributes some components into all product flows. This separation task requires a high recovery of the first and second components at the top, and a sloppy split for the remaining components in both product streams. In a refinery, the prefractionated product streams of this sloppy split would typically be further purified in a side-stripper or siderectifier. Figure 4 shows liquid composition profiles for the three feasible designs obtained using our method. An excellent performance of our inverse design methodology was observed independently of the product requirements, demonstrating that the BPD algorithm can be extended to mixtures with many species. 3.3. Case Study 3sRigorous Network Design with Prefractionators for an Ideal Mixture. A main objective of systematic separation synthesis is energy efficiency,19 yet this paper mainly illustrates the procedure to ensure realizable separations, especially the design of realistic prefractionating networks. A rigorous global energy minimization was not attempted at this point. The purification of a quaternary alkane mixture of equimolar fractions in pentane, hexane, heptane, and octane with vapor-liquid data given in Table 5 into products with a purity of at least 99% was studied. From the 18 structurally distinct flowsheets, to realize this separation

Table 6. Detailed Network Design Specifications and Operating Conditions Obtained by Solving the Inverse Design Problem to Separate a Nonideal Quaternary Mixture of Methanol, Ethanol, n-Propanol, and Acetic Acid (NW1), and Three Complex Networks Containing Prefractionating Columns to Purify an Alkane Mixture of Pentane, Hexane, Heptane, and Octane Using Temperature Collocation Combined with Minimum BPD Design Algorithms into Pure Products (99% of purity) NW1

NW2

NW3

NW4

feed stage product stages

15 10 17 15 1 1 1 1 37 12 26 19 total stages 37 12 26 19 column I feed flows (kmol/h) 100 100 100 100 product flows (kmol/h) 25.025 63.536 35.740 51.113 74.975 36.464 64.260 48.887 reflux 4.651 2 0.5 0.4 feed stages 10 28 15 7 23 product stages 1 1 1 1 22 30 34 14 26 total stages 22 30 34 26 column II feed flows (kmol/h) 74.975 63.536 64.260 51.113 48.887 product flows (kmol/h) 41.374 43.388 21.019 32.742 33.601 20.148 43.241 23.404 43.854 reflux 1.5 0.5 0.52 0.9 feed stages 10 6 5 7 42 20 19 28 36 35 47 product stages 1 1 1 1 28 14 13 21 45 25 26 33 47 43 49 total stages 45 47 43 49 column III feed flows (kmol/h) 41.374 43.388 35.740 32.742 33.601 20.148 21.019 23.404 36.464 43.241 43.854 product flows (kmol/h) 25.050 25.140 25.240 25.041 24.820 24.732 25.804 24.720 25.105 24.958 24.093 24.943 25.170 24.863 25.296 reflux 3 3.5 4.3 3.9

task suggested by Agrawal,30,31 we choose to discuss detailed designs for only the three prefractionating networks given in Figure 5 due to space limitations. For each network, feasible design specifications to achieve the final purity constraints were discovered with the help of the minimum BPD algorithm. Feasible column profiles were determined solving eq 2 by temperature collocation, and specifications were automatically adjusted until feasibility was achieved with the network BPD criterion.19 All flowsheets in Figure 5 produce the same product purity using completely different configurations and operating conditions. The separation networks, NW2 and NW3, use two simple columns as prefractionators in which the light and heavy components of each mixture entering the first two columns are

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Figure 5. Liquid composition profiles of the temperature collocation method in the temperature stage domain for prefractionating networks NW2, NW3, and NW4. The composition profiles on the top of a are corresponding to the first column, S1; the second row to the second column, S2 or C2, respectively; and the bottom profiles to the third column, C3, for each network.

completely separated. At least one of the intermediate components distributes into several product streams. The prefractionating columns require only one complex column to break the quaternary mixture into almost pure components. In NW4, the first column operating as a prefractionator distributes the two intermediate components into two intermediate streams, which are subsequently fed to the second column, which yields one pure product between each pair of adjacent side-product streams. The third column consists of six complex column sections generating four product streams at once, top-middle1middle2-bottom products.

Figure 5 shows the composition profiles in the bubble point temperature domain of all columns in each network with continuous connections between all pairs of adjacent column sections in each distillation tower. For each column section, intermediate product purities and refluxes were adjusted to satisfy the network BPD feasibility criterion. Global BPD minimization was performed with a stochastic global search19 using the molar composition of the distributed components at the products and the reflux ratio, r, as the remaining search variables. The main objective was to ensure realizable separations for the desired purity targets demonstrating the


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Figure 6. Liquid composition profiles of a nonideal mixture of methanol (A), ethanol (B), n-propanol (C), and acetic acid (D) separated in the complex column network NW1. All composition profiles are described in the stage domain of the temperature collocation model (left) and AspenPlus simulation (right). The behavior of the nonideal mixture was modeled using NRTL.

applicability of temperature collocation to design realistic prefractionating networks. 4. Validation of Synthesized Designs with Rigorous Flowsheet Simulation The ideal proof for the validity of the designs obtained by temperature collocation would be to build different design solutions of column networks at an industrial site. Unfortunately, it is impractical to do this even on the laboratory scale. Therefore, we propose to validate the composition profiles of the separation networks obtained by temperature collocation against rigorous mass, equilibrium, summation, and heat (MESH) equations. Accordingly, we demonstrate validation of the synthesized flowsheets with a state-of-the-art rigorous flowsheet simulator, AspenPlus. In addition to the design of

flowsheets that satisfy the MESH equations, our methodology can be used to initialize AspenPlus simulations for entire separation flowsheets. Practicing engineers may appreciate a fully automatic algorithm capable of initializing rigorous trayby-tray simulations. 4.1. Validation of Complex Network Design to Separate Nonideal Mixtures. Table 6 gives all design and problem specifications from the inverse design solution using temperature collocation with a BPD criterion. We used these design results to initialize an AspenPlus flowsheet equivalent to the network NW1. With this input to the simulator, it converges in a few iterations to the desired product purities without the need for any further user adjustments. Figure 6 compares the predicted column profiles of each column of network NW1 from temperature collocation with the AspenPlus profiles. The frames in

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Figure 6 show the trajectories of all liquid composition profiles in terms of stage numbers. The product specifications used to solve the inverse design are very close to the ones obtained by running the process simulation in AspenPlus. In the middle of some column sections, the rigorous flowsheet simulator results obtained by forward simulation (performance problem) deviate somewhat from the inverse design results. These differences reflect the fact that AspenPlus uses the discrete MESH equations, while temperature collocation adopts continuous profile equations. However, at the product nodes, the two methodssdespite their different mathematical basessagree reasonably well with differences of less than 0.5% in these applications. This comparison confirms the excellent match between the temperature collocation design and industrial rigorous simulator AspenPlus even for nonideal mixtures. 4.2. Validation of 10-Component Mixture. The degree of complexity in finding feasible designs to separate a multicomponent mixture with sharp or nonsharp split product requirements becomes harder as the dimension of the search space increases even for simple distillation towers. Since our inverse

Table 7. Column Design Specifications for a Simple Distillation Column to Split a 10-Component Mixture Obtained by Temperature Collocation Combined with Minimum BPD Algorithm

feed stage total stages feed flow (kmol/h) product flows (kmol/h) reflux

direct

indirect

sloppy

13 59 100 10.113 89.887 18.7

35 41 100 90.005 9.995 3.934

20 38 100 26.650 73.350 2

procedure targets the product purity, the feasible design for separating the 10-component mixture is determined by the minimum BPD of each component liquid profile. Again, inverse design solutions are ideally suited to initialize AspenPlus simulation. In this demonstration, stage numbers, feed stage, product flow rates, and reflux ratios of example 3.2 were input into AspenPlus. Table 7 gives all design specifications necessary to set up rigorous Aspen simulations at each product requirement. The accuracy of our approach brings Aspen simulations to converge in a few iterations to the desired product quality

Figure 7. Initialization of AspenPlus by the temperature collocation approach to validate the 10-component hydrocarbon mixture. The middle column shows the liquid composition profiles of all 10 components along the stage number using the temperature collocation model. The right column describes the liquid composition profiles in the stage number respectively found by AspenPlus for each design scenario.


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Figure 8. Liquid composition profiles in the stage domain of the temperature collocation model (left) and AspenPlus simulation (right) for prefractionating complex column network NW2.

without any manipulation or extra work. Figure 7 shows all composition profiles in the column stage domain of the three different product purity requirements virtually matching the composition evolution and products compositions of all 10components computed by AspenPlus forward process design. 4.3. Validation of Prefractionating Complex Networks. The design of promising energy efficient networks which employ only prefractionators involve finding design parameters that cannot be estimated by idealized methods. These methods often are restricted to pinched columns with infinite trays without information about the network stages, localization of the multifeed and multiproduct flowsheet, specific operating conditions such as vapor and liquid flows, or composition profiles. Accordingly, the limiting assumptions of these conventional design methods cannot help in validating process designs with rigorous flowsheet simulations. Aspects such as multifeed location and the position of side-product trays are critical information that any process simulator needs for its basic setup.

Section 3.3 showed all liquid and vapor profiles resulting from the predicted operating conditions, total tray numbers, the location of product removal, and feed streams for all columns in complex prefractionating networks for the separation of a quaternary alkane mixture with temperature collocation. Table 6 collects all necessary information to initiate all three prefractionating networks NW2, NW3, and NW4 in AspenPlus. Figures 8, 9, and 10 depict the liquid composition profiles in the column stage domain for all distillation columns of each complex network. Each diagram compares the liquid composition trajectories for each component obtained by temperature collocation on the left with the solution from AspenPlus shown on the right. In all cases, Aspen converged rapidly to virtually the same final product purities. There are some minor differences between the trajectories of the two design approaches. However, the excellent agreement in the initial and final points of the liquid composition trajectories demonstrates the equivalence between inverse design results and rigorous MESH forward flowsheet simulations.

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5. Discussion The automatic synthesis of multicomponent mixture separation especially for nonideal solution models remains an unsolved problem for conventional design methodologies. The inverse design methodology based on temperature collocation elucidates the topology of the thermodynamic search space in separation problems to identify feasible operating conditions for multicomponent separation problems including nonideal mixtures, prefractionating column configurations, and high-order multicomponent mixtures. The conventional method to design multicomponent mixtures often uses pseudocomponents by lumping components in the feed to reduce the size of the system. This approach reduces the mixture to be separated into a set of pseudocomponents upon which algorithms of minimum reflux conditions can be applied for sharp separations or related species mixtures.42 However, lumping rules are limited to grouping components with similar

volatilities. Usually, nonkey components are lumped, leading to sharp split assumptions for nonkey compounds which do not achieve realistic results. It also assumes that components’ molar fractions in the lumped feed and product cuts are equal. Lumping criteria should not be applied in cases such as the following: (1) individual concentration in a pseudocomponent such as a highly toxic material (e.g., sulfur compounds in crude oil like H2S43) should be purified, (2) the relative volatility of the individual components to be lumped are very different, and (3) the amount of species to be grouped in the pseudocomponent in the feed are not exactly the same as the internal composition in the pseudocomponent in the product. In these situations, rigorous and complete simulation of multicomponent mixtures is advisible. Temperature collocation drastically reduces the problem size for separation design, so that species lumping is not necessary.


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The design of complex networks to separate a quaternary nonideal mixture showed the versatility of temperature collocation for the most common types of phase equilibrium relationships. We provided examples in which heat integrated complex networks using prefractionators actually decreased the heat load for the entire network. However, these designs require sloppy intermediate products which cannot be identified with existing design algorithms. All designs obtained by temperature collocation were validated by the commercial flowsheet simulation. Initializing AspenPlus flowsheet simulations converged in a few iterations, achieving essentially the same product requirements without the need to adjust the design specifications obtained by our inverse design methodology. The results show that designs obtained by temperature collocation are equivalent to rigorous MESH

solutions. Alternatively, the design method can be considered a robust and rigorous process to initialize AspenPlus flowsheet simulations. 6. Conclusion This article demonstrates the applicability and stringency of the minimum bubble point distance criterion combined with temperature collocation to design complex separation networks for nonideal mixtures, prefractionating networks, and mixtures with many different species. The article focused on introducing the methodology to design entire feasible networks. A description on how to perform global energy minimization with the new methodology will be the subject of a follow-up paper. The minimum BPD metric was robust and efficient enough to find detailed network specifications, guaranteeing feasibility by

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enforcing exact profile intersections between all column sections in the network. The thermodynamically motivated bubble point temperature transformation is extended to any number of multicomponent mixtures with the most commonly used phase equilibrium relationships including constant volatility and ideal and nonideal mixtures. This method applies to both sharp and sloppy splits. Temperature transformation requires a one-to-one mapping between compositions and temperature and thus implies temperature profile monotonicity. Some authors point out that, in highly nonideal as well as in reactive distillations, monotonicity may not hold.44–47 However, collocation on generalized column profiles can also be employed in terms of tray numbers as in eq 1 without temperature transformation. Thus, the proposed design method is not limited to the temperature integration of eq 2. Even though the practical significance of separation trains with nonmonotonic temperature profiles has yet to be shown, the existence of special cases where temperature effects may occur is duly noted. Several complex networks were designed rigorously to separate a quaternary nonideal mixture. Also, three prefractionating complex networks were studied in detail, showing solutions with exact intersection between all pairs of adjacent sections in any column of each network. The general applicability of the minimum BPD feasibility test for purifying mixtures with a large number of species without the need for lumping was illustrated by the rigorous column design to split a 10-component mixture into a set of desired product cuts. Finally, the last section demonstrated a remarkable agreement between the inverse design solutions and rigorous forward flowsheet simulations. All optimal, feasible designs were validated by initialization of AspenPlus simulations, which converged in a few iterations, practically establishing the agreement between our method and rigorous tray-by-tray MESH equations. The forward product specifications obtained by Aspen simulations match closely the results obtained by the inverse design with temperature collocation. The methodology is ready to address current unresolved design problems such as the computer-aided design of energyefficient separations, the design of biorefineries, or new process designs for carbon sequestration. Acknowledgment Financial support by DOE Grant DE-FG36-06GO16104 (PI Dr. Rakesh Agrawal) is gratefully acknowledged. We acknowledge Dr. Chau-Chyun Chen for his support in providing an Aspen software research license. Literature Cited (1) Linninger, A. A.; Stephanopoulos, E.; Ali, S. A.; Han, C.; Stephanopoulos, G. Generation and assessment of batch processes with ecological considerations. Comput. Chem. Eng. 1995, 19, S7–S13. (2) Linninger, A. A.; Ali, S. A.; Stephanopoulos, G. Knowledge-based validation and waste management of batch pharmaceutical process designs. Comput. Chem. Eng. 1996, 20, S1431–S1436. (3) DOE Technical Topic Description. www.doe.gov (accessed Apr 2010). (4) Grossmann, I. E.; Aguirre, P. A.; Barttfeld, M. Optimal synthesis of complex distillation columns using rigorous models. Comput. Chem. Eng. 2005, 29 (6), 1203–1215. (5) Harwardt, A.; Kossack, S.; Marquardt, W. Optimal column sequencing for multicomponent mixtures. Comput.-Aided Chem. Eng. 2008, 25, 91–96. (6) Engelien, H. K.; Skogestad, S. Minimum energy diagrams for multieffect distillation arrangements. AIChE J. 2005, 51 (6), 1714–1725.

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ReceiVed for reView January 9, 2010 ReVised manuscript receiVed April 14, 2010 Accepted April 20, 2010 IE1000532

Rigorous Separation Design. 1. Multicomponent Mixtures, Nonideal

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