Ultimate Reaction Selectivity Limits for Intensified ReactorâSeparators

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Kinetics, Catalysis, and Reaction Engineering

Ultimate Reaction Selectivity Limits for Intensified Reactor-Separators Jeffrey A. Frumkin, Lorenz Fleitmann, and Michael F. Doherty Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b04143 • Publication Date (Web): 14 Nov 2018 Downloaded from http://pubs.acs.org on November 23, 2018

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Industrial & Engineering Chemistry Research

Ultimate Reaction Selectivity Limits for Intensified Reactor-Separators Jeffrey A. Frumkina , Lorenz Fleitmanna,1 , and Michael F. Dohertya,∗ a Department of Chemical Engineering, Engineering II, University of California Santa Barbara, Santa Barbara, CA 93106, United States November 10, 2018

Abstract The Continuous Flow Stirred Tank Reactor (CFSTR) Equivalence Principle, developed by Feinberg and Ellison, proves that any and every reaction/mixing/separation process is equivalent to a process comprising at most R+1 CFSTRs and a perfect mixer-separator, where R is the number of linearly independent chemical reactions. Frumkin and Doherty showed the CFSTR Equivalence Principle can be used together with global optimization to find the maximum selectivity of a chemistry independent of process design. These selectivity targets are useful in the context of process intensification as they represent ultimate selectivity improvements that can be achieved by combining multiple unit operations into a single device. In this work, the model is reformulated as a mixed-integer nonlinear program to solve this nonlinear and non-convex optimization problem. We implement a more robust, deterministic global optimization using a spatial branch-and-bound algorithm (BARON) to investigate the selectivity limits for the production on acrolein, a chemistry which has thirteen components and seventeen reactions. We find maximum selectivities that are lower than the stoichiometric selectivity limit and can be used as a target for process intensification. Keywords: attainable regions, reactor design, reactor optimization, reaction selectivity

The authors declare no competing financial interest.

∗ To whom correspondence should be addressed. Phone: +1 (805) 893-5309, e–mail: [email protected], Address: 3323 Engineering II University of California, Santa Barbara, Santa Barbara, CA 93106-5080 1 Institute of Technical Thermodynamics, RWTH Aachen University, Aachen, Germany

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1

Introduction

Knowledge of the maximum attainable selectivities for a given chemical reaction network would be very useful during process intensification and design. In addition to serving as a benchmark for conventional process design, upper bounds on reaction selectivity also represent the ultimate selectivity that can be achieved through process intensification. However, until recently a methodology to obtain such information for chemistries with more than two or three reactions was not readily available. Horn1 first introduced the concept of attainable regions (AR) in 1964 in an attempt to find optimal process designs for a given chemistry. Since then, much creativity and effort has resulted in methodologies to enumerate ARs using geometric arguments for systems involving reaction and mixing2–8 as well as systems involving reactions, separations, and mixing.9 Unfortunately, all of these geometric methods are limited to a small number of components and reactions, which limits their utility. The Continuous Flow Stirred Tank Reactor (CFSTR) Equivalence Principle, developed by Feinberg and Ellison,10 abandons the geometric approach altogether. This principle allows any and every reactor-mixerseparator system (RMS system) to be decomposed into an equivalent system comprising at most R + 1 CFSTRs and a perfect mixer-separator (a “Feinberg Decomposition”, or FD), where R is the number of linearly independent chemical reactions (see Figure 1).

Feinberg and coworkers10, 11 showed that the

CFSTR Equivalence Principle can be used to find the maximum attainable production flow rate of some desired product from a given chemistry for a given feed and reaction volume. This is done by maximizing the flow rate of the specified component from the FD (with a given feed and total reactor volume) generated by the CFSTR Equivalence Principle. The utility of this method results from the fact that an objective function can be optimized over the AR without having to first identify the AR or its boundary. Frumkin and Doherty12 showed that the CFSTR Equivalence Principle together with the chemical kinetics and a global optimization routine can be used to find the maximum selectivity of a chemistry entirely independent of process design. They also show that constraints on overall reaction conversion per pass and molar flow rates within the FD can be enforced to obtain bounds on selectivity that may be more practical for certain classes of chemistries than the unconstrained case. In this work, the problem is formulated as a nonlinear program to solve the optimization problem. The problem is nonlinear and non-convex, and we implement a more robust, deterministic global optimization using a spatial branch-and-bound algorithm (BARON)13 implemented in the General Algebraic Modeling System (GAMS).14 (See SI for software and hardware implementation details.) We demonstrate the utility of the approach to investigate the selectivity limits for the production of acrolein, a chemistry which has eleven components and seventeen reactions. We find upper bounds on selectivity that are less than the

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Figure 1: Depiction of the CFSTR Equivalence Principle. The FD exactly models all potential reactor-mixerseparator system configurations for a given chemistry of interest. stoichiometric selectivity limit and provide ultimate targets for process intensification.

2

Optimization of the Feinberg Decomposition

Due to a nonlinear objective function and many nonlinear constraints, the optimization of the FD is formulated as a nonlinear program (NLP) with the general form:15 min x

f (x)

s.t. h(x) = 0

(1)

g(x) ≤ 0 x ∈ Rn Here, f (x) is the objective function to be minimized, h(x) are the equality constraints and g(x) are the inequality constraints; x represents the continuous variables. Feinberg and coworkers10, 11 used the CFSTR Equivalence Principle to find the maximum attainable flow rate of a desired product. Our goal, however, is to find the maximum attainable selectivity for one or more components. Thus, in general the objective function is a weighted average of the selectivity of one or more products. (One may wish to maximize the selectivity of more than one product, or minimize the selectivity of an undesired product.) The optimization variables include the concentrations, temperatures, and pressures in each CFSTR, the volume of each CFSTR, and possibly the molar feed flow rates to each CFSTR. The equality constraints imposed on the FD include material balances, equations of state, and chemical kinetic rate laws. The inequality constraints bound the temperatures, pressures, and compositions



Figure 2: A depiction of the FD with certain process variables shown. within each CFSTR of the FD as well as restrict the total reaction volume and possibly the molar flow rate to each CFSTR (if the Frumkin and Doherty approach is adopted). The full set of model equations is given by Equations 2 to 21. Figure 2 depicts the location of certain process variables within the FD. Equation 2 gives the objective function as a weighted average of selectivities (normally most of the weights will be zero), which are defined in Equation 3. The steady-state material balance constraints on the FD are given by Equation 4 for R reference components, and Equation 5 gives the extent of reaction material balance constraints. R of the equations given by Equations 4 and 5 are redundant; however, including these redundant reactions (instead of just using C constraint equations of the form of Equation 4) results in faster convergence and tighter bounds, as will be discussed in the next section. Equation 6 constrains the limiting reactant (LR) conversion per pass of the FD to take a specified value. Equation 7 defines the volumetric occurrence rate function of reaction r in CFSTR i while Equation 8 defines the rate expression for component j in CFSTR i. The effluent flow rates from each CFSTR are calculated using Equation 9, and Equations 10 and 11 allow for the calculation of the inlet concentrations associated with the given (c, T, P) state, volume, and feed flow rate of each CFSTR. As shown by Frumkin and Doherty,12 constraining the feed flow rates to a CFSTR may make certain composition-temperature-pressure (c, T, P) states infeasible for a given volume (concentrations become less than zero or greater than the components molar densities). Therefore, we must calculate the inlet concentrations to ensure only feasible (c, T, P) states are permitted. Equation 12 defines the density of component j in CFSTR i (from either an equation of state or other constitutive relation), Equation 13 ensures that the concentrations of all components sum to the total concentration (this is especially important for gas-phase chemistries where ctot = P/RT ), and Equation 14 ensures that the volume fractions in each CFSTR sum to unity. Equation 15 constrains the concentrations of 3 ACS Paragon Plus Environment

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all components to lie between zero and their pure component molar density. Equations 16 and 17 constrain the temperatures and pressures of the CFSTRs within the set bounds while 18 ensures the total reaction volume constraint is respected. Equations 19 restrict the flow rates to and from each CFSTR to only positive values and Equation 20 enforces positive effluent flow rates from the FD. Finally, Equation 21 ensures that flow rates to each CFSTR are less than a specified multiple α of the total feed flow rate to the FD. When α is specified to be a large number (e.g., 1000) then the algorithm becomes equivalent to the Feinberg and Ellison theory.

max

cj,i ,Vi ,Ti ,Fi

C X

σj Sj

(2)

j=1 j6=LR

Pj − Fj j = 1, ..., C FLR XLR R+1 X rj,i Vi = 0 {j} = R reference components Fj − Pj +

s.t. Sj =

(3) (4)

i=1

Fj − Pj + ν Tj ξ = 0 XLR =

j = 1, 2, . . . C

FLR − PLR FLR

rˆr,i = g(ci , Ti , Pi )

(5) (6)

i = 1, ..., R + 1, r = 1, ..., R

(7)

νj,r rˆr,i

i = 1, ..., R + 1, j = 1, ..., C

(8)

C X

i = 1, ..., R + 1

(9)

i = 1, ..., R + 1

(10)

i = 1, ..., R + 1, j = 1, ..., C

(11)

i = 1, ..., R + 1, j = 1, ..., C

(12)

i = 1, ..., R + 1

(13)

C X cj,i =1 ρ j=1 j,i

i = 1, ..., R + 1

(14)

0 ≤ cj,i , c0j,i ≤ ρj,i

i = 1, ..., R + 1, j = 1, ..., C

(15)

Tmin ≤ Ti ≤ Tmax

i = 1, ..., R + 1

(16)

Pmin ≤ Pi ≤ Pmax

i = 1, ..., R + 1

(17)

rj,i =

R X r=1

Pi = Fi +

rj,i Vi

j=1

qi0 =

Pi ctot,i

qi0 c0j,i =

−

C X rj,i Vi ρ j=1 j,i

cj,i Pi − rj,i Vi ctot,i

ρj,i = f (ci , Ti , Pi ) ctot,i −

C X

cj,i = 0

j=1



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Figure 3: A visualization of the BARON Algorithm adapted from Ryoo and Sahinidis17 and published with permission. (a): The original problem is relaxed into a convex optimization problem. (b): The convex problem is solved to find a lower bound (LB) on the objective function. The solution of the relaxed problem is used as a starting point in the local solution of the original problem to find an upper bound (UB). (c) The problem is broken down into two regions, each of which are relaxed to find their upper and lower bounds. Region 2 can be excluded because its lower bound is greater than the upper bound of Region 1. R+1 X

Vi ≤ Vmax

(18)

i=1

Pi , Fi , qi0 ≥ 0

i = 1, ..., R + 1

(19)

Pj ≥ 0

j = 1, ..., C

(20)

Fi ≤ αF

i = 1, ..., R + 1

(21)

The optimization of the FD is achieved using the deterministic global optimization solver BARON13 in conjunction with GAMS.14 Using branch-and-bound and range-reduction techniques,16 BARON divides the feasible solution space into successively smaller regions and eliminates those that do not provide a better objective function value than the current best. Within each region a lower bound on the objective function to be minimized is determined by relaxing the original formulation within that region into a convex problem and solving the relaxed problem (Figure 3a). In relaxing the problem, the equality constraints are made linear by fixing the nonlinear terms. Note that in relaxing the constraints, equations can lose their meaning as the optimization variables are varied. For example, in chemical processes involving reaction, relaxing constraints leads to material balances that fail to close. The solution to the relaxed problem is then used as a starting point to find a local solution of the original problem which is used as an upper bound for the region (Figure 3b). The region is divided into subregions which are also relaxed to obtain lower and upper bounds. (Figure 3c). Sections of the solution space can be removed from consideration by comparing the bounds of regions and subregions. If the lower bound on the objective function of some subregion is greater than the best known upper bound on the objective function, then clearly the subregion will not provide a more favorable objective function value. For example, in Figure 3c, the lower bound in Subregion 2 is greater than the upper bound in Subregion 1; therefore, we know the optimal objective function value does not lie in Subregion 2. In optimizing the FD we are looking for a maximum selectivity; however, the problem can


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be reformulated as the minimization of (−1)

PC

j=1

σj Sj without loss of generality.

An important practical advantage of using BARON is that it does not require any user intervention once the optimization is initiated. This is in constrast with the MultiStart method18 used previously, which would frequently require restarts. The goal of the Frumkin and Doherty application of the CFSTR Equivalence Principle is to generate curves of ultimate selectivity versus conversion. In using the MultiStart method, the selectivity versus conversion solution typically contains outliers that do not lie on the true, and smooth, utimate selectivity versus conversion curve. these outliers inform the user that the algorihm has converged to a local optimum instead of the global optimum. The user then must re-initiate the algorithm at the conversions corresponding to the outlying selectivities. If this methodology is to be used in general purpose software (e.g., AspenPlus), the software must give consistently correct results ithout requiring user intervention. Thus, in this context, we see that BARON is a more robust and superior optimization method.

3

Obtaining Tight Bounds

Branch-and-bound algorithms become more efficient as the gap between the original problem and its relaxation become tighter. This allows the algorithm to quickly discard subregions, leading to faster convergence. Unfortunately, the relaxation process often results in loss of the physical meaning of constraints.19 As a result, material balances (Equation 4) may fail to close, which is problematic from a physical sense. Also problematic is that non-closure of material balances results in absurdly large objective function upper bounds (e.g., a very large gap between the original problem and its relaxation), leading to very long convergence times. For example, in one chemistry we optimized, the upper bound on selectivity was found to be on the order of 105 . There is, in fact, a stoichiometric upper bound on reaction selectivity that is independent of kinetics and is of order O(1), as discussed by Frumkin and Doherty.12 In this section, we outline how we can reformulate the problem to obtain tighter bounds that match the stoichiometric selectivity limit, increase the efficiency of the algorithm, and preserve the physical nature of the constraints. The constraints for which we want to preserve the physical meaning are the material balances of every component around the FD (Equation 4). These constraints can be written in vector form as

F − P + rV = 0

(22)

where F is the vector of feed flow rates to the FD, P is the vector of effluent flow rates from the FD, r is a C × (R + 1) matrix of generation rates in the CFSTRs, and V is a (R + 1) × 1 vector of CFSTR volumes. For a specified feed to the FD, the only unknowns are P because V is specified as a part of the optimization



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algorithm and r is calculated from the (c, T, P) states of the R + 1 CFSTRs. Equation 22 is nonlinear in the rV term, so this is the term that must be relaxed. However, these relaxations result in violation of the underlying stoichiometry of the chemistry, which results in absurdly large upper bounds on selectivity. As demonstrated by Ruiz and Grossmann,19 introducing redundant constraints, especially redundant linear constraints (which do not need to be relaxed) can help recover the physical meaning and improve the relaxation. We recover the physical meaning by using only R (nonlinear) material balances of the form of Equation 4 and incorporating C extent of reaction material balances (all of which are linear), R of which are redundant. The C extent of reaction material balances are given by

Fj − Pj + ν Tj ξ = 0

(23)

F −P +Vξ =0

(24)

where ν Tj is the 1 × R vector of stoichiometric coefficients of component j in every reaction, V is a C × R matrix of stoichiometric coefficients of all C components in all R reactions, and ξ is a R × 1 vector of extents of reaction within the FD. For a specified vector of feed molar flow rates to the FD (F ), the C equations given by Equation 24 has C + R unknowns (the C effluent flow rates Pj and the R extents of reaction ξr ); therefore, there are (C + R) − C = R degrees of freedom corresponding to the R extents of reaction. (Recall that V depends only on the stoichiometry of the chemistry.) The extents of reaction in the FD are not arbitrary; rather, they are directly related to the molar generation rates in the CFSTRs within the FD. Therefore, we can solve for the R extents of reaction using R material balances (Equation 4). We will call the R components whose material balances are used to solve for the reaction extents the “reference components”. Then, we can solve for the C effluent flow rates using C reaction extent balances (Equation 24). Thus, instead of using C material balances of the form of Equation 4, we use a new set of C + R material balances given by

Fj − Pj +

R+1 X

rj,i Vi = 0,

{j} = R reference components

(25a)

i=1

Fj − Pj + ν Tj ξ = 0,

j = 1, 2, . . . C

(25b)

F ref − P ref + r ref V = 0

(26a)

F −P +Vξ =0

(26b)

or in vector form as


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Figure 4: Reaction network for the partial oxidation of propylene to acrolein. Solid blue arrows correspond to reactions promoted by water and dashed blue arrows correspond to reactions inhibited by water. (Redrawn with permission from Bui L, Chakrabarti R, Bhan A. ACS Catalysis. 2016; 6(10): 6567-6580.21 Copyright 2016 American Chemical Society.) We note that the R reference components must be chosen such that the matrix Vref is invertible. This is discussed in the SI. Thus, by incorporating redundant material balances in this manner, we can improve convergence.

4

Partial Oxidation of Propylene to Acrolein

Acrolein is an intermediate in the production of methyl methacrylate and is also used in the production of acrylic acid, 1,3-propanediol, pyridines and certain flavors and fragrances.20 One synthesis route to acrolein is through the partial oxidation of propylene over a Bi2 Mo3 O12 catalyst.21 As is typical of partial oxidation chemistries, many side reactions occur, producing acetic acid, acrylic acid, acetaldehyde, ethylene, carbon monoxide, and carbon dioxide. The reaction network21 is given in Figure 4. A full set of balanced chemical reactions was derived from the work by Bui et al.21 and can be found in the SI. This chemistry has 17 reactions; however, only nine of them are linearly independent. Therefore, the FD for this chemistry requires at most 10 CFSTRs. Reaction kinetics for this chemistry were derived by Bui et al.21 in a recirculating batch reactor at 623K and 116 kPa with initial feed partial pressures of propylene, water, air, and helium being 6 kPa, 7 kPa, 34 kPa, and 69 kPa, respectively. The kinetic rate expressions are given by

rˆ1 = k1 PC3 H6


(27)


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rˆ2 = k2 PC3 H6

(28)

rˆ3 = k3 PC2 H3 CHO PH2 O

(29)

rˆ4 = k4 PCH3 CHO PH2 O

(30)

rˆ5 = k5 PCH3 CHO

(31)

PC3 H4O2 P H2 O

(32)

rˆ6 = k6

rˆ7 = k7 PCH3 COCH3

(33)

rˆ8 = k8 PC3 H6 PH2 O

(34)


(35)


(36)

rˆ11 = k11 PC2 H3 CHO

(37)

rˆ12 = k12

PC2 H3 COOH PH2 O

(38)

rˆ13 = k13 PC2 H3 COOH PH2 O

(39)

rˆ14 = k14 PC3 H6

(40)

rˆ15 = k15

PCH3 COOH PH2 O

(41)

rˆ16 = k16 PC3 H6

(42)

rˆ17 = k17 PC2 H3 CHO

(43)

Because the experiments were carried out at a single temperature and pressure, the FD and the conventional reactors to which it is compared will operate isothermally and isobarically. Note, however, that this methodology is perfectly valid for temperature- and pressure-varying systems, provided a kinetic model is available over the given temperature and pressure range. The reaction rates have a first order dependence on the partial pressure of the reacting hydrocarbon. Reactions represented by solid blue arrows in Figure 4 additionally have a first order dependence on the partial pressure of water, and the reactions represented by the dashed blue arrows have an inverse first order dependence on the partial pressure of water. The experiments by Bui et al.21 were conducted only up to a conversion of 25%. We assume the kinetics are valid up to a conversion of 100% to potentially motivate additional experiments at other molar ratios and at higher conversions. We further assume that the kinetics are valid in the absence of water and helium. As can be seen from the reaction network, the stoichiometric selectivity limit for acrolein is unity (if only reaction 1 proceeds). With only propylene, oxygen, and nitrogen fed, only reactions 1, 2, 14, and 16 can initially proceed at a conversion near zero. Therefore, the selectivity at a conversion near zero is equal to 9 ACS Paragon Plus Environment

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Figure 5: Comparison of selectivities. The selectivity of acrolein in a conventional CFSTR, and conventional PFR, and the optimized FD with α = 1 and for the unconstrained (α = ∞) flow rate case. Note that the α = 1 and α = ∞ cases are nearly identical. the rate of reaction 1 divided by the sum of the reaction rates for these reactions,

0 Sacrolein =

k1 PC3 H6 (k1 + k2 + k14 + k16 )PC3 H6

(44)

0 which gives Sacrolein = 0.807. Acrolein is produced only from propylene, and the by-product forming reactions

are irreversible; therefore, we cannot achieve a selectivity greater than 0.807. Figure 5 shows the acrolein selectivity for a conventional CFSTR, a conventional PFR, and the FD for both the α = 1 and the unconstrained (α = ∞) case. For the given reaction kinetics, process intensification involving any combination of reaction, separation, and mixing cannot surpass the ultimate selectivity bound given by the FD. The feed flow rate of propylene for all three reactor configurations is FC3 H6 = 10 mol/min, and the molar ratio of air to propylene is 19. The reactor temperature and pressure are specified to be 623K and 116 kpa, respectively and are maintained constant in every reactor. The temperatures and pressures are identical to the experimental conditions reported in Bui et al.21 As expected, the acrolein selectivity in a conventional CFSTR and PFR is 80% at a conversion near zero; however, selectivity decreases rapidly with conversion. At a conversion of 80%, the selectivity in a CFSTR and PFR are 8% and 19%, respectively. In contrast, the constrained FD maintains a high selectivity at all conversions, decreasing by less than one percent at a conversion of unity. Furthermore, the constrained FD performs almost as well as the unconstrained FD. Thus, we see that a better reactor design than the conventional CFSTR or PFR may



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Table 1: Molar ratio of propylene to oxygen for the overall feed stream and the average feed streams to the Feinberg reactors for conversions per pass of XLR = 0.05, XLR = 0.45 and XLR = 0.99. The mole fractions of the feed streams to the ten Feinberg CFSTRs are averaged to obtain one representative feed composition. Component C3 H6 O2 N2 FC3 H6 FO2

x0j 0.05 0.20 0.75 0.25

x0j,i (XLR = 0.05) 0.997 0.003 332.7

x0j,i (XLR = 0.45) 0.986 0.014 71.7

x0j,i (XLR = 0.99) 0.968 0.032 30.6

exist. Inspecting the details of the CFSTRs within the FD (see SI), we observe that the FD is able to achieve higher selectivities than the conventional reactors by operating with significantly different feed conditions to the reactors within the FD. This is shown in Table 1. The mole fractions in the feed stream F to the conventional reactors (and to the FD) are given by x0j while the mole fractions in the feeds to the FD CFSTRs are given by x0j,i . The feeds to the conventional reactors have a 4 to 1 molar ratio of O2 to propylene while the CFSTRs in the FD operate with a large excess of propylene. The amount of O2 fed to the FD reactors is precisely the amount needed to allow for the prescribed conversion of the FD. That is, the O2 is completely consumed. Furthermore, nitrogen is not fed to the CFSTRs within the FD. Thus, we see that an optimization of the FD reveals that operating with an excess of propylene (instead of oxygen) results in much better selectivities. Using this information, we can adjust the feeds to the conventional reactors to potentially obtain better selectivities. Figure 6 compares the selectivities of the conventional PFR with both the original feed composition and modified feed composition. The total feed flow rate of the modified feed is still 200.5 mol/min, but the molar ratio of air to propylene is now about 1.2 (instead of the previous value of 19), making oxygen the limiting reactant. Changing the feed composition changes the limiting reactant; therefore, in Figure 6 the abscissa is the absolute rate of propylene converted instead of conversion. We see that with this improved feed composition the selectivities in the conventional PFR are significantly improved. In fact, the selectivity in the improved PFR is only slightly lower than the Feinberg Selectivity Limit. Furthermore, the selectivity in the PFR with the improved feed no longer decreases to zero. While these results are promising, a number of steps must be taken moving forward. First and foremost, the kinetic model must be modified to explicitly include the partial pressure of oxygen in the rate laws. Bui, et al.21 found the rate laws to be independent of the partial pressure of oxygen (probably due to oxygen being present in 4:1 excess). By making oxygen the limiting reactant, the reactors will be operating at much higher conversion of oxygen, and thus a new kinetic model with explicit oxygen dependence is required.


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Figure 6: Improved conventional PFR selectivities. The selectivity of acrolein in a conventional PFR with both the original and improved feed compositions compared to the Feinberg Selectivity limits correspond to the α = 1 and α = ∞ cases plotted against the rate of conversion of propylene. Note that the α = 1 and α = ∞ cases are nearly identical. With the new kinetics, the optimization of the FD can be completed again to check if the new kinetics yield different results. Finally, we can analyze the results of the FD to potentially gain insight about how one can better design a conventional process, either through process intensification or otherwise.

5

Conclusions

Optimization of the FD given by the CFSTR Equivalence Principle yields a useful target or benchmark for process intensification and design. By utilizing the BARON algorithm, in conjunction with redundant linear constraints, we can achieve a more robust and efficient optimization of the FD. In conclusion, the CFSTR Equivalence Principle is a useful engineering tool that can be applied to obtain important and practical information about reactor design and process intensification.

Acknowledgments We are grateful to Dr. Robert Huss, Eastman Chemical Company, for a very illuminating discussion.



Funding This work was supported by the Chemical Life Cycle Collaborative at UCSB, the US EPA [Grant number RD835579], and the German Academic Exchange Service (DAAD) through a scholarship within the AachenCalifornia Network (ACalNet).

Nomenclature Symbols C

Number of components

cj

Molar concentration

c

C × 1 vector of molar concentrations

F

Inlet molar flow rate

F

C × 1 column vector of inlet molar flow rates

kr

Temperature dependent rate constant of reaction r

P

Effluent molar flow rate

P

Pressure

P

C × 1 column vector of effluent molar flow rates

q

Volumetric flow rate

rj

volumetric species formation rate function of component j

r

C × 1 vector of volumetric species formation rate functions

r

C × (R + 1) matrix of generation rates in the CFSTRs within the FD

R

Number of reactions

Sj

Selectivity of component j

T

Temperature

V

Reactor volume

V

(R + 1) × 1 vector of CFSTR volumes within the FD

V

C × R matrix of stoichiometric coefficients of all C components in all R reactions

X

Conversion of limiting reactant

Greek Letters α

Flow rate constraint ratio

νj,r

Stoichiometric coefficient of component j in reaction r


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ν Tj

1 × R vector of stoichiometric coefficients of component j in every reaction

ξ

R × 1 vector of extents of reaction within the FD

ρj

Molar density of component j

Subscripts and Superscripts 0

Denotes reactor inlet variable

i

Denotes CFSTR number

j

Denotes component index

r

Reaction number Supporting Information: kinetic model for acrolein synthesis from propylene, detailed results for

acrolein synthesis from propylene, explanation of how to use extents of reaction to restrict the stoichiometric AR to reflect irreversible reactions, explanation of rules for choosing reference components, software and hardware details

References [1] Horn FJM. Attainable and non-attainble regions in chemical reaction technique. In Chemical Reaction Engineering (Proceedings of the Thrid European Symposium) 1964. [2] Glasser D, Hildebrandt D, Crowe CA. A geometric approach to steady flow reactors: The attainable regions and optimization in concentration space. Ind Eng Chem Res. 1987; 26: 1803-181. [3] Hildebrandt D, Glasser D, Crowe CM. Geometry of the attainable region generated by reaction and mixing: With and without constraints. Ind Eng Chem Res. 1990; 29(1): 49–58. [4] Hildebrandt D, Glasser D. The attainable region and optimal reactor structures. Chem Eng Sci. 1990; 45(8): 2161–2168. [5] Feinberg M, Hildebrandt D. Optimal reactor design from a geometric viewpoint—I. Universal properties of the attainable region. Chem Eng Sci. 1997; 52(10): 1637–1665. [6] Feinberg M. Optimal reactor design from a geometric viewpoint. Part II. Critical sidestream reactors. Chem Eng Sci. 2000; 55(13): 2455–2479. [7] Feinberg M. Optimal reactor design from a geometric viewpoint—III. Critical CFSTRs. Chem Eng Sci. 2000; 55(17): 3553–3565. 14 ACS Paragon Plus Environment


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