Ultimate bounds on reaction selectivity for batch reactors

Accepted Manuscript Ultimate Bounds on Reaction Selectivity for Batch Reactors Jeffrey A. Frumkin, Michael F. Doherty PII: DOI: Reference:

S0009-2509(18)30807-8 https://doi.org/10.1016/j.ces.2018.11.030 CES 14618

To appear in:

Chemical Engineering Science

Please cite this article as: J.A. Frumkin, M.F. Doherty, Ultimate Bounds on Reaction Selectivity for Batch Reactors, Chemical Engineering Science (2018), doi: https://doi.org/10.1016/j.ces.2018.11.030

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Ultimate Bounds on Reaction Selectivity for Batch Reactors 1

Jeffrey A. Frumkin and Michael F. Doherty1,∗ Department of Chemical Engineering, University of California Santa Barbara, Santa Barbara, CA 93106 November 10, 2018

Abstract Targets and benchmarks are useful in chemical process design as they provide an objective, quantitative assessment of a proposed process flowsheet. In addition, with target bounds on reaction selectivity one can also explore the sustainability limits for a chemistry of interest. Unfortunately, targets for reaction selectivity are difficult to obtain using conventional design methods. In 2001, Feinberg and Ellison developed the Continuous Flow Stirred Tank Reactor (CFSTR) Equivalence Principle, providing a methodology to obtain ultimate bounds on steady-state productivity for a chemistry of interest entirely independent of process design. In previous articles, we showed how the CFSTR Equivalence Principle can be used to obtain bounds on reaction selectivity independent of process design for steady-state processes. In this article we prove that the CFSTR Equivalence Principle is also applicable to batch and semi-batch processes, thus providing a unifying framework to obtain an ultimate target for reaction selectivity applicable to all candidate processes for a chemistry of interest. This limit also applies to systems with periodic and chaotic operations. We demonstrate the method with an example for the production of lactic acid through the alkaline conversion of fructose. Keywords: attainable regions, reactor design, reactor optimization, reaction selectivity, batch reactors

Declarations of interest: none

∗ To

whom correspondence should be addressed. Phone: +1 (805) 893-5309 e–mail: [email protected] This article is dedicated to Professor Rakesh Agrawal in honor of his many outstanding achievements in chemical engineering.

1

Introduction

With a growing world population and an ever-increasing standard of living, the task of sustainably producing chemical products is becoming increasingly important. One aspect of sustainable chemical production is using raw materials as efficiently as possible. The design metric that best represents raw materials efficiency is chemical reaction selectivity. In this work, we present a methodology that allows one to determine ultimate bounds on reaction selectivity for chemistries of interest entirely independent of process design. With these target reaction selectivities, we can explore the sustainability limits for chemical production. When developing a new process or seeking improvement of an existing process it is difficult to know when a design is sufficient and no further improvement is necessary, or perhaps even possible, without some benchmark with which to compare. An ultimate bound (or target) on reaction selectivity would be very useful in process design; however, such a bound cannot be obtained using conventional design methods. In some circumstances “ultimate” refers to a maximum selectivity (e.g., for a desired product), and in other instances refers to a minimum selectivity (e.g., for an undesired component). In either case, a target bound on reaction selectivity would assist in determining when a design should be finalized. In previous articles (Frumkin and Doherty, 2018; Frumkin et al., Submitted. 2018) we showed how to obtain practical bounds on reaction selectivity for steady-state processes using only the reaction kinetics for a chemistry of interest by building on the work of Feinberg and Ellison (Feinberg and Ellison, 2001). In this work, we expand the applicability of the methodology to batch and semi-batch processes. The idea of a target bound on selectivity is related to the concept of Attainable Regions (ARs), first introduced by Horn (Horn, 1964). The general idea behind ARs is that for any given chemistry and associated reaction kinetics there is some attainable region of output states (e.g., concentrations, effluent flow rates, etc.) that encompasses all candidate process designs for that chemistry. Thus, if the true and complete AR for a chemistry is known, then the maximum attainable selectivity of a desired product for a chemistry must correspond to some point within the AR. In this article we define the selectivity of some component as the moles of that component generated due to reaction divided by the moles of limiting reactant consumed by reaction. Over the last several decades, much effort has been invested in AR theory, but all fall short of providing a unifying methodology that generates a single bound on reaction selectivity that cannot be outperformed by any process. Traditional inside-out, geometry-based AR methods (Glasser et al., 1987; Feinberg and Hildebrandt, 1997; Feinberg, 2000a,b; Hildebrandt et al., 1990; Hildebrandt and Glasser, 1990) (for steadystate processes involving only reaction and mixing) seek to build the AR starting from a single point (or points), working outward until no further expansion is possible. Unfortunately, these methods are not

1

guaranteed to find the entire AR as no sufficiency condition exists (Glasser et al., 1987; Hildebrandt et al., 1990; Hildebrandt and Glasser, 1990; Feinberg, 2002). Therefore, the ARs produced using such methods will not necessarily include the maximum attainable selectivity. Furthermore, due to their geometric nature these inside-out methods are limited to relatively small systems (typically three to four components), which reduces their utility. Other AR methods seek to bound the AR from the outside (Abraham and Feinberg, 2004; Ming et al., 2010); however, these methods do not permit separations (which can only serve to enlarge the AR) and thus may not include the maximum attainable selectivity. Recently, Ming, et al. (Ming et al., 2013) demonstrated how traditional AR methods can be applied to batch processes; however, no sufficiency condition exists to guarantee that the entire AR has been found. Similarly, flowsheet optimization methods (Dowling and Biegler, 2015; Biegler and Grossmann, 2004; Lakshmanan and Biegler, 1996; Duran and Grossmann, 1986; Türkay and Grossmann, 1996; Achenie and Biegler, 1990; Kokossis and Floudas, 1991, 1994) may be able to treat batch reactors; however, there is no guarantee that the best process flowsheet is even included in the problem formulation. Thus, the need for an ultimate bound on selectivity that is applicable to all candidate processes (batch and continuous) for a chemistry is yet unmet. In 2001, Feinberg and Ellison (Feinberg and Ellison, 2001) made a major breakthrough and abandoned the geometric approach to AR theory entirely. Using a new and unexpected concept called the Continuous Flow Stirred Tank Reactor (CFSTR) Equivalence Principle, they proved that any arbitrary, steady-state reactor-mixer-separator system (RMS system) with a total reaction volume V > 0 can be exactly modeled by a new system with the same effluent molar flow rates as the original system (Feinberg and Ellison, 2001). This new system comprises at most R+1 well-mixed CFSTRs (where R is the number of linearly independent reactions) with total reaction volume no greater than V , each coupled to a perfect separator system (i.e., perfect split blocks). Furthermore, each of these CFSTRs corresponds to some local reaction rate from within the original process, so the compositions, temperatures, and pressures of these CFSTRs are no more extreme that those in the original process. We refer to this system of R + 1 CFSTRs and perfect separator system as a Feinberg Decomposition (FD). As shown by Feinberg and Ellison (Feinberg and Ellison, 2001), the FD for a chemistry can be modeled using Equation 1

F −P +

"R+1 X i=1

#

r(ci , Ti , Pi )Vi = 0,

where 0 ≤ Vi ≤ V and

R+1 X

Vi = V

(1)

i=1

where F is the C × 1 vector of feed flow rates into the FD, P is the C × 1 vector of effluent flow rates from the FD, r is the C × 1 vector of molar generation rate functions, (ci , Ti , Pi ) is the composition-temperaturepressure state of CFSTR i, Vi is the reaction volume of CFSTR i, and V is the total reaction volume of the FD.

2

Figure 1: The Feinberg Decomposition represents a model encompassing all candidate reactor-mixerseparator systems for a chemistry of interest. By allowing the local reaction rates in each CFSTR to take any allowed value consistent with bounds on composition, temperature, and pressure (and subjected to the specified equations of state and valid ranges of kinetics), the FD of a chemistry can be used to replace any and every candidate steady-state RMS system for that chemistry, both those RMS systems we can imagine and those we cannot. (See Figure 1). Therefore, the FD of a chemistry can be used to explore the ultimate steady-state production limits of a given chemistry. Feinberg and Ellison (Feinberg and Ellison, 2001) and Tang and Feinberg (Tang, 2005; Tang and Feinberg, 2007) optimized the FD of several chemistries to find target bounds on the molar production rates of desired products. Frumkin and Doherty (Frumkin and Doherty, 2018) demonstrated that the optimization of the FD can be reformulated to find target bounds on reaction selectivity as a function of overall conversion and that meaningful flow rate capacity constraints can be placed on the FD to yield more useful bounds on selectivity. Feinberg and Ellison proved the CFSTR Equivalence Principle (Feinberg and Ellison, 2001) for steadystate processes; however, many industrially significant processes are carried out in a batch or semi-batch fashion. Examples include biochemical processes (Krahe, 2000), fine chemicals (Pollak and Vouillamoz, 3

2000), hydrogenation of fats and oils (Puri, 1980), many polymerizations (Tobita and Hamielec, 2000), polymethacrylates (Albrecht et al., 2000), acrylonitrile butadiene styrene (Maul et al., 2000), styrene-acrylonitrile copolymers (Maul et al., 2000), α-methyl styrene-acrylonitrile copolymers (Maul et al., 2000), vinyl methyl ether (Rinno, 2000), polyvinyl chloride (Fischer et al., 2000), pharmaceuticals (Moran and Henkel, 2000), polyehteramides (Ouhadi et al., 2000), and polycarbonates (Abts et al., 2000). Thus, a methodology to generate target bounds on reaction selectivity for batch and semi-batch processes would have great value. In this article, we prove that the CFSTR Equivalence Principle also holds for batch and semi-batch systems. Therefore, the FD of a chemistry can be used to model any and every RMS system, not just those at steady-state. (This includes not only batch processes, but also continuous processes that operate in a transient fashion.) Thus, by optimizing the FD of a chemistry, we obtain an ultimate bound on reaction selectivity that is applicable to all conceivable RMS systems (as well as those beyond our imagination). Additionally, we show that the methodology developed here can be modified to find the minimum required reaction volume and batch time needed to meet certain selectivity goals. Finally, we conclude with an example and analysis of results.

2

Applicability to Batch Processes

To prove that the CFSTR Equivalence Principle is applicable to batch systems, we must re-derive its proof, starting by considering an arbitrary batch RMS system with any number and configuration of batch reactors. The approach is to divide space (i.e., reactor volume) and time into discrete intervals; an approximation that becomes exact as the number of intervals approahces infinity. Carathéodory’s Theorem of convex sets assures that the number of such spatio-temporal intervals can be reduced to R + 1, where R is equal to the number of linearly independent reactions in the chemistry of interest. As a consequence, the FD is independent of the number and size of spatio-temporal intervals introduced in the derivation. At time t = 0 the number of moles of each of the C components in the reaction chemistry is given by N 0 , a C × 1 vector. Note that these N 0 moles may be divided up between the batch reactors in any arbitrary way. That is to say, all N 0 moles need not start within one batch reactor. After some batch time tf , the number of moles has changed by ∆N to N . This is depicted in Figure 2. We can write a vector material balance equation around this arbitrary batch process, which has the form N 0 − N + ∆N = 0

(2)

We define the total molar production rate, R (a C × 1 vector with units of moles/time), as change in moles

4

Figure 2: An arbitrary batch RMS system. divided by tf . R=

∆N tf

(3)

Substituting this definition into Equation 2, we can write a vector material balance equation explicitly in terms of batch time N 0 − N + Rtf = 0

(4)

We now wish to introduce reaction volume, V , into Equation 4 so that we can limit the total reaction volume in our model. We do this by defining a global-average volumetric molar production rate vector, r † , a C × 1 vector with units of moles/volume/time. r † is related to R and ∆N by

r† =

∆N R = V V tf

(5)

With this definition, the overall material balance becomes N 0 − N + r † V tf = 0

(6)

We now have a general material balance that can be applied to any batch system and explicitly includes the total reaction volume and the total batch time. Equation 5 gives an expression for r † in terms of ∆N , V , and tf ; however, we still need a way to express ∆N for an arbitrary process. To do so, we need to partition the reaction system spatially into many control volumes and temporally into many time steps such that in each control volume and within each time step the local composition-temperature-pressure state is constant. We first break up ∆N for the whole batch system into an arbitrary number, p, of local control volumes

5

Figure 3: A batch reactor divided into a number of control volumes. (i.e., a cell model decomposition), where p may be a very large number,

∆N =

p X

∆nl

(7)

l=1

where ∆nl is the change in moles within control volume l over the entire time span 0 to tf . By summing over all control volumes, we obtain the overall change in moles. The reaction system is divided up such that in each control volume there is no spatial variation in composition, temperature or pressure at a given time (i.e., well mixed). As an example, a stirred batch reactor may have a relatively well-mixed center with imperfect mixing near the corners and walls resulting in different reaction rates at different points within the batch reactor. Thus, we could envision dividing a batch reactor into a number of control volumes as shown in Figure 3. We now define ∆nl,k to be the change in moles of control volume l within time step δtk , where δtk = tk − tk−1 and tk is the running time after k steps. With this division, ∆nl can be rewritten as

∆nl =

q X

∆nl,k

(8)

k=1

The number and duration of the time steps are chosen such that within each control volume and during each time step the composition-temperature-pressure state is uniform. As a result, the number of time steps, q,

6

may be very large. Substituting Equation 8 into Equation 7, we obtain " q p X X

∆N =

l=1

∆nl,k

k=1

#

(9)

Because we have broken the reaction system into control volumes and time steps such that within each control volume and time step the composition-temperature-pressure state is constant, the local molar production rate at the state (cl,k , Tl,k , Pl,k ), r(cl,k , Tl,k , Pl,k ), will also be constant within each control volume and during each time step. As a result, the change in moles in control volume l in time step δtk is equal to

∆nl,k = r(cl,k , Tl,k , Pl,k )Vl δtk

(10)

where r is a C × 1 vector of rate expressions (e.g., mass action kinetics, Langmuir-Hinshelwood kinetics, etc.) for each of the C components at the composition-temperature-pressure state (cl,k , Tl,k , Pl,k ) in control volume Vl within the time step δtk . Similarly, by summing Equation 10 over all time steps, we can obtain the total change in moles in control volume l over the entire time span from t = 0 to t = tf , which is given by ∆nl =

q X

r(cl,k , Tl,k , Pl,k )Vl δtk

where

q X

δtk = tf

(11)

k=1

k=1

Finally, the total change in moles for the entire reaction process can be obtained by summing Equation 11 over all control volumes, giving

q p X X

∆N =

r(cl,k , Tl,k , Pl,k )δtk Vl

(12)

l=1 k=1

Substituting Equation 12 into Equation 5, we obtain

r† =

p X q X

r(cl,k , Tl,k , Pl,k )

l=1 k=1

δtk Vl tf V

(13)

Thus, we see that r† is the time- and volume-weighted average of all local molar production rates r(cl,k , Tl,k , Pl,k ) at all time steps δtk and within all control volumes Vl within the RMS system. In the limit as the number of inverals approaches infinity, Equation 13 becomes the integral

r† =

Z

0

V

Z

tf

r(cl,k , Tl,k , Pl,k )dtdV

(14)

0

Let us briefly consider the form of r† for a few examples. Imagine a batch RMS system comprising a single, perfectly mixed batch reactor. In this case, r† is the time-weighted average of the r(c(t), T (t), P(t)) 7

states from t = 0 to t = tf , as shown in Equation 15. (There is only a single control volume because the reactor is perfectly mixed.) r† =

q X

r(ck , Tk , Pk )

k=1

δtk tf

(15)

Similarly, for a RMS system composed of two perfectly mixed batch reactors, r† is equal to the volumeweighted average of the time-weighted average of the two batch reactors, which is given in Equation 16 # " q q X X 1 δtk δtk r = V1 r(c1,k , T1,k , P1,k ) f + V2 r(c2,k , T2,k , P2,k ) f V1 + V2 t t †

k=1

(16)

k=1

An imperfectly mixed batch reactor can be divided up into an arbitrary number, p, of control volumes such that in each control volume l and within each time step δtk the value of r(ck , Tk , Pk ) is uniform within the time-step. In such a case, r† is given by Equation 13. The case of more than one imperfectly mixed batch reactors reduces to the case of Equation 13 in which p can be expected to be a larger number. As a note, the case of a fed batch reactor is also included in this methodology. In pure batch systems (no fed batch reactors), the local (c, T, P) states change with time due to the reactions that occur. In fed batch systems, the the local (c, T, P) state also changes due to the continuous (or discrete) addition of new material to the reactor. The cause of the change to the local (c, T, P) state does not matter so long as each control volume can still be temporally divided into time-steps small enough that (c, T, P) may be approximated as uniform within the time-step. This is always possible (although it might require infinitesimally small time-steps); therefore the methodology holds even for fed batch systems. Returning to Equation 13 and the proof, we see that by defining the constants αl = Vl /V and βk = δtk /tf we can re-express r † as r† =

p X q X

αl βk r(cl,k , Tl,k , Pl,k )

(17)

l=1 k=1

where αl and βk have the properties

0 ≤ αl ≤ 1

0 ≤ βk ≤ 1

p X

αl =

l=1

q X

βk = 1

(18)

k=1

Furthermore, we can define γl,k = αl βk , which also has the properties

0 ≤ γl,k ≤ 1

and

q X p X k=1 l=1

γl,k =

q X p X k=1 l=1

8

αl βk =

p X l=1

αl

!

q X

k=1

βk

!

=1

(19)

For ease of notation, we define a new index i = 1, 2, . . . , qp, which allows us to rewrite Equation 17 as

r† =

qp X

γi r(ci , Ti , Pi )

(20)

i=1

where i represents a given control volume at a given time-step. Equation 20 represents the global average volumetric molar production rate vector as a weighted average of the value of the reaction rate vector at all points (ci , Ti , Pi ) within the reaction system. Thus, with Equations 6 and 20, we see that we can model a specified arbitrary batch RMS system. For the task of finding an upper bound reaction selectivity, we need a model that can be used to represent every candidate batch process for a chemistry. An arbitrary process could theoretically contain all (c, T, P) states consistent with equations of state and bounds on temperature and pressure (which are often dictated by the valid ranges for the kinetic rate expressions). The set of all such (c, T, P) states is called the constraintconsistent reactor state set, designated Ω (Feinberg and Ellison, 2001). The set of all possible local molar production rates r(c, T, P) is obtained by operating on Ω with r(·) and is denoted r(Ω). To keep our model as general as possible (so that it is applicable to all processes), we define r † to be a weighted average of all possible of all points within r(Ω).

r† =

I X

γi r(ci , Ti , Pi ),

r(ci , Ti , Pi ) ∈ r(Ω)

(21)

i=1

(Note that in general c, T , and P will be continuous, and thus Ω will contain an infinite number of points.) If a particular state (ci , Ti , Pi ) is not present in a particular batch RMS system to be modeled, its associated weight γi will be equal to zero. r(Ω) contains all possible local molar generation rates that are attainable in any process; therefore, Equation 21 can be used to model the global-average volumetric molar production rate vector for any batch process – those we can imagine and those we cannot. Substituting Equation 21 into the overall material balance given by Equation 6, we obtain a material balance equation that can be used to model any batch process for a chemistry of interest, given by Equation 22.

0

N −N +Vt

f

"

I X i=1

#

γi r(ci , Ti , Pi ) = 0,

where 0 ≤ γi ≤ 1 and

I X

γi = 1

(22)

i=1

Defining effective inlet and outlet flow rates to the batch RMS system as F eff = N 0 /tf and P eff = N /tf and applying Carathéodory’s Theorem of Convex sets, the overall material balance for our arbitrary batch

9

process becomes

F eff − P eff +

"R+1 X i=1

#

r(ci , Ti , Pi )Vi = 0,

where 0 ≤ Vi ≤ V and

R+1 X

Vi = V

(23)

i=1

The term r(ci , Ti , Pi ) is equivalent to the generation term corresponding to a CFSTR operating at the condition (ci , Ti , Pi ). Thus, from Equation 22 we see that the overall material balances for any batch RMS system can be expressed in terms of a weighted average of R+1 steady-state CFSTRs. Equation 23 is identical to the model derived by Feinberg and Ellison (Feinberg and Ellison, 2001) for steady-state processes (see Equation 1) and the model we used in previous work (Frumkin and Doherty, 2018). Thus, we see that the a single Feinberg Decomposition model for a given chemistry is applicable to both steady-state and batch processes for that chemistry. This includes processes that operate in a periodic or chaotic fashion. This is true because each state point in the time series output of any process (steady-state, batch, periodic, and chaotic) lies in the constraint-consistent reactor state set, which in turn means the process’s average local molar production rate, r † can be expressed as a convex combination of no more than R + 1 points from r(Ω). As a result, the same optimization of the FD of a chemistry can be used as a bound for both steady-state and batch processes and is applicable to all systems that process N 0 moles of reactants in time tf . Details regarding the application of Carathéodory’s Theorem can be found in Feinberg and Ellison (Feinberg and Ellison, 2001). Furthermore, a brief, qualitative explanation can be found in the “CFSTR Equivalence Principle” section of Frumkin and Doherty (Frumkin and Doherty, 2018) as well as a more detailed explanation in Appendix B of Frumkin and Doherty. (Frumkin and Doherty, 2018) The SI of Frumkin and Doherty (Frumkin and Doherty, 2018) additionally discusses certain special cases including the constant production rate formulation (in contrast to the constant feed flow rate formulation) and the use of activitybased kinetics. Details of how to perform this optimization can be found in Frumkin and Doherty (Frumkin and Doherty, 2018) and Frumkin, et al. (Frumkin et al., Submitted. 2018) We now address reaction capacity constraints on the FD. From Equation 1 (the mathematical representation of the FD for continuous processes), we see that the reaction capacity is the total reaction volume V . However, in Equation 22 (the mathematical representation of the FD for batch processes), the reaction capacity term is V tf . In comparing the selectivity of a real process to the ultimate selectivity, we must ensure a fair comparison is made. This means the reaction capacity allotted to the FD should be equivalent to the reaction capacity of the real process. In comparing the FD to a real, continuous process, the reaction capacity allotted to the FD is the volume V of the real, continuous process. In comparing the FD to a real, batch process, the reaction capacity allotted to the FD is V tf , where V is the volume of the real, batch reactor, and tf is the total reaction time of the real, batch process. 10

Figure 4: (a): A batch process. (b): A “continuous” analogue of the batch process achieved through the use of surge tanks. In going from Equation 22 to Equation 23, we defined effective flow rates into and out of the the arbitrary batch process. In a real process, this could achieved through the use of surge tanks without changing any fundamental details of the process, as shown in Figure 4. Note that the batch process does not need to be operated in this way for the theorem to hold; this is only meant to help explain why the same FD can be used for both batch and steady-state processes. Consider a batch process with an initial charge N 0 which after some time, tf , is converted to a product mix N , as shown in Figure 4a. To make this batch process “continuous”, we instead meter into a reactant surge tank the initial charge N 0 at a flow rate F eff = N 0 /tf over a time interval equal to the batch time tf . This is depicted in Figure 4b. At the start of each batch (every tf units of time), the initial charge, N 0 , is loaded from the reactant surge tank to the batch reactor all at once. At the other end of the “continuous” batch process, the product mix N is discharged all at once at the end of each batch (every tf units of time) to a product surge tank. The product surge tank is then slowly discharged at a rate of P eff = N /tf over a time interval equal to the batch time tf .

3

Results

To demonstrate the methodology presented here, we consider the alkaline conversion of fructose to lactic acid. We use the stoichiometric and kinetic models developed by Cabassud, et al. (Cabassud et al., 2005)

11

who utilize lumped pseudo-components to simplify the chemistry. The associated reactions are given by k

1 F + 2NaOH −→ 2LA

k

2 P F −→

(24)

k

3 2F + 3NaOH −→ 3A

where F represents fructose and other hexoses, LA represents lactic acid, A represents other lumped acids, and P represents other products. The associated kinetic rate expressions for the three reactions follow elementary kinetics and are given by rˆ1 = k1 (T )cF c2NaOH

k1 (T ) = 2.42 × 103 e(

−65960 ) RT

rˆ2 = k2 (T )cF

k2 (T ) = 2.00 × 106 e(

−64698 ) RT

rˆ3 = k3 (T )c2F c3NaOH

k3 (T ) = 2.70 × 10−4 e

(25)

( −78156 ) RT

where rˆ1 , rˆ2 , and rˆ3 have units of mol/(m3 · s), and k1 , k2 , and k3 have units of m6 /(mol2 · s), 1/s, and m12 /(mol4 · s), respectively. Here, R has units of J/mol/K, and T is in Kelvin. This chemistry has R = 3 linearly independent reactions; therefore, the Feinberg Decomposition requires at most 4 CFSTRs. The upper bound on selectivity is calculated by performing a global optimization on the Feinberg Decomposition. The objective function is the selectivity of lactic acid, and the optimization variables are the concentrations, temperatures, and volumes of each reactor. Pressure is not included in the optimization variables because the chemistry is carried out in the liquid phase. The optimization is performed without constraints on the inlet flow rates to the CFSTRs. (That is, α = 108 in the problem formulation; see Frumkin and Doherty (Frumkin and Doherty, 2018) or Frumkin, et al. (Frumkin et al., Submitted. 2018) for details.) We generate plots of maximum selectivity versus conversion by performing the optimization repeatedly while constraining the overall conversion of the FD to take values between 0 and 100% in 5% increments. More information about the selectivity optimization algorithm can be found in Frumkin and Doherty. (Frumkin and Doherty, 2018) Figure 5 compares the selectivity obtained in a conventional batch reactor with the maximum selectivity as calculated from optimizing the Feinberg Decomposition (the Feinberg Selectivity Limit). The conventional batch reactor operates at 40o C and initially contains only fructose and NaOH, with initial concentrations of 139 and 2000 mol/m3 , respectively. In optimizing the FD, we constrain the temperatures in the 4 CFSTRs to be between 40o C and 100oC and the concentrations of fructose, NaOH, lactic acid, A, and P to be between [0, 0, 0, 0, 0] mol/m3 and [1110.12, 2000, 2220.24, 1665.18, 1110.12] mol/m3 to be in the valid ranges of the 12

Figure 5: Comparison of selectivity in a conventional batch reactor and the Feinberg Selectivity Limit found by optimizing the Feinberg Decomposition. kinetic model. To ensure a fair comparison is made, the capacity allotted to the batch representation of the FD is V tf , where V is the volume of the conventional batch reactor to which we are comparing the FD, and tf is the total reaction time required to obtain a specific conversion in the conventional batch reactor. To proces 500 kg (2,775 moles) of fructose at an initial concentration of 139 mol/m3 , the conventional batch reactor requires a volume of V = (2,775 moles )/(139 moles/m3 ) = 20 m3 . Therefore, the total capacity allotted to the FD is given by 20m3 × tf . The batch time tf at each value of overall conversion is equal to the time required to reach that conversion in the conventional batch reactor (obtained by solving the batch reactor design equations). In Figure 5, we see that the conventional batch reactor performs well below the maximum attainable selectivity given by the Feinberg Selectivity Limit. Thus, there is significant room for improvement. In turn, the Feinberg Selectivities are much lower than the stoichiometric limit on selectivity, which is equal to 2. Thus, if we wish to obtain a higher selectivity than the Feinberg Selectivity Limit we cannot do so with the given kinetics — no matter how creative of a design we develop. Rather, we must focus our resources on improving the reaction kinetics. If we look at the details of the optimized FD, we can actually gain insight as to how to achieve the maximum possible selectivity in a conventional reactor. Let us consider the results for the lactic acid chemistry at an overall conversion of 95%, the details of which can be found in the SI. At this conversion, all of the reaction capacity is alloted to a single reactor, despite the fact that the FD contains four CFSTRs. 13

Figure 6: Comparison of required batch times for a conventional batch reactor, the Feinberg Decomposition operating at the maximum possible selectivity, and the Feinberg Decomposition operating at 99%, 99.5%, and 99.9% of the maximum possible selectivity. Therefore, the maximum possible selectivity can be obtained using a single, conventional CFSTR. This CFSTR operates at a temperature of 40o C and a conversion of 99.8%. (The discrepancy between the CFSTR conversion and the overall conversion is due to a small by-pass stream of fructose.) The concentrations of fructose, NaOH, lactic acid, P, and A are 0.756, 2000, 898, 162, and 532 mol/m3 , respectively. To achieve this maximum selectivity, a CFSTR reactor volume of 9.4×105 m3 is required, which translates to a batch capacity of (20 m3 ) × (tf seconds). The optimization of the Feinberg Decomposition can also be re-formulated to obtain the minimum batch time required to achieve a given selectivity. Such an optimization was performed, and the results are shown in Figure 6. The solid black curve corresponds to the time required for a batch reactor to achieve a given conversion of fructose. This curve also corresponds to the reaction time allotted to the FD at each conversion. We see that to obtain the maximum possible selectivity (depicted in Figure 5), we require all of the permitted batch time. Figure 6 also shows the minimum required batch time to achieve 99%, 99.5%, and 99.9% of the maximum possible selectivity. As we see, if slightly less than the maximum selectivity is acceptable, a significantly shorter batch time is required. To achieve the maximum selectivity at a conversion of 0.9999 the FD requires a 22 h batch time. However, if we are content with 99% of the maximum selectivity, the required batch time is only 3 hours, an 86.5% reduction. With a simple change of variables, this capacity minimization holds for continuous processes as well. We

14

previously saw that a 9.4×105 m3 CFSTR is required used to obtain the maximum selectivity at an overall process conversion of 95%. From Figure 6, we see that at 95% conversion the maximum selectivity requires a batch time of 6.5 h. To achieve 99% of the maximum selectivity, a batch time of only 2.2 hours is required — a 66% reduction. Although we are considering a continuous process the percentage of the capacity reduction remains the same as for the batch process. Thus, 99% of the maximum possible selectivity can be obtained using only one third of the reaction volume required to obtain the maximum possible selectivity.

4

Conclusions

Using methods similar to Feinberg and Ellison (Feinberg and Ellison, 2001) we have proved that the CFSTR Equivalence Principle is applicable to both batch and semi-batch processes. Thus, a single optimization of the Feinberg Decomposition is all that is needed to obtain bounds on batch and steady-state candidate processes for a chemistry in addition to those that operate in a periodic or chaotic fashion. We have used this methodology to show that the maximum possible selectivity of lactic acid in the alkaline conversion of fructose is significantly higher than what can be achieved in a conventional batch reactor. Furthermore, by reformulating the optimization we are able to determine the minimum batch time required to achieve a given selectivity. In summary, the CFSTR Equivalence Principle is an incredibly powerful design tool which can be used in a number of ways to assist in chemical engineering design.

15

Acknowledgments This work was supported by the Chemical Life Cycle Collaborative at UCSB and the US EPA, Grant # RD835579.

Nomenclature Symbols C

Number of components

c

C × 1 vector of molar concentrations

F

C × 1 column vector of inlet molar flow rates

F eff

C × 1 column vector of effective inlet molar flow rates

kr

Temperature dependent rate constant of reaction r

N0

C × 1 column vector of initial moles

N

C × 1 column vector of moles after time tf

∆nl

C × 1 column vector of change in moles within control volume l over the entire time span 0 to tf

∆nl,k

C × 1 column vector of change in moles within control volume l within time step δtk

∆N

C × 1 column vector of total change in moles for the entire batch process over the entire time span 0 to tf

P

Pressure

P

C × 1 column vector of effluent molar flow rates

P eff

C × 1 column vector of effective effluent molar flow rates

r†

C × 1 vector of volume-averaged species generation rates

r

C × 1 vector of volumetric species formation rate functions

r(Ω)

Volumetric species formation rate function set; r taken at all points in Ω

R

Universal gas constant

R

C × 1 vector of total generation rates

R

Number of reactions

t

Reaction time

tf

Final reaction time

δtk

Length of time step between tk − 1 and tk

T

Temperature

V

Reactor volume

16

X

Conversion of limiting reactant

Greek Letters α

Flow rate constraint ratio

Ω

Constraint-consistent reactor state set

Subscripts and Superscripts i

Denotes CFSTR number

j

Denotes component index

Appendix A: Semi-Batch Processes In the main body of the article, we proved that the CFSTR Equivalence Principle is applicable to batch processes. We now show that semi-batch processes are also covered by the CFSTR Equivalence Principle. By a semi-batch process, we mean a process which contains both batch reactors and steady-state, continuous reactors. For a semi-batch (s.b.) process, we first divide the global-average volumetric production rate vector, r †s.b. , into two parts — one corresponding to the batch sections of the process and one corresponding to the steady-state (s.s.) sections of the process, denoted r †batch and r †s.s. , respectively. r †s.b. = r †batch + r†s.s.

(A.1)

As in the previous section, we divide r †batch into an arbitrary number of control volumes and time-steps. (See Equation 13.) Feinberg and Ellison show us that for a steady-state process, the global-average molar production rate vector takes the form

r†s.s. =

R+1 X

λi r(ci , Ti , Pi ),

where 0 ≤ λi ≤ 1 and

R+1 X

λi = 1

(A.2)

i=1

i=1

Thus, Equation A.1 becomes

r†s.b. =

R+1 X

γm r(cm , Tm , Pm ) +

m=1

R+1 X i=1

17

λi r(ci , Ti , Pi )

(A.3)

Let us define a new set of coefficients µn , n = 1, 2, . . . 2R + 2

where 0 ≤ µn ≤ 1 and summation as

P2R+2 n=1

 γn   2 µn = λ   n−(R+1) 2

1≤ n ≤ R+1 (A.4) R + 2 ≤ n ≤ 2R + 2

µn = 1. With this definition, Equation A.3 can be rewritten with a single

r†s.b. =

R+1 X

µn r(cn , Tn , Pn ) +

n=1

r†s.b. =

2R+2 X

2R+2 X

µn r(cn , Tn , Pn )

(A.5)

n=R+2

µn r(cn , Tn , Pn )

(A.6)

n=1

Equation A.6 shows that r †s.b. can be written as a convex combination of 2R + 2 points, where all points r(cn , Tn , Pn ) come from r(Ω). We can therefore apply Carathéodory’s Theorem to Equation A.6, which tells us we only need R + 1 of the points, giving

r †s.b. =

R+1 X

µn r(cn , Tn , Pn )

(A.7)

n=1

Equation A.7 takes the exact same form as the global-average volumetric production rate vector for steadystate and batch processes. Therefore, we see that the FD used for steady-state and batch processes can also be used to model semi-batch processes for a chemistry. Thus, we see that the CFSTR Equivalence Principle is a unified framework allowing us to obtain bounds on reaction selectivity that are valid for any candidate process for a chemistry of interest, including those that operate in a periodic or chaotic fashion.

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*Highlights

Highlights:

  

The steady-state CFSTR Equivalence Principle is also applicable to batch processes Selectivity limits of batch, semi-batch, and continuous processes can be obtained This model can give minimum batch times required to achieve a given selectivity

Graphical Abstract