Performance Improvements of Parallel− Series Reactions in Tubular

Ind. Eng. Chem. Res. 2004, 43, 969-979

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Performance Improvements of Parallel-Series Reactions in Tubular Reactors Using Reactant Dosing Concepts Sascha Thomas,† Subramaniam Pushpavanam,‡ and Andreas Seidel-Morgenstern*,†,§ Institut fu¨ r Verfahrenstechnik, Otto-von-Guericke-Universita¨ t Magdeburg, D-39106 Magdeburg, Germany, Department of Chemical Engineering, Indian Institute of Technology, Madras, 600036 Chennai, India, and Max-Planck-Institut fu¨ r Dynamik komplexer technischer Systeme, D-39106 Magdeburg, Germany

In this paper, the possibility of enhancing selectivity in parallel-series reaction networks using different reactant feeding strategies is investigated theoretically. Isothermal tubular reactors are considered where reactants can be introduced at the entrance and also added over the wall. The latter method of dosing can be realized, for example, in a membrane reactor. The control variable considered is the dosing profile of a reactant along the wall. As a typical objective function, the mole fraction of a desired intermediate product at the reactor outlet is maximized. The optimal profile is calculated analytically under some assumptions using Pontryagin’s maximum principle. The results enable us to understand how the different variables determine the control policy. Insight from this approach is subsequently used to determine numerically the optimal profiles using sequential quadratic programming under more general conditions. Introduction In the field of chemical reaction engineering, considerable effort has been devoted to the problem of improving the selectivity with which an intermediate can be produced in coupled parallel-series reactions. Typical examples are partial oxidations, where the formation of undesired side products or the total oxidation products significantly reduces the process efficiency.1 Despite the progress achieved in catalysis, many industrially important reactions still suffer from poor conversionselectivity relations.2 It is well-known that, in parallel-series reaction networks, optimal local reactant concentrations are essential to obtain a high selectivity toward a certain product.3 If undesired consecutive reactions can occur, it is usually advantageous to avoid backmixing. This is one of the main reasons why partial hydrogenations or oxidations are preferentially performed in tubular fixedbed reactors.4 Typically, all reactants enter such reactors together at the reactor inlet (co-feed mode). To influence the reaction rates along the reactor length, the temperature can be used as a control parameter. Edgar and Himmelblau5 determined optimal temperature profiles to maximize selectivity and yield at the reactor outlet. An alternative attractive possibility that can influence the course of complex reactions in tubular reactors is to relax the restriction of the co-feed mode and to allow for more complex feeding regimes. It is possible to add one or several of the reactants to the reactor in a distributed manner. This concept allows for a large variety of options that differ mainly in the positions at which one or several components are dosed. This method * To whom correspondence should be addressed. Tel.: +49391-67-18643. Fax: +49-391-67-12028. E-mail: anseidel@ vst.uni-magdeburg.de. † Otto-von-Guericke-Universita¨t Magdeburg. ‡ Indian Institute of Technology. § Max-Planck-Institut fu¨r Dynamik komplexer technischer Systeme.

can be advantageous for certain dependencies of reaction rates on concentration. In particular, it is wellknown that the reaction orders with respect to the dosed component in the different reactions are of essential importance.6,7 Aside from dosing one or several components into a fixed-bed reactor at discrete positions, the possibility also exists to realize a distributed reactant feed over the reactor wall. This can be conveniently achieved using tubular membrane reactors. A series connection based on the connection of several membrane reactor segments is also a feasible option. The aspect of improving the product selectivities in parallel-series reactions by feeding one reactant through a membrane tube into the reaction zone has been studied, for example, by Lafarga et al.,8 Coronas et al.,9,10 ten Elshof et al.,11 Zeng et al.,12 Lu et al.,13 Diakov et al.,14 Al-Juaied et al.,15 and Klose et al.16 In most of these works, a single tubular membrane was used to distribute one reactant in the inner volume, which was filled with catalyst particles. In this paper, results of calculating optimal profiles for dosing one reactant in a tubular reactor are presented. To describe the kinetics, different cases of parallel-series reactions and simplified rate equations are considered. At first, Pontryagin’s maximum principle is used to analyze ideal situations analytically.17 Subsequently, distributed dosing over one or several stages along the reactor axis is analyzed for more general conditions exploiting a sequential quadratic programming (SQP) optimization algorithm.18 Theory To study optimal reactant feeding in tubular reactors, an isothermal plug-flow model was developed to describe a reactor of volume V and length L (cross-sectional area A and perimeter P). Two dosing possibilities were considered. A reactant i can be distributed continuously (concept A) or in a stagewise manner (concept B) (Figure 1). Assuming steady-state conditions and the occurrence of M reactions, the mass balance for component i in

10.1021/ie020878u CCC: $27.50 © 2004 American Chemical Society Published on Web 01/16/2004

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Case II. In the second case, a parallel reaction scheme with two reactants is considered.

A+BfD

(desired reaction delivering the target product D at rate rD)

A+BfU

(undesired parallel reaction at rate rU)

Rate expressions with dependencies on species B similar to those used in case I were assumed, i.e.

Figure 1. Schematic illustration of continuous and stagewise (three stages) dosing in tubular reactors.

(5)

rU ) kUxAxB

(6)

Case III. In the third case, the desired product (D) can react in a further consecutive reaction to give an undesired component (U).

concept A is given by

dni

xB2 rD ) kDxA RxB2 + 1

M

νijrj ∑ j)1

) Pji(z) + A

i ) 1, ..., N

(1)

A+BfD

(desired reaction delivering the target product D at rate rD)

For concept B, the corresponding mass balance equation for stage k is

D+BfU

(undesired consecutive reaction at rate rU)

dz

dnki k

dz

M

νijrj ∑ j)1

) Pjki + A

i ) 1, ..., N; k ) 1, ..., S (2)

In the above equations, S is the number of equally sized stages and ji represents the flux of species i into the reactor through the reactor wall. To analyze the potential of an optimal distribution of one reactant along the reactor axis, three typical reaction systems (cases I, II, and III) are considered. Case I. In the first case, one reactant (e.g., A) can undergo two parallel reactions to yield the desired product (D) and the undesired product (U). This is the prototype of equimolar parallel reactions. Examples of this class of reactions are isomerizations.

AfD

(desired reaction delivering the target product D at rate rD)

AfU

(undesired parallel reaction delivering the undesired product U at rate rU)

In this study, the following rate expressions were assumed for these two parallel reactions 2

xA rD ) kD RxA2 + 1

(3)

rU ) kUxA

(4)

Thus, depending on the size of the parameter R, the desired reaction has an order between 0 and 2 with respect to the mole fraction of component A, whereas the undesired reaction is always first-order in A. In view of the wide range of adsorption isotherm models offered by statistical thermodynamics,19 eq 3 can be considered as a conceivable general rate expression for a heterogeneously catalyzed reaction.

The following rate laws were assumed

xB2 rD ) kDxA RxB2 + 1

(7)

rU ) kUxBxD

(8)

These three reaction schemes were analyzed according to two different approaches: (A) Pontryagin’s principle and (B) discrete optimization using SQP. The results of approach (A) are presented first. (A) Pontryagin’s Principle. This principle has been described extensively, see ref 17. Below the method is applied to all three cases. Case I. In this case, component A is the only reactant. The problem of determining the optimal control policy involves calculating the profile of the flux jA along the reactor axis required to meet a specified objective. The control variable jA occurs linearly in the governing equations, thereby rendering the control problem singular. The optimal control policy is hence made of three segments (bang-singular-bang17)

1. jA ) jmax A

2. jA ) jsing A

3. jA ) jmin (9) A

Here, jmax and jmin are upper and lower bounds, reA A denotes a spectively, of the control variable jA. jsing A profile that varies with z and is bounded between jmin A and jmax A . The equations determining the spatial variation of the molar flow rate, ni, of species i for this case are

dnA ) (-rD - rU)A + jAP dz dnD ) rDA dz

for component A

for component D

(10) (11)

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dnU ) rUA dz dnI )0 dz dntot ) jAP dz

for component U

(12)

for a possible inert component (13)

λA

x˘ A ) [(-rD - rU)A + PjA(1 - xA)]/ntot ) fA (15) x˘ D ) (rDA - PjAxD)/ntot ) fD

(16)

n˘ tot ) jAP ) ftot

(17)

Now, we solve the free length problem, i.e., we determine the length of the reactor that results in an optimal profile subject to a constraint on ntot, namely

ntot(L) ) ntot(0) + ∆ntot

(18)

This constraint determines the total number of moles of A that we can process in the reactor. The Hamiltonian for eqs 15-17 is defined as

H ) λAfA + λDfD + λtotftot

λA

λD(L) ) 1

λA(L) ) 0

(20)

The adjoint variables evolve according to the equations

λ˙ A ) -λA

∂fA ∂fD ∂ftot - λD - λtot ∂xA ∂xA ∂xA

(21)

λ˙ D ) -λA

∂fA ∂fD ∂ftot - λD - λtot ∂xD ∂xD ∂xD

(22)

∂fA ∂fD ∂ftot - λD - λtot ∂ntot ∂ntot ∂ntot

(23)

λ˙ tot ) -λA

Because we have a three-dimensional system with free length, we can analytically solve for the singular surface in the phase space xA-xD-ntot. Along the singular trajectory, we have

∂H )I)0 ∂jA

(-rD - rU) rD + λD )0 ntot ntot

(27)

Equations 25-27 form a system of three linear equations in the adjoint variables. We obtain a nontrivial solution only when

()

d rD )0 dxA rU

(28)

This equation has to be satisfied along the singular sing arc that consists of the trajectory xA ) xsing A , where xA is the solution of eq 28. In other words, along the singular arc, the mole fraction of A is such that the ratio of the two reaction rates rD and rU is always maximized. For the special case of eqs 3 and 4, we obtain

) xopt xsing A A )

1 xR

(29)

The value of the corresponding control parameter jA is given by

) jsing A

|

[rD(xA) + rU(xA)]A P(1 - xA)

(30)

xA ) xsing A

(19)

Because the length of reactor is not fixed, we have

H(L) ) 0

)

∂rD ∂rU ∂rD + (1 - xA) - λD (1 - xA) ) 0 (26) ∂xA ∂xA ∂xA

Because H ) 0 along the singular trajectory, it follows that

where ntot is the molar flow rate (14)

Our objective is to maximize the mole fraction xD of the desired product at the reactor exit. Because rD and rU depend only on mole fraction xA, it is sufficient to consider the three eqs 10, 11, and 14, which can be reformulated using the relations nA ) xAntot and nD ) xDntot

(

(24)

The optimal control policy maximizing xD is hence determined by the feed concentration of A as follows: min 1. If xA(0) > xsing A , we first maintain jA ) jA ) 0 until sing sing xA reaches xA . We then dose A at jA until the molar flow rate ntot equals the desired specified value in the reactor (eq 18). We then set jA ) 0. This ensures full conversion of A and a further increase in concentration xD. This control policy also automatically satisfies the boundary conditions and, hence, is optimal. max 2. If xA(0) < xsing until xA A , we first maintain jA ) jA sing sing reaches xA . Then, we maintain jA ) jA until ntot satisfies the constraint imposed. We finally set jA ) 0 until all A is consumed. The optimal policy is hence

(i)

min jmax A -singular-jA

if xA(0) < xsing A

(ii)

min jmin A -singular-jA

if xA(0) > xsing A

(iii)

singular-jmin A

if xA(0) ) xsing A

Case II. We next consider the case of parallel reactions consuming two reactants

A+BfD A+BfU

where

1 - xA xD - λD + λtot ) 0 I ) λA ntot ntot

(25)

Because of eq 25, we must also have I˙ ) 0 along this trajectory. This yields

Reactant A is assumed to be fed at the reactor inlet, and reactant B is dosed along the reactor. The variation of the molar flow rate of each component along the reactor is given by

n˘ A ) ntot x˘ A ) (-rD - rU)A ) fA

(31)

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n˘ B ) ntot x˘ B ) (-rD - rU)A + jBP ) fB

(32)

n˘ D ) ntot x˘ D ) rDA ) fD

(33)

In contrast to case I discussed above, in this case, we have neglected the variation of the total molar flow, ntot, along the reactor. The assumption of treating ntot as a constant helps in the analysis below. It renders the problem three-dimensional and facilitates the derivation of an analytical solution. The assumption of constant ntot is valid when the feed containing A is diluted, i.e., is mixed with significant amounts of inerts. The Hamiltonian is given by

H ) λAfA + λBfB + λDfD

(34)

Along the singular arc, we have

∂H )0 ∂jB

or

λB ) 0

(35)

Consequently, along the arc, the following relation holds

λ˙ B ) 0

-λA

or

∂fA ∂fD - λD )0 ∂xB ∂xB

or

λAfA + λDfD ) 0

(37)

Equations 35-37 are linear in the adjoint variables λA, λB, and λD. A nonzero solution can be found if

()

∂ fA )0 ∂xB fD

()

|

[rD(xA,xB) + rU(xA,xB)]A P

xB)xBsing

if xB(0) < xsing B

(ii)

jmin B -singular

if xB(0) > xsing B

(iii)

singular

if xB(0) ) xsing B

Case III. We now extend the analysis to the third case, i.e., to consecutive reactions

A+BfD D+BfU Here, D is the desired product. We assume that A is fed at the reactor entrance, while B is dosed along the reactor. We again neglect the changes in the total molar flow rate ntot, i.e., we treat ntot as a constant. This assumption would be valid if, for example, A were diluted, i.e., mixed with inerts at the inlet. Then, the system can be described by the following set of three equations

rDA ) fA ntot

(41)

(rDA + rUA - jBP) ) fB ntot

(42)

(-rD + rU)A ) fD ntot

(43)

x˘ A ) x˘ B ) -

x˘ D ) -

(38)

(39)

∂ fA )0 ∂xB fD

We solve this equation for xB using eqs 5 and 6. The ensures that we maximize the ratio of the solution xsing B reaction rates (rD/rU). The corresponding flux of B amounts to

) jsing B

jmax B -singular

The evolution of xU does not have to be considered because rD and rU are independent of xU. Our objective is again to maximize xD(L), for which jB(z) and L have to be determined. Following the procedure outlined above for case II for this case also a nonzero solution is obtained if

This condition is identical to

∂ rD )0 ∂xB rU

(i)

(36)

Because we again consider the free length problem, we finally also have

H)0

The optimal policy is hence

()

(44)

The singular trajectory is such that it maintains xB ) xsing B , which results from the solution of eq 44. The

(40)

This is not a constant, as xA varies along the reactor and rD and rU depend on xA. Thus, jsing A (z) can only be calculated numerically incorporating the local xA. The optimal policy is determined by the concentration of B at the reactor inlet. max 1. If xB(0) < xsing until xB ) B , we dose with jB ) jB sing sing xB . We then maintain jB ) jB (xA,xB) until all A is consumed (this defines the reactor length). sing 2. If xB(0) ) xsing B , we dose with jB ) jB (xA,xB) until all A is consumed. min ) 0 3. If xB(0) > xsing B , we first maintain jB ) jB sing sing until xB ) xB . We then maintain jB ) jB (xA) until all A is consumed.

Figure 2. Dependence of the ratio of desired and undesired reaction rates on the mole fraction of component A for different values of R (case I with kD ) kU, eqs 3 and 4).

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Figure 3. Results of Pontryagin’s method (case I): (a) mole fractions and (b) overall flow rate versus dimensionless length for 0 R ) 4, kD ) kU ) 1000 mol/(s‚m3), ntot ) 0.1 mol/s, ∆ntot ) 10 mol/ ) 0.17 and L ) 4.42 m (V ) s, x0A ) 0.7. The results are xout D 0.0442 m3), end of dosing at 18.1% of the total reactor length.

Figure 4. Results of Pontryagin’s method (case I): (a) mole fractions and (b) overall flow rate versus dimensionless length for 0 R ) 10, kD ) 100 000 mol/(s‚m3), kU ) 0.1kD, ntot ) 0.1 mol/s, ∆ntot ) 23 mol/s, x0A ) 0.7. The results are xout ) 0.572 and L ) 0.706 m D (V ) 0.007 06 m3), end of dosing at 27% of the total reactor length.

Thus, the optimal policy is corresponding control parameter jsing is given by B

|

[rD(xA,xB) + rU(xB,xD)]A jsing ) B P

xB)xBsing

(45)

Because rD and rU contain the mole fractions xA, xB, and xD, jsing varies along the reactor length to ensure that B xB ) xsing B . The optimal dosing profile needs a numerical calculation of eq 45 considering the local xA and xD. The optimal control policy is hence again determined by the concentration of B at the reactor inlet. max 1. If xB(0) < xsing until xB B , we first dose with jB ) jB sing ) xB . We then dose along the reactor with jB ) jsing B (xA,xB,xD) until the optimal condition rD - rU ) 0 is satisfied. This condition determines the length of the reactor. 2. If xB(0) ) xsing B , we dose along the singular arc until rD - rU ) 0. min ) 0 until x 3. If xB(0) > xsing B B , we first set jB ) jB sing decreases to xB . We then dose along the reactor with jB ) jsing B (xA,xB,xD) until the optimality condition rD - rU ) 0 is attained.

(i)

jmax B -singular

if xB(0) < xsing B

(ii)

jmin B -singular

if xB(0) > xsing B

(iii)

singular

if xB(0) ) xsing B

(B) Discrete Optimization. The application of Pontryagin’s principle discussed above leads to instructive solutions. However, the analytical solutions derived are valid only under some assumptions such as no change in the overall molar flow inside the reactor. To be more general, a stagewise optimized dosing was examined numerically through the application of sequential quadratic programming (SQP).18 To model the configuration of concept B shown in Figure 1, eq 2 and the following initial conditions were used

n1i (z1)0) ) nfeed i (zk-1)1) nki (zk)0) ) nk-1 i

i ) 1, ..., N; k ) 1

(46)

i ) 1, ..., N; k ) 2, ..., S (47)

, The optimal stagewise feeding of one reactant, jk,opt i was determined using an SQP algorithm available

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Figure 5. Results of Pontryagin’s method and optimized dosing as predicted by the SQP algorithm (case I) for R ) 100, kD ) 100 000 0 mol/(s‚m3), kU ) 0.1kD, L ) 1.79 m (V ) 0.0179 m3), ntot ) 0.1 mol/s, x0A ) 0.7: (a) mole fractions and (b) overall flow rate versus dimensionless length according to Pontryagin’s method (∆ntot ) 23 mol/s, end of dosing at 77.1% of the total reactor length, xout D ) 0.323); (c) mole fractions and (d) overall flow rate versus dimensionless length according to SQP (S ) 10, ntot(L) ) 23.3 mol/s, xout D ) 0.323).

within the DIVA software developed at the University of Stuttgart.20 This software platform also provides efficient ODE solvers required to solve eq 2. As for the solutions described above, in all numerical optimizations performed, the molar fraction of the desired product D at the reactor exit, xD(L), was chosen as the objective function to be maximized. Further, in the analysis performed, it was assumed that the products D and U are not fed to the reactor. Discussion Case I. In Figure 2, the ratio of the desired and the undesired reaction rates, rD/rU, is depicted versus the mole fraction of reactant A for different values of R and for kD ) kU. For R ) 0, a linear dependence of rD/rU on xA holds with the slope equivalent to the ratio of the reaction rate constants (kD/kU ) 1). In this case, the highest value of rD/rU is obtained for the maximal xA, i.e., for xA ) 1. For R > 0, at first, the ratio increases with increasing xA. After passing through a maximum, the ratio decreases as xA is further increased. Because of the relative reduction of the desired reaction rate rD with growing R, in the whole range, the absolute value of the ratio rD/rU decreases. The optimum value, xopt A , can be determined using eq 29. Thus, the optimal mole

fractions of A to maximize rD/rU are xopt A ) 0.5, 0.316, and 0.1 for R ) 4, 10, and 100. Obviously, the ratio of the two rate constants kD and kU influences the ratio rD/rU but has no influence on its concentration dependence. The optimal mole fraction profile, xA; the resulting profile of component D, xD; and the overall flow rate, ntot, are shown as functions of the dimensionless reactor length for R ) 4 and for kD ) kU ) 1000 mol/(s‚m3) in Figure 3. The optimization was done by applying Pontryagin’s method. The parameters used are summarized in the figure caption. The mole fraction of A drops rapidly from the initial condition (here, x0A ) 0.7) to the optimal amount of xopt A ) 0.5 (Figure 3a). In the first small part of the reactor (not visible at this scale), no A is dosed (jA ) jmin A ) 0). After reaching the optimal value of xA ) 0.5, the dosing of A starts with jA ) jsing A . Because rD and rU depend only on the mole fraction of A, the optimal flux of A is constant. It can be calculated using eq 30 with xopt A ) 0.5. The constant feeding of A leads to a linear increase in the overall molar flow rate, as illustrated in Figure 3b. If the specified change in the total number of moles is reached (here, ∆ntot ) 10 mol/s), the dosing of A stops, and the mole fraction of the reactant decreases toward the reactor outlet. With

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Figure 6. Results of Pontryagin’s method and optimized dosing as predicted by the SQP algorithm (case II) for R ) 100, kD ) 100 000 0 mol/(s‚m3), kU ) 0.1kD, L ) 1.0 m (V ) 0.04 m3), ntot ) 10 mol/s, x0A ) 0.9, x0B ) 0.1: (a) mole fractions and (b) overall flow rate versus dimensionless length according to Pontryagin’s method (xout D ) 0.299); (c) mole fractions and (d) overall flow rate versus dimensionless length according to SQP (S ) 10, xout D ) 0.315); (e) flux of B for Pontryagin’s method (dashed line) and for SQP (solid line).

this optimal performance for the example considered, an outlet mole fraction of the desired product of xD ) 0.17 is achieved. For the assumed value of ∆ntot ) 10 mol/s, the length where the mole fraction of A drops to ) 0.01) was determined to a specified value (here, xmin A be 4.42 m. Hence, 18.1% of the reactor length is used for the optimal dosing of reactant A; the rest is needed for the consumption of A. Figure 4 displays the mole fraction profiles of reactant A and the desired product D, as well as the overall flow rate, versus the dimensionless reactor length for parameters similar to those used in Figure 3 but for R ) 10. To compensate for the decrease of the desired reaction rate rD, the kinetic constants were also changed [kD ) 100 000 mol/(s‚m3), kU ) 0.1kD]. As shown in the figure, immediately after the reactor inlet, the optimal reactant mole fraction of xopt A ) 0.316 (eq 29) is

reached. For the assumed value of ∆ntot, 27% of the reactor length is needed to feed reactant A in an optimal manner; 73% is subsequently required for further conversion to reach xmin ) 0.01. A In Figure 5a and b are presented optimization results obtained by applying Pontryagin’s method for R ) 100. For this value, the reactant mole fraction xopt A ) 0.1 provides the best performance with respect to maximizing the outlet amount of D. For the optimal distribution of A, 77.1% of the reactor length is used. That is, to consume the remaining amount of A, only approximately one-fifth of the reactor length is necessary. However, compared to the situation for R ) 10 (Figure 4), a lower molar fraction of D at the reactor outlet is achievable, and a longer reactor is found to be optimal. Up to this point, continuous dosing (concept A, Figure 1) has been considered. Concerning possible industrial

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applications, it is also interesting to examine stagewise reactant dosing (concept B, Figure 1). To realize such a reactant supply, a series connection of several membrane reactors is conceivable.16 In Figure 5c and d the corresponding optimal mole fraction profiles of A and D, as well as the resulting overall flow rate, versus the dimensionless length are depicted for 10 stages (S ) 10, eq 2). These results were obtained using the SQP algorithm. The parameters are the same as used in Figure 5a and b, including the length determined in solving the free length problem. It can be observed that, immediately after the inlet, the optimal reactant mole fraction xopt A ) 0.1 is established. Then, the constant dosing period starts over seven stages. Approximately the same amount of A as in the solution presented in Figure 5a and b is dosed, leading to an outlet flow rate of 23.3 mol/s. In the calculation presented in Figure 5a and b, the position of stopping the dosing was found to be 1.38 m. This point is indeed situated within the eighth stage. The determined dosing flux in this stage is decreased slightly in comparison with the stages before it. The final 20% of the reactor is needed to consume the remaining component A, and the SQP algorithm determines that no more A needs to be dosed in these sections. With this approach, the same outlet fraction of the desired component D (xD ) 0.323) is reached as was determined by the bang-singular-bang strategy found using Pontryagin’s principle. For case I, the optimal concentration of the dosed reactant is constant. According to eq 30, the dosing flux, jA, also does not change along the reactor length. Consequently, the dosed reactant can be distributed using only one stage. To avoid losses in selectivity, the reactant should be fed already to the reactor inlet with its optimal value. A second subsequent section of the reactor where no dosing is performed is needed to reach sufficient conversion of A. Case II. In Figure 6, the results of Pontryagin’s method and of the SQP algorithm are depicted for case II. The parameters assumed are summarized in the figure caption. The inlet mole fraction of the dosed component B corresponds to its optimal value of 0.1 for R ) 100 (solution of eq 39). The mole fractions of Figure 6a were calculated using the maximum principle. As the flux of the dosed B is equal to the sum of the reaction rates at each reactor position (eq 40), the mole fraction of B is constant (Figure 6a). Because of the stoichiometry of case II and the immediate consumption of all dosed B, the overall molar flow rate remains constant (Figure 6b). How component B is fed along the reactor can be further observed from Figure 6b. In Figure 6c and d the mole fractions, the overall flow rate and the amount of B dosed are shown for the optimized dosing found from the SQP algorithm for 10 stages. In this calculation, we did not restrict the total flow rate to be constant. It can be seen that, until the fourth stage, component B is dosed in such a manner that its mole fraction is optimal, xopt B ) 0.1. After the fourth stage, no more B is supplied. This can be attributed to the fact that the outlet mole fraction of D can be further increased by decreasing the overall flow rate while also reducing the local concentration of B. In Figure 6e, we compare the fluxes of B using the two methods. The dashed line represents the results of the maximum principle under the assumption of constant overall flow rate, and the solid line is for the case of using SQP for 10 stages when this assumption is relaxed. The optimal

Figure 7. Results of optimized dosing (case II) as predicted by the SQP algorithm: (a) mole fractions and (b) overall flow rate versus dimensionless length for R ) 0, kD ) 10 000 mol/(s‚m3), kU ) 0.1kD. Stagewise dosing (solid lines): S ) 10, L ) 1.0 m (V ) 0 0.04 m3), ntot ) 1.0 mol/s, x0A ) 1.0, x0B ) 0, xout D ) 0.564. FBR 0 (dashed lines): L ) 1.0 m (V ) 0.04 m3), ntot ) 2.05 mol/s, x0A ) 0.487, x0B ) 0.513, xout D ) 0.7.

value of the mole fraction using the maximum principle is 0.299. This compares favorably to the value obtained using SQP, which is 0.315. In the remainder of this discussion we will depict only results obtained with the SQP algorithm for stagewise reactant dosing. Optimal mole fraction profiles of case II for R ) 0, k1 ) 10 000 mol/(s‚m3) and k2 ) 0.1k1 are shown in Figure 7a. Reactant A is fed at the inlet of the first stage. The second reactant, B, is introduced in an optimized manner over the 10 wall segments. In the first segment, the fraction of A decreases because of the dilution effect of feeding B and because of the reaction. Component B increases in the first stage and then decreases. It can be noticed from Figure 7b, which contains the amount of dosed B and the overall flow rate, that reactant B is dosed only in the first stage. The optimal outlet mole fraction of D amounts to 0.564. If all of the B that is dosed in the first stage is fed together with reactant A at the reactor inlet (i.e., if classical fixed-bed reactor (FBR) operation is considered), the mole fraction profile of D represented by the dashed line in Figure 7a arises. It can be observed that, by using such a classical fixed-bed reactor, the outlet amount of D can be increased by around 20% to a value of 0.7. This result can be attributed to the influence of the concentration of B on the ratio rD/rU. For the value R ) 0, the ratio increases with increasing mole fraction

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Figure 8. Results of optimized dosing (case II) as predicted by the SQP algorithm: (a) mole fractions and (b) overall flow rate versus dimensionless length for R ) 100, kD ) 100 000 mol/(s‚ m3), kU ) 0.1kD. Stagewise dosing (solid lines): S ) 10, L ) 1.0 m 0 (V ) 0.04 m3), ntot ) 10.0 mol/s, x0A ) 1.0, x0B ) 0.0, xout D ) 0.31. 0 FBR (dashed lines): L ) 1.0 m (V ) 0.04 m3), ntot ) 20.2 mol/s, x0A ) 0.495, x0B ) 0.505, xout D ) 0.217.

of component B. In the part of the fixed-bed reactor where 80% of the target product is formed, the mean mole fraction of B amounts to xjB ) 0.261 compared to xjB ) 0.163 in the corresponding reactor part for a stagewise optimal dosing of B. It can be concluded that the fixed-bed reactor provides the best performance if the effective reaction order of the dosed reactant in the expression rD/rU is positive (here it is 1 for R ) 0). In Figure 8, the results for a higher R value (R ) 100) are shown. The corresponding profiles of the mole fractions can be seen in Figure 8a. At the end of the first stage, the dosed component B has reached its optimal mole fraction of 0.1. This optimal value follows directly from the derivative of the ratio rD/rU with respect to the mole fraction of B (eq 39). The solution is identical to the one considered for case I leading to eq 29, i.e.

xopt B )

1 ) 0.1 xR

(48)

After the fourth stage, no new B is introduced, and its mole fraction decreases according to the stoichiometric ratio along with the fraction of component A toward the reactor outlet. Because of the change in the reaction rate with decreasing mole fraction of A, the

Figure 9. Results of optimized dosing (case II, modified reaction rate rD eq 49, optimal dosing of component A) as predicted by the SQP algorithm: (a) mole fractions and (b) overall flow rate versus dimensionless length for RA ) 100 and RB ) 0, kD ) 20 000 mol/ 0 (s‚m3), kU ) 0.1kD, S ) 10, L ) 1.0 m (V ) 0.005 m3), ntot ) 1.0 0 0 out mol/s, xA ) 0.0, xB ) 1.0, xD ) 0.142.

dosing flux of B has to be lowered as the length is increased to hold xB constant. With this optimal distribution of B, the outlet mole fraction of D can be maximized to 0.31. If all dosed B is fed at the reactor inlet together with component A (i.e., fixed-bed reactor (FBR) operation), component D reaches only a mole fraction of xout D ) 0.217 at the reactor outlet (dashed line in Figure 8a). The remarkable difference can be attributed to the differences in the mole fraction profiles of B. If B is fed over the reactor length, it is possible to optimize locally the mole fraction of the dosed component. In contrast, in the fixed-bed reactor, there exists only one position (at the feed point) where an optimal reactant concentration can be set. To clarify further which component has to be dosed in a parallel reaction network with two reactants, the kinetics of case II was modified. For this modification, compared to eq 5, the rate of the desired reaction was changed with respect to the influence of reactant A as follows

xA2

xB2

rD ) kD RAxA2 + 1 RBxB2 + 1

(49)

The rate expression for rU, eq 6, was unchanged. Now, the influence of the feeding mode of the two reactants on the ratio rD/rU and thus on the desired product

978

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Figure 10. Results of optimized dosing (case II, modified reaction rate rD eq 49, optimal dosing of component B) as predicted by the SQP algorithm: (a) mole fractions and (b) overall flow rate versus dimensionless length for RA ) 100 and RB ) 0, kD ) 20 000 mol/ 0 (s‚m3), kU ) 0.1kD, S ) 10, L ) 1.0 m (V ) 0.005 m3), ntot ) 1.0 ) 0.081. mol/s, x0A ) 1.0, x0B ) 0.0, xout D

fraction xD could be examined by varying RA and RB. In Figure 9a, the mole fraction profiles are illustrated for RA ) 100 and RB ) 0 with an optimal distribution of component A. In the first stage, component A is inserted in the quantity needed to reach the optimal mole fraction. Then, the dosing amount is lowered to hold a constant concentration of A along the reactor until the dosing stops. In the last 30% of the reactor, no A is introduced, and the mole fraction of A decreases toward the outlet. Here, the maximum outlet mole fraction of D is 0.142. In contrast to this situation, Figure 10 contains the results for the same parameters with an optimal dosing of the reactant B. Because of the enhancement of rD/rU with increasing xB, the best performance is provided now by the fixed-bed reactor. Thus, in the series connection of 10 stages, all B should be dosed in the first stage. However, the obtained outlet mole fraction of the target product D is 40% less than the value that can be achived by employing the optimal distribution of component A (compare Figures 9a and 10a). Also for case II (i.e., for two parallel reaction with two reactants and the assumed kinetic expressions), it holds that the optimal concentration of the dosed component is constant. In contrast to case I, because of the decreasing mole fraction of the second reactant fed at the reactor inlet, the dosed flux needs to be adjusted

Figure 11. Results of optimized dosing (case III, optimal dosing of component B) as predicted by the SQP algorithm: (a) mole fractions and (b) overall flow rate versus dimensionless length for R ) 100, kD ) 3000 mol/(s‚m3), kU ) kD/6, S ) 10, L ) 1.0 m (V ) 0 0.04 m3), ntot ) 1.0 mol/s, x0A ) 1.0, x0B ) 0.0, xout D ) 0.169.

locally. This could be done by using a small number of stages connected in series (e.g., Figure 6e). A final stage without dosing is needed to consume the remaining reactant. Case III. Finally, in Figure 11 is depicted a selected result for optimized stagewise reactant dosing in the case of two consecutive reactions with two reactants (rate eqs 7 and 8). In the example presented here, for S ) 10, component B is distributed over almost the entire reactor length. In the first stage, the dosing flux is highest to raise the mole fraction of B to the optimal amount of xopt B ) 0.1 (R ) 100). From the second to the eighth stage, B is then introduced with nearly constant flux to hold this optimal concentration level. The dosing is stopped in the last two stages. Obviously, for this case, a series connection of three reactor segments of adjusted lengths appears to be attractive. Further results using lower numbers of stages will be presented elsewhere.21 Conclusions The possibility of distributed dosing of one reactant over the wall into a tubular reactor was studied for parallel-series reactions. Pontryagin’s maximum principle and an SQP algorithm were applied to determine optimal dosing profiles. The results agree well with each

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other and also confirm analytical solutions derived recently for several special cases.22 Although the calculations were performed for selected situations in terms of stoichiometry, reaction rates, and parameters assumed, the results obtained allow some general conclusions to be drawn. When the reaction orders of the individual reactions fulfill certain requirements, an intelligent dosing scheme can lead to significant improvements. The determination of which reactant might be efficiently dosed can be made by analyzing the ratio of the rate of the desired reaction, rD, to the rate of the undesired reaction, rU. Preferably, a reactant for which optimal concentrations (leading to a maximum of rD/rU) exist should be dosed along the reactor. Obviously, in the case of two consecutive reactions, the component participating in both reactions should be distributed to adjust optimal local concentrations. A distributed dosing scheme possesses certain advantages compared to the conventional co-feed fixed-bed reactor mode. It might be conveniently implemented using membranes as the reactor walls. A rather low number of stages (