Novel Reactor Temperature and Recycle Flow Rate Policies for

Ind. Eng. Chem. Res. 2005, 44, 6729-6740

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Novel Reactor Temperature and Recycle Flow Rate Policies for Optimal Process Operation in the Plantwide Context Jeffrey D. Ward, Duncan A. Mellichamp, and Michael F. Doherty* Department of Chemical Engineering, University of California, Santa Barbara, California 93106-5080

Ward et al. (Ward, J. D.; Mellichamp, D. A.; Doherty, M. F. Ind. Eng. Chem. Res. 2004, 43, 3957) investigated the effect of process chemistry on the selection of the operating policy for plants with recycle. This paper extends that work to consider the possibility of reactor temperature as a degree of freedom in plantwide process operation. It is possible to predict, based on the process chemistry, when it may be appropriate to implement a variable-temperature operating policy, and, alternatively, when a constant-temperature operating policy is appropriate in the face of a production rate change or other disturbances. An interesting and nonobvious result is also developed: For so-called bounded chemistries, it is usually optimal to operate the process with the reactor at its high-temperature constraint, even if the activation energy of the undesired reaction is greater than the activation energy of the desired reaction. This means that the plantwide operating policy is the same for almost all bounded chemistries. In contrast, for nonbounded chemistries, the operating policy follows conventional wisdom and changes depending on the relative magnitude of the activation energies. Implications of the optimization analysis for control structure design are also discussed. 1. Introduction The influence of reactor temperature on the selectivity and yield of a reactor network is well-known and is discussed by many authors and in many textbooks on reactor design; e.g., Fogler2 and Levenspiel.3 For example, for the case of one desired reaction and one undesired reaction, where the rates of reaction are temperature dependent via the Arrhenius equation

k ) k0e-EA/RT

(1)

and the desired reaction has an activation energy greater than the undesired reaction, then both selectivity and yield are improved by operating the reactor at high temperature. If the activation energy condition is reversed, then selectivity is improved by operating at lower temperature, but yield may be reduced because both reactions will proceed more slowly. What is less well-known, but even more important, is what role the reactor temperature should play as an independent variable in plantwide optimization. There has been considerable discussion in the literature about the choice of a plantwide inventory/flow operating policy (and control structure) for plants with recycle. As discussed by Ward et al.1 the chemistry of the process under investigation plays a critical role in the proper selection of a plantwide operating policy. This work considered plants which are operated with a constant reactor temperature. A critical question in the selection of a plantwide operating policy is how to accommodate a production rate change. Luyben4 recommends that recycle flow rates be held constant, Skogestad and coworkers5,6 recommend that reactor holdup be held constant, and Yu and co-workers7,8 recommend that both the reactor level and recycle flow rates be permit* To whom correspondence should be addressed. mfd@ engineering.ucsb.edu.

ted to change. Several authors have noted6,8 that another possible way of accommodating a production rate change is to vary the reactor temperature. Monroy-Loperena et al.9 recently considered the use of reactor temperature as a degree of freedom for a parallel control structure. However, the chemistry under investigation by all of these authors (A f B) does not represent the most general case, because there is no downside to a temperature increase. If a production rate increase is desired, then the reaction rate constant k can simply be increased (via the reactor temperature) so that the rate of production of the desired product is increased while maintaining the same reactor holdup and recycle flow rates. In many cases, however, there is a downside to increasing the reactor temperature, namely that an undesired reaction may experience a rate increase that is greater than that for the desired reaction. In this case, the issue of how reactor temperature should be employed as part of a plantwide operating policy is not so clear. Furthermore, in some cases the reactor temperature operating policy may influence the level/flow rate policy, because lower temperatures generally require longer reactor residence times and therefore greater reactor holdup. The purpose of this paper is to extend the analysis of Ward et al.1 to include the use of reactor temperature as an independent variable in the selection of an operating policy. Because this work is an extension of results presented in a previous paper, the reader may find it useful to refer to that paper for background information. It is shown in the present work that the problem of selecting a plantwide operating policy is different from the problem of designing a reactor network, and in some instances the intuition that is developed from the point of view of plantwide operation is radically different from the intuition that is developed from the reactor network design perspective. For ex-

10.1021/ie0491589 CCC: $30.25 © 2005 American Chemical Society Published on Web 07/13/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005

process optimizer may be required to supervise the decentralized regulatory control structure. 2. Methodology 2.1. Review of Previous Results. This section briefly reviews results that were presented in Ward et al.1 The reader may wish to refer to this paper for further details. Consider the process flow diagram shown schematically in Figure 1, and the following process chemistry, hereafter called chemistry 1 (Table 1)

Figure 1. Generic process flow diagram. Table 1. Chemistries 1 2 3 4 5 6 7 8 9 10

A+BfC A+AfD A+BfC A+CfD A+BfC C+CfD A+BfC CfD+E A+BfC A+AfD C+CfE AfC CfD AfC A+AfD A + C f 2C A+AfD A+AfC A+CfD A + C f 2C A+AfD C+CfE

r0 ) k0[A][B] r1 ) k1[A]2 r0 ) k0[A][B] r1 ) k1[A][C] r0 ) k0[A][B] r1 ) k1[C]2 r0 ) k0[A][B] r1 ) k1[C] r0 ) k0[A][B] r1 ) k1[A]2 r2 ) k2[C]2 r0 ) k0[A] r1 ) k1[C] r0 ) k0[A] r1 ) k1[A]2 r0 ) k0[A][C] r1 ) k1[A]2 r0 ) k0[A]2 r1 ) k1[A][C] r0 ) k0[A][C] r1 ) k1[A]2 r2 ) k2[C]2

ample, it is shown here that for bounded chemistries, it is usually optimal to operate the process with the reactor at its high-temperature constraint, even if the activation energy of the undesired reaction is greater than the activation energy of the desired reaction. The operating policies that are developed specify which process constraints will be active, which will be inactive, and which may switch during operation as the production rate is changed. Maarleveld and Rijnsdorp10 were among the first authors to recognize that it is often the case that the optimal operating point for a process is at a constraint rather than at the “hilltop”. Recently, self-optimizing control has been advocated by several authors11,12 as a basis for designing distributed feedback control structures for chemical processes. The idea behind self-optimizing control is to select variables for control which, when kept near their setpoints, keep the process near to its economic optimum over the expected range of disturbances. The analysis presented in this paper provides insight into whether a self-optimizing control structure will be adequate to achieve good economic performance of the process, and if so how such a control structure should be designed. If the optimal operating point for a plant always lies at the intersection of the same constraints, then it is straightforward to design a self-optimizing control structure. The process variables selected for control are those that lie on the constraints. If one or more optimization variables lie away from constraints and move considerably as properties of the process change, or if the active constraints are expected to shift over the range of process operation, then a centralized

A+BfC

r0 ) k0[A][B]

desired

A+AfD

r1 ) k1[A]2

undesired (2)

The independent variables for the economic optimization are the recycle flow rates of species A and B, RA and RB. Therefore, it is necessary to express the byproduct production rate PD in terms of these degrees of freedom. From a global steady-state material balance, it is seen that

PC ) k0[A][B]V

(3)

PD ) k1[A]2V

(4)

The concentrations of species A and B in the reactor can be replaced by recycle flow rates by introducing the reactor effluent volumetric flow rate q

q ) RAvA + RBvB + PCvC + PDvD

(5)

where vA is the molar volume of species A, etc. Thus

V PC ) k0RARB 2 q

(6)

V PD ) k1R 2A 2 q

(7)

Dividing the expression for the undesired reaction by the expression for the desired reaction gives

k1R 2A k1RA PD ) ) PC k0RARB k0RB

(8)

Flow rates are made dimensionless by dividing them by the production rate of the desired product, PC. Reaction rate constants are made dimensionless by dividing by the rate constant of the desired reaction. Therefore, in dimensionless form

PD′ ) k1′

RA′ RB′

(9)

where k1′ ) k1/k0, PD′ ) PD/PC, RA′ ) RA/PC, and RB′ ) RB/PC. For this chemistry, PD f 0 as RA f 0, reflecting the fact that byproduct production rate becomes small in the limit of high per-pass conversion of species A, or as RB f ∞, because a large excess of species B suppresses the undesired reaction.

Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6731 ′ )/T′ k1′ ) k1/k0 ) k1,0 ′ e(1-EA,1

By contrast, the result for chemistry 3

A+BfC

r0 ) k0[A][B]

desired

C+CfD

r1 ) k1[C]2

undesired (10)

is

PD′ )

(

k1′ k1′ 1-2 RA′ RB′ RA′ RB′

)

-1

≈

k1′ RA′ RB′

(11)

In this case, species A and B are symmetric (interchangeable). PD f 0 if either RA f ∞ or RB f ∞. The per-pass conversion of both species should be kept low to dilute the product species C and suppress the undesired reaction. These results have important implications for the plantwide optimization problem. This methodology can be generalized in a number of ways, including to undesired reactions with a different overall reaction order than the desired reaction, multiple undesired reactions, equilibrium reactions, and plug flow reactors (Ward et al.1). The simplest model for the economic potential of a chemical plant that will capture the tradeoff between recycle costs and byproduct production costs is

C ) CC PC + CD PD(RA, RB) + CR(RA + RB) (12) where C is the cost of operating the plant (smaller values of C imply larger profit), CC is the negative of the revenue from the production of one mole of desired product (CC < 0), CD is the cost of producing one mole of undesired byproduct (including raw materials costs and separations costs) (CD > 0), and CR is the cost of separating and recycling one mole of reactant species (CR > 0). Because PC is fixed, the term CC PC is constant and its value will affect the value of the total profit but not the location of the minimum cost. Therefore, the following dimensionless cost objective function is defined

C′ )

C - CC′ ) CD′ PD′ + (RA′ + RB′) PCCR

k1′ ) k1/k0 )

k1,0 -(EA,1-EA,0)/RT e k0,0

Vk0 v2C PC

(14)

(15)

Activation energies and Arrhenius preexponential constants are rendered dimensionless by dividing by the corresponding quantity for the desired reaction. Therefore, eq 14 becomes

)

1 q′2 RA′RB′

(17)

where vC is the molar volume of the desired product C, and q′ is the dimensionless reactor effluent volumetric flow rate

q′ )

q PC vC

) vA′RA′ + vB′RB′ + 1 + vD′PD′

(18)

where v′i ) vi/vC. If the reaction rate constant is temperature dependent via the Arrhenius equation, then

Vk0,0 e-1/T′ 2

)

vC PC

1 q′2 RA′RB′

(19)

and, therefore, the dimensionless reactor volume is

(13)

It is assumed here that all EA,i are temperature independent. A dimensionless temperature is defined based on the activation energy of the desired reaction

T′ ) RT/EA,0

′ > 1, the exponent in eq 16 will If EA,1 > EA,0 then EA,1 have a negative value, and k1′ will increase with increasing temperature. Therefore, conventional wisdom suggests that increasing the reactor temperature favors the production of byproduct species. Note that for many liquid-phase chemistries, the activation energy is an order of magnitude or more greater than the average thermal energy of the molecules in solution, i.e., EA/(RT) > 10. The explanation is that only a small fraction of intermolecular collisions have enough energy to overcome the energetic barrier and react to form products. One consequence is that a typical value of the dimensionless temperature T′ is small, usually T′ < 0.1 2.3. Reactor Volume with Variable Reactor Temperature. In our previous paper, the following result was developed for chemistry 1

V′ )

where CD′ is the ratio of the cost of producing one mole of byproduct to the cost of recycling one mole of reactant species. Typically, CD′ is a relatively large number, on the order of 100. When C ′ is minimized, the plant is at its most profitable operating point. 2.2. Byproduct Production Rates with Variable Reactor Temperature. If the reactor temperature is available as an operational degree of freedom, then k1′ will be a function of temperature

(16)

Vk0,0 vC2PC

)

e1/T′ 2 q′ RA′RB′

(20)

Note that with variable reactor temperature, this process has three independent operational degrees of freedom: RA′, RB′, and T′. In eq 20, q′ is a function of T′ (as well as RA′ and RB′) because PD′ is a function of T′. However, it is usually the case for profitable process operation that PD′ , 1. If this is the case, then q′ will be essentially independent of temperature, and dimensionless reactor volume V′ will depend on the dimensionless temperature T′ only via the exponential term (and not by q′). In this case, the reactor holdup required to achieve a given production rate will decrease as the reactor temperature is increased. Note also that for most liquid-phase chemistries the Arrhenius factor k0 is typically a large number. Therefore the dimensionless reactor volume defined in this way is also a large number, typically V′ > 106. 3. Results This section presents the optimization landscapes for a variety of process chemistries. It is important to bear in mind the nature of the constraints which may become active during process operation. Generally speaking, every process variable has an upper and lower bound.

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Table 2. Summary of Derived Heuristics for Plantwide Operating Policy Based on Active/Inactive Constraints for Processes with One Recycle Stream chemistry:

bounded

constraint:

Vmax

RA, max

isothermal EA,1 ′ >1 EA,1 ′ 1 before production rate increase. × marks the optimal operating point.

Figure 7. Chemistry 6 (nonbounded chemistry) chemistry with EA,1 ′ > 1 after production rate increase. × marks the optimal operating point.

Figure 8. Chemistry 7 (bounded chemistry) with EA,1 ′ > 1 before production rate increase. × marks the optimal operating point.

high temperature constraint) and the ratio of the recycle flow rate to the production rate. However, such a control structure would only work properly in the region of production rates where the maximum recycle flow rate constraint was inactive. A supervisory control structure with a mechanism for constraint handling may be required to operate the process most profitably over the entire range of production rates. Figure 4 shows the optimization landscape for chemistry 7, a bounded chemistry. The optimal operating point lies at the intersection of the high temperature constraint and the reactor volume constraint. When the production rate is increased (Figure 5), the location of the optimal operating point shifts to the new intersec-

Figure 9. Chemistry 7 (bounded chemistry) with EA,1 ′ > 1 after production rate increase. × marks the optimal operating point.

Figure 10. Chemistry 8 (bounded chemistry) with EA,1 ′ > 1 after production rate increase. × marks the optimal operating point. Table 3. Values of Dimensionless Groups in Figures figure

chemistry

CD′

k1,′ 0

2 3 4 5 6 7 8 9 10

6 6 7 7 6 6 7 7 8

100 100 100 100 100 100 100 100 100

0.08 0.08 0.1 0.1 1 1 1 1 1

EA,1 ′ 0.9 0.9 0.9 0.9 1.2 1.2 1.2 1.2 1.2

Vmax ′ 3 × 105 2 × 105 3 × 107 2 × 107 1.8 × 108 0.9 × 108 3 × 107 2 × 107 2 × 107

R A′ ,max 9 4.5 0.8 0.53 4.5 2.25 0.8 0.53 0.53

tion of the reactor volume constraint and the high temperature constraint. This situation corresponds to an operating policy in which both the reactor temperature and the reactor holdup are kept constant at their maximum values at all times. For chemistries of this type, a self-optimizing control structure would be relatively easy to design and implement. The control structure would keep the reactor temperature constant (at the high-temperature constraint), and the reactor holdup constant (at the maximum value). Such a control structure would keep the process at its optimal operating point over the entire possible range of production rates. ′ > 1, Nonbounded Chemistry. If EA,1 ′ > 1, 3.2. EA,1 then low reactor temperatures will minimize the reaction rate constant ratio k1′. Furthermore, for a nonbounded chemistry, it is not desirable to maximize the per-pass conversion of any species. However, as the temperature is decreased with fixed recycle flow rates, the reactor holdup must increase, as argued earlier (eq 20). Therefore, at all times either the low-temperature constraint or the maximum reactor holdup constraint

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will be active, but not both. It is possible that the active constraint (low temperature or reactor holdup) may switch during operation as the production rate is changed or other disturbances are encountered. Figures 6 and 7 show the optimization landscape for chemistry 6, with EA,1 ′ ) 1.2, before and after a production rate increase. Before the production rate increase (Figure 6), the optimal operating point lies on the minimum reactor temperature constraint, away from the reactor volume constraint. Because the reactor volume and recycle flow rate constraints are nondimensionalized by dividing by the production rate PC, an increase in the production rate causes these constraints to shift. In contrast, the reactor temperature is not nondimensionalized by the production rate, and the high and low-temperature constraints do not shift. If there is a significant change in the production rate, then the reactor volume constraint may become active, while the minimum temperature constraint becomes inactive, as shown in Figure 7. Generally, as the production rate is increased, the reactor volume constraint will “sweep” the optimal operating point to higher temperatures (and recycle flow rates), until the high temperature constraint or recycle flow rate constraint becomes active and finally the process becomes inoperable as the feasible region shrinks to zero and the desired production rate can no longer be achieved. In this case, it would be difficult to design a selfoptimizing control structure, because there are two constraints which may switch during the course of process operation. There are no obvious process variables that can be kept constant to ensure most profitable process operation when the production rate is changed. Therefore, a supervisory control structure may be necessary. 3.3. EA,1 ′ > 1, Bounded Chemistry. As before, since EA,1 ′ > 1, low reactor temperatures will minimize the reaction rate constant ratio k1′. However, because the process chemistry is bounded, it is also optimal from a selectivity point of view to maximize the per pass conversion of one or more species. Reducing the reactor temperature will reduce the conversion of all species. Since the maximum reactor holdup is fixed, these two competing effects give rise to a surprising result that contradicts the conventional wisdom regarding reactor temperature operating policy. Consider chemistry 7

AfC

r0 ) k0[A]

desired

A+AfD

r1 ) k1[A]2

undesired (22)

(21)

and the case where EA1 ′ ) EA1/EA0 > 1. Because this chemistry is reactor-volume bounded, it is expected that the optimal operating policy will require operation with the reactor completely full at all times. However, it is not immediately clear what the optimal policy should be with regard to the reactor temperature, as a result of competing contributions to the selectivity. Because EA,1 > EA,0, decreasing the reactor temperature will improve the ratio of the reaction rate constants (decrease k1′). However, lowering the temperature will decrease the rate of both reactions and therefore limit the per-pass conversion in the reactor, inhibiting the ability of the system to suppress the undesired reaction by driving the concentration of the reactor-volume bounded species (A) to zero.

From previous results, we can write

RA′ q′

(23)

e1/T′ q′ RA′

(24)

PD′ ) k1′ V′ )

Again, because this chemistry is reactor-volume bounded, the process should be operated with the reactor volume completely full at all times. Under this assumption, we can write

RA′ e1/T′ ) q′ Vmax ′

(25)

and

PD′ ) k1′

e1/T′ Vmax ′

(26)

however note that ′ )/T′ ′ e(1-EA,1 k1′ ) k1,0

(27)

and therefore

PD′ )

k1,0 ′ (2-EA,1 ′ )/T′ e V′max

(28)

Thus, unless EA,1 ′ > 2, which is unlikely, the byproduct production rate will decrease with increasing temperature when the reactor is operated completely full at all times. This result is exactly the opposite of what would be expected on the basis of activation energies alone. The undesired reaction is suppressed more effectively at higher temperatures because higher temperatures maximize conversion and reduce the concentration of the bounded species, which reduces the rate of byproduct formation. For bounded chemistries, it is usually best to maximize the per-pass conversion of the bounded species (minimize the recycle flow rate of that species) even at the expense of operating at high temperature, which increases the reaction rate constant ratio k1′. This is in contrast to the case of a nonbounded chemistry, where it is not desirable to make the perpass conversion as high as possible, and therefore it is not desirable to operate on the high-temperature constraint. Figures 8 and 9 show the optimization landscape before and after a production rate change. The optimum lies at the intersection of the high-temperature constraint and the reactor volume constraint for all production rates. An increase in the production rate of 50% (Figure 9) shrinks the feasible region, however the same constraints remain active. In this case, as for the case of a bounded chemistry with EA,1 ′ < 1, it is straightforward to design a self-optimizing control structure. Both the reactor temperature and reactor holdup should be kept constant at their maximum allowable values. In the remainder of this section, analytical results are developed for several other process chemistries in this category (EA,1 ′ > 1, bounded chemistry). It was just a coincidence that all of the terms of RA′ and q′ canceled out in the expression for PD′ (eq 26). For an example where such cancellation is not exact,

Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6735

(T′1 )(R ′)

consider chemistry 8

-m

Vmax ′ ) exp

A + C f 2C

r0 ) k0[A][C]

desired

A+AfD

r1 ) k1[A]2

undesired (30)

(29)

A

RA′ ) (Vmax)-1/m exp

V′ )

(31)

e1/T′ 2 e1/T′ q′ ) (R ′ + 1 + 2PD′)2 RA′ RA′ A

(41)

Solving this expression for RA′ with the assumption that the reactor holdup is at its maximum value gives

then

PD′ ) k1′RA′

(q′)m

1 (mT′ )q′

(42)

And substituting this result into the expression for the byproduct production rate (eq 40) yields

(

(32)

PD′ ) k1,0 ′ exp

)

1 - EA,1 ′ (Vmax ′ )-(n-m)/m T′ n-m exp (q′)n-m(q′)m-n (43) mT′

(

)

where it is assumed for the last equality that the mass density of all species is the same. Because this chemistry is reactor-volume bounded, the optimal operating policy will have RA as small as possible. Therefore as a simplifying assumption, we consider the case that RA′ , 1. A consequence of operating the process near to this limit is that PD′ , 1 and q′ ≈ 1. Therefore

n ′ - EA,1 m ′ exp (Vmax ′ )(m-n)/m PD′ ) k1,0 T′

e1/T′ V′ ≈ RA′

Therefore, to maximize selectivity, a reactor with this chemistry should be operated on the low-temperature bound iff

(33)

(

and, as before, we can write

RA′ ≈

1/T′

e Vmax ′

(34)

and

k1,0 ′ (2-EA,1 ′ )/T′ e PD′ ≈ Vmax ′

(35)

Figure 10 shows the optimization landscape for chemistry 8, without the simplifying assumption of eq 33, using the same values for dimensionless groups (EA,1 ′ , k0,0 ′ ) as in Figure 8. The shape of the cost contours and reactor volume constraint are qualitatively quite similar to those of Figure 8 over the domain 0 < RA′ < 1, which suggests that the approximation RA′ , 1 is valid over this region. Now consider the general chemistry

AfC

r0 ) k0[A]m

desired

(36)

AfD

r1 ) k1[A]n

undesired

(37)

where n > m for a bounded chemistry and EA,1 > EA,0 and m and n are not necessarily integers. We follow previous analysis to obtain

( (

) )

PC ) k0,0 exp -

EA,0 (RA)m(q′)-mV RT

(38)

PD ) k1, 0exp -

EA,1 (RA)n(q′)-nV RT

(39)

Also from Section 2.3

(

)

1 - EA,1 ′ (RA′)n-m(q′)m-n T′

(44)

EA,1 n > EA,0 m

(45)

EA,1 >1 EA,0

(46)

not

Otherwise it should be operated on the high-temperature bound. 4. Case Study: Benzene Chlorination This section presents results from a case study based on a process to produce chlorobenzene. More details are provided in the Supporting Information. The benzene chlorination process was the subject of a previous case study by Ward et al.1 Here, the case study is expanded to consider the situation involving nonisothermal reactor operation. This section first reviews the previous results for the isothermal case, and then presents new results for the nonisothermal case. 4.1. Description of the Process. Chlorination is often employed to introduce reactive sites on organic molecules. For example, one route in the production of phenol (C6H5OH) from benzene is via the intermediate chlorobenzene (C6H5Cl). In this case care must be taken to minimize the production of higher chlorinated benzenes. The reactions are

benzene + Cl2 f chlorobenzene + HCl

(47)

chlorobenzene + Cl2 f dichlorobenzene + HCl (48)

Dividing eq 39 by eq 38 gives

′ exp PD′ ) k1,0

)

(40)

These reactions can be carried out in the liquid phase at temperatures between 25 and 70 °C. Dichlorobenzene has no value and must be disposed of safely. Silberstein et al.13 report kinetic data for these reactions. The kinetic rate expressions they suggest are third-order overall, being first-order each in the concentration of catalyst (stannic chloride, SnCl4), chlorine, and benzene (or chlorobenzene). Because the catalyst

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Figure 11. Process flow diagram for benzene chlorination.

Figure 12. Optimization of the flexible benzene chlorination process at the nominal production rate of 50 kmol/h. × marks the optimal operating point.

concentration and chlorine concentration influence both reaction rates in the same manner, their values do not influence the selectivity-conversion profile. Therefore, it is assumed that these parameters are fixed and not available as degrees of freedom: [SnCl4] ) 0.030 mol/L and [Cl2] ) 0.25 mol/L. With these assumptions, the reaction rates are given by

r0 ) k0 [benzene]

k0 ) k0,0e-Ea,0/RT

r1 ) k1[chlorobenzene]

k1 ) k1, 0e-Ea,1/RT (49)

where k0,0 ) 2.39 × 105 min.- 1, k1,0 ) 2.30 × 105 min.-1, EA,0 ) 34.8 kJ/mol, and EA,1 ) 42.7 kJ/mol. The process was designed according to the methods of Douglas.14 The process flow diagram at level 4 in Douglas’ hierarchy is shown in Figure 11. The distillation columns were designed and costed using the methods of Doherty and Malone.15 4.2. Results: Isothermal Operation. Figure 12 shows the optimization landscape for the process after it has been built. The solid line shows the operating economic potential of the flexible process as a function of the single operational degree of freedom, the recycle flow rate of benzene. The operating economic potential of the plant after it has been built includes the fixed capital cost associated with the process equipment as well as the variable operating costs. The vertical dashed lines show the reactor volume constraint (on the left) and the recycle capacity constraint (on the right). Again as expected, there is a maximum in the economic potential function. However, the location of the maximum has shifted now that capital costs are fixed, and lies outside the feasible region. As is sometimes the case for nonbounded chemistries, the optimal operating point lies on the recycle capacity constraint, at 290 kmol/h. This operating point corresponds to a reactor holdup of 72% of the maximum value. The economic potential of the process at this point is $3.1 million/year. Note that this is somewhat less than the maximum achievable economic potential at the design stage, because this value reflects the additional cost for oversizing the

Figure 13. Optimization of the flexible benzene chlorination process after production rate decrease of 50%. × marks the optimal operating point.

process equipment. An alternative, inferior operating policy would be to operate with the reactor completely full. The economic potential of the process with this inferior operating policy is $2.2 million/year, a loss of nearly 30% compared to the maximum achievable economic potential. Now consider a decrease in the production rate of 50%. The optimization landscape is shown in Figure 13. Because much less product is being produced, the economic potential of the process is significantly reduced. Another consequence is that the optimal recycle flow rate has been reduced by 50%, and now lies within the process constraints, at 202 kmol/hr, where the economic potential of the process is $494 000/year and the reactor holdup is at 34% of the maximum value. The process is within the region of operation where the optimal operating policy is to scale the reactor holdup and recycle flow rate linearly with production rate. In contrast, the process incurs a loss of $1.6 million/year if it is operated with the reactor completely full. The economic potential of the process when it is operated on the recycle capacity constraint is $458 000/year, corresponding to a loss of 7.3% relative to the maximum achievable economic potential. 4.3. Results: Nonisothermal Operation. Up to this point, the design and operation of the benzene chlorination process has assumed that the reactor will be operated at a constant temperature T ) 333 K, in which case the process has a single degree of freedom, the recycle flow rate of benzene RB. Now we consider the case where the reactor temperature also can be adjusted. In this case, the process has two operational degrees of freedom, and the optimization landscape can be represented using a contour plot diagram. There is a lower bound on the reactor temperature that is determined by the minimum temperature difference required for heat exchange with the cooling water: T g 308 K (T g 35°C). Figure 14 shows the optimization landscape for the case where the process is operated at 50% of the base case production rate. Because the activation energy of the undesired reaction is greater than the activation energy of the desired reaction and the process chemistry is nonbounded, it is expected (based on the analysis of Ward et al.1) that the reactor should be operated at the lowest possible temperature, where either the reactor low-temperature constraint or the reactor volume constraint is active. This is a different result from the case where a process with the same chemistry and the same values of the activation energies is operated isothermally. In this case, we expect the reactor volume constraint not to be active. The difference is that with this relative magnitude of reaction rate constants, in the nonisothermal case it is beneficial to operate at the lowest possible temperature, which requires the greatest


Figure 14. Optimization landscape for benzene chlorination process at 50% of the nominal production rate. × marks the optimal operating point.

Figure 16. Optimization landscape for benzene chlorination process at 80% of the nominal production rate. × marks the optimal operating point. Table 4. Summary of Figures 12-16 figure 12 13 14 15 16 a

Figure 15. Optimization landscape for benzene chlorination process at 65% of the nominal production rate. × marks the optimal operating point.

possible reactor holdup. Therefore, the process should be operated at the reactor volume constraint (or the lowtemperature constraint) for the purpose of achieving low temperature even though it is not beneficial to operate with the greatest possible per-pass conversion of reactant species. Because the production rate is small (50%) compared to the nominal production rate, it is possible to reduce the reactor temperature to the minimum value without violating the reactor holdup constraint. This situation is reflected in Figure 14 by the location of the optimal operating point X on the low-temperature constraint and away from the reactor volume constraint. Now consider how the operation of the process should change as the production rate is increased from this point. Figure 15 shows the optimization landscape when the production rate is at 65% of the nominal value. At this production rate, it is not possible to cool the reactor all the way to the low-temperature constraint without violating the reactor volume constraint. This circumstance is reflected in the figure by the optimal operating point now situated on the reactor volume constraint. Thus, the active constraint has shifted from reactor temperature to reactor holdup. The optimal recycle flow rate is still away from the constraint; its value must be determined by solving a nonlinear optimization problem. Finally, Figure 16 shows the optimization landscape when the production rate is at 80% of the nominal value. At this production rate, the optimal recycle flow rate has shifted to sufficiently high values that the recycle flow rate constraint is active. Therefore the optimal

conditions PC ) P h Ca PC ) 0.5P hC PC ) 0.5P hC PC ) 0.65P hC PC ) 0.8P hC

T)T ha T)T h T ) variable T ) variable T ) variable

active constraints RB ) RB,max none T ) Tmin V ) Vmax V ) Vmax, RB ) RB,max

Note: P h C ) 50 kmol/hr, T h ) 60 °C.

operating point lies at the intersection of the reactor volume constraint and the recycle flow rate constraint. This constraint switch is the last one that can take place for a chemistry of this type as the production rate is increased. As the production rate is further increased, it remains optimal to operate the process at the intersection of these constraints until the feasible region shrinks to zero and the process is no longer operable. 4.4. Case Study Conclusions. Table 4 summarizes the process conditions and active constraints illustrated in the optimization landscapes presented in this case study. If the process is operated isothermally, because the benzene chlorination chemistry is nonbounded, then the recycle flow rate constraint can shift from active to inactive, but the reactor holdup constraint is inactive as is usually the case for nonbounded chemistries. Because the activation energy of the undesired reaction is greater than that for the undesired reaction, when the process is operated with variable reactor temperature both the reactor low temperature constraint, the reactor volume constraint, and/or the recycle flow rate constraint may be active, but the high temperature constraint is always inactive. These results are exactly consistent with the derived heuristics for optimal plantwide operation given in Table 2. 5. Case Study: Novel High Molecular Weight Ethers for Gasoline Blending This section presents results from a case study based on a process to produce 2-methoxy-2-methylheptane. Full details are available in a web-published supplement to this document. 5.1. Description of the Process. Alkyl ethers are used as fuel additives in gasoline to improve combustibility and octane number and to meet legislative requirements for oxygenate content. In the early 1990s, methyl tert-butyl ether (MTBE) was thought to be a valuable additive for this purpose. However, one critical drawback of MTBE is that it is moderately soluble in water: 4.3 wt %. Since its introduction, MTBE has been detected in the groundwater in many communities.

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MH + H2O f MHOH

Figure 17. 2-methoxy-2-methylheptane. Table 5. Values of Kinetic Parameters k0,0 ) 7.5 × 1011 mol/(kg cat s) k1,0 ) 2.1 × 1011 mol/(kg cat s)

EA,0 ) 90 kJ/mol EA,1 ) 102.6 kJ/mol

Consequently, MTBE is being phased out as a gasoline additive in California, and other states may follow. The solubility of ethers in water decreases with increasing molecular weight, so one possible way of overcoming this drawback is to use ethers with a higher molecular weight. Recently, Krause and co-workers16-19 have investigated the kinetics of the production of high molecular weight ethers from C6 to C8 alkenes and C1 to C4 alcohols. In this case study, we consider the production of 2-methoxy-2-methyl heptane (MMH, Figure 17) from 2-methyl-1-heptene (MH) and methanol. It is assumed that the hydrocarbon feed to the process is a mixture of alkenes and alkanes. This situation is represented in the case study as a stream which is 80 mol % 2-methylheptane and 20 mol % 2-methyl-1heptene. Because of the difficulty of separating the alkane from the alkene, unconverted alkene is not separated and recycled back to the reactor, but rather is fed along with the alkane to a downstream blending process. A requirement for the blending process is that the hydrocarbon stream contain no more than 2 mol % alkene. Therefore, it is necessary to achieve an alkene conversion of approximately 90%, but there is no incentive to go beyond this value. Unfortunately, higher molecular weight olefins are considerably less reactive than isobutene in etherification reactions. To achieve this high conversion with a reasonable reactor residence time, it is necessary to operate the process at a higher temperature than the MTBE process, and with a molar excess of methanol. Both of these considerations will tend to promote the undesired reaction and make selectivity losses a greater concern. Kinetic data are taken from papers by Krause and co-workers cited earlier. They generally develop sophisticated kinetic models for these types of reactions based on the activity of the species in solution and assuming either a Langmuir-Hinshelwood or Eley-Rideal type reaction mechanism. As is the case for the MTBE process, the reactions are typically equilibrium limited and high conversion can be achieved by employing multiple reactors with distillation columns between to remove the desired product.20,21 However, such a reactor network is too complicated for the pedagogical purpose of this case study. Therefore, the reactions are assumed here to be kinetically limited over the range of conversion of interest. Data reported by Krause and co-workers and/or data generated from the models suggested by Krause and co-workers were fitted to pseudo-homogeneous models of the form

MH + MeOH f MMH r0 ) k0xMHxMeOH

k0 ) k0,0e-EA,0/RT

2MeOH f DME + H2O r1 ) k1xMeOH2

k1 ) k1,0e-EA,1/RT

r2 ) fast

(50)

where x is the mole fraction of species. The values of the parameters are given in Table 5. Because the perpass conversion of the alkene is fixed, the process chemistry is effectively of the type A f C, A + A f D, where A is methanol, C is the desired ether, and D represents undesired byproducts. Therefore, the etherification chemistry is a bounded chemistry. It is assumed that the acid resin catalyst employed in these reactions costs $10/kg and will deactivate at temperatures above 120 °C. These values are representative of typical properties of commercial acid resin catalysts such as Amberlyst. Figure 18 shows the process flow diagram at Douglas’ level 4. 5.2. Results. Figure 19 shows the economic optimization landscape. All of the process constraints are shown as dashed lines. The reactor volume constraint prohibits low temperatures and low recycle flow rates. The capacity of the separations system prohibits high recycle flow rates. The properties of the catalyst impose a hightemperature constraint: the catalyst cannot be used over 120 °C. The feasible operating region then is the region of space that is bounded by these curves. The location of the base-case design is marked with an “X”. However it is clear from the contour plot, as expected based on the analysis in this paper (Table 2) for most bounded chemistries with EA,1 > EA,0, that it is best to operate the reactor at the high-temperature constraint, the opposite of what would be expected on the basis of the activation energies of the reactions alone. The reason for this result is that increasing the reactor temperature allows for a greater per-pass conversion of the bounded species A, which suppresses the undesired reaction even though the ratio of the rate constants (k1/ k0) increases. The optimal operating point is marked with an “O”. The operating economic potential of the process at the optimal operating point is $2.41 million/ year, while the operating economic potential at the base case design point is $2.14 million/year, which would correspond to a loss of 11% relative to the greatest possible economic potential. Figure 20 shows the byproduct production rate versus reactor temperature when the reactor is operated completely full. Although the activation energy of the undesired reaction is greater than the activation energy of the desired reaction, selectivity nevertheless improves when the process is operated at higher temperatures. The reason for this is that it is possible to achieve a higher conversion at higher temperatures, which suppresses the undesired reaction by decreasing the concentration of the methanol, the bounded species. If the production rate were increased, the feasible region would shrink, but the optimal operating point would remain at the intersection of the reactor volume constraint and the upper reactor temperature constraint. 5.3. Case Study Conclusions. This case study also illustrates results which are summarized in Table 2. For a bounded chemistry with the activation energy of the undesired reaction greater than that of the desired reaction, it is usually best to operate the process at the highest possible temperature, which is the opposite of what would be expected on the basis of the activation energies alone. The optimal operating point is always at the intersection of the reactor volume constraint and


Figure 18. Etherification process at Douglas14 level 4.

Figure 19. Optimization landscape for the etherification process with reactor volume constraint. × marks the design operating point, O marks the optimal operating point.

Figure 20. Byproduct production rate vs reactor temperature along the reactor volume constraint.

the high temperature constraint. Such a process is a good candidate for self-optimizing control, because the process can be kept at its economic optimum simply by controlling the reactor holdup and reactor temperature so that they always lie on their constraints. 6. Conclusions This paper presents an analysis of the problem of determining the optimal plantwide operating policy for plants with recycle, when reactor temperature is also available as a degree of freedom. It is shown that the optimal reactor temperature operating policy in the plantwide context is sometimes radically different from what conventional wisdom based on reactor design would suggest. Process chemistries are classified as bounded or nonbounded and as having dimensionless ′ greater than or less than one. On activation energy EA,1 the basis of this classification, processes are assigned an operating policy that is defined in terms of the

constraints that are active, inactive, or may switch during the course of operating the process. The analysis also provides insight as to whether it may be possible to design a self-optimizing control structure for the process. If the optimal point of operation of the process is always at the intersection of the same constraints, then it is straightforward to design a self-optimizing control structure: the process should be controlled so that those constraints are always active. By contrast, if constraints shift during operation it may be difficult to design a self-optimizing control structure. All of the analysis presented in this paper and the one which preceded it has focused on liquid-phase reactions with pseudo-homogeneous kinetics, and almost all of the analysis has been limited to the case where the reactor network consists exclusively of a single ideal CSTR. The reason for this is that it is possible to obtain analytical results in this case whereas it is not possible to obtain analytical results in many other cases. It is worthwhile to consider how the insight which is developed from this simple case could be applied to other cases, such as nonhomogeneous kinetics, gas-phase reactions, more complex reactor networks, and even the case where the only information available about the process chemistry is the stoichiometry and some selectivity-conversion data from a pilot plant or laboratory experiment. What are here termed “bounded” chemistries constitute a subset of all chemistries for which selectivity increases with conversion for at least one reactant species, and what are here termed “nonbounded” chemistries constitute a subset of all chemistries for which selectivity is decreasing with conversion for all reactant species. As a heuristic, one could argue that if pilot plant data suggest that selectivity is increasing with conversion, then the chemistry is bounded, whereas if it is decreasing then the chemistry is nonbounded. This heuristic, coupled with good engineering judgment, could be used to extend the results developed in these papers to many other, more complicated cases and in particular to reaction systems where a formal kinetic rate expression is not available. Appendix A. Multiple Recycled Species and Multiple Undesired Reactions All of the results presented in this paper are for the case of a single undesired reaction. The case of multiple undesired reactions could be the subject of a separate paper. However we present here just a few comments and observations about that situation. All of the framework and methodology presented in this paper and

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Ward et al.1 can be generalized to the case of multiple undesired reactions. With multiple undesired reactions, there are multiple EA,i ′ , one for each undesired reaction. If all EA,i ′ < 1 or all EA,i ′ > 1, then the process chemistry falls into the same classification as EA,1 ′ < 1 or EA,1 ′ > 1, respectively. If some EA,i < 1 and some EA,i > 1, i.e., then the activation energy of the desired reaction is between the activation energies of the undesired reactions, it is wellknown from reactor design that there is an intermediate temperature which balances the losses from the reactions and represents the point of maximum selectivity. If this point lies above the maximum temperature constraint, then the process behaves as if EA,1 ′ < 1; if it lies below the minimum temperature constraint, then the process behaves as if EA,1 ′ > 1. If the point of maximum selectivity lies between the minimum and maximum temperature constraints, and if the process chemistry is nonbounded, then it is generally desirable to operate at a temperature between the constraints. However, if the process chemistry is bounded, it may be optimal nevertheless to operate at the highest possible temperature in order to suppress one or more of the undesired reactions by maximizing the per-pass conversion of the bounded species. When multiple recycled species are involved and reactor temperature is variable, then the optimization landscape cannot be represented on a contour plot. In this situation, the benefit of classification of chemistries as bounded or nonbounded is clear. In general, if even one reactor volume bounded species is present (which means that the overall process chemistry is classified as bounded) then it is usually optimal to maximize the per-pass conversion by maximizing the reactor holdup and reactor temperature. Otherwise, these constraints may shift depending on the production rate. Supporting Information Available: More details regarding a process to produce chlorobenzene. This material is available free of charge via the Internet at http://pubs.acs.org. Nomenclature C ) cost ($/h) CP ) cost associated with stream P ($/mol) kn ) reaction rate constant for reaction n k0,n ) Arrhenius pre-factor for reaction n EA,n ) activation energy for reaction n P ) production rate of species (mol/h) q ) reactor effluent volumetric flow rate (L/h) r ) specific reaction rate [mol/(L h)] R ) recycle flow rate (mol/h) vA ) molar volume of species A (L/mol) V ) reactor volume (L)

Literature Cited (1) Ward, J. D.; Mellichamp, D. A.; Doherty, M. F. Importance of Process Chemistry in Selecting the Operating Policy for Plants with Recycle. Ind. Eng. Chem. Res. 2004, 43, 3957.

(2) Fogler, H. S. Elements of Chemical Reaction Engineering, 3rd ed.; Prentice-Hall: Upper Saddle River, NJ 1999. (3) Levenspiel, O., Chemical Reaction Engineering, 3rd ed.; John Wiley and Sons: New York 1999. (4) Luyben, W. L. Snowball Effects in Reactor/Separator Processes with Recycle. Ind. Eng. Chem. Res. 1994, 33, 299. (5) Larsson, T.; Govatsmark, M. S.; Skogestad, S.; Yu, C. C. Control Structure Selection for Reactor, Separator and Recycle Processes. Ind. Eng. Chem. Res. 2003, 42, 1225. (6) Larsson, T.; Skogestad, S. Plantwide Control- -A Review and a New Design Procedure. Model Ident. Control 2000, 21, 209. (7) Wu, K. L.; Yu, C. C. Reactor/Separator Processes with Recycle- -1. Candidate Control Structures for Operability. Comput. Chem. Eng. 1996, 20, 1291. (8) Wu, K. L.; Yu, C. C.; Luyben, W. L.; Skogestad, S. Reactor/ Separator Processes with Recycles- -2. Design for Composition Control. Comput. Chem. Eng. 2002, 27, 401. (9) Monroy-Loperena, R.; Solar, R.; Alvarez-Ramirez, J. Balanced Control Scheme for Reactor/Separator Processes with Material Recycle. Ind. Eng. Chem. Res. 2004, 43, 1853. (10) Maarleveld, A.; Rijnsdorp, J. E. Constraint Control on Distillation Columns. Automatica 1970, 6, 51. (11) Skogestad, S. Plantwide control: the search for the self-optimizing control structure. J. Process Contr. 2000, 10, 487. (12) Zheng, A.; Mahajanam, R. V.; Douglas, J. M. Hierarchical Procedure for Plantwide Control System Synthesis. AIChE J. 1999, 45, 1255. (13) Silberstein, B.; Bliss, H.; Butt, J. B. Kinetics of Homogeneously Catalyzed Gas-Liquid Reactions: Chlorination of Benzene with Stannic Chloride Catalyst. Ind. Eng. Chem. Fundam. 1969, 8, 366. (14) Douglas, J. M.; Conceptual Design of Chemical Processes; McGraw-Hill: NewYork, 1988. (15) Doherty, M. F.; Malone, M. F. Conceptual Design of Distillation Systems; McGraw-Hill: New York, 2001. (16) Karinen, R. S.; Linnekoski, J. A.; Krause, A. O. I. Etherification of C5 and C8 alkenes with C1 to C4 alcohols. Catal. Lett. 2001, 76, 81-87. (17) Karinen, R. S.; Krause, A. O. I. Reactivity of some C8-alkenes in Etherification with Methanol. Appl. Catal. A-gen. 1999, 188, 247-256. (18) Kiviranta-Paakkonen, P. K.; Struckmann, L. K.; Linnekoski, J. A.; Krause, A. O. I. Dehydration of the Alcohol in the Etherification of Isoamylenes with Methanol and Ethanol. Ind. Eng. Chem. Res. 1998, 37, 18-24. (19) Karinen, R. Etherification of Some C8 Alkenes to Fuel Ethers Sc. D. Diss. Helsinki University of Technology, Espoo, Finland 2002. (20) Bitar, L. S.; Hazbun, E. A.; Piel, W. J. MTBE production and economics. Hydrocarb. Process. 1984, 63, 10, 63-66. (21) Scholz, B.; Butzert, H.; Neumeister, J.; Nierlich, F. Methyl Tert-Butyl Ether. Ullmann’s Encyclopedia of Industrial Chemistry; Verlagsgesellschaft: Weinheim, Germany, 1990. p 543550.

Received for review September 3, 2004 Revised manuscript received May 4, 2005 Accepted May 16, 2005 IE0491589

Novel Reactor Temperature and Recycle Flow Rate Policies for

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