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Ind. Eng. Chem. Res. 2003, 42, 3294-3303
Nonlinear Behavior of Reactor-Separator Networks: Influence of Separator Control Structure K.-P. Zeyer,*,† S. Pushpavanam,‡ and A. Kienle†,§ Max-Planck-Institut fu¨ r Dynamik komplexer technischer Systeme, Sandtorstrasse 1, D-39106 Magdeburg, Germany, Department of Chemical Engineering, Indian Institute of Technology, Chennai (Madras) 600036, India, and Lehrstuhl fu¨ r Automatisierungstechnik, Otto-von-Guericke-Universita¨ t, Universita¨ tsplatz 2, D-39106 Magdeburg, Germany
In this paper we analyze the behavior of a coupled reactor-separator system with reactant recycle. The reactor is represented by a continuously stirred tank reactor, and the separator is represented by a flash unit. The reactor is operated isothermally and sustains a first-order reaction A f B. The individual units always possess a unique, stable, and feasible steady state. Surprisingly, even for the simple model system considered here, more complex patterns of behaviorsinvolving infeasibility, multiple steady states, and limit cyclesscan be observed when the recycle is closed. It is shown that the behavior crucially depends on the flow and the flash control strategy. Stability criteria are derived for different flow and flash control strategies. They depend on the operating conditions and on the basic physicochemical properties of the mixture. Potential sources for instability are systematically identified and illustrated. 1. Introduction Chemical process plants typically consist of reactors and separators. In the reactor, reactants are partially converted to desired products, and in the subsequent separator, unreacted reactants are separated from the products and recycled into the reactor. While the nonlinear dynamic behavior of stand-alone reactors (see, e.g., ref 1-3 for an overview) and separators (see, e.g., ref 4 for an overview) was studied extensively in the past, now there is a growing interest in the dynamics of coupled reactor-separator systems with mass and energy recycles. It is a well-known fact that recycles can increase the plant time constant considerably. Further they lead to instability or to sensitivity with respect to disturbances.5,6 The behavior strongly depends on the flow control strategy. As a rule of thumb, one of the internal streams must be flow-controlled in the reactor-separator system to reduce sensitivity to plant disturbances.5 For a recent review on control structure selection for simple reactor-separator systems, we refer to ref 7. Most of the work mentioned so far is based on linear reactor-separator systems. Recent work has also focused on nonlinear reactor-separator systems. Pushpavanam and Kienle8 have studied a continuously stirred tank reactor (CSTR) with a first-order exothermic reaction coupled to an isothermal isobaric flash. They found that the nonlinear behavior strongly depends on the flow control strategy. If the fresh feed to the reactor is fixed, the behavior of the coupled system is completely different from that of the standalone reactor. Large parts of the steady-state solution branches were either unstable or infeasible, leading to static or oscillatory instability. Instead, if one of the * To whom correspondence should be addressed. Tel.: +49/ (0)391/6110-187. Fax: +49/(0)391/6110-516. E-mail:
[email protected]. † Max-Planck-Institut fu¨r Dynamik komplexer technischer Systeme. ‡ Indian Institute of Technology. § Otto-von-Guericke-Universita¨t.
internal streams (the recycle to the reactor, for example) is fixed, the behavior is very similar to the stand-alone reactor and at least one feasible and stable steady-state solution exists for all operating conditions. Steady-state multiplicity and instability in this example is caused by the exothermicity of the chemical reaction. Complex kinetics is an additional source for complex behavior in reactor-separator systems. This was studied for an isothermal autocatalytic system in ref 9. Again, the focus was on isothermal isobaric flash operation and the influence of different flow control strategies. Different isothermal reaction systems with increasingly complex kinetics, which are representative for polymerization systems, were studied in ref 10. It was shown that consecutive reaction systems can have multiple steady states. A sharp separation between reactants and products was assumed with constant product compositions. In practice, a constant product composition of the separator can be obtained by some suitable control. Further work on the nonlinear behavior of reactorseparator systems was reported in the Russian literature. Boyarinov and Duev11 studied a CSTR with a bimolecular reaction A + B h 2C, perfect separation and recycle of reactants. They have shown that in such a system an infinite number of steady states may exist. Extensions to plug-flow reactors and a general reaction mechanism with complete recycle of unreacted reactants are given in refs 12 and 13. Extensions to ideal but finite separation, with a distillation column in the limit of infinite length and infinite reflux, are given for reactions A + B h C in ref 14 and for the parallel reactions A + B f C and A + C f D in ref 15. In all of these case studies mentioned so far, emphasis is on the nonlinearity introduced by the reactor. In addition, in some cases, the influence of the underlying flow control structure was taken into account. However, a systematic study on the influence of the separator control structure is missing so far. To close this gap, we focus on a simple model system as illustrated in Figure 1. The system consists of a CSTR and a flash unit. The reactor is assumed to be operated
10.1021/ie020768n CCC: $25.00 © 2003 American Chemical Society Published on Web 06/11/2003
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Figure 1. Schematic diagram of the reactor-separator system being considered. Table 1. Summary of Different Control Configurations Investigated in This Study flow control strategy F0 fixed F fixed F and F0 fixed
flash control strategy Tfl fixed Qfl fixed V fixed section 3.1 section 3.2 section 3.3
section 4.1 section 4.2 section 4.3
not possible section 5 not possible
isothermally and sustains a first-order reaction of the form A f B. Reactant A, which is assumed to be the heavy component, is recycled with the liquid phase from the flash to the reactor. Three different modes of isobaric flash operation are considered to be the following: (i) fixed flash temperature Tfl; (ii) fixed heating rate of the flash Qfl; (iii) fixed vapor flow rate of the flash V. In practice, each of them is easily implemented with some suitable control strategy for the heating rate of the flash unit. In the first case, Tfl is used as the controlled variable, in the second case Qfl, and in the third case V. Throughout the paper, ideal control will be assumed. The well-known adiabatic flash, which is often used in practice (see, e.g., ref 16), is included for Qfl ) 0 as a special case in mode ii. In addition, different flow control strategies are possible for the plant in Figure 1. In particular, the following three control strategies will be considered: (i) fixed flow rate F0 in the inlet of the reactor; (ii) fixed flow rate F from the outlet of the reactor; (iii) fixed flow rates F and F0 in the inlet and outlet of the reactor. In the first case, F0 is flow-controlled and the level in the reactor is controlled with F. In the second case, F is flow-controlled and the level in the reactor is controlled with F0. In the third case, F and F0 are flow-controlled and the level in the reactor is variable. Ideal flow and level controllers will be assumed throughout the paper. The outline of the paper is as follows. First the underlying model equations for the reactor and the separator units are discussed. Afterward the different control structures are analyzed and compared with each other. The focus is on the feasibility, uniqueness, and stability of steady states. Because both aspects, the flow and the separator control strategy, are equally important, all possible combinations will be considered, as indicated in Table 1. It will be shown that even for the simple model system considered here surprisingly complex behavior can be induced by the coupling between the reactor and separator. 2. Model Equations Figure 1 depicts a schematic representation of the system under investigation. F0 represents the fresh feed
stream to the system. The reactor holdup is represented by MR, and the mole fraction of the reactant in the reactor is given by z. The molar flow rate leaving the reactor is F, and the molar recycle flow rate is L. The equilibrium reactant compositions in the liquid and vapor of the flash unit are given by xe and ye, respectively. The liquid holdup in the flash is Mfl, and its pressure and temperature are given by pfl and Tfl, respectively. 2.1. Reactor. The reactor is assumed to sustain a first-order reaction of the form A f B. This assumption allows us to treat all streams as being binary mixtures. Throughout this paper, we assume that the reactor is operated isothermally. With these assumptions, the model equations governing the evolution of the molar holdup MR and the mole fraction z are obtained from
Material balance of reactant A d(MRz) ) F0xf + Lxe - Fz - MRkz dt
(1)
where k ) k0 exp(-E/(RTR))
Total material balance dMR ) F0 + L - F dt
(2)
For constant reactor holdup, the above material balances simplify to
MR
dz ) F0xf + Lxe - Fz - MRkz dt 0 ) F0 + L - F
(3) (4)
For variable reactor holdup, the reactant material balance can be simplified by substituting eq 2 into eq 1.
MR
dz ) F0(xf - z) + L(xe - z) - MRkz dt
(5)
It is worth noting that the stand-alone reactor with an isothermal first-order reaction represents a linear system, which always has a unique and stable steady state. 2.2. Flash. We assume that the flash is operated under thermodynamic equilibrium, under conditions of constant pressure and constant liquid holdup. The energy balance is assumed to be quasi-static. The latter assumption implies that temporal changes in enthalpy are governed by the concentration dynamics. It should be noted that this is a reasonable assumption because the temperature in the flash is the equilibrium temperature which depends algebraically on the composition. With these assumptions, the model equations are
Material balance for component A Mfl
dxe ) Fz - Vye - Lxe dt
(6)
Total material balance 0)F-V-L
(7)
(Quasi-static) energy balance of the flash 0 ) Fhf - Vhv - Lhl + Qfl
(8)
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The total derivative dye/dxe is obtained by differentiation of eq 11
dye ∂ye ∂ye dTfl ) + dxe ∂xe ∂Tfl dxe
(15)
and substitution of dTfl/dxe from eq 12. Graphically, dye/ dxe represents the slope of the equilibrium line in the well-known McCabe-Thiele diagram. For thermodynamically stable mixtures, this slope is always positive. The stability and uniqueness of steady states of the stand-alone flash are shown in the appendix for the three different flash control strategies to be discussed subsequently. Figure 2. Vapor-liquid equilibrium diagram of a binary mixture.
In view of hv ) hl + ∆h, hf - hl ) cp(TR - Tfl), and the total material balance (7), the energy balance (8) is finally obtained as
0 ) Fcp(TR - Tfl) - V∆h + Qfl
(9)
In the remainder, a linear dependency of ∆h, the enthalpy of vaporization, on the concentration ye is assumed.
∆h ) ye∆hA + (1 - ye)∆hB
(10)
Here, ∆hA and ∆hB are the enthalpies of vaporization of the pure components A and B, which are assumed to be constant. Finally, the vapor-liquid equilibrium is described by the relations
ye ) KA(Tfl,xe) xe
(11)
1 ) KA(Tfl,xe) xe + KB(Tfl,xe) (1 - xe)
(12)
assuming an ideal vapor phase. For illustration purposes, especially
Ki ) pis(Tfl)/p, i ) A, B
(13)
will be used, where in addition an ideal liquid phase is assumed. The saturation pressures pis are calculated with the Clausius-Clapeyron relation
ln
(
)
pis ∆hi 1 1 ) p R Ti Tfl
(14)
Here, Ti is the boiling point temperature of component i at pressure p and pis is the saturation pressure at Tfl. The reason for taking the Clausius-Clapeyron equation is that all parameters have a direct physical meaning and thereby greatly help with the interpretation of the results. Nevertheless, most of the results to be discussed subsequently are also valid for the more general relations (11) and (12). In the remainder, also the derivatives of the equilibrium relations (11) and (12) are required. The derivative dTfl/dxe can be obtained by implicit differentiation from eq 12. In what follows, the focus is on binary nonazeotropic mixtures, where component A is the heavy boiling component. For this class of mixtures, the derivative dTfl/dxe is always positive, as shown in Figure 2.
3. Analysis for Fixed Flash Temperature Tfl First, we will focus on the flash being operated at a fixed temperature Tfl. As pointed out above, this can be achieved by suitable control with the flash heating rate Qfl. The required heating rate follows from the energy balance (9). However, the energy balance can be neglected if one is not interested in the required heating rate Qfl. Further, because pfl and Tfl are fixed for this configuration, xe and ye are also constant because we have a binary mixture (see Figure 2). Hence, the lefthand side in eq 6 is equal to zero. We now discuss the different control strategies for the flow rates and compare their behavior with each other. Namely, we focus on three different cases, i.e., F0, F, or F0 and F fixed by some suitable flow control. The latter is only possible when we allow the reactor holdup to vary, whereas a constant molar holdup can be maintained in the first two cases by using the other flow rate as the manipulated variable. The corresponding problem with a nonisothermal reactor was already treated in ref 8. For the nonisothermal reactor-separator network, multiple steady states and oscillatory behavior may arise as a result of the temperature dependence of the reaction rate. In contrast to this, the isothermal reactor-flash unit always has a unique and stable steady state, as will be shown below. However, bounds on the feasible operation are introduced in some cases because of the isothermal operation of the flash. We will use the constant Tfl mode of operation as a base case and compare it with the other flash control strategies with fixed heating rate and fixed vapor flow rate. 3.1. F0 Is Fixed. For fixed F0 and constant holdups in the reactor and the flash, the vapor flow rate V is equal to the fresh feed rate F0 according to the overall material balance of the plant. F and L are obtained from the material balances of the flash (6) and (7) with xe ) constant for fixed Tfl (see also Figure 2)
F ) F0
xe - y e z - ye , L ) F0 xe - z xe - z
(16)
Upon substitution of F and L into the material balance of the reactor (3), a single differential equation for the concentration in the reactor is obtained.
MR
dz ) F0(xf - ye) - MRkz dt
(17)
This is a linear equation in z. The corresponding eigenvalue is λ ) -k. Consequently, this equation always has a unique and stable steady state, which,
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Figure 3. F0 and Tfl control configuration: dependence of xe (black line), ye (red line), and z (green line) on F0. Tfl ) 350 K; other parameters are as given in Table 2.
A typical bifurcation diagram for fixed F and Tfl is shown in Figure 4. Again, feasibility requires xe > z > ye. From eq 19, we find that z approaches xe only asymptotically for F f ∞. Hence, infeasibility only arises for z < ye at low values of F. According to eq 18, this corresponds to a negative liquid flow rate. 3.3. F0 and F Are Fixed. For fixed F and F0, the unknown flow rates L and V are calculated again from the flash equations with xe ) constant according to
z - ye xe - z L)F , V)F x e - ye x e - ye
Figure 4. F and Tfl control configuration: dependence of xe (black line), ye (red line), and z (green line) on F. Tfl ) 350 K; other parameters are as given in Table 2.
however, is only feasible in the range xe > z > ye. As shown in Figure 2, this condition implies that the flash is operated in the two-phase region with positive vapor and liquid flow rates. This requirement is only satisfied if the fixed flash temperature Tfl is compatible with the reactor concentration z. Outside the feasible range, steady-state operation is not possible because of permanent accumulation in the system. For a detailed discussion, we refer to ref 8. A typical bifurcation diagram for fixed feed rate and fixed flash temperature is shown in Figure 3. Infeasibility due to z < ye corresponding to a negative liquid flow rate L according to eq 16 is observed for low values of F0. For high values of F0, infeasibility occurs for z > xe corresponding to negative flow rates F and L in eq 16. As z approaches the upper boundary xe, F and L tend to infinity according to eq 16. In summary, we find that feasible operation of the reactor-separator network for this control configuration is bounded on both sides in Figure 3. 3.2. F Is Fixed. For fixed flow rate F, the holdup in the reactor is adjusted with the fresh feed rate F0, which therefore depends on the operating conditions. According to the overall material balance of the plant, the vapor flow rate V is again equal to F0 and therefore also varies with the operating conditions. The unknown flow rates F0, V, and L are obtained from the material balances of the flash (6) and (7) with xe ) constant.
xe - z z - ye F0 ) V ) F , L)F xe - ye xe - ye
(18)
Upon substitution of F0 and L into the material balance of the reactor (3) again, a single differential equation for the concentration in the reactor is obtained.
(
)
xe - z z - ye dz ) F xf + xe - z - MRkz (19) MR dt xe - ye xe - y e Again, this is a linear equation in z. The corresponding eigenvalue is λ ) -F(xf - ye)/[MR(xe - ye)] - k, which is always negative. Therefore, this configuration always has also a unique stable steady state.
(20)
Fixing F0 and F simultaneously is only possible for variable reactor holdup. For variable reactor holdup, material balances (2) and (5) have to be considered and the model equations are represented by two nonlinear differential equations for the holdup MR and the reactant concentration z in the reactor according to
MR
(z - ye)(xe - z) dz ) F0(xf - z) + F - MRkz (21) dt xe - ye xe - z dMR ) F0 - F dt xe - ye
(22)
At steady state, the second equation can be uniquely solved for z and the first equation can be uniquely solved for MR. The stability of the unique steady state follows from the corresponding Jacobian A)
(
[
]
(z - ye)(xe - z) F0 1 F (xe - z) - (z - ye) F0(xf - z) + F + -k xe - y e MR MR xe - ye MR2 F 0 xe - ye -
)
(23)
By substitution of the steady-state material balances (21) and (22), the Jacobian is finally obtained as
(
kz F z - ye -k MR M R x e - ye A) F 0 xe - y e -
)
(24)
Necessary and sufficient conditions for the stability of a second-order system are (e.g., ref 17, p 311)
det(A) > 0
(25)
trace(A) < 0
(26)
In the present case, it is readily shown that both stability conditions are always satisfied. Further, the steady-state liquid flow rate from the flash L ) F - F0 is always positive for admissible parameters F > F0. Hence, all steady states are feasible for feasible parameters, provided the tank is large enough to admit the required holdup MR. Feasibility is the great advantage of this control configuration
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compared to the previous two configurations discussed above. A typical bifurcation diagram is shown in Figure 5. 4. Analysis for Fixed Heating Rate Qfl In a second step, the focus is on a fixed flash heating rate Qfl. This can be achieved by direct control of the heating rate. As a special case, the adiabatic flash with Qfl ) 0 is included, which is frequently used in industry. A typical example is the industrial process for the production of acetic acid studied in ref 16. In the remainder, some surprising and unexpected patterns of behavior will be shown for this standard mode of flash operation and will be compared to those of the isothermal case, which was analyzed in the previous section. Like in the isothermal case, the focus will be on the three different flow control strategies where either F0, F, or F0 and F are fixed by some suitable control. Again, constant reactor holdup will be assumed in the first two cases, whereas variable holdup will be considered in the third case. 4.1. F0 Is Fixed. For fixed F0 and constant holdups in the reactor and flash drum, the vapor flow rate V has to be equal to the flow rate F0 because of the overall material balance of the plant. F, the liquid flow rate in the outlet of the reactor, can be obtained from the energy balance (9)
F)
F0∆h[ye(xe)] - Qfl
(27)
cp[TR - Tfl(xe)]
and the liquid flow rate from the flash L is
L ) F - V ) F - F0
(28)
It is important to note that both F and L can be interpreted as functions of xe only because ye and Tfl can also be interpreted as functions of xe only according to the equilibrium relations (11) and (12). In view of eqs 27 and 28 and V ) F0, the component material balances for the flash and the reactor (3) and (6) can be written as
MR
dz ) F0xf + [F(xe) - F0]xe - F(xe)z - MRkz (29) dt
Mfl
dxe ) F(xe) z - F0ye(xe) - [F(xe) - F0]xe dt
(
(30)
)
Stability follows from the Jacobian of eqs 29 and 30.
[
]
F 1 dF -k L + (xe - z) MR MR dxe A) F dye 1 dF -L - F0 - (xe - z) Mfl Mfl dxe dxe -
[
]
(31)
From eq 27, we find that
[
]
dye dTfl dF 1 ) F0(∆hA - ∆hB) + Fcp dxe cp(TR - Tfl) dxe dxe
(32) where both derivatives dye/dxe and dTfl/dxe are positive, as discussed in section 2.2.
Figure 5. F, F0, and Tfl control configuration: dependence of xe (black line), ye (red line), and z (green line) on F. F0 ) 15.0 mol/s; Tfl ) 350 K; other parameters are as given in Table 2.
Figure 6. F0 and Qfl control configuration: dependence of xe (black line), ye (red line), and z (green line) on F0. xf ) 0.9; ∆hA ) 105 J/mol; ∆hB ) 25 080 J/mol; TA ) 320 K; Qfl ) 15.0 × 105 W; other parameters are as given in Table 2.
In view of the stability conditions (25) and (26) and the sign of the derivative dF/dxe according to eq 32, stability can only be proven for Tfl < TR and ∆hA > ∆hB corresponding to dF/dxe > 0 in eq 32. In all other cases, i.e., Tfl > TR or ∆hA < ∆hB, instabilities are possible. The first stability condition, Tfl < TR, corresponds to a situation where the flash is operated at a markedly lower pressure than the reactor. The second stability condition, ∆hA > ∆hB, depends only on the properties of the mixture and not on the operating conditions. The requirement that the enthalpy of vaporization of the heavy component is bigger than the enthalpy of vaporization of the light component is satisfied for most but not all mixtures. A classical counterexample is a mixture of acetic acid (normal boiling temperature TA ) 391.1 K; enthalpy of vaporization ∆hA ) 5660 kcal/kmol) and water (TB ) 373.2 K; ∆hB ) 9717 kcal/kmol). The physical properties have been taken from ref 18. These kinds of mixtures also play an important role in industrial production processes such as acetic acid production or esterification processes. An example with Tfl < TR for all operating conditions and ∆hA > ∆hB is shown in Figure 6. Like the corresponding isothermal flash problem (see section 3.1), a unique and stable steady state exists for all values of F0. However, the present flash control strategy turns out to be more flexible because only a lower bound for feasible operation exists. Like in the previous section, only the part of the bifurcation where the green curve lies between the red and black curves according to ye < z < xe is feasible. In contrast to the previous section, the temperature of the flash and therefore also concentrations xe and ye vary with the operating conditions. An example with Tfl < TR for all operating conditions and ∆hA < ∆hB, which violates the stability conditions, is shown in Figure 7. Here, static instability according to det(A) < 0 occurs at a limit point and dynamic instability according to trace(A) > 0 at a Hopf point. Both are in the feasible range. Like in Figure 6, stability is only limited by the boundary z ) ye. Hence, only the part of the unstable solution branch with z < ye is infeasible in Figure 7. In contrast to this, the entire stable branch is feasible. In the vicinity of the limit
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Figure 7. F0 and Qfl control configuration: dependence of xe (black line), ye (red line), and z (green line) on F0. Solid line: stable steady state. Dashed line: unstable steady state. Square: Hopf bifurcation. Circles: unstable periodic solutions. xf ) 0.9; ∆hA ) 25 080 J/mol; ∆hB ) 105 J/mol; TA ) 320 K; Qfl ) 15.0 × 105 W; other parameters are as given in Table 2.
point, multiplicity of feasible steady states can be observed. The Hopf point is subcritical and gives rise to a branch of unstable periodic solutions. For a fixed set of parameters in Figure 7, therefore the following patterns of behavior can occur: (a) no steady state at all; (b) single stable steady state; (c) two unstable steady states; (d) a stable and an unstable steady state. Additional phenomena may arise through the interaction of the bifurcation points with the boundaries z ) xe and z ) ye. A rigorous classification of all different kinds of bifurcation diagrams using singularity theory is straightforward.19,20 However, we will not do this here because the focus is on qualitative rather than quantitative behavior. A third example with Tfl > TR and ∆hA > ∆hB which shows stable periodic oscillations is shown in Figure 8. A branch of stable oscillations emerges at a supercritical Hopf point. The branch of periodic solutions ends at a feasibility boundary, where the periodic solution branches for z and ye cross each other. In summary, we find surprisingly complex patterns of behavior even for a simple model system. The behavior can be much more complicated than the corresponding isothermal mode of operation in section 3.1. Hence, this control configuration should be avoided if ∆hA < ∆hB or if Tfl > TR is likely to occur during the operation of the reactor-separator network. In the present case, multiplicity and instability is introduced exclusively by the specific coupling between the reactor and separator for this control configuration. As discussed in the previous section and the appendix, the individual units have a unique and stable steady state. A more detailed discussion of the physical sources for multiplicity and instability will be given in the concluding section. 4.2. F Is Fixed. Instead of F, now the vapor flow V can be obtained from the energy balance (9).
1 V) {Q + Fcp[TR - Tfl(xe)]} ∆h[ye(xe)] fl
(33)
From the total material balance of the plant, we find that the variable feed flow rate F0, which is now used
Figure 8. F0 and Qfl control configuration: dependence of xe (black line), ye (red line), and z (green line) on F0. Solid line: stable steady state. Dashed line: unstable steady state. Square: Hopf bifurcation. Asterisks: stable periodic solutions. xf ) 0.9; TR ) 310 K; ∆hA ) 105 J/mol; ∆hB ) 25 080 J/mol; Qfl ) 15.0 × 105 W; other parameters are as given in Table 2.
to keep the holdup in the reactor constant, has to be equal to the vapor flow rate V. The liquid flow rate from the flash L is
L)F-V
(34)
Similar to the previous case, V from eq 33 and thereby L from eq 34 are interpreted as functions of xe only because ye and Tfl are functions of xe according to the equilibrium relations (11) and (12). In view of eqs 33 and 34 and F0 ) V, the component material balances for the flash and the reactor (3) and (6) can be written as
MR
dz ) V(xe) xf + [F - V(xe)]xe - Fz - MRkz (35) dt Mfl
dxe ) Fz - V(xe) ye(xe) - [F - V(xe)]xe (36) dt
Stability follows from the Jacobian of eqs 35 and 36
A)
(
[
]
F 1 dV -k F - V + (xf - xe) MR MR dxe dye F dV 1 -(F - V) - V + (xe - ye) Mfl Mfl dxe dxe -
[
]
)
(37)
with
[
]
dye dTfl dV 1 )V(∆hA - ∆hB) + Fcp dxe ∆h dxe dxe
(38)
It should be noted that dV/dxe is always negative for ∆hA > ∆hB and that dV/dxe can be negative or positive if ∆hA < ∆hB. However, it can be shown that the Jacobian (37) with respect to eqs 10 and 38 always satisfies the stability conditions (25) and (26) no matter what sign dV/dxe has. Hence, in contrast to the previous
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Figure 9. F and Qfl control configuration: dependence of xe (black line), ye (red line), and z (green line) on F. Qfl ) 2.5 × 105 W; other parameters are as given in Table 2.
case we always have a unique and stable steady state. This is analogous to the corresponding isothermal flash operation, which was treated in section 3.2. A typical example is shown in Figure 9. Like for the corresponding isothermal configuration in Figure 4, infeasibility only occurs for small values of F. 4.3. F0 and F Are Fixed. For fixed F0 and F, the vapor flow V can be obtained from the energy balance again according to eq 33. Because of the variable holdup in the reactor, the vapor flow is only equal to the feed flow rate F0 at steady state but is usually different from F0 during the transient state. The liquid outflow of the flash drum L can be calculated from eq 34 again because the holdup in the flash drum is still constant. Hence, the material balances of the reactor (2) and (5) and the flash (6) can be written as
MR
dz ) F0(xf - z) + [F - V(xe)](xe - z) - MRkz (39) dt dMR ) F0 - V(xe) dt Mfl
(40)
dxe ) Fz - V(xe) ye(xe) - [F - V(xe)]xe (41) dt
This is now a third-order system because the dynamic total material balance of the reactor has to be considered for the variable holdup case. The Jacobian is
(
A)
[
]
F0 dV kz 1 F-V F - V - (xe - z) -k MR MR dxe MR MR dV 0 0 dxe dye F 1 dV - (F - V) - V + (xe - ye) 0 Mfl Mfl dxe dxe -
[
]
)
(42)
From the Routh-Hurwitz stability condition, the following necessary and sufficient conditions for stability can be derived for this third-order system (e.g., ref 17, pp 157 and 412)
trace(A) < 0
(43)
det(A) < 0
(44)
det(A) - trace(A) (a11a33 - a13a31) < 0
(45)
It is important to note that det(A) ) 0 corresponds to a real bifurcation and det(A) - trace(A) (a11a33 - a13a31) ) 0 to a Hopf bifurcation. At a real bifurcation, the number of steady-state solution changes and the Hopf bifurcation gives rise to oscillatory behavior. In the present case, it can be shown that the trace is always
Figure 10. F, F0, and Qfl control configuration: dependence of xe (black line), ye (red line), and z (green line) on F0. Solid line: stable steady state. Dashed line: unstable steady state. F ) 21.53 mol/ s; ∆hA ) 65 000 J/mol; ∆hB ) 85 900 J/mol; Qfl ) 14.0 × 105 W; other parameters are as given in Table 2.
Figure 11. F, F0, and Qfl control configuration: dependence of xe (black line), ye (red line), and z (green line) on F0. F ) 21.53 mol/s; ∆hA ) 85 900 J/mol; ∆hB ) 65 000 J/mol; Qfl ) 14.0 × 105 W; other parameters are as given in Table 2.
negative but the determinant can change sign depending on the sign of dV/dxe according to eq 38. In particular, we find that the determinant is always negative for dV/dxe < 0 (i.e., ∆hA > ∆hB). In contrast to this, the determinant can be positive or negative for dV/ dxe > 0. Further, it can be shown that a11a33 - a13a31 > 0 always holds. Hence, static instability and therefore steady-state multiplicity is possible for ∆hA < ∆hB. An example with a turning point is shown in Figure 10. As pointed out above, in most cases ∆hA > ∆hB, but for some practically interesting cases, also the reverse behavior, i.e., ∆hA < ∆hB, can be observed. In contrast to the corresponding isothermal flash configuration, both feasibility boundaries, i.e., z ) ye and z ) xe, can be crossed, cutting off a part of the stable and a part of the unstable solution branch. Hence, additional scenarios are possible compared to section 4.1, which, however, will not be discussed in detail here. In contrast to this, for ∆hA > ∆hB a unique stable steady state exists. A typical example is shown in Figure 11. Again, infeasibility occurs for low and large values of F0. A deeper inspection of condition (45) indicates that in both cases no Hopf bifurcations can occur. This hypothesis is supported by numerical investigations. However, a rigorous proof was not possible because of the complexity of the underlying equations. Finally, it should be noted that for isothermal flash operation the variable holdup configuration was most suitable with respect to the uniqueness, stability, and feasibility of steady states. In contrast to this, for constantly heated flash operation treated in this section, instability, multiplicity, and infeasibility may occur for the variable holdup configuration. 5. Analysis for Fixed Vapor Flow Rate V In this section, a third type of flash operation will be considered. For this type of flash operation, the vapor flow rate V is fixed by some suitable control with the heating rate Qfl.
Ind. Eng. Chem. Res., Vol. 42, No. 14, 2003 3301 Table 2. Parameters Used for All Figures unless Indicated Otherwise in the Respective Figure Captions parameter
meaning
value
MR xf k0 E R TR ∆hA ∆hB TA TB Mfl cp
molar holdup CSTR mole fraction of A in the feed preexponential factor activation energy gas constant temperature CSTR heat of vaporization of A heat of vaporization of B boiling point A boiling point B molar holdup of flash molar heat
100.0 1.0 1.0 × 106 40000.0 8.3144 320.0 25 080.0 100 000.0 400.0 300.0 20.0 209.836
unit mol 1/s J/mol J/K/mol K J/mol J/mol K K mol J/K/mol
At steady state, the overall material balance for the plant in Figure 1 reads
F0 ) V
(46)
Therefore, F0 and V cannot be fixed independently and the corresponding control structures are not feasible. Hence, only the control structure with fixed F and fixed V has to be considered. Again, the reactor holdup is assumed to be constant, corresponding to perfect level control with F0. The model equations for this control configuration are given by
MR
dz ) Vxf + (F - V)xe - Fz - MRkz dt dxe Mfl ) Fz - Vye - (F - V)xe dt
(47) (48)
The energy balance fixes the heating rate which is required for achieving constant V. Because we are not interested in the value of Qfl, the energy balance can be neglected. At steady state, eq 47 can be solved uniquely for z according to
z)
Vxf + (F - V)xe F + MRk
(49)
Upon substitution into eq 48, the following relation is obtained:
ye(xe) )
1 FVxf - (F - V)MRkxe V F + MRk
(50)
According to this relation, ye lies on a straight line with a negative slope in the ye-xe plane. Further it has to satisfy the equilibrium conditions (11) and (12), which are represented by the well-known McCabe-Thiele
Figure 12. F and V control configuration: dependence of xe (black line), ye (red line), and z (green line) on F. V ) 15.0 mol/s; other parameters are as given in Table 2.
diagram in the ye-xe plane. For thermodynamically stable vapor-liquid equilibrium with dye/dxe > 0, these curves always have a unique intersection corresponding to a unique steady state. Stability follows from the eigenvalues of the Jacobian
(
F F-V -k MR MR J) F F-V V dye Mfl Mfl Mfl dxe -
)
(51)
These are always stable according to conditions (25) and (26). Further the solutions are always feasible for feasible F > V. A typical example is shown in Figure 12. 6. Discussion and Concluding Remarks The results of the above analysis are summarized in Table 3. From this table, it is found that for the first column corresponding to the isothermal flash operation always a unique and stable steady state exists. Hence, we conclude that multiplicities and instabilities which were found in ref 8 for the corresponding problem with a nonisothermal reactor are primarily caused by the exothermicity of the chemical reaction. However, in both cases, i.e., for the isothermal as well as the nonisothermal reactor, because of the coupling between the reactor and separator, bounds of feasible operation of the reactor-separator system can arise. Beyond these boundaries, no feasible steady state at all is possible, leading to a monotonic increase of the recycle flow rate as described in ref 8. In that respect the control strategy with fixed F, F0, and T is most suitable because always a unique, stable, and feasible steady state exists. Equally well is the control strategy with fixed F and V in the third column of Table 3. In contrast to this, major problems may rise if the heating rate of the flash is fixed (column 2 in Table 3). The popular adiabatic flash is a special case of this control strategy with Qfl ) 0. Particular care has to be taken with this flash control strategy if the enthalpy of vaporization of the heavy boiling component is smaller
Table 3. Summary of the Results for Different Control Configurations Investigated in This Study flow control strategy
Tfl fixed
F0 fixed
unique, stable infeasible for small and large F0
F fixed
unique, stable, infeasible for small F
F and F0 fixed
unique, stable, always feasible
flash control strategy Qfl fixed (i) Tfl < TR and ∆hA > ∆hB: unique, stable, infeasible for small F0. (ii) all other cases: multiple steady states and limit cycles possible unique, stable, infeasible for small F (i) ∆hA > ∆hB: unique, stable, infeasible for small and large F0. (ii) ∆hA < ∆hB: multiple steady states possible
V fixed not possible
unique, stable, always feasible not possible
3302
Ind. Eng. Chem. Res., Vol. 42, No. 14, 2003
than the enthalpy of vaporization of the light boiling component (for fixed F0, and fixed F and F0) or the temperature of the flash can be higher than the reactor temperature (for fixed F0). This, however, is unlikely for an adiabatic flash when the pressure of the flash is much lower than the pressure of the reactor. Instability and multiplicity for the heated flash are caused by the energy balance of the flash in combination with the recycle to the reactor. This has a remarkable analogy to a similar problem that was studied in distillation by Jacobsen and Skogestad.21 In particular, they have shown that the energy balance can induce instability even in binary distillation if the enthalpy of vaporization of the heavy boiling component is smaller than the enthalpy of vaporization of the light boiling component. Multiple steady states were even reported for a one-stage column with a condenser. For vanishing reflux, this configuration is equivalent to the heated flash considered here. As shown in the appendix, the stand-alone flash always has a unique and stable steady state. Hence, we conclude that a finite reflux of the condensed vapor is a necessary condition for multiplicity and instability of a single-stage distillation column. Similarly, a finite recycle of the liquid via the reactor turned out to be a necessary condition for multiplicity and instability of the reactor-separator system involving a heated flash. Multiplicity and instability of the reactor-separator system is also most likely to occur when the enthalpy of vaporization of the heavy boiling component is smaller than the enthalpy of vaporization of the light boiling component. It is worth noting that the variable holdup configuration (F and F0 fixed) should also be avoided in this case. More complex patterns of behavior can be expected if the constantly heated or adiabatic flash in a reactorseparator system is replaced by a distillation column. This would imply a recycle of the vapor at the top of the column via the condenser and a recycle of the liquid at the bottom of the column via the reactor for heavy boiling reactants. Throughout the paper, ideal controllers with instantaneous control were assumed. The same steady states are obtained for PI controllers with finite dynamics. However, the stability may change depending on the parameters of the controllers. In particular, the location of the Hopf bifurcation points in Figures 7 and 8 will change. The above results were obtained for a fairly simple model system. However, we expect that they also provide some insight and understanding for more complex systems and thereby add “another piece to the puzzle”.
the graphical construction in Figure 2. Because xe and ye are constant for given Tfl and pfl, the flash is a purely static system and instability is not possible. 2. Qfl Is Fixed. For fixed heating rate Qfl, the flash equations are given by eqs 6 and 7 and the corresponding equilibrium relations. The energy balance (9) can be solved explicitly for the vapor flow rate V according to eq 33. After substitution of eqs 7 and 33 into eq 6, a single equation in xe is obtained according to eq 36. The stability of the stand-alone flash follows from the derivative of this equation with respect to xe for fixed F and z. This derivative is equivalent to element A2,2 in eq 37 and represents the eigenvalue of the stand-alone flash. By means of eqs 10 and 38, it is readily shown that A2,2 in eq 37 is always negative and therefore stable. 3. V Is Fixed. For fixed vapor flow rate V, the flash equations are given by eqs 6 and 7. The energy balance determines the heating rate, which is required to adjust the desired vapor flow rate and need not be considered here. The system of eqs 6 and 7 is completely equivalent to the single-stage flash distillation which was investigated by Doherty and Perkins.22 By means of a Liapunov function, they proved that this problem always has a unique and stable steady state. Notation cp ) molar heat capacity [J/mol/K] E ) activation energy [J/mol] F0 ) feed to the reactor [mol/s] F ) feed to the flash [mol/s] h ) molar enthalpy [J/mol] ∆h ) enthalpy of vaporization [J/mol] k ) reaction rate constant [1/s] Ki ) equilibrium constant L ) flow rate of the liquid from the flash [mol/s] λ ) eigenvalue M ) molar holdup [mol] p ) pressure [Pa] pis ) vapor pressure of pure component i [Pa] Q ) heating rate [W] R ) gas constant [J/mol/K] t ) time [s] T ) temperature [K] Ti ) boiling point temperature of component i [K] V ) flow rate of the vapor from the flash [mol/s] xe ) mole fraction of reactant A in the liquid from the flash xf ) mole fraction of reactant A in the feed ye ) mole fraction of reactant A in the vapor from the flash z ) mole fraction of reactant A in the liquid from the reactor Super- and Subscripts
Acknowledgment The authors thank the Volkswagen-Stiftung for financial support under Grant I/77 311. Appendix. Stability of the Stand-Alone Flash In the following, the stability of the stand-alone flash with fixed feed F and z will be discussed for the three different modes of operation discussed in this paper. 1. Tfl Is Fixed. For fixed temperature Tfl, the flash equations are given by the material balances (6) and (7) and the corresponding equilibrium relations. In the binary case considered here, these equations always have a unique steady-state solution, as illustrated by
i ) component v ) vapor phase l ) liquid phase f ) feed fl ) flash R ) reactor
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Received for review September 24, 2002 Revised manuscript received April 2, 2003 Accepted April 15, 2003 IE020768N
