Ind. Eng. Chem. Res. 2006, 45, 1019-1028
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PROCESS DESIGN AND CONTROL Nonlinear Behavior of Coupled Reactor-Separator Systems with Azeotropic Vapor-Liquid Equilibriums (VLEs): Comparison of Different Control Strategies Sree Rama Raju Vetukuri,† S. Pushpavanam,†,* K. P. Zeyer,# and A. Kienle#,§ Department of Chemical Engineering, Indian Institute of Technology, Madras, Chennai-600036, India, Max-Planck-Institut fu¨r Dynamik Komplexer Technischer Systeme, Sandtorstrasse 1, D-39106 Magdeburg, Germany, and Lehrstuhl fu¨r Automatizierungstechnik/Modelbildung, Otto-Von-Guericke-UniVersitat, UniVersitatsplatz 2, D-39106 Magdeburg, Germany
In this paper, we discuss the behavior of a coupled reactor-separator system. The reactor is modeled as a continuously stirred tank reactor (CSTR) sustaining an elementary first-order reaction and the separator as a single-stage flash unit. The coupling occurs via the recycling of the bottoms stream from the downstream separator to the upstream reactor. The emphasis in this paper is on studying the behavior of the coupled system when the vapor-liquid equilibrium (VLE) of the mixture has an azeotrope. Consequently, the bottoms stream can be reactant-rich or reactant-lean. Different control strategies for the operation of the reactor and separator are discussed. The stability of the steady states is analyzed analytically for each strategy. Physical explanations for the stabilty results are presented. Generally, the recycle of the reactant lean stream is observed to cause instability. 1. Introduction Coupled reactor-separator systems are ubiquitous in the chemical industry. In a typical chemical industry, the reactants are partially converted to products, and in the subsequent separator, unconverted reactants are separated from the products and recycled to the reactor. In the separator, the feed mixture is separated into two streams: a reactant-rich stream and a product-rich stream. The nonlinear behavior of the individual units has been studied extensively in the past (for reactors, see Uppal et al.;1 for separators, see Kienle et al.2). The coupling of the units through the recycle stream induces constraints in the performance of the individual units, and, consequently, the behavior of the coupled system can differ drastically from the individual units. Moreover, the coupling introduces several feedback effects that result in interesting behavior. Pushpavanam and Kienle3 studied an exothermic reaction in a continuously stirred tank reactor (CSTR). The CSTR is coupled to a downstream flash unit, which is operated under isothermal isobaric conditions. For a binary mixture, this uniquely fixes the compositions of the streams leaving the separator. They established that the couples system can have a maximum of two steady states when the fresh feed rate to the reactor is fixed. For low Damkohler number (Da) values, the system does not possess a steady state. Bildea et al.4 have shown the existence of a critical Da value below which no solution exists for coupled reactor-separator networks. In their analysis, * To whom correspondence should be addressed. Tel.: +91-4422574161. Fax: +91-44-22570545. E-mail:
[email protected]. † Department of Chemical Engineering, Indian Institute of Technology. # Max-Planck-Institut fur Dynamik Komplexer Technischer Systeme. § Lehrstuhl fur Automatizierungstechnik/Modelbildung, Otto-vonGuericke-Universitat.
they assume the separator to be perfect. Zeyer et al.5 analyzed the behavior of an isothermal CSTR sustaining a first-order reaction coupled to a downstream flash. Their analysis was restricted to the case of ideal vapor-liquid equilibriums (VLEs) of the mixtures. They have shown that the nonlinear behavior is crucially dependent on the flow and flash control strategies. They investigated how the multiplicity is dependent on the operating conditions and basic physicochemical properties of the mixture. Vamsi Krishna6 analyzed the behavior of coupled reactor-separator systems when the mixtures exhibited an azeotrope in the VLE. That group has conducted a detailed analysis of the isothermal isobaric operation of a flash for two different modes of an isothermal reactor operation. In the first mode of operation, the fresh feed flow rate is fixed and the reactor effluent flow rate is used to control the reactor holdup. Here, they observed two different branches, corresponding to the two regions of the VLE. The steady state possessed by the system is always stable. In the second mode of operation, the effluent flow from the reactor is fixed and the fresh feed flow rate is used to control the reactor holdup. Here, they observed two possible coexisting steady states for a range of Da values: one of them is stable, and the other is unstable. Here, for a range of Da values, there can be one feasible unstable steady state, whereas for higher Da values, there are no steady-state solutions. In this work, the influence of an azeotrope in the VLE on the behavior of a coupled nonlinear reactor-separator system is analyzed. The reactor is modeled as an isothermal CSTR that sustains an irreversible first-order reaction of the form A f B. The separator is modeled as a flash. The effluent from the reactor is fed into a downstream separator. We have assumed that the heats of vaporization of the two components are equal, to focus on the effects induced by an azeotropic VLE exclusively. Consequently, the heat of vaporization of the mixture is
10.1021/ie0502382 CCC: $33.50 © 2006 American Chemical Society Published on Web 01/04/2006
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Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006 Table 1. Summary of Different Control Configurations Investigated in This Study flow control strategy
Tfl fixed
Flash Control Strategy Qfl fixed
V fixed
F0 fixed F fixed F0 and F fixed
a a see section 3
see section 4.1 see section 4.2 see section 4.3
not possible see section 5 not possible
a
Figure 1. Schematic diagram of the reactor-separator system being considered.
Figure 2. Vapor-liquid equilibrium (VLE) diagram of a binary mixture exhibiting a minimum boiling azeotrope.
independent of its composition. This is a valid assumption for many liquids if the boiling points of the pure components do not differ greatly (Trouton’s rule7). We have considered two different systems: one that exhibits a minimum boiling azeotrope, and another that exhibits a maximum boiling azeotrope. The VLE is generated using Porter’s model.8 Numerical simulations are performed only for the case of the minimum boiling azeotrope, and the stability and bifurcation analysis is extended to the case of the maximum boiling azeotrope analytically. 1.1. Model Assumptions. In this work, we focus on the simple model system illustrated in Figure 1. The system consists of a CSTR and a flash unit. Here, we analyze the system under the following assumptions: (1) The reactor is operated isothermally and it sustains a firstorder reaction A f B (the liquid-phase reaction). This assumption allows us to treat all streams as binary mixtures. (2) The flash is operated under thermodynamic equilibrium, under conditions of constant pressure and constant holdup, Mfl. (3) The energy balance is assumed to be quasi-static. By virtue of this assumption, the temperature in the flash is the equilibrium temperature, which is determined algebraically by the composition. (4) The binary feed mixture to the separator forms a minimum boiling azeotrope (see Figure 2). Consequently, there are two feasible ranges for z, which is the mole fraction of A in the reactor, for which the flash can be operated in the two-phase
Investigated by Vamsi Krishna.6
regime: when z ∈ IIL, we have xe e z e ye, and when z ∈ IIR, we have ye e z e xe. Here xe (ye) is the mole fraction of A in the liquid (vapor) stream leaving the separator. (5) The two units are energetically decoupled in the recycle stream, i.e., there is no energy which is recycled from the separator to the reactor. This is achieved through use of the heat exchanger (HX1), as shown in Figure 1. In this paper, we analyze the situation of a minimum boiling azeotrope in detail. The behavior of a maximum boiling azeotrope is discussed, emphasizing the similarities and differences in behavior with the minimum boiling azeotrope. This study analyses different control configurations of reactor and flash operation. Three different modes of isobaric flash operation are considered: (i) fixed flash temperature (Tfl), (ii) fixed heating rate of the flash (Qfl), and (iii) fixed vapor flow rate of the flash (V). The different flow control strategies for the reactor considered are described as follows: (i) fixed fresh feed rate (F0) to the reactor and the level in the reactor is controlled with F; (ii) fixed effluent flow rate F from the reactor and the level in the reactor is controlled with F0; and (iii) F and F0 are flow controlled and the level in the reactor is variable. Because both aspects (the flow and the separator control strategies) are equally important, all possible combinations are considered, as indicated in Table 1. Some of the combinations are not feasible, because they are contradictory. Thus, when F0 is fixed, we cannot fix V independently, because it is constrained to be equal to F0. Earlier studies have focused on the isothermal-isobaric operation of the flash or they have used a perfect separation in the flash (Bildea et al.4). An important feature in the work presented here is in the use of the energy balance of the flash to determine the performance of the system. This allows for a variation in the composition of the product stream as we vary the operating conditions. We remark that not all steady solutions of the coupled system are feasible. When the flash feed composition and the equilibrium flash temperature lie in region I (see Figure 2), no vapor flow leaves the system and the feed to the flash is completely returned to the reactor. For F0 fixed at a nonzero value, it is therefore not possible to operate at a steady state, because the reactor holdup and the internal flow rates would increase continuously without ever reaching a steady state. When the flash feed composition and the equilibrium flash temperature lie in region III (see Figure 2), the feed is fully vaporized and there is no recycle. In this region, the system behaves similar to a single CSTR without recycle. 2. Model Equations 2.1. Reactor. The model equations for the reactor are as follows. In regard to the material balance of reactant A:
d(MRz) ) F0xf + Lxe - Fz - MRkz dt where k ) k0 exp[-E/(RTR)].
(1)
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006 1021
Equation 2 gives the total material balance:
dMR ) F0 + L - F dt
(10a)
RTfl ln γB ) Rx2e
(10b)
(2)
For variable reactor holdup, the component material balance can be simplified by substituting eq 2 into eq 1. Thus, we obtain
MR
RTfl ln γA ) R(1 - xe)2
dz ) F0(xf - z) + L(xe - z) - MRkz dt
(3)
Equation 3 also holds for the case of constant molar holdup, and, for this case, eq 2 becomes
0 ) F0 + L - F
(4)
where R is the Porter parameter. The choice of R determines whether the system exhibits a minimum boiling azeotrope or a maximum boiling azeotrope. If we increase R to higher values on the positive side, the system exhibits a minimum boiling azeotrope (Figure 3). As we increase R in the negative direction, we get a maximum boiling azeotrope (Figure 4). For our simulations, we use a value of R ) 5500 J/mol; this ensures a minimum boiling azeotrope. The boiling-point condition is given by
∑yi ) 1
2.2. Flash. The model equations for the flash are as follows. The material balance for component A is given as
(11)
Together with eq 7, it follows that
dxe Mfl ) Fz - Vye - Lxe dt
(5) 1)
Assuming constant hold up in the flash, the total material balance is
0)F-V-L
(6)
psat psat A B γAxe + γB(1 - xe) p p
(12)
The parameters chosen for generating the minimum boiling azeotropic VLE in our simulations are R ) 5500 J/mol, TA ) 355.5 K, TB ) 337.7 K, and ∆hA ) ∆hB ) 35 334.6 J/mol.
The quasi-static energy balance of the flash yields
0 ) Fcp(TR - Tfl) - V∆h + Qfl
(7)
2.3. Vapor-Liquid Equilibrium (VLE). The vapor-liquid equilibrium (VLE) is described (assuming an ideal vapor phase) by the relation
yep ) xeγApsat A
(8)
where p is the pressure maintained in the flash. sat The saturation pressures psat A and pB are calculated using the Clausius-Clapeyron relation,
ln
( )
(
∆hv,i 1 psat i 1 ) p R Ti Tfl
)
(9)
where Ti is the boiling point of the pure component i at a pressure p and ∆hV,i is the latent heat of vaporization of pure component i. For our simulations, we assume ∆hv,A ) ∆hv,B ) ∆h. 2.4. Porter’s Model. The regular solution model can be used for an approximate description of nonideal mixtures (according to Malesi’nski9). Mixtures are signified as regular if the molecules are nonpolar and similar in size. For regular solutions, the excess entropy vanishes and, therefore, the free excess enthalpy is equal to the mixing enthalpy. One advantage of the regular solution model is that the number of necessary interaction parameters is low. For mixtures of n compounds, the number of interaction parameters is equal to half the number of off-diagonal elements of the n × n interaction matrix, because all elements of the main diagonal are zero and the matrix is symmetrical. Thus, for binary mixtures, only one interaction parameter is needed, and, for ternary mixtures, only three interaction parameters are needed. In this work, only binary mixtures are investigated. Therefore, one interaction parameter (R) is sufficient. This special case of the regular solution model is also called Porter’s model.7 This is a single-parameter model that can capture the presence of an azeotropic VLE. The activity coefficients (γi) are described in Porter’s model as
Figure 3. Influence of R (in units of J/mol) on the VLE diagrams (for a minimum boiling azeotrope).
Figure 4. Influence of R (in units of J/mol) on the VLE diagrams (for a maximum boiling azeotrope).
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Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006
in the Monsanto acetic acid production (Waschler et al.10). We will now establish the stability of the system analytically. For fixed F and F0, the unknown flow rates L and V are calculated from the flash equations, according to
L)F
( ) ( )
V)F
z - ye xe - ye
(14a)
xe - z x e - ye
(14b)
Substitution of these expressions for L and V into the reactor model equations results in
Figure 5. F, F0, and Tfl configuration, for Tfl ) 335.0 K: (s) stable steady state and (- - -) unstable steady state.
MR
(z - ye)(xe - z) dz ) F0(xf - z) + F - MRkz (15) dt xe - ye xe - z dMR ) F0 - F dt xe - ye
3. Analysis for Fixed Flash Temperature (Tfl) First, we will focus on the flash being operated at a fixed temperature Tfl. This can be achieved via suitable control with the flash heating rate Qfl. The required heating rate that is required to maintain a constant flash temperature can be calculated using the quasi-static energy balance of the flash (eq 7). However, the energy balance can be neglected if one is not interested in the heating rate (Qfl). We will consider the case where F0 and F are flow-controlled and Tfl is maintained constant. 3.1. F0 and F are Flow-Controlled. Here, the reactor holdup is not controlled at a fixed value. It is a state variable for this control strategy. Setting the right-hand side to zero in eqs 1, 2, and 5, and using eq 6, we find that the steady-state behavior of the reactor-separator network is described by
0 ) xf - ye - Daz
(13)
Here, the parameter Da is defined as
Da )
MRke-E/(RTR) F0
Because the pressure and the temperature of the flash are fixed, we have two different values for ye, corresponding to the two regions of the VLE. Figure 5 shows the dependence of z on Da. The two branches of the bifurcation diagram result because of the two regions of the VLE. The extent of these branches is determined by the condition that the reactor composition lies in the two-phase region of the flash operation. We see that there are two different intervals of Da values (0.6790 e Da e 1.2790 and 6.4178 e Da e 18.52) where steady-state solutions exist. The steady-state values of the reactor composition, corresponding to the two different intervals, are 0.0467 e z e 0.1348 (reactant-lean condition) and 0.4263 e z e 0.845 (reactantrich condition). Outside this range, the coupled system is infeasible; i.e., a small disturbance in z may result in either vanishing or accumulating reactor holdup. The steady states possessed by the system are not always stable. The steady-state solutions are stable when the reactant-rich stream is recycled to the reactor and unstable when the reactant-lean stream is recycled to the reactor. It can be clearly observed that the branches do not overlap in the feasible region. A situation where the recycle stream is lean of reactant A can occur if the recycle is used for the recovery of a nonvolatile catalyst and not for the recovery of reactant A. Such a situation occurs, for example,
(16)
The coupled system is has been reduced effectively to a twodimensional system. The Jacobian J is obtained as
J)
(
-
F z - ye -k MR xe - ye
F xe - ye
0
kz MR
)
(17)
The necessary and sufficient conditions for the stability of a second-order system are
det(J ) > 0
(18)
trace(J ) < 0
(19)
It is readily shown that the condition on the trace (J ) is always satisfied for the entire composition range because (z - ye)/(xe - ye) is positive on both sides of the azeotropic composition. The determinant of J is given by
det(J ) )
kz F x e - y e MR
(20)
This is positive only if xe - ye > 0, which corresponds to operation in region IIR. For compositions in region IIL, the determinant is negative, because xe - ye < 0. Therefore, for this control strategy, the system is stable only when the reactantrich stream is recycled to the reactor, i.e., in region IIR. We now discuss the interaction of the different physical variables of the system and how they determine stability at the steady state. This gives us a physical interpretation for the stability result that we have established mathematically. Consider a steady state where the reactor composition and the flash temperature lie in region IIR (see Figure 2). We consider the system to be initially at a steady state of z ) 0.6356. The other operating conditions are given in caption of Figure 6. A positive disturbance is given in z, so that is attains a value of z ) 0.8. The evolution of the system from z ) 0.8 is shown in Figure 6. We can see that the system relaxes back to the steady state, confirming its stability. The flash is operated at constant temperature and pressure; therefore, the effluent compositions from the flash are fixed. The recycle flow rate is obtained using the lever rule (eq 14a). The lever rule gives the relative amounts of vapor and liquid in the flash or the ratio of liquid flow rate
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006 1023
Figure 6. Positive disturbance in effluent composition from the reactor, for z ) 0.8 (z ∈ IIR): TR ) 330 K, Tfl ) 335 K, F ) 30 mol/s, z ) 0.6356, and Da ) 0.9018.
Figure 7. Positive disturbance in effluent composition from the reactor, for z ) 0.12 (z ∈ IIL): TR ) 330 K, Tfl ) 335 K, F ) 30 mol/s, z ) 0.090676, Da ) 9.53486.
to the flash feed flow rate. In the region z ∈ IIR, ye e z e xe. Therefore, a positive (negative) disturbance in z, results an increase (decrease) in the recycle stream. This is also clearly evident in Figure 2. A positive disturbance in z increases the recycle stream, leading to an increase in the reactor holdup. This results in an increase in the reactor residence time and an increase in the conversion. This decreases z, and then the initial disturbance is negated and the system is stabilized. Similarly, a negative disturbance in z reduces the flow rate of the recycle stream, leading to a decrease in reactor holdup. This results in a decrease in the reactor residence time and a consequent reduction of conversion. This increases z and the state turns out to be stable. Now consider the reactor composition and the flash temperature at the steady state to be in region IIL (see Figure 2). We consider the system to be initially at a steady state of z ) 0.09076. The other operating conditions are given in the caption of Figure 7. In the region z ∈ IIL, xe e z e ye. Therefore, a positive (negative) disturbance in z results in a decrease (increase) in the recycle stream, again from the lever rule. This is also clearly evident in Figure 2. A positive disturbance is given only in z, so that it attains a value of z ) 0.12. All other variables are assumed to be at the steady state. The evolution of the molar reactor holdup of the system is shown in Figure 7. We can see that the reactor holdup goes to zero after a time of 27 s, confirming the steady state to be unstable. A positive disturbance in z (Figure 7) reduces the recycle stream, thereby
decreasing the reactor holdup. This results in lower residence times and reduces conversion, leading to accumulation of the reactant, which further reduces the recycle stream. This continues and leads to a vanishing reactor holdup. Similarly, the effect of a negative disturbance in z can also be analyzed. A negative disturbance in z increases the recycle stream, thereby increasing the reactor holdup. This result in higher residence times and higher conversion, leading to a further increase in recycle flow. This continues and leads to an accumulating reactor holdup. Consequently, the steady states in this region are unstable. We now extend the analysis to systems that exhibit a maximum boiling azeotrope. The coupled system is always dynamically stable for the entire range of compositions. This is because the trace of the Jacobian is always negative for the entire composition range. From eq 20, the determinant is positive only when xe - ye > 0, which corresponds to the compositions less than the azeotropic compositions. Therefore, the system is statically stable (unstable) for compositions that are less (more) than the azeotropic compositions. Therefore, for this control strategy, the system is stable (unstable) when the reactant-rich (reactant-lean) stream is recycled to the reactor. 4. Analysis for Fixed Heating Rate (Qfl) In the second step, the focus is on fixed flash heating rate (Qfl). This can be achieved via direct control of the heating rate. In this step, we focus on three different flow control strategies around the reactor, where either F0, F, or F0 and F are fixed by some suitable control. 4.1. F0 is Flow-Controlled. In this strategy, the reactor holdup is maintained constant by manipulating the molar outflow of the CSTR. The dependent variables are F, V, L, Tfl, z, xe, and ye, and the equations that determine these are eqs 3-8 and eq 12. When the VLE of the binary mixture is ideal, instabilities were observed for this control strategy when Tfl > TR (from Zeyer et al.5). For the ideal VLE case, the reactant is assumed to have the higher boiling point. It is always stable for Tfl < TR. Our aim is to investigate the stability of the system when the VLE of the system has an azeotrope, so we have investigated only the case when the temperature of the reactor was greater than the flash temperature (Tfl < TR). Now any change in the behavior can be attributed to the azeotrope in the VLE. This condition can be satisfied when the flash is operated at a markedly lower pressure than the reactor. Specifying a reactor temperature greater than the boiling points of the pure components can ensure this when the VLE of the system exhibits a minimum boiling azeotrope. The dependence of the steady-state reactor composition (z) and the vapor and liquid composition from the flash (ye, xe) on the feed flow rate to the CSTR is obtained from steady-state continuation studies (Figure 8). The solution branches that correspond to the two regions of the VLE have been determined to be always stable. The two branches are present for different intervals of fresh feed flow rate. Except for low flow rates, the steady state of the system is always feasible. There is a lower bound on the feed rate for feasible operation. Below this lower bound, z > ye. This corresponds to the case where the feed to the flash is fully vaporized, which results in negative recycle flow rates. This is observed when the feed composition to the flash is less than the azeotropic composition. For compositions greater than the azeotropic compositions, the system steady state is always feasible. Hence, at most, a single, stable, feasible steady-state solution is possible for this control strategy. We will now establish the stability of the system analytically.
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Figure 8. F0 and Qfl control configuration: TR ) 400 K, Qfl ) 15 × 105 W; other parameters are as given in Table 2. The solid line represents the stable steady state.
Figure 9. F and Qfl control configuration: TR ) 400 K; other parameters are as given in Table 2. The solid line represents the stable steady state.
The vapor flow rate (V) must be equal to F0, given the overall material balance of the plant. F, which is the effluent from the CSTR, can be obtained from the energy balance presented in eq 7,
From eq 26, the sign of dF/dxe is same as the sign of dTfl/ dxe. Moreover, the product (xe - z) dF/dxe is always positive throughout the range of compositions. It is readily shown that the trace of the Jacobian is always less than zero and the determinant of the Jacobian is always greater than zero. The necessary and sufficient conditions are again always satisfied; therefore, the system always has a unique steady state that is always stable. In other words, we can work on either side of the azeotrope just by manipulating the fresh feed rate to the reactor. In this mode of operation, the transition as we go from one side to another is smooth. When the system exhibits a maximum boiling azeotrope, the slope dTfl/dxe is positive (negative) for compositions less (more) than the azeotropic composition. From eq 26, the sign of dF/ dxe is same as the sign of dTfl/dxe. Therefore, the product (xe z) dF/dxe is always positive throughout the range of compositions, because the quantity xe - z is positive (negative) for compositions less (more) than the azeotropic composition. It is readily shown that the necessary and sufficient conditions for stability (eqs 18 and 19) are always satisfied for the entire range of flow rates. Therefore, the system always has a unique stable steady state, similar to the result for the systems exhibiting a minimum boiling azeotrope. 4.2. F is Flow-Controlled. In this strategy, the molar holdup in the reactor is kept constant by manipulating the fresh feed flow rate. The dependent state variables are F0, V, L, Tfl, z, xe, and ye, and the equations that determine these state variables are eqs 3-8 and eq 12. The dependence of the steady-state reactor composition (z), and the vapor and liquid composition from the flash (ye, xe), on the exit molar flow rate to the CSTR is obtained from the steadystate continuation studies (see Figures 9 and 10). Stability cannot be guaranteed for this control strategy. For the parameters listed in Table 2 and TR ) 400 K, the system is always stable (Figure 9). The system is infeasible for low exit molar flow rates to the CSTR, i.e., there is a lower bound for feasible operation below which we violate the condition z < ye (see Figure 9). For the combinations of the parameters listed in caption of Figure 10, static instability is observed. Here, z is very close to xe. The system does not exhibit dynamic stability, because the trace of the Jacobian is always negative. Stability is guaranteed when the reactant-rich stream is recycled to the reactor. The analytical basis for dynamic stability is now described.
F)
F0∆h - Qfl
(21)
cp[TR - Tfl]
and the liquid flow rate from the flash (L) is given as
L ) F - F0
(22)
The component and material balances can be written as
MR
dz ) F0xf + [F - F0]xe - Fz - MRkz dt
(23)
dxe ) Fz - F0ye - [F - F0]xe dt
(24)
Mfl
By elimination of these variables, the coupled system is effectively a two-dimensional system. The Jacobian is given by
J)
(
[
]
F -k MR
1 dF L + (xe - z) MR dxe
F Mfl
dye 1 dF -L - F0 - (xe - z) Mfl dxe dxe
[
])
(25)
Together with eq 21, we get
dTfl F dF ) dxe TR - Tfl dxe
(for TR > Tfl)
(26)
The necessary and sufficient conditions for the stability of a second-order system are
det(J ) > 0
(18)
trace(J ) < 0
(19)
We know that dye/dxe is always positive for thermodynamically stable systems and dTf/dxe is positive for compositions z ∈ IIR and negative for compositions z ∈ IIL (see Figure 2).
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006 1025 Table 2. Parameters Used for All Figures, Unless Indicated Otherwise in the Respective Figure Captions parameter
meaning
value
MR xf k0 E R ∆hA ∆hB TA TB Mfl cp Qfl R
molar holdup of the continuously stirred tank reactor (CSTR) mole fraction of A in the feed pre-exponential factor activation energy gas constant heat of vaporization of A heat of vaporization of B boiling point of A boiling point of B molar holdup of the flash molar heat heating rate of the flash Porter’s model parameter
100.0 mol 1.0 1.0 × 106 s-1 40 000.0 J/mol 8.3144 J K-1 mol-1 35 334.6 J/mol 35 334.6 J/mol 355.5 K 337.7 K 20.0 mol 209.836 J K-1 mol-1 15 × 105 W 5500 J/mol
The vapor flow rate V can be obtained from the flash energy balance described by eq 7:
V)
1 [Q + Fcp(TR - Tfl)] ∆h fl
(27)
The liquid flow rate from the flash L is
L)F-V
(28)
From the total material balance of the plant, F0 ) V. The component material balances for the flash and the reactor now become
MR
dz ) Vxf + [F - V]xe - Fz - MRkz dt
(29)
dxe ) Fz - Vye - [F - V]xe dt
(30)
Mfl
The Jacobian of this two-dimensional system, which determines the stability, is given as
(
-
J)
F -k MR
F Mfl
[
1 dV F - V + (xf - xe) MR dxe
[
]
]
dye 1 dV -(F - V) - V + (xe - ye) Mfl dxe dxe
)
(31)
Together with eq 27, we find that
dTfl 1 dV )Fc dxe ∆h p dxe
( )
(32)
From eq 32, it is clear that the sign of dV/dxe is just opposite to that of dTfl/dxe. Moreover, the product (xe - ye) dV/dxe is negative throughout the entire range of compositions. From this, it is readily shown that the trace is always negative. Therefore, the system cannot exhibit dynamic instability. However, the determinant can change sign, depending on the parameters of the system under consideration, and so we can have some branches that are stable and some that are unstable. The system exhibits a limit point when the reactor temperature TR is sufficiently low. We now consider the systems exhibiting a maximum boiling azeotrope. We can again only guarantee dynamic stability for this control strategy. Equation 32 clearly shows that the sign of dV/dxe is just opposite to that of dTfl/dxe. Moreover, the product (xe - ye) dV/dxe is negative throughout the entire range of compositions, because the term xe - ye is positive (negative) for compositions less (more) than the azeotropic composition. From this, we can readily show that the trace is always negative,
Figure 10. F and Qfl control configuration: TR ) 320 K; other parameters are as given in Table 2. The solid line represents the stable steady state, and the dashed line represents the unstable steady state.
thereby guaranteeing dynamic stability. We cannot guarantee static stability for this control strategy, because the determinant can change sign, depending on the parameters of the system under consideration, similar to that observed for the minimum boiling azeotrope. 4.3. F0 and F are Flow-Controlled. This control strategy results in a variable holdup in the reactor. The dependent state variables are MR, L, V, Tfl, z, xe, and ye, and the equations that govern these are eqs 1, 2, 5-8, and 12. The dependence of the steady-state reactor composition (z), and the vapor and liquid composition from the flash (ye, xe), on the changing fresh feed flow rate to the CSTR is obtained from steady-state continuation studies (Figure 11). It is observed that the system is always dynamically stable but statically unstable for a range of flow rates. For the set of parameters in Figure 11, the following patterns of behavior can occur, depending on the range of flow rates: (a) no steady state at all, for high and very low feed flow rates; (b) a single, stable steady state, where only the branch corresponding to the recycle of the reactantrich stream exists; and (c) a stable and an unstable steady state. The unstable (stable) branch corresponds to the recycle of the reactant-lean (reactant-rich) stream to the reactor. When the system has no steady state, this corresponds to the operation of the flash outside the two-phase regime. For intermediate flow rates, the coupled system exhibits two steady states: one stable and the other unstable. When the coupled system is operated at an unstable steady state, an infinitesimal disturbance will result in the system evolving to the coexisting stable steady state. For example, we consider the system to be initially at a steady state
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Figure 11. F0, F, and Qfl control configuration: F ) 150 mol/s, TR ) 400 K; other parameters are as given in Table 2. The solid line represents the stable steady state, and the dashed line represents the unstable steady state.
Figure 12. F0, F, and Qfl control configuration: F ) 150 mol/s, TR ) 400 K; other parameters are as given in Table 2. Disturbance in z from 0.0895 to 0.09.
of z ) 0.0895. The other operating parameters are given in caption of Figure 12. A positive disturbance is given in z, so that it attains a value of z ) 0.09. The evolution of the system from z ) 0.09 is shown in Figure 12. The system finally evolves to the stable steady state after a period of ∼16 min. Furthermore, the steady-state liquid flow rate from the flash is always positive (F > F0). Hence, all the steady states are feasible, provided the tank is large enough to admit the required holdup. Feasibility is the great advantage of this control configuration, compared to the previous two configurations. The derivation of analytical condition for stability is now detailed. The vapor flow rate V can be obtained from the energy balance of the flash described by eq 7:
The necessary and sufficient conditions for the stability of a third-order system are
(27)
The liquid flow rate from the flash L is
L)F-V
(28)
The component and total material balances for the reactor and flash can be now written as a third-order system:
dz ) F0(xf - z) + [F - V](xe - z) - MRkz dt dMR ) F0 - V dt Mfl
(33) (34)
dxe ) Fz - Vye - [F - V]xe dt
(35)
The Jacobian of the three-dimensional system is given by
(
J ) -
F0 F - V -k MR MR
0
F Mfl
-
kz MR
[
dV 1 F - V - (xe - z) MR dxe dV dxe
]
0
-
0
dye 1 dV - (F - V) - V + (xe - ye) Mfl dxe dxe
[
(37)
det(J ) < 0
(38)
det(J ) - trace(J )(a11a33 - a13a31) < 0
(39)
(xe - ye) dV/dxe is negative for the entire range of compositions. It can be readily shown that the trace of the Jacobian is always negative. However, the determinant can show a change of sign,
det(J) )
1 V ) [Qfl + Fcp(TR - Tfl)] ∆h
MR
trace(J ) < 0
]
)
(36)
( )
dV F kz dxe Mfl MR
(40)
From eq 32, we find that the sign of dV/dxe is just opposite to that of dTfl/dxe. The determinant is negative (positive) for compositions greater (less) than the azeotropic compositions, z ∈ IIR (z ∈ IIL). Therefore, the determinant changes sign, and this corresponds to a real bifurcation. A deeper inspection shows that the condition described by eq 39 determines whether a Hopf bifurcation can occur. Our numerical investigations show that Hopf bifurcations do not occur for this control strategy. We now consider systems exhibiting a maximum boiling azeotrope. The coupled system is always dynamically stable for the entire range of flow rates, because (xe - ye) dV/dxe is negative for the entire range of compositions. Therefore, from this observation, we can readily show that the trace of the Jacobian is always negative. Together with eq 32, we can conclude that the sign of dV/dxe is just opposite to that of dTfl/ dxe. Therefore, from eq 40, we can readily show that the determinant is negative (positive) for compositions less (greater) than the azeotropic compositions. Therefore, for this control strategy, the system is stable only when the reactant-rich stream is recycled to the reactor for systems exhibiting a maximum boiling azeotrope. 5. Analysis for Fixed Vapor Flow Rate (V) The vapor flow rate V is fixed using a suitable control with the heating rate Qfl. In view of the overall material balance for the plant at steady state, F0 ) V. Therefore, F0 and V cannot be
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006 1027 Table 3. Summary of the Results for Different Control Configurations Investigated in This Study flow control strategy F0
fixeda
Two branches are observed, corresponding to the two regions in the VLE diagram. Both branches have a unique, stable steady state for a range of Da. No steady state exists for certain ranges of Da. Possible steady states: (i) 0, for high Da; (ii) 1, unstable, for intermediate Da; (iii) 2, one is stable and the other is unstable, for low Da. Instability is due to the recycling of the reactant lean stream.
F fixeda
F and F0 fixed
a
Flash Control Strategy Qfl fixed
Tfl fixed
Two branches are observed, one of which is stable and the other is unstable. The stable (unstable) branch corresponds to recycling of the reactant-rich (lean) stream.
V fixed
Unique, stable steady state observed. Infeasible for small F0.
Not possible.
Dynamic stability is always guaranteed. The system is statically stable for some sets of parameters, and for some parameters, the system is statically unstable. Static stability is always guaranteed when the recycle stream is reactant-rich. Possible steady states: (i) 0, for high and very low F0; (ii) 1, stable, for intermediate F0; and (iii) 2, one is stable and the other is unstable, for high F0. All the observed steady states are feasible.
Unique, stable steady state. Infeasible for high V.
Not possible.
Investigated by Vamsi Krishna.6
(
The Jacobian is given by
J)
Figure 13. F and V control configuration: F ) 1000 mol/s, TR ) 400 K; other parameters are as given in Table 2. The solid line represents the stable steady state.
F -k MR
F-V MR
F Mfl
-
F-V V dye Mfl Mfl dxe
)
(43)
It can be readily shown that the stability conditions that are described by eqs 18 and 19 are always satisfied, because, for a thermodynamically stable system, dye/dxe is always positive. Therefore, for this control strategy, there is, at most, a single stable feasible steady state. The trace and the determinant of the Jacobian are dependent only on the slope dye/dxe. For a thermodynamically stable system, dye/dxe is always positive. Therefore, similar to the systems exhibiting a minimum boiling azeotrope, the coupled system, at most, has a single stable steady state when the VLE has a maximum boiling azeotrope. 6. Conclusions
fixed independently, and, thus, the only feasible control strategy is when F is fixed. 5.1. F is Flow-Controlled. Molar reactor holdup is maintained constant, corresponding to a perfect level control with F0. The energy balance of the flash (see eq 7) can be neglected if one is not interested in the heating rate Qfl. The state variables that describe this system are F0, L, Tfl, z, xe, and ye, and the equations that govern these are eqs 3-6, 9, and 12. The dependence of reactor composition (z), and the vapor and liquid composition from the flash (ye, xe), on the changing vapor flow rate from the flash is obtained from the steady-state continuation studies (Figure 13). It is observed that the solution branches are always stable and the branches do not overlap in the feasible region. Hence, at most, a single, stable, feasible steady-state solution is possible for this control strategy. Infeasibility is observed at higher vapor flow rates. The analytical explanation for stability is now described: The component material balances for the reactor and flash are given by
MR
dz ) Vxf + (F - V)xe - Fz - MRkz dt
(41)
dxe ) Fz - Vye - (F - V)xe dt
(42)
Mfl
In this paper, we have analyzed the case of a coupled reactorseparator system for five different control configurations. The reactor was assumed to be isothermal, and a detailed analysis is performed when the vapor-liquid equilibrium (VLE) is assumed to have a minimum boiling azeotrope. The presence of an azeotrope results in two different values of xe and ye, corresponding to the two different regions in the VLE diagram. Thus, the coupled system can admit two branches of steadystate solutions. These two branches are such that one corresponds to a reactant-rich stream and the other corresponds to a reactant-lean stream. We are interested in understanding the behavior of the process; hence, we have restricted our analysis to that of the open-loop system. An appropriate control strategy can be designed that will help us determine the pairing of the manipulated variable and the control variable by analyzing the different control schemes analyzed here. For some of the different control configurations considered, we can prove that the system does not exhibit any dynamic instability. This could be proven analytically for most of the cases considered. The system could exhibit static instability, depending on the control configurations and operating parameters. The results of the system behavior in the different configurations are summarized in Table 3. Static instability generally occurs when there is a recycling of the reactant-lean
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stream. The results for the systems exhibiting a maximum boiling azeotrope were also similar. The results of the analysis in this paper have implications in regard to the design of control systems. Acknowledgment The authors acknowledge financial support for this research work from VW-Stiftung (under Grant No. 1/77 311). Notations cp ) molar heat capacity [J mol-1 K-1] Da ) Damkohler number E ) activation energy [J/mol] F0 ) fresh feed to the reactor [mol/s] F ) feed to the flash [mol/s] ∆h ) molal latent heat of vaporization [J/mol] k ) reaction rate constant [s-1] L ) recycle flow rate from the flash to the reactor [mol/s] M ) molar hold up [mol] p ) pressure [Pa] psat i ) vapor pressure of pure component i [Pa] Q ) heating rate [W] R ) gas constant [J mol-1 K-1] t ) time [s] T ) temperature [K] Ti ) boiling point temperature of component i [K] V ) flow rate of vapor from the flash [mol/s] xe ) mole fraction of liquid A in the liquid from the flash xf ) mole fraction of reactant A in the feed ye ) mole fraction of liquid A in the vapor from the flash z ) mole fraction of reactant A in the liquid from the reactor R ) Porter’s parameter [J/mol] γ ) activity coefficient Subscripts A ) component A
B ) component B i ) component R ) reactor fl ) flash Literature Cited (1) Uppal, A.; Ray, W.; Poore, A. On the dynamics behavior of continuous stirred tank reactors. Chem. Eng. Sci. 1974, 29, 967-985. (2) Kienle, A.; Groebel, M.; Gilles, E. D. Multiplicities and instabilities in binary distillationsTheoretical and experimental results. Chem. Eng. Sci. 1995, 50, 2691. (3) Pushpavanam, S.; Kienle, A. Nonlinear behavior of an ideal reactor separator network with mass recycle. Chem. Eng. Sci. 2001, 56, 2837. (4) Bildea, C. S.; Dimian, A. C.; Iedema, P. D. Nonlinear behavior of reactor-separator-recycle systems. Comput. Chem. Eng. 2000, 24, 209215. (5) Zeyer, K. P.; Pushpavanam, S.; Kienle, A. Nonlinear Behavior of Reactor-Separator Networks: Influence of Separator Control Structure. Ind. Eng. Chem. Res. 2003, 42, 3294-3303. (6) Vamsi Krishna, M. Non-linear behaviour of Coupled Reactor Separator Sytems with an azeotropic VLE. M.S Thesis, Department of Chemical Engineering, IIT Madras, India, 2005. (7) Trouton, F. Philos. Mag. 1884, 18, 54. (8) Porter, A. W. On the vapour-pressures of mixtures. Trans. Faraday Soc. 1920, 16, 336-345. (9) Malesi’nski, W. Azeotropy and Other Theoretical Problems of Vapour-Liquid Equilibrium; Interscience Publications: London, 1965. (10) Waschler, R.; Kienle, A.; Anoprienko, A.; Osipova, T. Dynamic Plantwide Modelling, Flowsheet Simulation and Nonlinear Analysis of an Industrial Production Plant. In European Symposium on Computer Aided Process Engineering-12: 35th European Symposium of the Working Party on Computer Aided Process Engineering: ESCAPE-12; Grievink, J., van Schijndel, J., Eds.; Computer-Aided Chemical Engineering, Vol. 10; Elsevier: Amsterdam, 2002; pp 583-588.
ReceiVed for reView February 23, 2005 ReVised manuscript receiVed October 27, 2005 Accepted December 1, 2005 IE0502382
