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Effect of Conversion-Dependent Viscosity on the Nonlinear Behavior of a Reactor with Fixed Pressure Drop Manthena Vamsi Krishna,† Pushpavanam Subramaniam,*,† and Achim Kienle‡,§ Department of Chemical Engineering, Indian Institute of Technology, Chennai (Madras) 600036, India, Max-Planck-Institut fu¨ r Dynamik komplexer technischerSysteme, Sandtorstrasse 1, 39106 Magdeburg, Germany, and Lehrstuhl fu¨ r Automatisierungstechnik Otto-von-Guericke-Universita¨ t Magdeburg, Universita¨ tsplatz 2, 39106 Magdeburg, Germany
Polymerization reactions are characterized by changes in physical properties, such as viscosity, as the reaction proceeds. The effect of a conversion-dependent viscosity on the nonlinear behavior of a polymerization reactor is investigated in this paper. This involves considering the coupling between the momentum balance, mass balance, and energy balance of the reactor. It is shown that, when the viscosity increases with the conversion steady-state multiplicity can occur when the pressure drop across the reactor is fixed. The different steady states have different inlet flow rates and exit conversions. Two models of ideal continuous reactors, i.e., CSTR and PFR, are investigated. We analyze these reactors considering operation under a constant pressure drop. Our results indicate that multiple steady states can occur in systems such as polymerization reactors where viscosity significantly increases with conversion in the reactor. We show even an isothermal reactor can exhibit multiple steady states. The physical source, i.e., the positivefeedback effect, that causes this multiplicity is identified. The applicability of the predicted results is illustrated for a case study of the manufacture of low-density polyethylene. We postulate that operation of a reactor with a fixed pressure drop leads to decoupling of the reactor from a downstream separator in a coupled reactor-separator network. This can hence form a desirable control strategy for operating coupled systems. 1. Introduction Low-density polyethylene (LDPE) is produced in either a stirred autoclave or a tubular reactor. In industry, the two processes are equally popular.1 The stirred autoclave consists of a cylindrical vessel with a high length-to-diameter ratio. The vessel is usually stirred by a multiple-paddle arrangement. Initiator is added at several locations in the reactor. The reaction chamber is partitioned into one to five zones using baffles. This creates a group of reactors in series. Each zone has an independent initiator injection and temperature control arrangement. The tubular reactor, on the other hand, is essentially a long double-pipe heat exchanger. A cooling jacket is utilized to remove a portion of the reaction heat. Here also initiator is added at more than one position along the reactor. The tubular reactor hence resembles a series of alternating polymerization and cooling zones. A flow sheet for an LDPE manufacturing process is shown in Figure 1. Both autoclave and tubular reactors are operated at very high pressures on the order of 200300 MPa. To achieve such pressures, the feed mixed with the recycle streams from the separators downstream is compressed. In particular, the pressure at the inlet of the reactor is fixed using a compressor. There
Figure 1. Schematic flow sheet of reactor-separator network used in the LDPE process.
* To whom correspondence should be addressed. E-mail:
[email protected]. Fax: +91-44-22570509. Tel.: +91-4422578218. † Indian Institute of Technology. ‡ Max-Planck-Institut fu ¨ r Dynamik komplexer technischerSysteme. § Lehrstuhl fu ¨ r Automatisierungstechnik Otto-von-GuerickeUniversita¨t Magdeburg.
is a pressure release valve at the exit of the reactor that serves to reduce the pressure of the effluent stream to ensure separation in the high-pressure separator. From this high-pressure separator, the stream then goes to a low-pressure separator from which the unreacted reactants are recycled. The pressure at the exit of the reactor is manipulated using a valve at the outlet as desired. Hence, in LDPE manufacturing, the reactor can be
10.1021/ie0497792 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/24/2004
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thought of as essentially being operated at a constant pressure drop. Polymers in general are non-Newtonian fluids. LDPE reactors have been simulated for pressure drop in the past by treating them as pseudo-Newtonian fluids2 by incorporating a concentration- and temperature-dependent viscosity. In the reactor, the viscosity increases significantly with conversion as polymerization proceeds. It is well-known that, for gases, viscosity increases with temperature. Balakotaiah et al.3 considered the steady-state multiplicity of a packed-bed reactor operating under a fixed pressure drop. They concluded that the coupling between the momentum, energy, and species balances can lead to steady-state multiplicity when the viscosity increases as the reaction proceeds and the total pressure drop is specified. They analyzed an exothermic zeroth-order gas-phase reaction. As discussed above, the LDPE polymerization reactor can be viewed as being operated under a constant pressure drop. In this work, we study the behavior of the reactor operated at a constant pressure drop. We include the effect of viscosity as an increasing function of conversion. Reactors have been studied extensively for their steady-state and dynamic behavior in the literature. The processes occurring in these systems are characterized by nonlinear interactions. It is well-known that processes that are self-sustaining, i.e., that have interactions with a positive-feedback effect can give rise to interesting steady-state and dynamic behavior. It has been established that continuous stirred tank reactors (CSTRs) sustaining exothermic reactions4 and autocatalytic reactions5 exhibit multiple steady states and sustained periodic oscillations. However, these studies ignored the effect of momentum transport in the reactor; they analyzed only the influence of mass and energy balances in the reactor. Reactor-separator networks for polymerization systems have also been studied in the past. Bildea et al.6 showed that the LDPE reactor-separator system can exhibit steady-state multiplicity. The highly exothermic nature of the polymerization reaction was identified as being responsible for this behavior. Further, it was shown that a feasible steady state is obtained only if the volume of the reactor exceeds a critical value. They also concluded that the large heat of reaction of polymerization systems such as those used in LDPE manufacturing renders steady-state multiplicity highly probable, which could lead to a possible instability in the process. However, they ignored the effect of momentum transport in the reactor in their analysis. Their work implicitly assumed that the qualitative multiplicity features and dynamic behavior are not affected by the interaction between the momentum transport and the thermochemical kinetics. For the LDPE reactor, the effect of momentum transfer on the behavior can be quite significant. Our objective in this work is to determine the different kinds of qualitative behavior of the reactor in LDPE manufacture induced by this effect. The paper is organized as follows: First, we study the qualitative features of a first-order reaction of the form A f B in a stand-alone reactor, operated under a constant pressure drop with a conversion-dependent viscosity. We analyze the cases when the reactor is operated (i) isothermally and (ii) nonisothermally. The reactor is first modeled as a CSTR and then as a PFR
Figure 2. Schematic of the CSTR showing pressure drop concentrated across the valves.
(plug-flow reactor). The analysis of this conceptual model yields qualitative insight into the system behavior. Next, we study the behavior of a stirred autoclave reactor and plug-flow reactor operated under constant pressure drop using the kinetics of the LDPE process. The chain length distribution of the polymer is an important parameter that determines the mechanical properties of the polymer. The chain length distributions at the different possible steady states are analyzed. We conclude with the implications of our results for the operation of the coupled reactor-separator network of the LDPE process. 2. Elementary First-Order Reaction in a CSTR We consider the first-order irreversible reaction A f B in a CSTR. The reaction is assumed to occur in the liquid phase, and the product B is assumed to be more viscous than the reactant A. Hence, the fluid viscosity µ increases with the concentration of B or with conversion X, i.e., ∂µ/∂X > 0 The model developed is based on the following assumptions: (i) All streams can have at most two components, reactant A and product B. (ii) The fluid is incompressible, and all physical properties except viscosity are constant. (iii) The reactor coolant temperature is assumed to be equal to the fresh feed temperature. (iv) The molar holdup or volume of the reactor is a constant. A schematic diagram of such a reactor is shown in Figure 2. Here, we assume that the pressure drop across the reactor is concentrated in two valves, one at the inlet and the other at the outlet. The mean velocity of the inlet and outlet streams is vz, the feed concentration is CA0, and the concentration in the reactor is CA. The holdup in the reactor is V. 2.1. Isothermal Reactor. The mass balance equation for species A is
V
dCA ) vzAc(CA0 - CA) - ke-E/RTCAV dt
(1a)
Here, Ac is the cross-sectional area of the pipe carrying the reactant, and ke-E/RT is the rate constant, which has an Arrhenius dependency on temperature. The pressure drop across the reactor is the independent parameter that determines velocity vz. This is assumed to be given by
∆P ) (K1 + K2)Fvz2 ) Pin - Pout
(1b)
where K1 and K2 represent the resistances as K factors of the inlet and exit valves shown in Figure 2.
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We operate the reactor under a constant pressure drop. We consider velocity vz as a dependent variable and ∆P as an independent parameter. We assume that the resistance K2 is an exponentially increasing function of the conversion X of the form
K2 ) K1(e2γX - 1)
(2)
where the conversion is
X)1-
CA CA0
Physically, this effect of increasing K2 with conversion is caused by the increase in viscosity with conversion. For the assumed functional form in eq 2, K2 is zero when the conversion in the reactor is zero. The motivation for the choice of this functional form is that it makes the problem analytically tractable and it captures the qualitative behavior in which we are interested in this study. We choose
tch )
VxFK1 Acx∆P
as the characteristic time to make time dimensionless and define
R)
ke-E/RTfVxFK1 Acx∆P
which is the dimensionless inverse of the pressure drop. Eliminating vz from eqs 1a,b, we obtain the equation that determines the evolution of the conversion with dimensionless time (t* ) t/tch) as
dX ) -Xe-γX + R(1 - X) dt*
(3)
2.1.1. Steady-State Characteristics. The steady state of the isothermal CSTR is governed by the relation
F(X,R,γ) ) -Xe-γX + R(1 - X) ) 0
(4)
For our analysis, we consider R as the bifurcation parameter and γ as the auxiliary parameter. The highest-order singularity satisfied by the above function is
F)
∂3F ∂F ∂F ∂2F ) 2 ) 0, *0 * 0, ∂X ∂X ∂R ∂X3
According to Golubitsky and Schaeffer,7 the system has the normal form -X3 + λ ) 0. The universal unfolding of this normal form is -X3 + λ + X ) 0. This has co-dimension 1, as only one additional parameter is necessary to describe the different steadystate bifurcation diagrams. The steady-state dependency of X on R determined by eq 4 depends on the parameter γ. From the universal unfolding, we conclude that the system can have a region of three steady states for a range of R or it will have a unique steady state for all R. This is determined by the value of γ. For γ > 4, there is a range of R for which the system has multiple steady states. For γ
4 yields steady-state multiplicity. Figure 3 depicts the steady-state dependency of conversion on R for two different values of γ. One of the curves is drawn for γ ) 3, and the other is drawn for γ ) 5. The plot corresponding to γ ) 3 shows that the system exhibits a unique steady state for all R. The plot corresponding to γ ) 5 shows that the system admits multiple steady states for a range of R from 0.07 to 0.1. The stability of the steady-state solutions was determined from the eigenvalues of the linearized equations. This stability is also indicated in the bifurcation diagrams. Solid lines represent stable steady-state solution branches, and dashed lines represent unstable steady-state solutions. Steady-state multiplicity is caused in systems that experience an autocatalytic or a positive-feedback effect. It is well-known that CSTRs sustaining an exothermic reaction or an autocatalytic reaction exhibit steady-state multiplicity. The system investigated so far operates isothermally and yet shows steady-state multiplicity. We now describe the source of a positive-feedback effect prevailing in the system that we believe is responsible for the observed steady-state multiplicity. Consider the reactor across which the pressure drop is fixed. Consider a perturbation that causes the exit conversion to increase. The viscosity of the reacting fluid increases, and this leads to a decrease in the volumetric flow rate across the reactor as the pressure drop is fixed. This results in an increase of residence time, which leads to a further increase in conversion in the reactor. The initial perturbation of an increase in conversion hence gets amplified. This autocatalytic effect is the cause for the multiple steady solutions exhibited by the system and arises only when viscosity increases with conversion. 2.2. Nonisothermal CSTR. We now analyze the behavior of the nonisothermal reactor by considering the mass balance equation in conjunction with the energy balance equation and the momentum balance equation. Upon elimination of the velocity vz as explained before, the dimensionless equations governing the evolution of conversion X and dimensionless temperature θ are
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dX ) -Xe-γX + R(1 - X)eθ dt*
(5a)
dθ ) -θe-γX + BR(1 - X)eθ - βRθ dt*
(5b)
where R, t*, and γ are defined as for the isothermal reactor. B is defined as
B)
E(-∆H)CA0 RTf2CpF
β as
β)
ke
UA FCpV
-E/RTf
and θ as
θ)
Figure 4. Dependency of conversion X on R when the pressure drop across the adiabatic CSTR and molar holdup of the CSTR are fixed. (a) β ) 0, γ ) 1, and B ) 1; (b) β ) 0, γ ) 4, and B ) 2.
E (T - Tf) RTf2
In the above equations, we have made the exponential or Frank-Kamenetskii approximation for the Arrhenius temperature dependency. 2.2.1. Steady-State Characteristics. The steady state of the nonisothermal reactor is described by
F(X,R,γ,B,β) ) -Xe-γX + R(1 - X)exp
(
)
BX ) 0 (6) 1 + βReγX
For the adiabatic CSTR with β ) 0, this equation reduces to
-Xe-(γ+B)X + R(1 - X) ) 0
(7)
This is similar to eq 3 describing an isothermal reactor in form, with (γ + B) replacing γ. This system has multiple solutions for γ + B > 4 and a unique solution otherwise. The bifurcation diagrams for B ) 1, γ ) 1 and B ) 2, γ ) 4 are shown in Figure 4. For the plot corresponding to B ) 1, γ ) 1, the system admits a unique steady state for all values of R. For the plot corresponding to B ) 2, γ ) 4, the system admits multiple steady states for a range of R between 0.034 and 0.075. It can be verified that no Hopf bifurcation points and hence no dynamic instability can occur for the case of the adiabatic reactor operation. For the nonadiabatic case, β * 0, the hysteresis variety is calculated numerically by solving7
F)
∂F ∂2F ) )0 ∂X ∂X2
These equations are solved for B, R, and X for various values of β when γ ) 4. The hysteresis variety in the B-β plane (for γ ) 4) is shown in Figure 5a. This divides the B-β plane into two regions. The bifurcation diagrams in the two regions are shown in Figure 5b. The isola variety is also possible in the general nonadiabatic case (β * 0). This variety exists for the case γ ) 0, i.e., when the viscosity is independent of conversion,4 so we believe that this behavior arises because of the exothermic nature of the reaction and not because of the interaction between the hydrodynamics and kinetics.
Figure 5. (a) Hysteresis variety in β-B plane for γ ) 4 whenthe pressure drop across the CSTR and molar holdup of the CSTR are fixed. (b) Dependency of conversion X on R when the pressure drop across the CSTR and molar holdup of the CSTR are fixed. (a) β ) 0, γ ) 4, and B ) 5 (region 1); (b) β ) 10, γ ) 4, and B ) 5 (region 2).
For the nonadiabatic case, β * 0, Hopf bifurcation points can occur in the bifurcation diagram. Again,
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however, we believe this is caused by the exothermic nature of the reaction and not by the interaction between hydrodynamics and kinetics of reaction as they occur for γ ) 0.4 From the analysis of the first-order reaction in a CSTR, we conclude that the system can exhibit multiple steady solutions over a range of pressure drops for a sufficiently strong dependency of viscosity on conversion. 3. Tubular Reactor We now analyze the irreversible first-order reaction occurring in a tubular reactor, i.e., a PFR. The equations describing this reaction in a PFR are the mass balance
dX ) R*(1 - X)eθ dξ
(8a)
the energy balance
dθ ) eθBR*(1 - X) - βR*θ dξ
(8b)
and the momentum balance
dP* ) -δeγXR*-1.75 dξ
(8c)
Here, ξ is a dimensionless length scale defined as the ratio of the axial coordinate z and the total length of the PFR L. Here, X, B, and β are defined as for the CSTR in eq 5. In the above equations, the viscosity of the reacting fluid is modeled as varying exponentially with conversion as µ ) µ0e4γX. Thus, γ is a parameter that indicates the strength of the dependence of viscosity on conversion. We use the Blasius correlation8 and write the friction factor in terms of flow variables as f ) 0.0791/ Re0.25. This yields the dimensionless quantities
δ)
0.0791F0.75µ00.25L2.75k1.75e-1.75E/RTf D1.25Pin
and
Figure 6. Dependency of conversion X on 1 - P* when the pressure drop across the PFR is fixed: (a) β ) 6, γ ) 3, and B ) 4; (b) β ) 6,γ ) 7 and B ) 4.
γ ) 3 and γ ) 7. The bifurcation diagram corresponding to γ ) 3 shows unique steady states for all values of dimensionless pressure drops. The bifurcation diagram corresponding to γ ) 7 shows multiple steady states for a range of pressure drops. We conclude that, as in the case of the CSTR, there exists a critical value of γ above which the reactor admits multiple steady states. These multiple solutions are characterized by a different inlet velocities, which are characterized by different values of R* in the above model. Stand-alone PFRs are analyzed conventionally for a fixed inlet velocity. Under these conditions, the reactors do not exhibit multiple solutions because they are initial value problems. Fixing the pressure drop across the PFR renders the problem a boundary value problem. This can cause multiple steady states in a PFR for sufficiently strong dependency of viscosity on conversion. 4. Case Study of Low-Density Polyethylene
-E/RTf
R* )
kLAce q
P* in eq 8c is the dimensionless pressure, i.e., the ratio of the pressure at any point in the PFR to the inlet pressure Pin, and q is the volumetric flow rate of the fluid. 3.1. Steady-State Characteristics. The analysis of the CSTR has provided some insight into the different kinds of qualitative behavior that can be observed when the reactor is operated under a constant pressure drop. The system exhibits multiple steady states for a range of pressure drops when the viscosity dependence on conversion is sufficiently strong. We use this insight to analyze the steady-state behavior of a tubular reactor. A tubular reactor was simulated for various pressure drops by integrating the system of eqs 8a-c with β ) 6 and B ) 4. We varied R* and, for each R*, determined the pressure, conversion, and temperature profiles in the reactor. Bifurcation diagrams were drawn with dimensionless pressure drop given by 1 - P* as the bifurcation parameter and exit conversion as the state variable. Figure 6 depicts the bifurcation diagrams for
We use the insight gained by our analysis of firstorder reactions in the CSTR and PFR to investigate and analyze the behavior of polymerization reactors used in LDPE manufacture. Our objective is to determine whether the qualitative behavior we have established for the conceptual model in the earlier section can be exhibited by the reactor used in LDPE manufacture. To simplify the computations, in our analysis, we consider the initiator to be added only at the reactor inlet. 4.1. Reaction Scheme Considered. We consider only the initiation, propagation, and termination steps of the polymerization9
Initiation I2 f 2R* R* + M f R1 Propagation Rx + M f Rx+1
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Termination by disproportionation
∑rR x)1
r λ0 )
Rx + Ry f Dx + Dy
x
Termination by combination
) kp[M][R*] - (ktc + ktd)λ02 (11a) ∞
rµ0 )
Rx + Ry f Dx+y Here, I2 represents the initiator, M represents the monomer, R* represents the free radicals formed from the initiator, Rx represents the living polymer with chain length x, and Dx represents the dead polymer with chain length x. Assuming the reaction steps to be elementary, the rate expressions for the generation of initiator, monomer, and free radicals are
∑ rD x)1
) x
(
)
1 ktc + ktd λ02 2
(11b)
∞
r λ1 )
∑xrR
x)1
) kp[M][R*] + kpλ0[M] - (ktc + ktd)λ0λ1
x
(11c) ∞
rµ1 )
∑xrD x)1
x
) (ktc + ktd)λ0λ!
(11d)
∞
rI2 ) -kd[I2]
(9a)
rλ 2 )
x2rR ∑ x)1
x
) kp[M][R*] + kp[M]λ0 + 2kp[M]λ1 (ktc + ktd)λ0λ2 (11e)
∞
rM ) -kp[M]
∑[Rx] - kp[R*][M]
∞
x)1
) -kp[M]λ0 - kp[R*][M]
(9b)
rR* ) 2kd[I2] - kp[M][R*]
(9c)
The rates of generation of living and dead polymers chains are given by
rµ2 )
r Rx ) ∞
(ktc + ktd)[Rx]
∑[Rx]
x)1 ∞
1
x-1
∑[Rx] + 2ktc y)1 ∑ [Ry][Rx-y] x)1
rDx ) ktd[Rx]
∞
λn )
xn[Rx] ∑ x)1
µn )
∑xn[Dx] x)2
(10a)
∞
(10b)
where λi is the ith moment of the living polymer and µi is the ith moment of the dead polymer. By using the definitions of moments of living and dead polymers, we obtain the rate of generation of moments of living and dead polymers as
) (ktc + ktd)λ0λ2 + ktcλ12
(11f)
d[S] ) vzAc([S]0 - [S]) + rSV dt
(12)
where S is the species of interest and rS is the rate of generation of species S per unit volume. In our model, we consider the species balances for M, R*, I2, λ0, λ1, λ2, µ0, µ1, and µ2. The energy balance, which describes the evolution of temperature in the adiabatic reactor, is given by
VFCp
Here, δ represents the Kronecker delta function. kd, kp, ktc, and ktd represent the rate constants for dissociation of the initiator, propagation, termination by combination, and termination by disproportionation, respectively. We use the method of moments to simulate the polymerization reactor. A detailed discussion of the method of moments and derivation of the rate equations in terms of moments can be found in Kiparissides et al.9 The method of moments is based on representing the progress of the reaction in terms of the leading moments of the chain length distributions of the “live” and “dead” polymer chains. These moments are defined by the following equations
x
4.2. Model Equations for the Autoclave Reactor. The material balance in this case, which describes the evolution of the concentrations of species S in the reactor is given by
V kp[R*][M]δ(x - 1) + kp([Rx-1] - [Rx])[M] -
x2rD ∑ x)1
∑(-∆Hi)riV
dT ) FCpvzAc(Tf - T) + dt
(13)
The summation in eq 13 is carried out over all reactions. In our work, in the energy balance, only the propagation step is considered to be exothermic, and heat effects of all other reactions are neglected. The momentum balance is given by assuming a model similar to the one used to analyze CSTR discussed earlier for calculating the pressure drop
∆P ) (K1 + K2)Fvz2
(14)
Here, we assume that K2, the K factor across the exit valve, has a power-law dependency on viscosity, i.e., K2 ) K1*µfn, where µf is the viscosity of the fluid in the reactor. In the above model, it is again assumed that the pressure drop across the reactor is concentrated in two valves, one at the entrance to the reactor and the other at the exit of the reactor. The first one generates a pressure drop because of the flow of reactants from the compressor to the reactor. The second one corresponds to the effluent of the reactor passing through the valve present at the exit of the reactor. The pressure drop caused by the first valve is independent of conversion, whereas the pressure drop caused by the second valve is a function of the viscosity of the fluid exiting the reactor, which depends on conversion. Thus, in eq 14,
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Figure 7. Dependency of conversion on fixed pressure drop across the autoclave reactor used in LDPE process: (a) n ) 0.1 and (b) n ) 0.2.
K1 is assumed constant, and K2 is proportional to a power of viscosity. n indicates the strength of the dependence of the pressure drop across the valve on the viscosity. The parameter n used here is analogous to γ in the earlier analysis of the CSTR and PFR sustaining a first-order reaction. The viscosity is taken to be a function of the zeroth and first moments of the dead polymer and the temperature in the reactor. Following Dilhan et al.,2 we use
µf ) µmµr
0.2, the reactor admits multiple steady states for a range of pressure drops. This range of pressure drops is 280350 bar, which is around 10% of the inlet pressure. It would hence appear that steady-state multiplicity caused by this effect can be observed experimentally. We conclude that there is a critical value of n above which the system exhibits multiple steady states. This result is analogous to the result for the nonisothermal CSTR sustaining a first-order reaction when the viscosity increases with conversion. For that system, a critical value of γ exists above which steady-state multiplicity occurs for a given B. 4.2.2. Chain Length Distributions of LDPE Manufactured in an Autoclave Reactor. The different steady states in the reactor are characterized by different chain length distributions of the polymer. To calculate the chain length distribution, we use the Wesslau distribution, which has been discussed by Kiparisides et al.9 In the present study, we consider only the initiation, propagation, and termination steps. Hence, our polymer is assumed to have only linear chains. The Wesslau distribution is given by
W(x) )
1
x2πx2σ2
xm ) (15b)
µm ) 1.98 × 10-5 + 1.15*10-3T 2
(15c)
exp -
2σ2
]
where x denotes the degree of polymerization or chain length. The parameters of the distribution can be expressed explicitly in terms of the leading moments of the number chain length distributions, i.e.
(15a)
µ11.5 log µr ) 0.0313 0.5 µ0
[
(ln x - ln xm)2
() µ2 µ0
1/2
, exp(σ2) )
µ0µ2 µ12
The mean, variance, and mode of this distribution are given by
where µf is the viscosity of the fluid in the reactor, and µm and µr are contributions to µf from the temperature, concentration, and molecular weight distribution of the polymer. In eq 15c, the temperature T is in kelvin, and the moments µ0 and µ1 are in mol/m3. 4.2.1. Steady-State Characteristics of the Autoclave Reactor for the Production of Low-Density Polyethylene. The autoclave reactor was simulated with the model discussed above for different values of pressure drop. The density of the fluid was taken as 540 kg/m3 and the specific heat as 2700 J kg-1 K-1. The heat of reaction for the propagation reaction was taken as 95 000 J/mol. The volume of autoclave reactor was taken as 100 m3. The feed to autoclave reactor is characterized by [M]0 ) 250 mol/m3, [I2]0 ) 5 mol/m3, T0 ) 500 K, and P ) 3000 bar. Figure 7 shows the bifurcation diagram for n ) 0.1 and n ) 0.2. As explained above, n quantifies the strength of the dependence of viscosity on conversion. The bifurcation parameter used is the pressure drop across the reactor. The conversion (defined as X ) [M]0 - [M]/[M]0, where [M]0 is the monomer concentration at the inlet of the reactor) is used as the state variable. The plot corresponding to n ) 0.1 in Figure 7 shows that the autoclave reactor admits a unique steady state for all pressure drops. In the plot corresponding to n )
2
xj ) xmeσ /2 2
2
σ* ) xm2(e2σ - eσ ) and
mo ) xme-σ
2
respectively. Chain length distributions for the autoclave reactor at the two stable steady states for a pressure drop of 320 bar are shown in Figure 8. The distribution in curve a corresponds to a conversion of 0.13. The mean, variance, and mode of this distribution are given by 103, 2768, and 73, respectively. The distribution in curve b corresponds to a conversion of 0.53. The mean, variance, and mode of this distribution are given by 312, 534 090, and 19, respectively. It can clearly be seen that the chain length distribution at high conversion shows a peak at a lower x, i.e., chain length, whereas that at low conversion shows a peak at a higher x. This is caused by the fact that, at high conversion, the monomer is consumed to produce molecules with longer chain lengths. This is clearly evident in the large tail of the distribution corresponding to that of the high conversion and results in the significantly high value of variance observed.
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Figure 8. Chain length distribution of polyethylene produced in the autoclave reactor at a pressure drop of 320 bar for conversions of (a) 0.13 and (b) 0.53.
Figure 9. Dependency of conversion on fixed pressure drop across the tubular reactor used in the LDPE process.
4.3. Model Equations for the Tubular Reactor. The steady-state material balance for species S in this case is
d[S] rS ) dz vz
(16)
where [S] is the concentration of the species and rS is the rate of generation of species S. The steady-state energy balance in this case is given by
FCp
(
)
(kpMλ0)(-∆H) dT ) dz vz
(17)
Here again, only the propagation step is considered exothermic, and heat effects of other reactions are neglected. Assuming plug flow, the momentum balance is given by 2
fFvz dp )dz 2D
(18)
where f, the friction factor, is given by2
f ) 0.316Re-0.25
(19)
The viscosity used in determining the Reynolds number is µf, as defined in eqs 15a-c. The feed to the reactor is characterized by [M]0 ) 250 mol/m3, [I2]0 ) 5 mol/m3, T ) 500 K, and P ) 3000 bar. The density of the fluid was taken as 540 kg/m3 and the specific heat as 2700 J kg-1 K-1. The heat of reaction for the propagation reaction was taken as 95 000 J/mol. The length and radius of the PFR were taken as 50 m and 20 cm, respectively. 4.3.1. Steady-State Characteristics of the Tubular Reactor. The tubular reactor was simulated for various pressure drops across the reactor by integrating eqs 16-18. The material balances are written for M, R*, I2, λ0, λ1, λ2, µ0, µ1, and µ2. A bifurcation diagram was drawn with the pressure drop in the reactor as the bifurcation parameter and conversion (based on monomer) in the reactor as the state variable. The bifurcation diagram is shown in
Figure 10. Chain length distribution of polyethylene produced in the tubular reactor at a pressure drop of 510 barbar for conversions of (a) 0.022 and (b) 0.975.
Figure 9. For a range of pressure drops of 20-800 bar, the system admits multiple steady states. 4.3.2. Chain Length Distributions of Low-Density Polyethylene Manufactured in a Tubular Reactor. To calculate the chain length distribution, we used the Wesslau distribution. Again, we considered only the initiation, propagation, and termination reactions. Hence, the product molecules were assumed to have only linear chains. Chain length distributions for the tubular reactor at two stable steady states for a pressure drop of 510 bar are shown in Figure 10. The distribution in curve a corresponds to a conversion of 0.022. The mean, variance, and mode of this distribution are 71, 68 890, and 1.26, respectively. The distribution in curve b corresponds to a conversion of 0.975. The mean, variance, and mode of this distribution are 90, 17 754, and 15.7, respectively. Conclusions In this work, we have analyzed the behavior of a polymerization reactor when operated under a constant pressure drop. We show the existence of steady-state multiplicity in the reactor when the viscosity is a
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Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 rA ) rate of generation of species A (mol/m3/s) S ) general species in polymerization t ) time (s) T ) temperature (K) U ) overall heat-transfer coefficient (W m-2 K-1) V ) reactor volume (m3) W ) Wesslau distribution x ) degree of polymerization X ) conversion in the reactor Greek Letters
Figure 11. Schematic flowsheet capturing the essential interaction in a reactor-separator network.
strongly increasing function of conversion. This dependency generates a positive-feedback effect that is different from the effect caused by the exothermic nature of the reaction studied by earlier workers.4 We identify the physical cause of this positive feedback. We show that these multiple steady states can be exhibited by reactors used in LDPE manufacture. We show that both types of reactors used in LDPE manufacture, the autoclave reactor and the tubular reactor, can exhibit these multiple steady states. The different steady states are characterized by different flow rates through the reactor and different chain length distributions of the product. Fixing the pressure drop across the reactor essentially decouples the reactor from the separator in a coupled reactor-separator system (see Figure 11). The behavior of the coupled system is now similar to the behavior of a stand-alone reactor. When the fresh feed flow rate to a coupled system is flow controlled, Bildea et al. 6 have shown that there is a minimum critical reactor volume above which feasible steady states exist. In particular, for low volumes, the coupled system cannot operate at a steady state. When the pressure drop is fixed across the reactor, a feasible steady-state exists for all reactor volumes. This can hence be used as an effective control strategy for these coupled reactor-separator systems. Acknowledgment The authors acknowledge financial support for this research work from VW-Stiftung under Grant 1/77 311. Notation A ) surface area across which heat transfer occurs (m2) Ac ) cross-sectional area of inlet/exit pipes to reactor (m2) B ) dimensionless heat of reaction CA ) concentration of A in the reactor (mol/m3) CA0 ) concentration of A in the feed stream (mol/m3) Cp ) specific heat of the reaction mixture (kJ kmol-1 K-1) Dx ) dead polymer of chain length x E ) activation energy of reaction (kJ/kmol) f ) friction factor in tubular reactor ∆H ) heat of reaction (kJ/kmol) I2 ) initiator k ) reaction rate constant Ki ) K factor that determines pressure drop across valve M ) monomer P ) pressure (N/m2) ∆P ) pressure drop across the reactor (N/m2) R ) universal gas constant (kJ kmol-1 K-1) Rx ) live polymer of chain length x
R ) dimensionless inverse pressure drop β ) dimensionless heat-transfer coefficient γ ) empirical constant governing the dependency of K2 on X R* ) dimensionless residence time of the PFR δ ) dimensionless parameter of the PFR δ(x - 1) ) Dirac delta function θ ) dimensionless temperature ξ ) dimensionless axial distance along the PFR λn ) nth moment of living polymer µn ) nth moment of dead polymer µf ) viscosity of fluid µr, µm ) contributions to viscosity from temperature and molecular weight distribution F ) density of fluid Subcripts A ) species A in the reactor A0 ) species A in the feed stream D ) dissociation of initiator ex ) exit f ) fluid in ) inlet p ) propagation reaction of polymer tc ) termination reaction by combination td ) termination reaction by disproportionation
Literature Cited (1) Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed.; John Wiley & Sons: New York, 1996; Vol. 17. (2) Dilhan, K. M.; Chiou, Y. N.; Kovenklioglu, S.; Bouaffar, A. High-pressure polymerization of ethylene and rheological behavior of polyethylene product. Polym. Eng. Sci. 1994, 34, 804-814. (3) Balakotaiah, V.; Lee, J. P.; Luss, D. Steady-state multiplicity of a packed bed reactor operating under a fixed pressure drop. Chem. Eng. Sci. 1986, 41, 749-755. (4) Uppal, A.; Ray, W. H.; Poore, A. B. On the dynamic behavior of the continuous stirred tank reactors. Chem. Eng. Sci. 1974, 29, 967-985. (5) Gray, P.; Scott, S. K. Chemical Oscillations and Instabilities: Nonlinear Chemical Kinetics; Clarendon Press: Oxford, U.K., 1994. (6) Bildea, C. S.; Dimian, A. C.; Kiss, A.; Iedema, P. D. State multiplicity in PFR-separator-recycle polymerization systems. Chem. Eng. Sci. 2003, 58, 2973-2984. (7) Golubitsky, M.; Schaeffer, D. Singularites and Groups in Bifurcation Theory; Springer: Berlin, 1984; Vol. 1. (8) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960. (9) Kiparissides, C.; Pladis, P. A comprehensive model for calculation of molecular weight and long chain branching distribution in free radical polymerizations. Chem. Eng. Sci. 1998, 53, 3315-3333.
Received for review March 20, 2004 Revised manuscript received September 22, 2004 Accepted October 6, 2004 IE0497792
