Ind. Eng. Chem. Res. 2008, 47, 2905-2911
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Catalyst Particle Design for Optimum Polyolefin Productivity Phillip Hamilton† and Dan Luss* Department of Chemical and Bimolecular Engineering, UniVersity of Houston, Houston, Texas 77204
Local softening or melting of growing polymer particles in fluidized bed reactors may lead to formation of sheets, which require reactor shutdown. The kinetic properties of metallocene catalysts can be modified by alteration of the substituents on the cyclopentadienyl rings. A kinetic model which accounts for the initiation, propagation, and deactivation reactions is used to predict the impact of the kinetic parameters, catalyst properties, and residence time in the reactor on the maximum productivity attained while preventing particle overheating. The maximum productivity decreases with an increase of the deactivation reaction modulus, φd, and is an increasing function of the propagation reaction, φp. The value of φp depends on both the propagation rate constant and the catalyst loading. The residence time at which the maximum productivity is attained, tm, depends most strongly on the moduli for the initiation reaction (φi) and the deactivation reaction (φd). The maximum productivity is rather insensitive to an increase of the residence time above tm. Introduction Gas-phase polymerization of olefin feeds is conducted in fluidized bed reactors using small spherical supported porous catalysts (about 50 µm in diameter). The activity of the catalysts strongly depends on the reaction conditions, impregnation procedure, and support interaction.1-5 The produced polyolefin fragments the support, replicating the original catalyst morphology as the particle grows.6 The fresh catalyst particles are fed near the top of the reactor, and the grown polymer particles are removed from the bottom of the reactor. The polymerization heat is removed by a keeping a low first-pass conversion of the monomer and a high recycle rate of the externally cooled effluent monomer. The reactor temperature usually is kept below 90 °C as the melting temperatures of polyethylene and polypropylene are 120 and 140 °C, respectively.7-10 Additional cooling is attained in certain cases by injection of a volatile liquid, which is condensed in the external heat exchanger. The application of these heat removal steps is adequate to control the reactor bulk temperature but not to prevent local particle overheating. An important detrimental operation problem is caused by local overheating of some particles. When the temperature of a particle approaches that of melting or softening, the particle can stick to other particles. The agglomerates of these particles form polymer sheets that require shutdown and their removal before the process can be restarted. Determining the optimum kinetic parameters and catalyst loading of the support requires maximizing the rate of polymer production while keeping the maximum temperature below a specified upper bound. Several studies considered the temperature rise on a single growing polymer particle while accounting only for the rate of the propagation reaction.11-16 According to this model the total number of active sites is constant during the particle growth and their concentration decreases by dilution as the particle grows. Additional modeling efforts have accounted for particle deactivation.17-22 Song and Luss23 used a more detailed kinetic model that accounted also for the initiation and deactivation reactions for * To whom correspondence should be addressed. Tel.: (713) 7434305. Fax: (713) 743-4323. † Present address: Shell Global Solutions. Houston, TX.
predicting the temperature rise. Specifically, they considered the simplified reaction network,23
where C0 is the total concentration of the deposited catalyst sites, C* is the concentration of the active sites to which no polymer chain is attached, C*(pj) is the concentration of active sites to which a living polymer with chain length j is attached, and C* ∞ ) ∑j)0 C*(pj) is the total concentration of the active sites. Values of the reaction rates and the activation energies of these reactions have been reported in the literature.21-28 Song and Luss23 have shown that accounting for the initiation and deactivation reactions increases the region of operating conditions or the catalyst loading for which the maximum temperature rise remains below a prescribed bound. They also showed that this model predicts that parametric sensitivity may occur only above the melting temperature of the polymer. Therefore it need not be considered since the model is not valid above the melting temperature. The goal of this study is to enhance the understanding of the impact of the various kinetic parameters on both the polymer productivity (time average polymer production) and the temperature rise. This information is essential for a rational determination of the optimal catalyst loading and a guide for which catalyst property modification may increase its productivity without exceeding the bound on the temperature rise. Mathematical Model We consider a growing polymer particle using a lumpedthermal, uniformly distributed model which assumes that the catalyst site concentration remains uniform as the polymer particle grows and that the particle temperature is uniform, but different from that of the ambient gas. We account for the impact of intraparticle monomer concentration gradients
10.1021/ie070134m CCC: $40.75 © 2008 American Chemical Society Published on Web 04/05/2008
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by an isothermal effectiveness factor. The governing equations are
d(MVp) ) kcaV(Mb - M) - ηMkp(Tp)C* dt
d(C*Vp) ) [ki(Tp)C0 - kd(Tp)C*]Vp dt
(3)
Fpcp
dTp ) hfaV(Ta - Tp) + dt
2
ψ
x
kp(Tp)C* De
[ (
[ (
Le
dy 3Bim ) 2 (1 - y) + dτ g
dx* 1 1 ) φi2 exp γi 1 - x0 - φd2 exp γd 1 - x* dτ y y 1 2 pηφp exp γp 1 - xx*2 (15) y
[(
t)0 (9)
We define the following dimensionless variables and parameters:
[(
(-∆H)kc(0)Mb C0 ) , x , β) 0 0 0 hf(0)Tb C (0) C (0)
x
Bim )
[ (
)]
)]
(17)
where
η)
ki(Tb) , De φd ) R(0)
)]
[ (
C/j
x
[ ( )] [ ( )]
gr dgr 1 ) p ηφp2 exp γp 1 - xx* dτ 3 y
e
φp ) R(0)
)]
dx0 1 1 ) -φi2 exp γi 1 - x0 - pηφp2 exp γp 1 - xx*x0 dτ y y (16)
Tp R M t , y) g ) , τ) Mb Tb r R(0) R(0)2/D
kp(Tb)C0(0) , φi ) R(0) De
]
(Le)py 1 xx* 1 (14) y β
(8)
M ) C* ) 0; Tp ) Tb; C0 ) C0(0); Vp ) Vp(0);
x* )
)]
[ ( )] [
ηφp2β exp γp 1 -
The corresponding initial conditions are
x)
(12)
1 dx 3Bim ) 2 (1 - x) - ηφp2 exp γp 1 - xx*[1 - px] dτ y gr (13)
r
(l ) i, p, d)
(-∆H)
(6)
(7)
)]
]
cpTpMw
(5)
and
∆El 1 1 kl(Tp) ) kl(Tb) exp R g Tb Tp
[
The last terms in the brackets of eqs 11 and 12 account for the impact of the change in the growing particle size on the reaction rate. Under typical operating conditions MwM/Fp ≈ 5%, and cpTpMw/(-∆H) ≈ 10%. The corresponding dimensionless model equations are
where
3(ψ coth ψ - 1)
(11)
Similarly, by combining eqs 2 and 5, we get
(4)
dVp Mw ) ηVpMkp(Tp) C* dt Fp
]
M wM d(M) ) kcaV(Mb - M) - kp(Tp)C*Mη 1 dt Fp
(-∆H)kp(Tp)C*Mη 1 -
d(C0Vp) ) -ki(Tp)C0Vp dt
ψ)R
[
(1)
d(TpVp) F pc p ) hfaV(Tb - Tp) + ηM(-∆H)kp(Tp)C* (2) dt
η)
reaction rate. We did not make this simplification, and thus our model contains some terms not included in previous models. By combining eqs 1 and 5, we get
x
kd(Tb) De
kc(0) R(0) Fpcpkc(0) M wM b Le ) , , p) De Fp hf(0) ∆El γl ) l ) i, p, d (10) RgTb
Floyd et al.21 pointed out that the value of the gas Biot number is a constant for the typically low Reynolds numbers of the small growing polymer particles. Previous literature models23-26 ignored the impact of the change in particle volume on the
ψ ) gr
3(ψ coth ψ - 1)
xφ
(18)
ψ2 2 p
[ (
exp γp 1 -
1 x y *
)]
(19)
As the particle grows, dilution decreases the uniform concentration of the catalytic sites by a factor of gr-3. Thus, the Thiele modulus, ψ, is proportional to gr-0.5, and the value of the isothermal effectiveness factor is shifted toward unity as the particle grows. The corresponding initial conditions are
x* ) x ) 0; y ) 1; x0 ) x0(0); gr ) 1
at τ ) 0 (20)
Equations 13-20 were used to simulate the polymerization in a polyolefin catalyst.
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The total polymer production 〈P〉 is
〈P〉 )
∫0t kp(Tp)C*MVpη dt f
(21)
where tf refers to the final reactor residence time. The dimensionless total production is defined as
Pˆ )
〈P〉 Vp(0)Mb
(22)
and is rewritten in terms of dimensionless variables as
∫0τ φp2x*x{exp[γp(1 - 1y)]}gr3η dτ
(23)
Figure 1. Dependence of dimensionless temperature and productivity on particle residence time in the reactor for a case where the maximum temperature rise reaches the imposed bound of 1.11. φp ) 3.65, φi ) 0.025, φd ) 0.001, γp ) 14, γi ) 12, and γd ) 10.
We define the dimensionless productivity with units of inverse time, P, as
The corresponding ranges of the dimensionless parameters are
Pˆ )
f
γp, γi, and γd ) 4-20,
Pˆ P) 1000tf
(24)
The set of algebraic-differential equations was solved by the Limex solver.28 We define a maximum temperature set to be the parameter sets at which the maximum transient particle temperature attains the specified maximum value. This maximum transient temperature must be lower than the melting one, as the model is not valid for temperatures exceeding that of melting. The melting temperature set, defined by Song and Luss,26 is the one which specifies the melting temperature as the maximum temperature. Simulation Results Simulations were conducted to determine the impact of the catalyst kinetic parameters and loading on the maximum possible polyolefin productivity under the constraint that that the maximum temperature rise does not exceed a specified bound. To accomplish this, we tested the impact of the six parameters (φp, φi, φd, γp, γi, γd) on the productivity and temperature rise. While the kinetic parameters affect the values of the six parameters, the loading of the catalyst affects only the value of φp. The simulations were conducted with each parameter bounded within a reasonable range suggested by literature and industrial reports. The ranges of values we used were21-27,28-33
R(0) ) 5-90 µm,
De ) 0.00008-0.0005 cm2/s
kp(353 K) ) 60-1500 s-1, Ma ) 0.0004-0.0012 mol/cm3 ki(353 K) ) 5-50 h-1,
kd(353 K) ) 0.5-20 h-1
∆Ep, ∆Ei, and ∆Ed ) 12-65 kJ/mol,
Tb ) 343-353 K
(-∆H) ≈ 108 kJ/mol (propylene), (-∆H) ≈ 104 kJ/mol (ethylene) λf ) 0.00012-0.0003 J/(cm s K), λe ) 0.0008-0.002 J/(cm s K) Fpcp ≈ 1.26 J/(cm3 K), C0(0) ) 0.0005-0.333 g of (M)/g of (catalyst)
φi ) 0.004-0.04, β ) 2-6.5, Bih ) 0.1-0.3,
φp ) 0.5-5
φd ) 0.0005-0.02 Bim ) 10-60 Le ) 5-30
p ) 0.01-0.03 All the simulations were conducted subject to the constraint that the maximum transient particle temperature, ymax, will not exceed 1.11, and that the particle diameter increases to at least 10 times the initial diameter, i.e., at least a 1000-fold increase in particle volume. This minimum level of production is needed to justify the catalyst and process costs. A typical simulation of the dependence of the temperature rise and productivity on the residence time (Figure 1) shows that a very rapid temperature rise occurred in the initial period. In this example the temperature rose rapidly up to the upper bound of 1.11, which corresponds to a 38 °C increase above the gas temperature during the first 2.2 min of sojourn in the reactor, and then cooled down. In this and all simulations the value of τ has been converted to time by using R(0)2/De ) 0.1 s. This corresponds to De ) 0.002 cm2/s and R(0) ) 45 µm. The productivity reached a maximum value at a longer residence time of 10.33 min and exhibited a very moderate decrease for longer residence times. Simulations were conducted to gain insight into and understanding of the impact that the various kinetic parameters and catalyst loading had on the productivity and the maximum temporal temperature rise. This information is essential for design of catalysts, as the kinetic properties of metallocene catalyst usually can be modified by alteration of the substituents on the cyclopentadienyl rings. The simulations were conducted for a base set of the six parameters in the middle of the reasonable range of values, i.e., at
φp ) 3,
γp ) 14
φi ) 0.025,
γi ) 12
φd ) 0.001,
γd ) 10
In the simulations shown here, we varied a single parameter over its entire reasonable range while keeping the values of all the other parameters constant.
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Figure 2. Dependence of productivity on φp and residence time in the reactor. φi ) 0.025, φd ) 0.001, γp ) 14, γi ) 12, and γd ) 10.
Figure 3. Dependence of productivity on γp and residence time in the reactor. φi ) 0.025, φd ) 0.001, γp ) 14, γi ) 12, and φp ) 3.0.
Figure 2 shows the dependence of the productivity on the reaction modulus φp and the residence time. An increase in the value of φp may be accomplished by an increase in the value of either the propagation rate constant and/or the catalyst loading on the support. The simulations show that the productivity increases rapidly at short residence times, attains a maximum value for a rather short residence time in the reactor, and then decreases rather slowly as the residence time in the reactor exceeds tm, the time at which the maximum productivity is obtained. While the value of φp has a strong impact on the maximum productivity, it has a much smaller impact on the value of tm, which is a decreasing function of φp. The dependence of productivity on the activation energy of the propagation reaction, γp, is shown in Figure 3. Figure 3 shows that an increase in γp increases the catalyst productivity and has a minor impact on the time required to reach the maximum productivity. An increase in γp affects the productivity much less than an increase in φp since the range of temperature change is rather small and the catalyst temperature is close to that of the bulk temperature during much of the reaction time. The dependence of the productivity on the initiation modulus, φi, and the residence time in the reactor is described in Figure 4. It shows that an increase in φi causes a faster initial productivity increase and strongly decreases the time required to reach the temporal maximum productivity. However, as the residence time is increased to several hours, the sensitivity of the productivity to the value of the initial activity modulus decreases. Increasing of the deactivation parameter, φd, decreases the productivity for all residence times in the reactor. As Figure 5 shows, the increase of the value of φd shifts the entire production
Figure 4. Dependence of productivity on φi and residence time in the reactor. φp ) 3.0, φd ) 0.001, γp ) 14, γi ) 12, and γd ) 10.
Figure 5. Dependence of productivity on φd and residence time in the reactor. φp ) 3.0, φi ) 0.025, γp ) 14, γi ) 12, and γd ) 10.
Figure 6. Dependence of productivity on γi and γd and residence time in the reactor. φp ) 3.0, φi ) 0.025, γp ) 14, φd ) 0.001, γd ) 10, and γi ) 12 unless otherwise specified on the plot.
curve to a lower level, while simultaneously decreasing the residence time at which the productivity reaches a temporal maximum. While an increase in the value of the deactivation modulus φd causes the residence time at which the maximum productivity is obtained to decrease, its main effect is that it decreases the magnitude of the maximum productivity. Numerical simulations revealed that the maximal productivity was rather independent of the values of γi and γd. These parameters also had only a minor impact on the residence time required for maximum productivity. Figure 6 shows the dependence of the productivity on the residence time for extreme values of γi and γd (4 and 20 for each parameter). The small impact difference between the graphs computed for the limiting
Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 2909
Figure 8. Effect of kinetic catalyst parameters on residence time for maximum productivity. Figure 7. Effect of kinetic catalyst parameters on the maximum possible productivity.
values is indicative of the very minor impact that variation of these values has on the productivity. A global optimization search was conducted by testing all possible combinations of the parameters in the industrially relevant parameter range. The global optimization search revealed that under the above imposed constraints on the reasonable range of parameter values the highest productivity (P ) 1486) was generated when
φp ) 5,
γp ) 17.4
φd ) 0.0005,
γd ) 4
φi ) 0.0129,
γi ) 4
at a residence time of 0.95 h. At this maximum φp, γi, φd, and γd had to be assigned values on the boundaries of the parameter values we considered, while φi and γp are within the region. Figure 7 compares the dependence of the maximal productivity on the four parameters (φp, γp, φi, and φd) which have the strongest impact on it. As the maximal productivity was rather insensitive to the values of γi and γd, we do not show their impact in Figure 7. To enable a comparison in one figure over the different ranges of reasonable parameter values, we define
φi* ) 100φi, φd* ) 200φd, γp* )
γp 5
The simulations show that the maximum productivity, Pm, is a sensitive function of the values of φp and φd*. However, only modest changes in the maximum productivity are obtained upon variations in the values of γp* and φi*. Increasing the value of either φp, γp, or φi leads to a higher and faster initial temperature rise. Thus, along the curve describing the dependence of Pm on the values of either φp, γp, or φi, the maximum temperature is attained for the largest possible value of that parameter. For all smaller values of a parameter, the maximal temperature is lower than the upper specified bound. An increase in the deactivation modulus φd decreases the reaction rate. Hence, it decreases in the maximum temperature as well as the maximum productivity. Simulations (Figure 8) show that the catalyst parameters which have the strongest influence on the residence time corresponding to the maximum productivity are the rate constant for the initiation reaction, φi, and the rate constant for the deactivation reaction, φd. The value for the rate constant of the propagation reaction, φp, and its activation energy, γp, also affect the residence time for maximum productivity, but their effect
is not as strong as that of φi and φd. The residence time yielding the maximum productivity is not sensitive to the activation energy of the deactivation reaction, γd, or the activation energy of the initiation reaction, γi. Decreasing either φi or φd will increase the time at which the maximum productivity occurs. Increasing these parameters increases the sensitivity of the catalyst productivity to residence time. Discussion A deleterious particle overheating encountered during polyolefin polymerization in fluidized bed reactors may be caused by several effects such as electrostatic attraction, particle disintegration, and monomer sorption. This study focuses on the impact of kinetic parameters on this temperature rise and on providing catalyst developers guidance in their attempts to develop a catalyst with the highest possible productivity subject to the maximum temperature rise. The values of the various kinetic parameters can be manipulated to a certain extent by changing the substituents on the cyclopentadienyl rings of the metallocene catalyst. In addition to kinetic factors, particle overheating may be the result of electrostatic attraction, particle disintegration, and monomer sorption. This study focuses however on enhancing the understanding of the impact of kinetic parameters which if not managed properly will lead to particle overheating. The results reported here provide useful insights into the optimization of catalyst design for gas-phase polyolefin production. The simulations provide a catalyst designer with guidance on how to manipulate the catalyst properties in order to increase the productivity while maintaining the maximum temperature rise below a specified upper bound. The simulations reveal that the maximum productivity can be increased by increasing the values of either φp, φi, or γp, subject to the constraints on the maximum particle temperature. The maximum possible productivity is most sensitive to values of the parameters φp and φd (Figure 7). Each increase in the value of φp shifts the productivity curve to a higher level (Figure 2). Conversely, each increase in the value of φd shifts the productivity curve to a lower lever (Figure 5). The parameter φi has some effect on the productivity initially. However, as the residence time increases the impact of φi on the productivity decreases (Figure 4). In order to maximize productivity, φp should be maximized and φd minimized. Increasing the value of any of the parameters φp, φi, φd, or γp decreases the residence time at which the maximum productivity occurs. However, the residence time at which the maximum productivity occurs is most sensitive to the parameters φi and φd (Figure 7); the interaction of these two effects moduli is the main factor that determines tm. Increasing the value of
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either φi or φd also increases the sensitivity of productivity to the residence time (Figures 3 and 4). For moderate values of these parameters, the productivity is only slightly decreased by increasing the residence time beyond tm. This means that for a large part of the reasonable parameter space the catalyst productivity is not very sensitive to an increase of the residence time above tm. However, for catalysts with high rates of either initiation and/or deactivation the catalyst productivity is sensitive to the residence time. The simulations revealed that in order to maximize the productivity the value of φp should be as large as possible while the values of φd, γi, and γd should be as small as possible. When the value of φp is increased, the values of φi and γp have to be adjusted in order to prevent the maximum temporal temperature rise from exceeding the specified upper bound. When the largest value for φp is selected (φp ) 5), any value of γp exceeding 17.4 results in the particle temperature exceeding the set maximum temperature even when the smallest possible value of φi (φi ) 0.004) is used. Additionally, when the largest value for φp is selected, any value of φi greater than 0.0129 causes the maximum particle temperature to exceed the set maximum temperature. The interaction among these variables, primarily the interaction between φi and φd, determines the residence time which yields the maximum productivity, which is 0.95 h for the range of parameters we used.
∆Ea ) activation energy of the polymerization reaction, J/mol gr ) dimensionless particle diameter, defined by eq 10 hf ) heat-transfer coefficient, J/(cm2 s K) -∆H ) heat of polymerization, J/mol kc ) mass-transfer coefficient, cm/s kp ) polymerization reaction rate, cm3/(mol of sites s) Le ) Lewis number for lumped model, defined by eq 10 M ) monomer concentration, mol/cm3 Mw ) monomer molecular weight, g/mol p ) dimensionless parameter, defined by eq 10 R ) particle radius, µm Rg ) gas constant, J/(mol K) t ) time, s T ) temperature, K x ) dimensionless monomer concentration, defined by eq 10 x* ) dimensionless active-sites concentration, defined by eq 10 y ) dimensionless temperature, defined by eq 10
Conclusions
Subscripts
The design of commercial catalysts for gas-phase olefin polymerization aims to attain a maximum productivity while limiting the maximum temporal temperature below a specified bound in order to avoid sheet formation. The simulations predict that the catalyst parameters which have the strongest effect on the maximum productivity are the propagation reaction modulus, φp, and the modulus of the deactivation reaction, φd. The values of the modulus of the initiation reaction, φi, and the dimensionless activation energy of the propagation reaction, γp, also affect the maximum productivity, but their impact is not as strong as those of φp and φd. The value for the maximum productivity is not sensitive to the activation energy of either the deactivation reaction, γd, or the initiation reaction, γi. In order to maximize the productivity, it is essential to minimize φd and to maximize φp. The value of the catalyst loading affects the value of propagation reaction modulus φp. Figure 8 shows that an increase in the value of any of the four parameters (φp, γp, φi and φd) decreases tm, the time at which the maximum productivity is obtained. An increase in the values of either φp, φi or γp leads to a larger maximum temperature rise and a faster rise in temperature and production rate. Hence, it decreased the value of tm. The increase in the deactivation modulus φd decreased the initial increase in the temperature and the period in which the temperature rises. Hence, increasing its value decreases tm. The simulations show that the productivity for residence times exceeding tm is rather insensitive to this increase.
b ) ambient d ) deactivation f ) final i ) initiation p ) propagation m, max ) maximum value
Nomenclature aV ) ratio of surface area to volume, 1/cm Bim ) Biot number for mass transfer, defined by eq 10 C ) concentration, mol/cm3 cp ) heat capacity of polymer particle, J/(mol K) C* ) active-sites concentration, mol of sites/cm3 De ) monomer effective diffusivity, cm2/s dp ) particle diameter, µm
Greek Symbols γ ) dimensionless activation energy, defined by eq 10 η ) isothermal effectiveness factor, defined by eq 6 Fp ) particle density, g/cm3 τ ) dimensionless time, defined by eq 10 φ ) nonisothermal Thiele modulus, defined by eq 10
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ReceiVed for reView January 22, 2007 ReVised manuscript receiVed January 18, 2008 Accepted January 21, 2008 IE070134M
