Yield Prediction in a Continuous High-Density ... - ACS Publications

Ind. Eng. Chem. Res. 2006, 45, 7451-7461

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Yield Prediction in a Continuous High-Density Polyethylene Solution Polymerization Staged Reaction Process System Alberto Buchelli* and William G. Todd Equistar Chemicals, LP, a Lyondell Company, La Porte Complex, 1515 Miller Cut-Off Road, Houston, Texas 77536

The catalyst behavior in a continuous polyethylene polymerization train is of utmost importance in the economic performance of a high-density polyethylene (HDPE) unit. A mathematical model was developed to predict polymer catalyst yield in the HDPE solution polymerization as a function of the processing conditions in the reactor train, catalyst deactivation kinetics, the reactors’ flow characteristics, and the residence time distribution. A very large plant data set was used to perform nonlinear model parameter estimation and to compare the model predictions to measured polymer yield in the plant. Several models were tested, and correlation coefficients r2 as high as 0.91 were obtained. Introduction The catalyst behavior in a continuous polyethylene polymerization train is of utmost importance in the economic performance of a high-density polyethylene (HDPE) unit. To name a few process variables, the catalyst deactivation is affected by the operating temperature and pressure in the reactors. Also, the heat evolution in the reactors and whether the reactors are operated in an adiabatic versus isothermal fashion will influence the catalyst behavior and ultimately the polymer yield. In addition to this, the residence time distribution (RTD) in the reactors plays a key role in the final polymer yield that is attained. Because of all of these effects, it is important for optimization of the plant economics to fully understand the catalyst behavior in the reactors and how it influences the overall reactor train performance. The modeling work presented in this paper summarizes the effects of processing conditions on the catalyst deactivation kinetics in the reactor train and finally on the overall polymer yield. The main objective of this modeling effort was to relate via mathematical relationships the processing conditions in the reactor train to catalyst deactivation kinetics and to the reactors’ flow characteristics so that a polymer yield can be predicted and the overall economic performance of the unit can be optimized.

Figure 1. HDPE polymerization block flow diagram.

Process Description A detailed evaluation and description of the Du Pont solution polymerization process for producing HDPE is presented in Chen et al.1 The HDPE solution polymerization process that is the subject of this study is depicted in the block diagram scheme shown in Figure 1, which basically corresponds to the flowsheets presented in Chen et al.1 and Lundeen et al.2 Catalyst, solvent, fresh ethylene feed, recycle ethylene, and comonomers are fed to the reactor train. The reactor train consists of three reaction stages as shown in Figure 2. The first stage of reaction is comprised of two continuous stirred tank reactors (CSTRs) operating in parallel. The outlet stream from this first stage is then contacted with more catalyst and fresh feed in a second stage of reaction. This second stage of reaction is a CSTR whose outlet stream enters a plug-flow reactor that is the third stage * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: 713-336-5214. Fax: 713336-5391

Figure 2. Reaction train layout.

of reaction. Unlike the previous two stages of reaction, the third stage does not have any catalyst or fresh feed injection. The polymerization takes place in an adiabatic form in the three stages in the reactor train; therefore, the heat of polymerization raises the process stream temperature from the inlet to the outlet of each one of the reactors in the reaction train. Following the reactor train shown in Figure 2, the polymer solution is passed through an adsorber filled with activated alumina to remove catalyst. Then the stream pressure is let down, and the polymer and liquid solvent phases are separated as shown in Figure 1. The polymer stream is then sent to the extrusion area, and the solvent, comonomers, and unreacted ethylene are sent to the distillation area for separation and purification. The unreacted ethylene stream is compressed and purified by removing the excess hydrogen in a dehydrogenation

10.1021/ie0601302 CCC: $33.50 © 2006 American Chemical Society Published on Web 09/22/2006

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reactor system. The recycle stream is further purified and sent to the front end of the polymerization reactor train. Additional information regarding the reaction process scheme shown in Figure 2 is shown in the work of Cribbs et al.3 The modeling work herein developed deals with predicting the polymer yield in each of the reactors in the train as well as the overall polymer yield. Mathematical Model Development It is a well-known fact that the yield for a polymer in a polyolefin reactor decreases with time, and for a given catalyst, the rate of change of the yield with respect to time is equal to the polymerization rate. On the basis of this observation, the rate of change of polymer yield versus time can be expressed mathematically as follows:

dY(t)/dt ) r

(1)

In addition, the rate of change of the polymerization rate “r” with respect to time can be expressed mathematically as follows:

-dr/dt ) kra

(2)

In polyolefin catalysts the value of the “a” constant in eq 2 is typically a characteristic of the type of catalyst used in the polymerization and the composition of active components in it and several other variables. An excellent paper about the rate of decay of a titanium based supported catalyst in propylene polymerization is presented by Brockmeier and Rogan.4 The “a” kinetic parameter can be determined by experiment in a batch polymerization reactor. For one of the catalysts used at the Victoria HDPE plant and which is the subject of this investigation, the value of the “a” constant in eq 2 was found to be equal to 1, which indicates that the polymerization rate behaves as a first-order decay with respect to time. Taking “a” in eq 2 to be equal to one and integrating eq 2 versus time assuming that at time t ) 0 the polymerization rate is ri, we obtain the following equation for the time-dependent rate of polymerization:

r ) rie-kt

(3)

The inverse of the “k” value in eq 3 is a time constant, which represents the deactivation time constant for the polymerization catalyst. For such a case, the time constant for deactivation of the catalyst is, therefore, given as follows:

τc ) 1/k

(4)

Using eq 3, the half-life of the polymerization catalyst becomes:

t1/2 ) τc ln(2)

(5)

According to Buchelli et al.5 the initial rate of ethylene polymerization can be represented for polyolefins polymerization as follows:

ri ) KpCcC) 2M

(6)

The function Cc which is indicative of the initial concentration of active sites in the catalyst can be expressed in terms of the processing conditions in the polymerization reactor. One can assume such a relationship to be as follows: b e f ) c ) d Cc ) Ka(H2/C) 2 ) (Bu/C2 ) (C2f) (C8f) (MI)

(7)

The polymerization propagation rate constant Kp can be set to be a function of temperature and pressure using a modified Arrhenius expression according to Buchelli et al.5:

Kp ) Kp0 exp(-(Ea/R + VP/R)(1/T - 1/Tr))

(8)

In eq 8, the activation volume term is used to account for the effect of pressure on the reaction rate for reactions occurring at high pressures as the ones encountered in the HDPE solution process. For the ethylene polymerization propagation reaction, the value of V according to Buchelli et al.6 is equal to -0.0268 m3/(K mol). The rate constant increases with increasing pressure for reactions with negative activation volumes. Substituting eq 6 into eq 3 gives an expression for the polymerization rate as a function of process variables (Kp, C) 2, and t) as follows: -(t/τc) r ) KpCcC) 2 Me

(9)

Yield Calculation in a Batch Polymerization Reactor In a batch polymerization reactor in which all the catalyst particles have the same residence time, the polymer yield Y(t) defined as the mass of polymer produced per mass of catalyst fed to the reactor can be obtained by integrating eq 1 with respect to time and using the result of eq 9 for the rate of polymerization “r” one obtains.

Y(t) )

∫0tr dt

(10)

After substituting the definition of eq 9 into eq 10 we obtain the following:

Y(t) )

∫0tKpCcC)2 Me-(t/τ ) dt c

(11)

Further manipulation of eq 11 results in the following:

Y(t) ) KpCcC) 2M

∫0te-(t/τ ) dt c

(12)

The result of integrating eq 12 gives the following expression: -(t/τc) ) Y(t) ) KpCcC) 2 Mτc(1 - e

(13)

Using the initial rate of polymerization given by eq 6 we obtain the following:

Y(t) ) riτc(1 - e-(t/τc))

(14)

As shown by Buchelli et al.,4 eq 14 above implies that the yield in a batch polymerization reactor is a function of the catalyst time constant, the initial rate of polymerization, and the reaction time in the reactor. Yield Calculation in a Continuous Polymerization Reactor The yield calculation in a continuous polymerization reactor must account for the effect of the reactor flow characteristics. A typical polymerization reactor could behave as plug-flow, as a back-mixed system, or as anything in between, thereby affecting the final yield obtained in the polymerization. An excellent paper regarding the theoretical polymer yield modeling in a continuous stirred bed polyolefins reactor is given in the work of Caracotsios.7

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Levenspiel8 has shown a traditional way to model a reactor as a series of back-mixed tanks. One mixed tank corresponds to a back-mixed system whereas an infinite number of mixed tanks in series represent a plug-flow reactor system. According to the tank-in-series model and applying that model to a polymerization reactor, there will always be a distribution of particles that leave the reactor at different times and that have attained a given polymer yield. Therefore, the total polymer yield must be the value of an integral that accounts for the individual yields of a fraction of polymer particles of a given residence time, integrated over the RTD of the reactor system. In mathematical words, Buchelli et al.4 have expressed this as

Y(t) )

∫0∞Ei(t) Yi(t) dt

(15)

In eq 15, Ei(t) is the exit age distribution of particles that have spent a time t in the reactor, and Yi(t) is the polymer yield for particles that have spent a time t in the reactor. The value of Ei(t) can be determined from the tanks-in-series model in the work of Levenspiel8 and can be obtained from the following equations:

Eθ ) τrEi(t) Eθ ) (N(N(t/τr))

(N-1)

)/(N - 1)!e

(16a) -N(t/τr)

Ei(t) ) (1/τr)(N(N(t/τr))(N-1))/(N - 1)!e-N(t/τr) (16c) For a given set of particles that have spent a time t in the reactor, the value of Yi(t) can be determined from eq 14 and given as

(17)

Finally, substitution of eqs 16c and 17 into eq 15 permits the calculation of the polymer yield in a continuous polymerization reactor represented by “N” CSTRs in series. Therefore, eq 15 becomes

Y(t) ) ((riτc)/τr)(N/(N - 1)!)N(N-1)

∫0∞(1 -

Y(t) ) ((ri1τc)/τr1)(N/(N - 1)!)N(N-1)

e-(t/τc))(t/τr1)(N-1)e-N(t/τr1) dt (19) For one CSTR (N ) 1), the polymer yield in eq 19 above can then be simplified to

∫0∞(1 -

Y(1) ) ((ri1τc)/τr1)(1/(1 - 1)!)1(1-1)

e-(t/τc))(t/τr1)(1-1)e-(t/τr1) dt (20) After simplifications and integration we obtain the yield equation for the R1 or R2 reactor:

Y(1) ) ri1τcτr1/(τr1 + τc)

(21)

Including the initial rate of polymerization given in eq 6, we obtain

Y(1) ) Kp(Cc)1(C) 2 )1Mτc(τr1/(τr1 + τc))

(22)

(16b)

The value of Ei(t) can be determined by combination of eq 16a and eq 16b to give:

Yi(t) ) riτc(1 - e-(t/τc))

are assumed to be operating under an identical set of process conditions which is obviously an approximation in a real plant environment. Therefore, the equations that apply to the first stage of reaction (R1 or R2 reactors) are based on eq 18:

∫0∞(1 -

e-(t/τc))(t/τr)(N-1)e-N(t/τr) dt (18) Yield Modeling Equations for the HDPE Solution Process The HDPE process herein modeled consists of three stages of reaction connected in series as shown in Figure 2. The first stage has two CSTRs (R1 and R2 CSTRs) connected in parallel. The combined stream leaving these two parallel reactors (first stage) enters a second continuous stirred tank (R3) called the second stage of reaction. Finally, the stream leaving the second stage enters a plug-flow reactor (third reaction stage) that leads the polymer solution into the alumina adsorbers. A simplified process flow diagram is shown in Figure 2. The catalyst feed into the reactors is split between the first and second stage of reaction only. Yield Prediction in the First Stage of Reaction Prediction of polymer yield in the first stage, in either reactor R1 or reactor R2, can be done by considering R1 and R2 to be equivalent to one ideal CSTR each and assuming that the residence time in this reactor is τr1. Also, the R1 and R2 reactors

Equations 7 and 8 need to be included in eq 22. Also, Equations 22, 7, and 8 must be applied to the plant data set for the first reaction stage, and the values of the parameters must be determined to create the polymer yield predictive model. Parameters Estimation. The following parameters must be calculated: Ka, b, c, kH, kpo, Ea, and τc. The activation volume V for ethylene reaction and the universal gas constant R are known parameters. A reference temperature Tr can be arbitrarily chosen. The software package Microsoft Excel was used to accomplish the parameter estimation. Independent Variables. The following six independent variables must be used in the set of Equations 22, 7 and 8: (1) free hydrogen to “free” ethylene molar ratio; (2) co-catalyst to “free” ethylene molar ratio; (3) reactor pressure; and (4) reactor average temperature. Because the first stage of reaction is adiabatic, the reactor average temperature can be defined as follows:

Tav ) (Tin + (Tin + ∆Trise))/2

(23a)

Tav ) (2Tin + ∆Trise)/2

(23b)

(5) “Free” ethylene in the reactor, %, and (6) residence time in reactor R1 or R2, must also be used. Dependent Variable. The dependent variable is the polymer yield Y(1) in either reactor (R1 or R2), which is a plant-measured variable. Yield Prediction in the Second Stage of Reaction The yield that is measured in the R3 reactor in the plant is the result of the polymerization induced by the catalyst leaving reactors R1 and R2 and the fresh catalyst that is introduced into reactor R3. Therefore, in mathematical terms that yield could be determined as follows:

Y(3,m) ) X(Y(3,R1) + Y(3,R2)) + (1 - X)Y(3) (24) The catalyst yield in reactor R3 produced by the “old catalyst” coming from reactors R1 and R2 is assumed to be the same because reactors R1 and R2 are identical (volume, residence

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Table 1. Process Variables Range Used for Yield Model Development in the First, Second and Third Stages of Reaction reactors R1, R2 (first stage)

process variable reactor outlet temperature, °C adiabatic reactor temperature rise,°C reactor average temperature, °C melt index, g/10 min reactor pressure, psi free hydrogen, ppm free ethylene, % free octene, % co-catalyst flow, lb/h residence time, min catalyst fraction measured polymer yield, lb/lb

reactor R3 (second stage)

Table 4. Model Runs for Each Reactor Stage: Correlation Coefficients R2

plug-flow (third stage)

131-262

227-262

267-293.5

33-142

43.6-140.5

13.5-51

110-193

169-236

252.7-275.3

0.34-57 2367-3000 0.03-35 1.3-10 0-9.6 0-5 1-3.7 0.07-0.47 1612-13889

0.34-57 2259-2856 0.2-53 2.3-6.7 0-4.7 0-11.4 0.4-0.98 0.07-0.87 1597-7647

0.34-57 2363-2791 0.2-53 1.6-5.3 0.56-3.96 0-11.4 0.3-1.05 1.0 564-2547

Table 2. Statistics for Linear Models: Polymer Yield versus Combinations of Process Variables for the First Stage of Reactiona polymer yield process variable combinations

R2

SSreg

SSresid

pressure temperature hold-up time free ethylene pressure-temperature temperature-free ethylene temperature-hold-up time pressure-free ethylene temperature-pressure pressure-hold-up time hold-up time-free ethylene pressure-temperature-hold-up time pressure-temperature-free ethylene temperature-hold-up time-free ethylene pressure-temperature-hold-up time-free ethylene

0.0017 0.5370 0.0110 0.5189 0.5794 0.6000 0.5629 0.6060 0.5790 0.5800 0.5180 0.5800 0.6000 0.6000

650.00 289.87 675.00 280.10 312.75 323.96 303.87 327.55 312.75 313.10 280.10 313.10 327.55 323.96

538.83 249.88 533.60 259.65 227.00 215.79 235.88 212.20 227.00 226.64 259.65 226.64 212.20 215.79

0.6060

327.55

212.20

SSreg ) regression sum of squares; SSresid ) residual sum of squares; R2 ) correlation coefficient; and total number of plant yield points considered ) 4042. a

Table 3. Model Runs for Each Reactor Stage: Independent Process Variables Sets independent variable

prediction no. 1

prediction no. 2

prediction no. 3

temperature, °C pressure, psi H2/C) 2 molar ratio co-catalyst/C) 2 molar ratio free ethylene, % free octene, % polymer melt index, g/10 min

yes yes yes yes yes no no

yes yes yes yes yes yes no

yes yes yes yes yes yes yes

time, and processing conditions of T, P) and the amount of catalyst going to each reactor is the same. In mathematical terms we have

Y(3,R2) ) Y(3,R1)

(25)

Using eq 25 and substituting into eq 24, we obtain the following:

Y(3,m) ) X(2Y(3,R1)) + (1 - X)Y(3)

(26)

reactor stage

prediction no. 1

prediction no. 2

prediction no. 3

first stage second stage third stage

0.707 832 0.834 852 0.624 892

0.914 196 0.824 18 0.698 784

0.912 456 0.891 576 0.800 516

Table 5. Polymer Yield Prediction ErrorsFirst Reactor Stage

prediction no. 1 prediction no. 2 prediction no. 3

average error, %

standard deviation of error

maximum error, %

minimum error, %

2.66 1.74 1.73

24.29 20.43 20.37

159.49 239.70 233.20

-56.63 -47.60 -48.30

Table 6. Polymer Yield Prediction ErrorsSecond Reactor Stage


average error, %


maximum error, %

minimum error, %

1.24 1.03 0.78

17.14 15.77 13.80

195.9 254.20 256.20

-37.80 -32.55 -30.03

Table 7. Polymer Yield Prediction ErrorsThird Reactor Stage


average error, %


maximum error, %

minimum error, %

2.87 2.49 2.12

25.66 24.41 22.02

174.70 210.39 122.64

-56.52 -52.40 -54.82

Table 8. Polymer Yield Prediction ErrorsOverall Plant


average error, %


maximum error, %

minimum error, %

1.55 1.31 1.07

15.47 14.15 13.06

157.88 206 213.77

-40.85 -31.73 -30.42

The values of the terms on the right-hand side of eq 27 are explained in the following. According to eq 18, for Y(3,R1,N ) 2) and N ) 2 CSTRs in series, we have

∫0∞(1 -

Y(3,R1,N ) 2) ) ((ri2τc)/τr2)[(2/(2 - 1)!)2(2-1)]

e-(t/τc))(t/τr2)(2-1)e-(t/τr2) dt (28) After integrating eq 28, we obtain the following equation:

Y(3,R1,N ) 2) ) ((ri2τc)/τr2)[4](τr22/4)(τr2 + 4τc)/(τr22 + 4τcτr2 + 4τc2) (29) And after simplification of eq 29, the following expression is obtained:

Y(3,R1,N ) 2) ) ri2τcτr2(τr2 + 4τc)/(τr22 + 4τcτr2 + 4τc2) (30) The value of Y(3,R1,N ) 1) is equal to eq 21. Consequently, we have the following:

The incremental catalyst yield Y(3,R1) can be calculated as follows:

Y(3,R1,N ) 1) ) ri1τcτr1/(τr1 + τc)

Y(3,R1) ) Y(3,R1,N ) 2) - Y(3,R1,N ) 1)

Therefore, the incremental catalyst yield Y(3,R1) in the R3

(27)

(31)


Figure 3. First reactor stagesmodel versus plant-measured polymer yield prediction no. 1 using the five independent variables in Table 3.

Figure 4. First reactor stagesmodel versus plant-measured polymer yield prediction no. 2 using the six independent variables in Table 3.

reactor is obtained by substituting eqs 30 and 31 into eq 27 to give

Finally, plugging eqs 34 and 32 into eq 26, one obtains the catalyst yield for the second stage of reaction:

Y(3,R1) ) ri2τcτr2(τr2 + 4τc)/(τr22 + 4τcτr2 + 4τc2) -

Y(3,m) ) 2X{ri2τcτr2(τr2 + 4τc)/(τr22 + 4τcτr2 + 4τc2) -

ri1τcτr1/(τr1 + τc) (32)

ri1τcτr1/(τr1 + τc)} + (1 - X)ri2τcτr2/(τr2 + τc) (35)

The yield in the second stage of reaction can be derived from eq 26. The yield for fresh catalyst being fed into stage reactor R3 is calculated according to eq 18.

In eq 35, the initial rates of reaction ri2 and ri1 must be calculated using eqs 6-8 and the appropriate set of operating conditions for each stage of reaction. The parameter estimation procedure is identical to that described for the first reactor stage.

Y(3) ) ((ri2τc)/τr2)(1/(1 - 1)!)1(1-1)

∫0∞(1 -

e-(t/τc))(t/τr2)(1-1)e-1(t/τr2) dt (33) After integrating and simplification of eq 33, we obtain the following:

Y(3) ) ri2τcτr2/(τr2 + τc)

(34)

Yield Prediction in the Third Stage of Reaction (Plug-Flow Reactor Section) The polymer leaving the second stage of reaction enters a long plug-flow adiabatic polymerization reactor equivalent to an extremely large number of CSTRs in series. Because

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Figure 5. First reactor stagesmodel versus plant-measured polymer yield prediction no. 3 using the seven independent variables in Table 3.

Figure 6. Second reactor stagesmodel versus plant-measured polymer yield prediction no. 1 using the five independent variables in Table 3.

of this, we can consider that all fluid particles from the point of entrance to the exit of the plug-flow reactor have spent the same amount of time in the reactor and, therefore, have a very uniform RTD. For such a case the yield obtained in the plugflow reactor can be described by an equation similar to eq 10 with the adequate integration limits as shown in eq 36. Therefore, the yield model equation for the plug-flow reactor becomes

Y(p) ) KpCcC) 2M

∫(τt

p r1

+ τr2)

e-(t/τc) dt

(36)

Integrating eq 36 gives -((τr1+τr2)/τc) - τce-(tp/τc)} Y(p) ) KpCcC) 2 M{τce

(37)

After simplifying eq 37, we obtain the following:

Y(p) ) ripτc{e-((τr1+τr2)/τc) - e-(τp/τc)}

(38)

In eq 38, the initial rate of reaction rip must be calculated using eqs 6-8 with the appropriate set of operating conditions for this third stage of reaction (P, T). The parameter estimation procedure is identical to that described for the first reactor stage. In eq 38, τr1 + τr2 is the cumulative residence time up to the outlet of the second stage of polymerization and τp is the total residence time in the reactor train. Analysis of Results To develop the polymer yield predictive model for each one of the reactors stages (first, second, and third) and the overall polymer yield model, each one of the eqs 22, 35, and 38 needed to be solved in conjunction with the set of equations 6-8 to determine the values of the adjustable parameters in the yield model. This process was done by means of analyzing a very large data set that included more than sixty different products made in the plant. These plant data sets were obtained for steadystate operation of the reactor train and included data that covered


Figure 7. Second reactor stagesmodel versus plant-measured polymer yield prediction no. 2 using the six independent variables in Table 3.

Figure 8. Second reactor stagesmodel versus plant-measured polymer yield prediction no. 3 using the seven independent variables in Table 3.

over one year of polymer production taken at about a 2-h time interval. The reactor train operating conditions and range of operation that were collected from a distributed control system are shown in Table 1. The measured polymer yield as shown in Table 1 is calculated in the plant based on the total polymer made in each reactor and knowing the catalyst flow fed to each reactor (R1, R2, and R3). The amount of polymer produced was determined based on an energy balance around each reactor and knowing the heat of polymerization according to the equation

Mp,ri ) mCp(To,ri - Ti,ri)/∆Hp

(39)

Therefore, the measured catalyst yield in each reactor stage would become

Ym,ri ) Mp,ri/Fc,ri

(40)

First Reactor Stage Linear Models. Linear models of the measured yield versus process variables (T, P, residence time, etc.) for the first reactor stage were done as a preliminary statistical test of the plant data set and to determine their respective correlation coefficients. The objective of doing this was to have a preliminary feel for the “goodness of fit” of the yield linear models and then to compare such correlation coefficients to the correlation coefficients obtained for the fundamental nonlinear polymer yield model developed in this work. However, these linear models were not developed for the second or third stages of reaction because the results for the first reaction stages would be representative of the results that would be obtained in the third and second stage of reactions. Table 2 shows the values of the correlation coefficients, regression, and residual sum of squares for different combinations of process variables for the linear models for the first reactor stage. Obviously, the highest correlation coefficient is obtained for two situations: (1) a polymer yield linear model

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Figure 9. Third reactor stagesmodel versus plant-measured polymer yield prediction no. 1 using the five independent variables in Table 3.

Figure 10. Third reactor stagesmodel versus plant-measured polymer yield prediction no. 2 using the six independent variables in Table 3.

in pressure and free ethylene as independent variables and (2) a polymer yield linear model in pressure, temperature, hold-up time, and free ethylene as independent variables. Nevertheless, even though we are dealing with plant data that are prone to having many errors and variability in the process measurements, the highest correlation coefficient R2 ) 0.6 is not that great statistically speaking, and the expectation was that by solving the complex nonlinear polymer yield model explained in the previous sections, the correlation coefficient would be improved to values at least as high as R2 ) 0.8 or better for stage and overall plant yield prediction. First, Second, and Third Reaction Stages via the Nonlinear Kinetic and RTD Yield Model. Among the independent process variables affecting the yield that were identified in eqs 6-8 were the following: reactor temperature, reactor pressure, molar hydrogen-to-ethylene ratio, cocatalyst-to-ethylene molar ratio, free ethylene in the reactor, free octane in the reactor, and polymer melt index. A series of three model predictions

were made for each reactor stage by varying the total number of independent process variables, fitting the adjustable parameters in the yield model, and determining the correlation coefficient for the fit. Table 3 shows the independent variables that were chosen for each model prediction for each reactor stage. The values of the correlation coefficients obtained for the three predictions for the first, second, and third reactor stages are shown in Table 4. It can be seen that, for any reactor stage, as the number of independent variables is increased from the five basic reactor process variables (temperature, pressure, hydrogen to ethylene molar ratio, co-catalyst to ethylene molar ratio, and free ethylene) to seven process variables by including the percent free octene and polymer melt index, the correlation coefficient for the parameter estimation for the yield model fit increases. Table 4 shows that the best correlation for any reactor stage prediction requires the use of the whole set of seven parameters shown in Table 3. It is important to notice that the correlation coefficients obtained via the fundamental yield model


Figure 11. Third reactor stagesmodel versus plant-measured polymer yield prediction no. 3 using the seven independent variables in Table 3.

Figure 12. Overall model versus plant-measured polymer yield prediction no. 1 using the five independent variables in Table 3.

in Table 4 for the first stage of reaction are much better than the correlation coefficients obtained for the linear model for the first stage of reaction shown in Table 2. Model-predicted versus plant-measured polymer yield (using eq 40) for the first, second, and third reaction stages are shown in Figures 3-11. Also, the overall model-predicted yield versus the plant-measured yields is shown in Figures 12-14. For any reactor stage, as the number of independent process variables is increased from five to seven according to Table 3, the visual appearance of the data is improved as it was suggested by the calculated values of the correlation coefficients shown in Table 4 and the values of the percent error in the predictions. The data shown in Figures 12-14 suggests that the best model is indeed the one that used prediction no. 3 in Table 4. Calculated polymer prediction errors and standard deviation of the error predictions for the first, second, third, and overall plant are shown in Tables 5-8, respectively. As can be seen from

the yield model prediction error, on average the predicted yield model does quite well when compared to the plant-observed values. The catalyst time constant and catalyst half-life values calculated through the parameter estimation process for the first, second, and third stages of reaction are shown in Table 9. It can be seen that those parameter values vary because for the first, second, and third reaction stages the pressure and temperature in those reactors are different. Values of the adjustable parameters (Ka, b, c, d, e, f) regressed in eq 7 are not shown in this work because they are relevant only to the actual catalyst used in this particular solution polymerization process. While operating conditions in a chemical process plant vary considerably, we have developed nevertheless in this work a yield prediction model based on fundamentals of chemical engineering that under most circumstances will adequately predict the HDPE plant yield performance. Hence, the polymer

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Figure 13. Overall model versus plant-measured polymer yield prediction no. 2 using the six independent variables in Table 3.

Figure 14. Overall model versus plant-measured polymer yield prediction no. 3 using the seven independent variables in Table 3. Table 9. Catalyst Time Constant and Half-Life in Each Reaction Stage

first stage second stage third stage

prediction no. 1

prediction no. 2

prediction no. 3

time constant (min)

halflife (min)

time constant (min)

halflife (min)

time constant (min)

halflife (min)

0.06 1.89 1.73

0.04 1.31 1.20

0.57 6.26 3.63

0.40 4.34 2.52

1.89 1.90 1.72

1.31 1.32 1.19

yield model will allow for optimization of process conditions to maximize polymer yield for existing and new products to be made in the plant. Conclusions The yield model developed for each one of the reactor stages (first, second, and third) in the reactor train and the overall

polymer yield model produce adequate predictions for engineering design, process simulation purposes, and for initial scouting of process conditions necessary for the manufacture of new products and overall plant optimization. The work presented herein shows that by developing a comprehensive nonlinear kinetic model that includes the reactor RTD and flow characteristics, one can tremendously improve the predicting capabilities of polymer yield when compared to a simplistic linear model or a model that includes only the catalyst deactivation kinetics. Acknowledgment The authors would like to thank Deborah Beran, head of the Equistar Cincinnati research center, for allowing the publication of this work. Thanks are also due to Mark Mack for reviewing the original manuscript.


Nomenclature a ) constant b ) adjustable parameter Bu/C) 2 ) molar co-catalyst-to-ethylene ratio in the reactor c ) adjustable parameter 3 C) 2 ) molar concentration of ethylene, mol/L (Cc)1 ) mathematical function dependent on the initial concentration of active sites in the catalyst as shown in eq 7 for the first stage or reaction (reactors R1,R2) C2f ) free ethylene in the reactor, % C8f ) free octene in the reactor, % Cc ) mathematical function dependent on the initial concentration of active sites in the catalyst as shown in eq 7 (C) 2 )1 ) molar concentration of ethylene in the first stage of reaction (reactors R1,R2) mol/L3 Cp ) specific heat of stream going to the ith reactor, kJ/(kg K) d/dt ) rate of change, 1/h d ) adjustable parameter e ) adjustable parameter Ei(t) ) exit age distribution of particles that have spent a time t in the reactor, 1/h Eθ ) dimensionless exit age distribution that has spent a time t in the reactor Ea ) activation energy, J/(kg-mol) f ) adjustable parameter Fc,ri ) mass flow rate of catalyst to the ith reactor stage, kg/h H2/C) 2 ) molar hydrogen to free ethylene ratio in the reactor k ) catalyst decay constant, 1/h Kp ) propagation rate constant, L/(h kg catalyst) kpo ) base propagation rate constant, L/(h kg catalyst) Ka ) constant m ) mass of stream going to the ith reactor, kg/h MI ) polymer melt index, g/(10 min) M ) ethylene molecular weight, kg/kg-mol Mp,ri ) mass of polymer produced in the ith reactor stage determined by eq 39, kg/h N ) number of continuous stirred tanks in series P ) absolute pressure, N/m2 R ) gas constant, J/(kg-mol K) r ) polymerization rate, kg/(h kg catalyst) ri ) initial rate of polymerization, kg/(h kg catalyst) ri1 ) initial rate of polymerization in reactor R1 or R2, kg/(h kg catalyst) ri2 ) initial rate of polymerization in reactor R3, kg/(h kg catalyst) t ) time, h t1/2 ) catalyst half-life, h To,ri ) outlet temperature from the ith reactor, K Ti,ri ) inlet temperature to the ith reactor, K T ) absolute temperature, K Tr ) reference temperature, K (Tr ) 1.0E35 K which makes the 1/Tr term negligible) τc ) catalyst time constant, h τr ) average reactor residence time, h τr1 ) average reactor residence time in first reactor stage R1 or R2, h

τr2 ) average reactor residence time in second reactor stage R3, h Tin ) inlet reactor temperature, °C ∆Hp ) heat of polymerization, kJ/kg ∆Trise ) temperature rise in the reactor, °C V* ) activation volume, m3/kg-mol X ) fraction of catalyst fed to the first stage of reaction Ym,ri ) measured polymer yield in the ith reactor stage according to eq 37, kg/(kg catalyst) Y(t) ) time function polymer yield, kg/(kg catalyst) Yi(t) ) polymer yield for particles that have spent a time t in the reactor, kg/(kg catalyst) Y(t) ) total polymer yield in the reactor, kg/(kg catalyst) Y(1) ) catalyst yield measured in R1 or R2 reactors, kg/(kg catalyst) Y(3,m) ) catalyst yield measured in reactor R3, kg/(kg catalyst) Y(3,R1) ) Incremental catalyst yield in reactor R3 produced by “old catalyst” from R1 reactor, kg/(kg catalyst) Y(3,R2) ) incremental catalyst yield in reactor R3 produced by “old catalyst” from R2 reactor, kg/(kg catalyst) Y(3) ) catalyst yield in R3 reactor produced by “fresh catalyst” introduced into R3, kg/(kg catalyst) Y(3,R1,N ) 2) ) total yield for R1 catalyst including reactors R1 or R2 and R3, kg/(kg catalyst) Y(3,R1,N ) 1) ) total yield for R1 catalyst including reactors R1 or R2, kg/(kg catalyst) Y(3) ) yield for fresh catalyst being fed into stage reactor R3, kg/(kg catalyst) Y(p) ) catalyst yield measured in plug-flow reactor, kg/(kg catalyst) Literature Cited (1) Chen, J. C. F.; Magovern, R. L.,; Sinclair, K. B. Low Density Polyethylene; Process Economics Program Report # 36B; SRI International: Menlo Park, CA, 1980; p 192. (2) Lundeen, A. J.; Feig, J. E. Process for the Manufacture of Linear Polyethylene Containing R-Alkene Comonomers. U.S. Patent 5,236,998, August 17, 1993. (3) Cribbs, L. V.; Etherton, B. P.; Mack, M. P.; Meas, J. H. Olefin Polymerization Process. U.S. Patent 6,127,484, April 29, 1999. (4) Brockmeier, N. F.; Rogan, J. B. Propylene Polymerization Kinetics in a Semi-Batch Reactor by use of a Supported Catalyst. Ind. Eng. Chem. Prod. Res. DeV. 1985, 24, 278. (5) Buchelli, A.; Caracotsios, M.; Corbin, G. A. Steady State and Grade Transition Characteristics of Amoco’s Gas Phase Reactors for the Manufacturing of Polypropylene Based Impact Copolymer Resins. Presented at Polymer Reaction Engineering Conference, An Industrial/Academic Interchange, February 13-18, 1994, St. Augustine, FL. (6) Buchelli, A.; Call, M. L.; Brown, A. L.; Bokis, C. P.; Ramanathan, S.; Franjione, J. Low-Density Polyethylene Model; final report of Equistar Chemicals LP, October 2001; Chapters 3 and 7 (unpublished results). (7) Caracotsios, M. Theoretical Modeling of Amoco’s Gas Phase Horizontal Stirred Bed Reactor for the Manufacturing of Polypropylene Resins. Chem. Eng. Sci. 1992, 47 (9-11), 2591. (8) Levenspiel, O. Chemical Reaction Engineering, 2nd ed.; John Wiley and Sons: New York, 1972; p 271, p 290.

ReceiVed for reView January 31, 2006 ReVised manuscript receiVed May 19, 2006 Accepted August 11, 2006 IE0601302

Yield Prediction in a Continuous High-Density ... - ACS Publications

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