Industrial Loop Reactor for Catalytic Propylene ... - ACS Publications

To predict the pressure change during polymerization and check the performance of pressure-relief devices in response to emergency accidents, a dynami...
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Ind. Eng. Chem. Res. 2010, 49, 11232–11243

Industrial Loop Reactor for Catalytic Propylene Polymerization: Dynamic Modeling of Emergency Accidents Zheng-Hong Luo,* Pei-Lin Su, and Wei Wu Department of Chemical and Biochemical Engineering, College of Chemistry and Chemical Engineering, Xiamen UniVersity, Xiamen 361005, China

In industrial loop reactors for catalytic propylene polymerization, the reactor pressure can influence the polymerization rate, feed flow rate, and product properties, especially safe operation of the reactor. To predict the pressure change during polymerization and check the performance of pressure-relief devices in response to emergency accidents, a dynamic model was developed based on mass and energy balances as well as thermodynamics and kinetics. In this model, the pressure was related to the temperature by a state equation for high-pressure liquid. The pipeline equations of the reactors described by the conservation of mechanical energy were incorporated into the model to better predict the pressure. In addition to the pressure, the model can also predict other variables, such as temperature, slurry density, and solids holdup in the reactors. Dynamic data from an industrial plant were used for model validation. Furthermore, the application of the model was demonstrated by simulating several typical emergency accidents. 1. Introduction Polypropylene is one of the most widespread polymers and can be produced by various technologies including the Hypol, Unipol, and Spheripol processes, among others.1 The last is certainly the most important at present. The key part of this technology consists of loop reactors and fluidized-bed reactors (FBRs).2 In the loop reactors, polymerization takes place in the liquid phase at a reactor pressure of 3.5-4.5 MPa, and the polymer matrix is produced as a solid suspension in the liquid stream. Propylene polymerization is a highly exothermic reaction, and the pressure of the reaction system is difficult to control.1-5 The reactor pressure can influence the polymerization rate, feed flow rate, and product properties.6-8 Accordingly, the reactor operating conditions are sensitive to fluctuations in the inlet and outlet flows. A fast rise of the reactor temperature and pressure could lead to reactor breakage and consequent release of toxic/flammable fluids.9-11 In reality, despite numerous safety precautions, accidents are typically due to unexpected failures of heat removal, originating from operating errors or equipment failure (namely, emergency accidents). Therefore, calculation of accident relief flow is necessary.10-13 In addition, such calculations are based on the quantitative description of the pressure change in the reactors and could be useful for controlling product properties and achieving safe production in reactors. Moreover, a dynamic model is useful for checking the performance of pressure-relief devices in response to emergency accidents. To develop a dynamic model, especially a dynamic pressure model, the kinetics and mass and energy balance equations have to be solved together with the thermodynamic state equation for the high-pressure liquid in the loop reactors. In addition, as described above, the change of the reactor pressure can influence the inlet and outlet flows of the loop reactors, which directly link to the mass and energy balance and kinetics equations.14,15 Therefore, the inlet- and outlet-pipeline equations should be included in the dynamic model to better predict the reactor pressure. * To whom correspondence should be addressed. E-mail: luozh@ xmu.edu.cn. Tel.: +86-592-2187190. Fax: +86-592-2187231.

To date, most articles published in the field of olefin polymerization have been concerned with the modeling of heat and mass transfer inside the polymer particles and with the reaction mechanism.1-5,14-20 The overall polymerization process combined with the pressure under dynamic conditions in the reactors has usually been neglected. Zacca et al. suggested a population balance approach to the modeling of multistage olefin polymerization using the catalyst residence time as the main coordinate.2 They considered dynamic operation in the loop reactors, but did not predict reactor pressure.2 Meier et al. studied particle mixing and segregation in a small-scale FBR for catalytic propylene polymerization using experimental and simulation methods.21 Temperature and catalyst concentration profiles in the FBR were obtained. However, the state equation for the studied system was not incorporated into their simulation work, and reactor pressure data were still absent from their work.21 Reginato et al. developed a dynamic mathematical model for liquid-phase polymerization in a loop reactor based on a nonideal continuous stirred-tank reactor (CSTR) model capable of dealing with multisite copolymerization of olefins.22 In their work, polymer moment balances were used to compute resin properties, such as average molecular weights, polydispersit, and melt flow index, but not reactor pressure. Gonzalez-Ruiz et al. studied slurry propylene polymerization catalyzed by the isospecific metallocene rac-Et(Ind)2ZrCl2/MAO in a semibatch reactor.23 In their work, a full factorial design with three temperatures (50, 65, and 75 °C) and four monomer partial pressures (1.5, 2.5, 3.2, and 3.8 atm) was performed. A corresponding polymerization kinetic model was also put forward. Although the effect of the pressure on the polymerization kinetics was investigated, the pressure was only the input parameter and was considered as constant when during simulation of the polymerization kinetics.23 Recently, Pinto et al. presented a dynamic distributed mathematical model for industrial liquid-phase loop polypropylene reactors and validated with actual industrial data for the first time.24 The model was able to represent the dynamic trajectories of production rates, the melt flow index, and the xylene solubility values during grade transitions within the experimental accuracy. Although their model was a distributed model and could predict some variable distributions in the loop

10.1021/ie1003784  2010 American Chemical Society Published on Web 10/05/2010

Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010

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Figure 1. Basell Spheripol-II loop process (China Lanzhou Petrochemical Company of China National Petroleum Corporation). A, catalyst; B, propylene; C, hydrogen; D, product; E, coolant; 1, pump P200; 2, pump P201; 3, pump P202; 4, prepolymerization reactor R200; 5, main polymerization reactor R201; 6, main polymerization reactor R202.

reactor, no reactor pressure data were included in their work. In addition, when considering the safety of the reactors and the corresponding calculation of accident relief flow in response to emergency accidents, one needs the concentrated/whole pressure instead of the distributed pressure in the reactor.10-13 In practice, the distributed model is more complicated and requires more input parameters to solve than the concentrated model. Asteasuain and Brandolin simulated a high-pressure ethylene polymerization reactor.25 Their model outputs along the reactor length included the complete molecular weight distribution and branching indexes, as well as monomer conversion, compositions, and reactor temperature and pressure. However, their model was a steady-state model. Based on the above discussion, it becomes clear that the early work/modeling efforts in this field were made to account for detailed aspects of the polymerization kinetics and to predict the polymer properties in reactors. It is also clear that few open reports were focused on a dynamic reactor pressure model along with the incorporation of the reactor pipeline equations into the kinetic and thermodynamic models. Moreover, most of the models proposed were not validated with any actual industrial data. In the present work, based on the mass and energy balances and thermodynamics and kinetics equations, as well as the pipeline equations of the reactors, a comprehensive dynamic model is developed for the prediction of the reactor variables, especially the pressure, in propylene polymerization loop reactors. Particular attention was paid to checking the performance of pressure-relief devices in response to emergency accidents and comparing the predicted results with the actual industrial data. 2. Process for Propylene Polymerization in the Loop Reactors The Spheripol technology is one of the most widespread commercial methods of producing polypropylene. Its core consists of three liquid-phase loop reactors (a prepolymerization reactor and two main polymerization reactors) and a gas-phase FBR. In this work, three industrial-scale loop reactors of the Spheripol technology from a chemical plant in China, shown in Figure 1, were selected as our studied object. Many articles relating to the Spheripol technology have appeared in the literature.2,3,14,16,22,25,26 Comprehensive reviews have also been published.27,28 Some points relating to this work are as follows: First, the loop reactors include a small prepolymerization reactor (R200) and two main polymerization

reactors of the same volume (R201 and R202). In practice, R200, R201, and R202 are sequentially linked together. Second, R200 consists of one continuous tubular reactor to form a closed loop. The two main polymerization reactors consist of two continuous tubular reactors connected in sequence with each other. In practice, the three loop reactors are all closed tubes as a whole, wherein the reacting slurry driven by recycling pumps circulate at high recycle rates, as depicted in Figure 1. Under these conditions, it is reasonable to regard the loop reactors as CSTRs with a constant volume, but with various slurry densities. The reaction slurry is assumed to be a mixture of a liquid phase (monomer and hydrogen) and a solid phase (polymer and catalyst).2,3,14,16,25,26 3. Dynamic Model 3.1. Polymerization Kinetics. Many articles on propylene polymerization kinetics are available.1-5,9,15-19 Zacca et al. developed a comprehensive kinetics of olefin polymerization.29 The comprehensive kinetic scheme consists of site-activation, chain-initiation, chain-propagation, chain-transfer, site-transformation, and site-deactivation reactions. Their kinetic mechanism gave complete elementary reactions by taking all chemical species into account except for the active impurities in the reaction environment. Recently, Luo et al.30 put forward a different kinetic scheme of propylene polymerization by considering the effects of the main active impurities on the propylene polymerization. In ref 30, the kinetic scheme consists of chain-initiation, chain-propagation, chain-transfer, and chaindeactivation reactions. The initiation reactions include reaction steps initiated by propylene, ethylene, acetylene, and propyne. Here, we propose another kinetic scheme. This kinetic scheme consists of chain-initiation, chain-propagation, chain-transfer, and chain-deactivation reactions, and the chain-initiation reaction is initiated only by propylene. In addition, it is well-known that hydrogen is used as the molecular-weight control agent to produce various grades of polypropylene in industry. Accordingly, only chain transfer to hydrogen is considered. We do not intend to review this field here. However, some points must be emphasized. First, the value of the rate constant for each step is independent of the chain length. Second, the value of the chain-initiation rate constant is equal to that of the chain-propagation rate constant. Finally, the chain-transfer reaction is assumed to occur spontaneously or through the reaction of an active chain with hydrogen. Therefore, the reaction mechanism for the polymerization is as shown in Table 1.

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Table 1. Kinetic Mechanism1-5,9,15-19 kiM

C* + M 98 PP* 1

initiation

kp

PP* 1 + M- 98 PP* 2

propagation

l kp

PP* i + M- 98 PP* i+1 ktr

PP* i + H2 98 PPi + C*

transfer reaction transfer to hydrogen deactivation reactions spontaneous deactivation

kd PP* i 98 PPi + Cd

kd

C* 98 Cd

Based on the above discussion and the pseudokinetic rate constant method, the following kinetic equations are employed herein: Propagation rate rp ) kp[M][C*] (1) Because propylene is mainly consumed by the propagation reaction, the polymerization rate is given by eq 1 in this work. Therefore, one can find that the polymerization rate is also described by rp. Transfer rate rtr ) ktr[H2]0.5[C*] (2) where [H2] represents the effective concentration of hydrogen. Some factors, including active-site deactivation and transformation and reactions between impurities and hydrogen, lead to a decrease of the effective concentration of hydrogen. Therefore, the chain-transformation rate follows eq 2 according to refs 16, 22, 27, and 28. Catalyst deactivation rate rd ) kd[C*] (3) The temperature dependence of above rate constants can be represented with the Arrhenius equation, as shown in eq 4. k ) k0e-E/RT

In the present study, attention is also paid to checking the performance of pressure-relief devices in response to emergency accidents. Accordingly, relief valves are included in the reactor model. In addition, the most important principle in establishing the reactor model is to represent the actual reactor as accurately as possible under the conditions of necessary assumptions and simplifications, which, as a result, also helps the simulation to converge efficiently and steadily to a reliable result. Therefore, the following assumptions were made based on the above descriptions and refs 2-5, 14-16, 22, and 25-31: (1) The loop reactors are considered as CSTRs. This implies that all of the state variables (e.g., temperature, pressure, slurry density) in the reactors are uniform and change only with the polymerization time. Furthermore, the axial flow pumps in the reactors can be handled as stirrers. Moreover, because, as described above, the work aims at directing the calculation of accident relief flow in response to emergency accidents, the whole pressure must be obtained, and corresponding CSTRs are applied. (2) The catalyst particles in the loop reactors always keep flowing because of the high recycle rates. (3) The effect of the reactor pressure on the polypropylene density is not considered. (4) Effects of the diffusion of the reactants between different phases are ignored. This means that the reaction system for every reactor is considered as a pseudohomogeneous phase. Therefore, according to the reaction steps shown in Table 1, the mass and energy balance equations can be established. For propylene in the ith loop reactor, the mass balance equation can be derived as Vi

(5)

For the catalyst particles in the ith loop reactor, a slightly different equation is used Vi

d[C*]i in out ) Qin + Qrelief )[C*]i - rd,iVi i [C*]i - (Qi i dt

(6) Similar equations can be derived for hydrogen and the solid polymer matrix Vi

d[H2]i ) GHin2,i /MH2 - (Qout + Qrelief )[H2]i - rtr,iVi i i dt

(7)

(4)

3.2. Reactor Model. To establish the proper reactor model, it is necessary to analyze the fluid flow in the loop reactors. During liquid-phase propylene polymerization in the loop reactors, small catalyst particles are continuously fed into R200. In the mean time, small quantities of propylene and hydrogen are also fed into R200. Accordingly, the polymerization reaction is initiated, and the catalyst particles are well filmed to prevent the polymer fragments from separating from the catalyst particles. Subsequently, the small catalyst particles coated with the thin polymer film in R200 are transported into R201 and react with the incoming monomers and hydrogen. In succession, the slurry of unreacted reactants and polymer in R201 is transported into R202, where they react with each other. The direct description of the fluid flow in the three loop reactors is shown in Figure 1.

d[M]i in out ) Qin + Qrelief )[M]i - rp,iVi i [M]i - (Qi i dt

Vi

dwi out ) Gin + Qrelief )wi + rp,iViMp i wi - (Qi i dt

(8)

For the ith loop reactor, the whole mass balance equation can be obtained as dmi out ) Gin + Qrelief ) i - (Qi i dt

(9)

According to the energy balance, the following equation can be obtained ViFiCp,i

dTi in in out ) Gin + Grelief )Cp,iTout + i Cp,iTi - (Gi i i dt (-∆Hr,i)rp,iVi - KAi(Ti - Tj,i) (10)

In eqs 5-10, i is the reactor number. Specifically, i ) 1, 2, and 3 correspond to R200, R201, and R202, respectively.

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In addition, each reactor model consists of eqs 5-10. Therefore, there are 18 equations to be used to constitute the reactor model because of the presence of three loop reactors in this work. 3.3. Thermodynamic State Equation for High-Pressure Liquid. As described above, the reactor pressure model is a necessary part of this work. It can be obtained by incorporating the thermodynamic state equation into the above reactor model. In practice, the reacting slurry is assumed to be a mixture of a liquid phase (monomer and hydrogen) and a solid phase (polymer and catalyst) at a pressure of 3.5-4.5 MPa.2,26-29 In addition, because of the very low concentrations of hydrogen and catalyst in the reactors, the reactor volumes are considered to be described in terms of the total volumes of the monomer and polymer. Furthermore, the effect of the pressure on the solid polymer density is not considered. However, the effect of the pressure on the liquid monomer density is obvious. Therefore, the reactor pressure can be described in terms of the pressure of the liquid propylene. Indeed, with the accumulation of liquid propylene in reactors with constant volumes, the increase of the propylene density is the essential reason for the pressure rise. Based on the above description, for the ith loop reactor, the monomer density can be expressed as FM,i )

mi(1 - wi) Vi - miwi /Fpp

(11)

For liquid propylene at a pressure of 3.5-4.5 MPa, an equation of state (EOS), namely, the Tait EOS, which is one of the most excellent state equations for high-pressure liquids,32-34 was applied, as shown in eq 12 Vi V0i

) 1 - D ln

Pi + EE P0 + EE

(12)

The pressure equation can be obtained by derivation of eq 12 and is shown in eq 13 Pr,i ) exp

[ ( )]

νi 1 1 - 0 (Pr0 + EE) - EE D ν

(13)

where Pi Pc

(14)

1 FM,i

(15)

Pr,i ) νi )

D ) exp(-2.5355 + 1.1690ξ + 0.1458Tbr) EE )

(

)

ν0 )

Table 2. Input Parameters in Eqs 11-24

0

P0 Pc

(19)

The saturation pressure (P ) can be obtained based on an improved Antoine equation, given by33-36

Pc (Pa) Tb (K) Tc (K) β ζ

4.665 × 106 225.3 365.6 0.14 0.0345

B T+C

)

(20)

where A ) ln 101325 + (B/Tc)(Tbr + C/Tc)

(21)

B ) Tc[(1 + C/Tc)(Tbr + C/Tc) ln(Pc /101325)]/(1 - Tbr) (22) C ) Tc(0.7F - 0.3Tbr)/(0.3 - F)

(23)

F ) [(1 - Tbr)(1 + β) ln 10]/[ln(Pc /101325)]

(24)

In eqs 12-24, Tr ) T/Tc, Tbr ) Tb/Tc. Equations 11-24 constitute the pressure model. In addition, these equations can determine the computational precision of the pressure, and therefore, the above model was first verified. Here, the state data obtained from NIST Chemistry WebBook37 and perturbed-chain statistical associating fluid theory (PC-SAFT)34,35,38,39 were used to test the Tait EOS. In addition, a set of reference values of the input parameters in eqs 11-24 were selected and are listed in Table 2. The results are reported in Appendix A. According to Appendix A, one can be satisfied that the Tait EOS can be used to predict the pressure data of the high-pressure liquid propylene. 3.4. Pipeline Model. Because the pressure is coupled to the inlet and outlet flows, it is necessary to establish the mathematical relations between the reactor pressure and the inlet and outlet flows. Here, the pipeline equations (inlet and outlet equations for three loop reactors) are introduced. According to Figure 1 and taking the first loop reactor (R200) as an example, we present the inlet-pipeline chart of R200 in Figure 2. For the pipeline from the plane of 1-1′ (1-1′) to the plane of 2-2′ (2-2′), taking 1-1′ as the reference plane, the conservation equation of mechanical energy can be obtained as40 P1-1′,1(i) ub,1-1′,1(i)2 + + We,1(i) ) gz2-2′,1(i) + + 2 F 2 P2-2′,1(i) ub,2-2′,1(i) + + hf,1(i) (25) 2 F

∑

where Z1-1′,1 ) Z2-2′,1 because of the nearly identical heights of 1-1′ and 2-2′ for R200, We ) 0 because of the absence of the transportation pump for the inlet-pipeline of R200, and P2-2′,1 is the pressure of R200

RTc exp{-(1.2310 + 0.8777ξ)[1 + (1 - Tr)2/7]} Pc (18) Pr0 )

value

(

(16)

(17)

parameter

P0 ) exp A -

gz1-1′,1(i)

45.1906 45.8849 - 1 exp(2.6583ξ + 3.7148 ln Tbr) Tr Tr2

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32-38

∑

(

hf,1(i) ) λ1(i)

L1(i) + d1(i)

∑

)

ζ

ub,2-2′,1(i)2 2

(26)

at 3 × 103 e λ1(i) e 3 × 106,

λ1(i) ) 0.0056 +

0.5 Re1(i)0.32 (27)

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Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010 Table 3. Kinetic Constants kinetic constant

computational equation, k ) k0 exp(-E/RT)

kiM

kiM ) 4.97 × 107 exp(-51900/RT)

kp

kp ) 4.97 × 107 exp(-51900/RT)

ktr

ktr ) 4.4 × 103 exp(-51900/RT)

kd

kd ) 7.92 × 103 exp(-51900/RT)

Figure 2. Inlet pipeline of R200 (A, storage tank D201; B, pumpP200; C, prepolymerization reactor R200).

where

Table 4. Plant Parametersa

Rei )

Fiub,2-2′,idi µ

(28)

By iteratively solving eqs 25-28, we can obtain ub for R200. In addition, ub can also calculated according to the equations Qin i

1 1 1 ) πdi2ub ) πd1-1′,i2ub,1-1′,i ) πd2-2′,i2ub,2-2′,i 4 4 4

(29) in in Gin i ) Fi Qi

(30)

In addition, unless otherwise noted, Fi ) FM,i and µ ) 5.54 × 10-5 Pa · s in eqs 25-30.41 Therefore, a quantitative relation between the pressure in R200 and the feed flow rate is obtained. Similar inlet-pipeline equations and corresponding relations can also be obtained for R201 and R202. In this article, we provide some simulated results in Appendix B, which confirms that the pressure is coupled to the flow. In addition to the above inlet equations, the outlet-pipeline equations for the loop reactors, which are similar to the inlet-pipeline equations, were obtained based on the conservation equation of mechanical energy. Therefore, the outlet-pipeline equations are not listed here because of limited space. 3.5. Pressure-Relief Model at Emergency Accidents. It is well-known that pressure-relief equipment must be installed on high-pressure reactors to protect the reactors and prevent safety accidents. Here, relief valves were selected as the relief equipment, and a relief model for emergency accidents using the relief valves is suggested. Furthermore, for liquid propylene polymerization, the reactors are full of slurries. During emergency accidents, the liquid propylene can be quickly gasified at the vents of the relief valves because of the high difference of pressures between the inside and outside reactors. Therefore, the leaking fluids can be considered as the gaseous propylene. Based on the above description and refs 9, 10, 42, and 43, at Pi g Pv, namely, when the relief valves are in unlocking status, eqs 31-33 can be used to constitute the pressure-relief model at

Patm 2 < Pi rr + 1

(

)

rr/rr-1

,

P F rr( rr +2 1)

Qrelief ) CCdSv i

i i

1+rr/1-rr

(31)

Vi (m3)

d1-1′,ib (m)

d2-2′,i (di)c (m)

Li (m)

R200 R201 R202

1 66 66

0.0833 0.1667 0.1667

0.1683 0.6096 0.6096

100 50 50

9.15 62.86 66.10

a In all cases, heat-transfer coefficient (K) ) 1600 W/(m2 · K). b d1-1′,i represents the diameter at 1-1′ for the ith reactor, that is, the outlet diameter of the pump corresponding to the ith reactor. c d2-2′,i represents the diameter at 2-2′ for the ith reactor, that is, the diameter of the ith reactor.

Patm 2 > Pi rr + 1

(

rr/rr-1

)

,

 [( ) ( ) ]

Qrelief ) CCdSv i

2PiFirr Patm rr - 1 Pi

2/rr

-

Patm Pi

(rr+1)/rr

(33)

Otherwise, namely, at Pi < Pv Qrelief )0 i

(34)

4. Model Implementation and Estimation of Kinetic Constants and Plant Data Equations 1-33 include a set of coupled nonlinear ordinary and algebraic equations for the dynamic polymerization. The ordinary differential equations were solved as an ODE23S function provided in Matlab 6.5 software. To solve the suggested model, the kinetic constants of the model must be estimated. The kinetic constants were estimated from the plant data. Our former model of the molecular weight distribution has already been thoroughly validated against data.16 The method of estimating the kinetic constants was also employed in our previous work.16 In addition, as described below, the catalyst used in the present work was the same as that reported in our previous work.16 Here, the same estimation method and similar values for the kinetic constants are applied. In addition, some thermodynamic property parameters and plant data related to the model must also be obtained in advance.44-46 The estimation of the main thermodynamic property parameters is included in Appendix C. The kinetic constants ultimately used are listed in Table 3, and the plant data are listed in Tables 4 and 5. Furthermore, the model solution procedure is illustrated in Figure 3. The plant process of the propylene polymerization is a classical slurry polymerization process using a CSTR in the presence of a fourth-generation Ziegler-Natta catalyst. 5. Results and Discussion

where rr ) Cp /Cv

reactor

jacket water temperature (Tj,i) (°C)

(32)

5.1. Comparison between Industrial Data and Simulated Data. The thermodynamic properties were first determined according to Tables 4 and 5 and Appendix C. By substituting

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a

Table 5. Plant Parameters

Gb,d (kg/h) reactor

T (°C)

R200 R201 R202

20 70 70

P1-1′,ic,d (Pa) 5.35 × 106 5.28 × 106 4.85 × 106

propylene

catalyst

H2

2105 2.418 × 104 1.037 × 104

0.551 -

1 0.51 0.51

a In all cases, sample grade ) T30S. b G represents the mass flow. P1-1′,i represents the pressure at 1-1′ for the ith reactor, that is, the outlet pressure of the pump corresponding to the ith reactor. d P1-1′,i and G can be directly presented from the control computer at the plant site. c

the thermodynamic properties and the kinetic constants (Table 3) for the related terms in eqs 1-33, the simulated results were obtained. Meanwhile, data were also collected from an industrial polypropylene plant. Here, we describe the simulation of the startup process of T30S. Figures 4-6 present comparisons between the plant and simulated data for the startup processes of T30S in different reactors, which show a good agreement between the plant data and the simulated results. Furthermore, Figures 4 and 5 show that the temperatures and pressures in the reactors both increase to maxima quickly and decrease to constant values thereafter. However, as shown in Figure 6, the slurry density in the reactors

Figure 3. Flowsheet of dynamic model solution.

Figure 4. Comparison of the temperature between the simulated data and the plant data in the startup process of T30S.

increases monotonically to a constant value. The simulated results confirm that the temperatures, pressures, and slurry densities in the reactors all change during the startup process. In practice, these changes are manifested in the propylene polymerization characteristics. As the polymerization process continues, the operating parameters, including the propylene

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Figure 5. Comparison of the pressure between the simulated data and the plant data in the startup process of T30S.

Figure 7. Simulated responses of state variables to the failure of heat removal in R202.

Figure 6. Comparison of the slurry density between the simulated data and the plant data in the startup process of T30S.

flow, the catalyst flow, and the hydrogen flow, always change before reaching a constant level, whereas the propylene conversion increases to a constant level, which means that the polymer concentration increases monotonically to a constant value as the polymerization proceeds during the startup period. Accordingly, the slurry density increases to a constant value. Moreover, propylene polymerization is a highly exothermic reaction, and it can lead to the increase of temperature and pressure to high values in the beginning of the startup process because the heat of polymerization cannot be efficiently removed through the heat-exchange cells installed on the reactors. As described above, as polymerization proceeds, the operating parameters always change to a constant level, and the heat exchange also reaches a steady state. Therefore, the temperature and pressure change to constant values. Based on the above discussion, it is clear that the end point of the startup process is a steady state. In addition, according to Figures 4-6, it is clear that the dynamic changes in the three reactors are similar and that the amount of change in R200 is least. Therefore, in the next section, only the dynamic changes in R202 are simulated and discussed. 5.2. Results and Discussion at Emergency Accidents. Several typical emergency accidents were simulated using the above models. In addition, to simplify the solution, the following ) 0. assumptions are made in this section: Pi < Pv, and Qrelief i 5.2.1. Failure of Heat Removal. Propylene polymerization is a highly exothermic reaction. If the huge amount of reaction heat cannot be removed in a timely manner during the polymerization, the reaction will be out of control. When the cyclic cooling water decreases, the temperature control system

is broken, or some other accident occurs, the temperature of the cooling water will increase as a result. Now, the system has been in the steady-state performance for a while, and this makes the temperature of the cooling water in the reactor R202 increase by 1 °C. The response of the system was simulated and is shown in Figure 7. According to Figure 7, one can know that, with an increase of the temperature of the cooling system, the reaction temperature and pressure rise very quickly, the reaction itself is accelerated, and the solids holdup of the slurry is increased in R202. 5.2.2. Fluctuation of Feed Propylene. Figure 8 shows the dynamic responses of the solids holdup of the slurry, the slurry density, the temperature, and the pressure in R202 during a step change in the propylene feed flow rate. According to Figure 8, the temperature and solids holdup both decrease when the propylene feed flow rate increases by 5% at 5000 s, mainly because the increase of the propylene feed flow rate results in a lower conversion rate and less exothermic heat. Because the capability of heat removal is the same, the reaction temperature decreases. In addition, the resistance term of the outlet-pipeline equation of R202 remains unchanged; that is, the outlet valve opening is the same. The temperature in the reactor will go up by more output and the same feed. As a result, the slurry density also increases. 5.2.3. Fluctuation of Feed Catalyst. Figure 9 shows the dynamic responses of the slurry solids holdup, slurry density, temperature, and pressure in R202 during a step change in the catalyst feed flow rate. When the catalyst feed flow rate increases, the catalyst concentration becomes larger, thus increasing the rate of propylene polymerization. On one hand, the higher rate results in larger consumption of propylene and reduces the concentration and accumulation of propylene. On the other hand, the resulting volume of polypropylene particles is smaller, and the volume of R202 is constant, which is equivalent to a larger size of propylene. Therefore, the slurry

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Figure 8. Simulated responses of state variables to the fluctuation of feed propylene in R202.

Figure 9. Simulated responses of state variables to the fluctuation of feed catalyst in R202.

density decreases, and so does the reaction pressure. At the same time, with the jacket’s same capabilities of heat removal, the larger amount of heat released by the accelerated reaction leads to an increase of the reaction temperature.

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Figure 10. Simulated responses of state variables to a change in the heattransfer coefficient in R202.

5.2.4. Change of Heat-Transfer Coefficient. Scale forms on the inside wall of a loop reactor after long-time use. Therefore, the heat-transfer coefficient of the reactor wall will become smaller. Here, we simulated this change and show the simulated results in Figure 10. The accumulated heat in R202 increases because of the smaller heat-transfer coefficient of the wall, and hence, the reactor temperature rises. Accordingly, this influences the rate of every elementary reaction, and the polymerization rate of propylene increases. This process easily leads to the emergence of temperature hot spots in the material in the reactor. To avoid obtaining unqualified products, the emergence of temperature hot spots should be curbed. 5.3. Results and Discussion of Pressure-Relief Model. The unsealing pressure of the relief valve is greater than the operating pressure and less than the design pressure (usually 1.05-1.1 times the operating pressure). As described above, the operating pressure of R202 is generally 4.184 MPa, and we set the unsealing pressure of the relief valve as 4.5 MPa. In addition, a relief valve with a discharging area of 0.002 m2 was chosen.9,10,42,43 We simulated the working conditions by increasing the temperature of the cooling water of the jacket of R202 by 3 °C. As shown in Figure 11, the corresponding simulated results show that all of the responses are mainly in the period of 4500-6000 s. After the step change of the cooling water temperature, the reaction pressure climbs very rapidly and exceeds 4.5 MPa, which results in the opening of the relief valve and the outlet of raw materials. However, the pressure does not decrease immediately because of the delay of the response. The pressure increases to a maximum and drops quickly thereafter. Once the pressure is lower than the unsealing pressure, the relief valve will be closed, and the pressure will go up again until it reopens the relief valve. This cycle fluctuates for a period of time. During the process, other variables, such as the temperature and the solids holdup of the slurry in R202, also fluctuate to some degree.

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Figure 11. Simulated responses of state variables to the failure of heat removal with the unsealing of the relief valve in R202.

between the maximum pressure and the pressure-relief area under other similar working conditions. This is helpful for the design of pressure-relief equipment or regulation and control of practical production. 6. Conclusions

Figure 12. Relationship between the maximum pressure and the discharging area of the relief valve in R202.

It is also necessary to investigate the relationship between the pressure maximum and the design pressure that the equipment can endure in the fluctuations. If the former is greater than the latter, the reactor will be out of control, and there will be leakage, explosion, and some other serious outcomes. Therefore, we must choose a relief valve with sufficient pressure-relief capabilities to ensure that the maximum pressure is less than the design pressure. Figure 12 shows the relationship between the maximum pressure and the discharging area in R202 when the temperature of the cooling water increases by 3 °C. Using the same model, we can calculate the relationship

A comprehensive dynamic model has been developed for the prediction of the reactor variables, especially the pressure, in propylene polymerization loop reactors. Based on the mass and energy balances and the thermodynamics, kinetics, and pipeline equations of the reactors, the model takes into account the flow type, flow rate, and polymerization kinetics as well as the liquid thermodynamics. Dynamic data from an industrial polypropylene plant were used for model validation. Furthermore, the application of the model was demonstrated by simulating several typical emergency accidents. Further studies in this field are in progress in our group. The following conclusions can be drawn based on the simulation results: (1) The predicted temperature, pressure, and slurry density data were found to agree with the plant data. (2) Upon failure of heat removal, the reaction temperature and pressure rise very quickly with the increase of the temperature of the cooling system. Upon fluctuation of the feed propylene, the temperature and solids holdup both decrease when a step change increases the propylene feed flow rate by 5% at 5000 s. Accordingly, the temperature in the reactor goes up by more output and the same feed. In addition, with an increase of the catalyst feed flow rate, the slurry density and the pressure both decrease, and the temperature increases. A

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change in the heat-transfer coefficient can lead to the emergence of temperature hot spots in the material in the reactor. (3) The pressure-relief model is helpful for the design of pressure-relief equipment or regulation and control of practical production. Nomenclature A ) well-defined parameter according to eq 21 Ai ) heat-exchange area, m2 B ) well-defined parameter according to eq 22 C ) well-defined parameter according to eq 23 C* ) activated catalyst site Cd ) deactivated catalyst site Cp ) heat capacity, kJ/kg Cv ) heat capacity, kJ/m3 CCd ) discharging coefficient of relief value [C*] ) activated catalyst site concentration, mol/L di ) pipeline diameter for the ith reactor, m D ) regression constant E ) activation energy, J/mol EE ) regression constant F ) well-defined parameter according to eq 24 g ) gravitational constant, 9.8 m/s2 GHin2,i ) influent mass flow of hydrogen for the ith reactor, g/s Giin ) influent mass flow for the ith reactor, g/s Giout ) effuent mass flow of the ith reactor, g/s Girelief ) relief mass flow of the ith reactor, g/s hf,i ) resistance value for the ith reactor, m2/s2 [H2] ) hydrogen concentration, mol/L k ) reaction rate constants corresponding to eqs 1-3 kd ) chain-termination rate constant, s-1 kiM ) monomer-initiation rate constant, L/(mol · s) kp ) chain-propagation rate constant, L/(mol · s) ktr ) rate constant of chain transfer to hydrogen, L/(mol · s) k0 ) pre-exponential factors corresponding to eqs 1-3 K ) heat-exchange coefficient, J/(m2 · T) Li ) loop length of the ith reactor, m M ) propylene MH2 ) molecular weight of hydrogen, g/mol Mp ) molecular weight of propylene, g/mol [M] ) monomer concentration, mol/L Patm ) atmospheric pressure, Pa Pc ) critical pressure, Pa Pi ) pressure in the ith reactor, Pa Pr,i ) reduced pressure of the ith reactor Pr0 ) saturation reduced pressure Pv ) unsealing pressure of relief valve, Pa P0 ) saturation pressure, Pa P1-1′,i ) pressure at 1-1′ for the ith reactor, Pa P2-2′,i ) pressure at 2-2′ for the ith reactor (that is, ith reactor pressure), Pa PPi (i ) 1, 2, 3, ...) ) polymer chain containing i segments PP*i (i ) 1, 2, 3, ...) ) active polymer chain containing i segments Qiin ) influent volume flow for the ith reactor, L/s Qiout ) effuent volume flow for the ith reactor, L/s Qirelief ) relief volume flow for the ith reactor, L/s R ) gas constant, 8.314 J/(mol · K) rd ) catalyst deactivation rate, mol/(L · s) rp ) polymerization rate, mol/(L · s) rp,i ) polymerization rate for the ith reactor, mol/(L · s) rtr ) chain-transfer rate, mol/(L · s) Rei ) Reynolds number for the ith reactor rr ) reduced heat capacity, m3/kg Sv ) discharging area of relief value, m2

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t ) time, s T ) temperature, K Tb ) boiling point, K Tbr ) reduced temperature factor, Tb/Tc Tc ) critical temperature, K Tr ) reduced temperature, K ub,1-1′,i ) flow rate at 1-1′ for the ith reactor, m/s ub,2-2′,i ) flow rate at 2-2′ for the ith reactor, m/s Vi ) volume of the ith reactor, L Vi0 ) saturation volume of the ith reactor, L wi ) mass fraction of solid-phase component in slurry in the ith reactor We,i ) potential energy supplied by the pump for the ith reactor, m2/s2 Z1-1′,i ) vertical height from 1-1′ to 1-1′ for the ith reactor, m Z2-2′,i ) vertical height from 2-2′ to 1-1′ for the ith reactor, m β ) eccentricity factor ∆Hr,i ) reaction exothermic heat in the ith reactor, J ζ ) configuration factor ζ ) resistance factor for the ith reactor λi ) resistance factor for the ith reactor defined by eq 27 µ ) fluid viscosity, Pa · s νi ) reciprocal of FM,i, L/g ν0 ) well-defined parameter according to eq 18 Fi ) slurry density of the ith reactor, g/L FM,i ) propylene density of the ith reactor, g/L FPP ) polypropylene density, g/L

Appendix A The state data obtained from NIST Chemistry WebBook37 and from perturbed-chain statistical associating fluid theory (PCSAFT)34,35,38,39 were used to test the Tait EOS. In addition, the polymerization system studied in this work was at about 4 MPa and 70 °C. Accordingly, the state data at 4 MPa and 70 °C were obtained and are presented in Figures 13 and 14, respectively. Figures 13 and 14 illustrate the comparisons among the state data obtained by the above methods at 4 MPa and 70 °C. The data obtained using the Tait EOS match well those obtained from PC-SAFT and NIST Chemistry WebBook, which shows that the Tait EOS can be used to predict the pressure data of the high-pressure liquid propylene. Appendix B By solving eqs 25-30, we obtain the relationship between the feed flow rate and the reactor pressure. The simulated result is

Figure 13. Comparisons of liquid propylene density data obtained by three methods at 4 MPa.

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Figure 16. Calculated path of enthalpy difference of a real fluid at different states.

Figure 17. Calculated path of enthalpy difference of propylene in this work. Figure 14. Comparisons of liquid propylene density data obtained by three methods at 70 °C.

effects of temperature and pressure on CpM. Therefore, Cpp ) 85 J/(mol · K).37 Estimation of ∆Hr,i. It is well-known that temperature and pressure can influence ∆Hr,i. Therefore, in a dynamic simulation, ∆Hr,i is not constant. The path for calculating the enthalpy difference of a real fluid at different states is given in Figure 16. According to Figure 16, the enthalpy difference from the state of T1 and P1 to the state of T2 and P2 can be described by eqs A3-A5 ∆H ) ∆H1 + ∆H2 + ∆H3 + ∆H4

(A3)

∆H3 ) 0

(A4)

∫

(A5)

where

∆H2 ) Figure 15. Relationship between the feed flow rate and the reactor pressure in three reactors.

shown in Figure 15, which confirms that the reactor pressure is coupled to the feed flow under dynamic operating conditions. Analogously, the reactor pressure is also coupled to the outlet flow. Therefore, it is necessary to establish the equations of the relationships between the reactor pressure and the inlet and outlet flows, namely, pipeline equations.

Cp,i ) (1 - wi)CpM + wiCpp

CpM ) R(A + BT + CT ) 2

0 ∆HM,T ) ∆HM + ∆HM,298 + ∆Hpp

(A6)

0 ∆HM,298 ) -86.35 kJ/mol

(A7)

∫

(A8)

where45,46

(A1)

where wi is the mass fraction of solid-phase component in slurry in the ith reactor, CpM is the heat capacity of liquid propylene, and Cpp is the heat capacity of polypropylene. CpM can be obtained according to the equation

(A2)

where A, B, and C can be obtained by data regression based on the database in the NIST Chemistry WebBook;37 their values were found to be 185.6343, -1.1669, and 0.0020, respectively. In addition, we assume that CpM is constant because of the weak

Cp dT

∆H1 and ∆H4 are isothermal enthalpy changes and can be calculated using the Lee-Kesler equation.44,45 Furthermore, in the present study, the enthalpy difference is that of propylene. The corresponding path for calculating the enthalpy difference of propylene is described in Figure 17.45 Based on eqs A3-A5 and Figure 17, we can obtain the calculated enthalpy difference of propylene (∆HM,T), namely, ∆Hr,i, as

Appendix C Here, we explain the estimations of Cp and ∆Hr,i. Estimation of Cp. For the ith reactor, Cp,i can be described by the equation

T2

T1

∆Hpp )

T

298

Cpp dT

In addition, ∆HM can be obtained according to eqs A3-A5. Acknowledgment The authors gratefully acknowledge China National Petroleum Corporation, China Lanzhou Petrochemical Company, Fujian Petrochemical Company of SINOPEC, and National Natural Science Foundation of China (No. 21076171) for supporting this work. We also thank Dr. Yao Z. (Department of Chemical Engineering and Biochemical Engineering, Zhejiang University)

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for valuable discussions in this work. The authors also thank the anonymous reviewers for comments on the original manuscript. Literature Cited (1) Luo, Z. H.; Su, P. L.; Shi, D. P.; Zheng, Z. W. Steady-state and dynamic modeling of commercial bulk polypropylene process of Hypol technology. Chem. Eng. J. 2009, 149, 370–382. (2) Zacca, J. J.; Debling, J. A.; Ray, W. H. Reactor residence time distribution effects on the multistage polymerization of olefinssI. Basic principles and illustrative examples, polypropylene. Chem. Eng. Sci. 1996, 51, 4859–4886. (3) Luo, Z. H.; Wen, S. H.; Zheng, Z. W. Modeling the effect of polymerization rate on the intraparticle mass and heat transfer during propylene polymerization in a loop reactor. J. Chem. Eng. Jpn. 2009, 42, 576–580. (4) Hatzantonis, H.; Goulas, A.; Kiparissides, C. A comprehensive model for the prediction of particle-size distribution in catalyzed olefin polymerization fluidized-bed reactors. Chem. Eng. Sci. 1998, 53, 3251–3267. (5) Yiannoulakis, H.; Yiagopoulos, A.; Kiparissides, C. Recent developments in the particle size distribution modeling of fluidized-bed olefin polymerization reactors. Chem. Eng. Sci. 2001, 56, 917–925. (6) Heuvelsland, A.; Wichmann, S.; Schellenberg, J. Investigations of the initial state polymerization of propylene with Ziegler-Natta catalysts in slurry. J. Appl. Polym. Sci. 2007, 106, 354–359. (7) Mirzaei, A.; Vakili, M.; Tafi, N. Prepolymerization of ethylene with a Ziegler-Natta catalyst. J. Appl. Polym. Sci. 2007, 105, 2703–2711. (8) Ahmed, T. S.; DeSimone, J. M.; Roberts, G. W. Continuous copolymerization of vinylidene fluoride with hexafluoropropylene in surpercirtical carbon dioxide: High-hexafluoropropylene-content amorphous copolymers. Macromolecules 2008, 41, 3086–3097. (9) Luft, G.; Broedermann, J.; Scheele, T. Pressure relief of high pressure devices. Chem. Eng. Technol. 2007, 30, 695–701. (10) Pellegrini, L.; Biardi, G.; Caldi, M. L.; Monteleone, M. R. Checking safety relief valve design by dynamic simulation. Ind. Eng. Chem. Res. 1997, 36, 3075–3080. (11) Fisher, H. G.; Melhem, G. A.; Levin, M. E.; Leung, J. C. 1,3-Butadiene: Thermal dimerization/trimerization, free-radical polymerization, and polymer decomposition reactions. In International Symposium on Runaway Reactions, Pressure Relief Design, and Effluent Handling; Melhem, G. A., Fisher, H. G., Eds.; American Institute of Chemical Engineers (AIChE): New York, 1998; pp 39-44. (12) Duh, Y. S.; Hu, K. H.; Chang, J. C.; Kao, C. S. Visualization of emergency viscous two-phase venting behaviors. J. Loss PreVent Proc. 2009, 22, 145–152. (13) Bell, K.; Morris, S. D. Flow visualization during top venting from a small vessel. J. Loss PreVent Proc. 1992, 5, 160–164. (14) Soares, J. B. P.; Hamielec, A. E. Copolymerization of olefin in a series of continuous stirred-tank slurry reactors using heterogeneous Ziegler-Natta and metallocene catalysts. I. General dynamic mathematical model. Polym. React. Eng. 1996, 4, 153–162. (15) Soares, J. B. P.; Hamielec, A. E. General Dynamic Mathematical Modeling of Heterogeneous Ziegler-Natta and Metallocene Catalyzed Copolymerisation with Multiple Site Types and Mass and Heat Transfer Resistance. Polym. React. Eng. 1995, 3, 261–324. (16) Luo, Z. H.; Zheng, Y.; Cao, Z. K.; Wen, S. H. Mathematical modeling of the molecular weight distribution of polypropylene produced in a loop reactor. Polym. Eng. Sci. 2007, 47, 1643–1649. (17) Khang, D. Y.; Lee, H. H. Particle size distribution in fluidized beds for catalytic polymerization. Chem. Eng. Sci. 1997, 52, 421–431. (18) Harshe, Y. M.; Utikar, R. P.; Ranade, W. A computational model for predicting particle size distribution and performance of fluidized bed polypropylene reactor. Chem. Eng. Sci. 2004, 59, 5145–5156. (19) Mckenna, T. F.; Dupuy, J.; Spitz, R. Modelling of transfer phenomena on heterogeneous Ziegler catalysts: Part 3. Modelling of intraparticle mass transfer resistance. J. Appl. Polym. Sci. 1997, 63, 315–322. (20) Veera, U. P. Mass transport models for a single particle in gasphase propylene polymerization. Chem. Eng. Sci. 2003, 58, 1765–1775. (21) Meier, G. B.; Weickert, G.; van Swaaij, W. P. M. FBR for Catalytic Propylene Polymerization: Controlled Mixing and Reactor Modeling. AIChE J. 2002, 48, 1268–1283.

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(22) Reginato, A. S.; Zacca, J. J.; Secchi, A. R. Modeling and simulation of propylene polymerization in nonideal loop reactors. AIChE J. 2003, 49, 2642–2654. (23) Gonzalez-Ruiz, R. A.; Quevedo-Sanchez, B.; Laurence, R. L.; Henson, M. A. Kinetic modeling of slurry propylene polymerization using rac-Et(lnd)2ZrCl2/MAO. AIChE J. 2006, 52, 1824–1835. (24) de Lucca, E. A.; Maciel, R.; Melo, P. A.; Pinto, J. C. Modeling and simulation of liquid phase propylene polymerization in industrial loop reactors. Macromol. Symp. 2008, 271, 8–14. (25) Asteasuain, M.; Brandolin, A. Modeling and optimization of a highpressure ethylene polymerization reactor using gPROMS. Comput. Chem. Eng. 2008, 32, 396–408. (26) Zacca, J. J.; Debling, J. A. Particle population overheating phenomena in olefin polymerization reactors. Chem. Eng. Sci. 2001, 56, 4029–4042. (27) Choi, K. Y.; Ray, W. H. Recent developments in transition metal catalysed olefin polymerizationsA survey. I. Ethylene polymerization. J. Macromol. Sci. C: Polym. ReV. 1985, 25, 1–55. (28) Choi, K. Y.; Ray, W. H. Recent developments in transition metal catalysed olefin polymerizationsA survey. II. Propylene polymerization. J. Macromol. Sci. C: Polym. ReV. 1985, 25, 57–97. (29) Zacca, J. J.; Ray, W. H. Modeling of the liquid phase polymerization of olefins in loop reactors. Chem. Eng. Sci. 1993, 48, 3743–3765. (30) Luo, Z. H.; Cao, Z. K.; Su, Y. T. Monte Carlo simulation of propylene polymerization (I) Effects of impurity on propylene polymerization. Chin. J. Chem. Eng. 2006, 14, 194–199. (31) Mattos, N. A. G.; Pinto, J. C. Steady-state modeling of slurry and bulk propylene polymerizations. Chem. Eng. Sci. 2001, 56, 4043–4057. (32) Hirschfelder, J. Q.; Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons: New York, 1954. (33) Tong, J. S. Chemical Engineering Thermodynamics; Tsinghua University Press: Beijing, 1995 (in Chinese). (34) Vargaftik, N. B. Tables on the Thermophysical Properties of Liquids and Gases; Hemisphere Publishing Corporation: New York, 1975. (35) Tong, J. S.; Ma, M. A modified equation of state for saturated liquids. J. Eng. Thermophys. 1985, 6, 11–16 (in Chinese). (36) Liang, Y. B.; Tong, J. S. A simple general vapour pressure equation and a new calculated method about heat of vapourization. J. Eng. Thermophys. 1998, 19, 673–676 (in Chinese). (37) NIST Chemistry WebBook; NIST Standard Reference Database Number 69; National Institute of Standards and Technology (NIST): Gaithersburg, MD, 2005; available at http://webbook.nist.gov/chemistry/ (accessed December 12, 2009). (38) Gross, J.; Sadowski, G. Perturbed-chain SAFT: An equation of state based on a perturbation theory for chain molecules. Ind. Eng. Chem. Res. 2001, 40, 1244–1342. (39) Gross, J.; Sadowski, G. Modeling polymer systems using the perturbed-chain statistical associating fluid theory equation of state. Ind. Eng. Chem. Res. 2002, 41, 1084–1093. (40) Chai, C. J.; Zhang, G. L. Chemical Engineering Fluid Flow and Heat Transfer; Chemical Industry Press: Beijing, 2000 (in Chinese). (41) Shi, D. P.; Luo, Z. H.; Zheng, Z. W. Numerical simulation of liquid-solid two-phase flow in a tubular loop polymerization reactor. Powder Technol. 2010, 198, 135–143. (42) Luan, C. Y.; Qiu, Q. Y. Calculation of rated discharge capacity of safety valve. Pressure Vessel Technol. 1991, 5, 30–33 (in Chinese). (43) Wang, D. Y.; Wang, S. L.; Ding, X. W. The comparison of four codes for pressure relief. Pet. Chem. Equip. 2002, 31, 40–43 (in Chinese). (44) Zhou, C. G.; Hu, Y. D.; Han, F. X. New correlations of real fluid departure function (I): Isothermal ethalpy departure. Chem. Eng. (China) 1989, 17, 32–36. (45) Khare, N. P.; Luca, B.; Seavey, K. C.; Liu, Y. A. Steady-state and dynamic modeling of gas-phase polypropylene processes using stirred-bed reactors. Ind. Eng. Chem. Res. 2004, 43, 884–900. (46) Brandrup, J.; Immergut, E. H. Polymer Handbook; Interscience: New York, 1975.

ReceiVed for reView February 19, 2010 ReVised manuscript receiVed August 29, 2010 Accepted September 15, 2010 IE1003784