Modeling Fouling Effects in LDPE Tubular Polymerization Reactors. 3

Andrew Bird,† Steve Hearn, and Joe Hannon. Performance Fluid Dynamics (PFD) Limited, 40 Lower Leeson Street, Dublin 2, Ireland. Fouling in a LDPE tu...
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Ind. Eng. Chem. Res. 2005, 44, 1493-1501

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Modeling Fouling Effects in LDPE Tubular Polymerization Reactors. 3. Computational Fluid Dynamics Analysis of a Reacting Zone Alberto Buchelli,* Michael L. Call, and Allen L. Brown Lyondell Chemical Company, Equistar Chemicals, LP, La Porte Complex, 1515 Miller Cut-Off Road, Houston, Texas 77536

Andrew Bird,† Steve Hearn, and Joe Hannon Performance Fluid Dynamics (PFD) Limited, 40 Lower Leeson Street, Dublin 2, Ireland

Fouling in a LDPE tubular polymerization reactor is caused by the polyethylene/ethylene mixture forming two phases inside the reactor. Some of the polymer-rich phase is deposited on the reactor’s inside wall, which considerably reduces heat-transfer rates. At a given reactor pressure, the reactor inside wall temperature is the critical parameter in determining when fouling occurs and this is controlled by the coolant stream temperatures. Fouling upstream of a reacting zone increases the inlet temperature to that zone. This increase in temperature results in high hotspot temperatures, low fractional conversion, and broader polymer molecular weight distribution. Introduction

Table 1. Reacting Section Simulations

Fouling in LDPE tubular polymerization reactors is caused by thermodynamically driven phase separation of polymer and ethylene. This phase separation occurs in the “cold-near” wall region of the tubular reactor. The polymer-rich phase then sticks to the reactor’s inside wall and causes a reduction of heat transfer through the wall, due to the low thermal conductivity of the deposited material. This can result in an increase in temperatures due to reduction of the rate of heat transfer and could potentially lead to dangerous ethylene decompositions in the reactor. Since plant data clearly show fouling behavior in the reactor, the objectives of this work were to be able to predict the reactor thermal performance and, in particular, the temperature profile in a reacting zone. To accomplish this objective, some modeling work based on computational fluid dynamics was undertaken to identify the effect of fouling on the polymerization kinetics in the reactor. Another objective was to determine the effect of fouling on polymer properties, fractional conversion, and reactor mixing performance. The details of the work undertaken to accomplish the aforementioned objectives are the subject of the remainder of this work. The process description of the plant and LDPE reactor is shown in (Bokis et al.1) and in part 1 of this set of papers regarding the modeling of fouling effects in LDPE tubular polymerization reactors. Influence of Fouling on Reactor Axial Temperature Profile Strategy. The aim of this part of this work was to investigate the effect of fouling on the process stream temperature profiles in a reacting zone of the reactor. * To whom correspondence should be addressed. Tel.: 713336-5214. Fax: 713-336-5391. E-mail: Alberto.Buchelli@ Equistarchem.com. † Tel.: +353 1 6612131. Fax: 353 1 6612132. E-mail: [email protected].

Reacting Section (Zone 3) process initiator simulation

type

conditions

Tin °F

1 2 3 4 5 6 7 8 9

DynoChem PFR default CFD CFDReaction DynoChem PFR default CFD CFDReaction DynoChem

clean clean clean clean fouled fouled fouled fouled defouling

464 464 464 464 500 500 500 500 520

coolant

Tin °F

Tin °F

Tout °F

394 394 394 394 394 394 394 394 394

N/A N/A 133 133 N/A N/A 133 133 N/A

N/A N/A 158 158 N/A N/A 158 158 N/A

The foulant layer was expected to reduce heat transfer through the tube wall and increase the inlet temperature to the reacting zone, thereby increasing the process stream temperature. It was then determined that fouling could lead to an increased risk of dangerous ethylene decomposition in the reactor. The kinetics of ethylene decomposition were not included in the models primarily for reasons of reducing the computational time for the CFD simulations. Part 1 of these series of papers shows that there are three reacting zones in the reactor. Zone 1, which is at the inlet, Zone 3, which follows the cooling section, and Zone 6, near the end of the reactor. Zone 3 was chosen for study, as it follows Zone 2. Also, Zone 2 shows fouling behavior so increased inlet temperatures to Zone 3 are observed when Zone 2 is fouling. Zone 1 will have no fouling at the inlet (no polymer conversion), so its inlet temperature will not be affected by fouling. Zone 6 may be affected by the “bumping” of the exit reactor valve, so Zone 3 was preferred. For the reacting section, Zone 3 was chosen and several simulation types were performed based on conditions shown in Table 1. As shown in Table 2, four parallel modeling approaches were adopted. In all simulations, the effect of process stream temperature, monomer conversion, initiator consumption (for each initiator component), and hot-spot temperature were predicted. In the DynoChem simulations,

10.1021/ie040159a CCC: $30.25 © 2005 American Chemical Society Published on Web 02/05/2005

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Table 2. Modeling Approaches and Software Characteristics DynoChem: One-dimensional modeling approach. Accounts for micromixing and mesomixing effects on the reaction rates, but has no heat-transfer effects built in. PRF: One-dimensional modeling approach. Assumes a premixed reactor and does not account for any mixing effects. It is a pure kinetic modeler that has no heat-transfer effects built in. Default CFD: Two-dimensional simulations which account for mixing by convection and turbulent diffusion. Does not include micromixing and mesomixing. Heat-transfer effects are modeled. CFDReaction: Two-dimensional simulations which account for mixing by convection and turbulent diffusion. Accounts for micromixing and mesomixing. Heat-transfer effects are modeled. Table 3. Initiator Stream Composition for Zone 3 initiator

molecular weight kg/kmol

weight %

concentration kmol/m3

TBPO TBPA DTBP hexane

216.3 132.2 146.2 86.0

14.9 3.3 0.8 81.0

0.462 0.167 0.037 6.320

the molecular weight distribution (number average, weight average, and polydispersity index) was also predicted (this was not done in the CFDReaction simulations for computational time limitations). Modeling Approaches. The modeling approaches for Dynochem and PFR simulations are described in Table 2. In addition to the main characteristics shown in Table 2, Dynochem runs the reactor adiabatically so the effect of heat transfer on the process stream temperature cannot be evaluated. Species concentrations, molecular weight distribution, and temperature are calculated as a function of distance along the reactor. By alteration of the inlet temperature of the reacting zone, useful simulations can be run to account for fouling behavior in the previous zone. The effect of mixing can also be investigated: (1) by running the simulation as a premixed plug flow reactor which does not account for mixing effects and (2) by running the simulation as a tubular reactor accounting for meso- and micromixing. The following simulations were performed: (1) Tubular reactor simulations of Zones 1, 3, and 5 accounting for mixing effects under clean conditions, (2) premixed plug-flow reactor simulations of Zone 3 under clean conditions, and (3) tubular reactor simulation of Zone 3 accounting for mixing effects under fouled conditions. Default CFD and CFDReaction simulations for the reaction scheme were also implemented in CFD, which

Table 4. Kinetic Parameters for the Initiation Reactions activation half-life pre-exponential energy at 40 kpsi at 464 °F* factor -1 cal/mol s s

initiator type tert-butyl peroctoate (TBPO) tert-butyl peracetate (TBPA) di-tert-butyl peroxide (DTBP)

5.75 × 1011 2.83 × 1015 1.81 × 1016

30090 34633 38454

0.22 0.16 1.06

Table 5. Kinetic Parameters for the Propagation and Termination Steps

reaction

pre-exponential factor m3 kmol-1 s-1

activation energy at 40 kpsi cal/mol

propagation termination

1.14 × 107 3.00 × 109

6138 3033

provides detailed information on the radial distribution of species and temperature. CFD solves partial differential equations, so it provides spatial information in two dimensions (axially and radially). Micro- and mesomixing effects are also accounted for using PFD’s CFDReaction software. In particular, radial profiles of temperature can be obtained, which are not available in DynoChem. In addition, CFD can account for the heat-transfer effects due to the fouling. The fouling model described earlier was used to increase the process side heattransfer resistance due to fouling. This should lead to more accurate temperature profiles in the reactor. Default CFD does not account for small-scale mixing effects (micro- and mesomixing). Two additional CFD runs using the default chemistry options were performed. These simulations account for the larger scale mixing by convection and turbulent diffusion, but assume no variation in concentration within a computa-

Figure 1. Process stream axial temperature profile in Zone 3 predicted by DynoChem under clean conditions, accounting for mixing effects. The solid line is the reaction zone temperature, the blue diamonds are the average process stream temperatures, and the black squares are the plant data.

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Figure 2. Initiator consumption in Zone 3 for the three initiators predicted by DynoChem under clean conditions. The concentrations are made dimensionless by dividing by the initiator inlet concentration.

tional cell. The following simulations were performed: (1) simulation of Zone 3 under clean conditions using CFDReaction and default CFD and (2) simulation of Zone 3 under fouled conditions using CFDReaction and default CFD. Reactor Geometry, Boundary Conditions, and Reaction Parameters Used. Geometry. For the CFD simulation, the geometry of the initiator injection port needed to be modeled. Details of the injector were included in the simulation and a computational mesh was constructed. The injector port was treated as a coaxial feed of diameter 0.021 in. This preserved the two-dimensionality of the geometry, and greatly reduces the simulation times, without significantly decreasing the accuracy of the simulations. A single block was modeled (30 ft in length) which was enough to capture any hot spot which would exist near the injection port. Boundary Conditions. For the CFD simulations, temperature boundary conditions were imposed on the outer tube wall, based on coolant flow temperatures. The coolant temperature at the injector was 158 °F, dropping linearly to 152 °F after 30 ft. Three initiators were used. These were tert-butyl peroctoate (TBPO), tert-butyl peracetate (TBPA), and di-tert-butyl peroxide (DTBP). All initiators were dissolved in hexane. The total initiator streamflow was taken as 55.3 lb/h and the initiator inlet temperature was estimated at 394 °F. The composition of the initiator stream is given in Table 3. Reaction Parameters. The reaction scheme adopted for free-radical polymerization of ethylene is given below: Initiation Steps:

ITBPO f 2A• kTBPO TBPO decomposition

(1)

ITBPA f 2A• kTBPA TBPA decomposition

(2)

IDTBP f 2A• kDTBP DTBP decomposition

(3)

A• + M f R1• kI chain initiation

(4)

Termination Steps:

Rn• + Rm• f Pn+m kt termination by coupling (6) Equations for global radical and polymer concentrations are solved: n

R)

Ri•; ∑ i)1

(5)

Pi ∑ i)1

(7)

Equations for initiator (I), initiator radical (A•), radical (R), monomer (M), and polymer (P) were solved. In addition to these equations, DynoChem equations describing the moments of the radical and polymer chain length distribution were solved. Moments of the radical chain length distribution: ∞

λ0 )

∑ n)1

Rn•, λ1 )

∞

∑ n)1

nRn•, λ2 )

∞

n2Rn• ∑ n)1

(8)

Moments of the polymer chain length distribution: ∞

µ0 )

∑ Pn, n)1

∞

µ1 )

∞

∑ nPn, n)1

µ2 )

n2Pn ∑ n)1

(9)

The moments allowed calculation of the following properties: Number average molecular weight (MWm is the monomer molecular weight):

µ1 NWD ) MWm µ0

(10)

Mass average molecular weight:

µ2 MWD ) MWm µ1

(11)

Polydispersity index:

Propagation Step:

Rn• + M f Rn+1• kp ∆Hp propagation

n

P)

ZP )

MWD µ2µ0 ) 2 NWD µ 1

(12)

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Figure 3. Monomer conversion in Zone 3 predicted by DynoChem under clean conditions.

Figure 4. Average molecular weights for Zone 3 predicted using DynoChem under clean conditions.

The reaction rate temperature dependency for each reaction was set according to a modified Arrhenius form:

( )

kj ) Aj exp -

Ej RT

(13)

The activation energies are functions of pressure and have all been evaluated at 40 kpsi. The kinetic parameters for the TBPA and DTBP were taken from Zhang et al.,2 and the parameters for TBPO were from Kiparissides et al.3 and are given in Table 4. The half-life of the first-order initiation reactions at 464 °F gives some indication of the relative speed of the reactions. TBPA has the shortest half-life of 0.16 s (this corresponds to approximately 9.5 ft in the reactor), and DTBP has the longest, 1.06 s (corresponding to approximately 63 ft in the reactor). TBPO and TBPA have similar half-lives. The other reactions (propagation and termination) are second-order reactions, and parameters were taken from Zhang et al.2 and are given in Table 5. These parameters were then used in the DynoChem and CFD simulations of Zone 3.

The heat release due to reaction was assumed to be solely from the propagation step, with a heat of reaction of -22690 cal/mol (-9.5 × 107 J/kmol) taken from Tsai and Fox.4 The heat release due to the initiator decomposition, initiation, and termination steps was neglected because these reactions would account for less than 1% of the total heat release according to Buchelli et al.5 For a typical LDPE molecule with 600 ethylene repetitive units, the decomposition reaction occurs once, the initiation reaction occurs twice, and the termination reaction occurs once, but the propagation reaction occurs 598 times. The heat release due to the short-chain branching (SBC) in the polymer is approximately about 4% of the heat release for the propagation reaction. However, the kinetic model developed in this study did not account for any branching or even beta-scission. It is believed that the assumption made that the heat of reaction is due solely from the propagation step is adequate for engineering calculations in the industrial polymerization reactor herein described.

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Figure 5. The effect of mixing on process stream temperature profiles for Zone 3 (clean conditions). Solid line ) PFR reaction zone temperature, blue diamonds ) PFR average process stream temperatures, line and hollow square ) premixed plug flow reactor temperature predictions, and solid squares ) plant data.

Figure 6. Velocity vectors colored with velocity magnitude (red ) 65 ft/s) near the injection nozzle for Zone 3 (clean conditions).

DynoChem Simulation - Clean Conditions. Simulations of all three reacting zones (Zones 1, 3, and 6) were performed, but the results for Zone 3 only are reported in this work. The main aim of this part of the work was to study the maximum process stream temperature in the reacting zone. DynoChem effectively divides the reactor into two regions, the reaction zone (which is initiator-rich) where reaction and heat release occur and the surrounding environment region. Mixing of these regions occurs at a rate calculated from the flow parameters and physical properties. This allows hotspot temperatures to be calculated (effectively reaction zone temperatures). The temperature profile for Zone 3 is given in Figure 1. Figure 1 shows the plant data under clean conditions (squares), the average process stream temperature

(diamonds), and the reaction zone temperature (line). The average process stream temperature shows good agreement with plant data. The reaction zone temperature matches the average temperature after about 5 ft of the reactor, indicating that micro- and mesomixing is complete by this stage. Nearer the injector, the reaction zone temperature exceeds the average stream temperature, and the difference between the two never exceeds 24 °F. The temperature 60 ft from the injector (548 °F) is warmer than the hot-spot temperature (488 °F), so runaway behavior is unlikely under clean conditions. Initiator consumption is shown in Figure 2. The plot shows radially averaged initiator concentration, scaled with the inlet radially averaged concentration to make the concentration dimensionless for comparison. The

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Figure 7. Temperature contours near the injector inlet (over the first 0.7 ft) in Zone 3 (clean conditions) using CFDReaction and default CFD. (Purple ) 470 °F, red ) 477 °F).

initiator blend used leads to production of free radicals over a length of about 40 ft. As indicated by the halflives, TBPA is consumed most rapidly, and DTBP at the slowest rate. All three initiators decompose at different rates, so each initiator decomposition step needs to be accounted for in the modeling. The monomer conversion predicted by the DynoChem model is shown in Figure 3. The monomer conversion in Zone 3 is predicted at 4.53%. The propagation reaction and chain initiation reactions both consume monomer, and these reactions happen once the initiator has decomposed to free radicals. Moment equations for the molecular weight distribution were also solved. These equations yield the number and weight average molecular weights and are shown in Figure 4. Both average molecular weights rise, and the polymer produced in Zone 3 has a predicted number average molecular weight of 26600 kg/kmol and

a predicted weight average molecular weight of 55,300 kg/kmol. This leads to a polydispersity index (Mw/Mn) of 2.1. Effect of Mixing on Reactions. To assess the importance of mixing on the process results, a premixed plug flow reactor (PFR) simulation was performed. This simulation did not account for mixing effects and simply modeled the kinetics of the reactions. Comparison of the temperature profiles for the PFR and the simulation with mixing (denoted tube) are shown in Figure 5. The PFR simulation yielded higher temperatures, particularly over the first 30 ft of Zone 3. The final temperatures attained by the two streams were similar (for the tube it was 548 °F, for the PFR it was 551 °F). The main difference was that hot spots cannot be picked up with the PFR approach, which may be more significant under other conditions. Mixing appears to delay the start of reaction, and the process temperature only rises after mixing is nearly complete (after about 4 ft). With the PFR run, the process stream temperature started to rise immediately. The conversion profile shows a similar trend, the PFR predicting a higher conversion of 4.94% (compared to 4.53% predicted earlier). This is consistent with the temperature profile. The number average molecular weight was 27900 kg/kmol for the PFR (26600 kg/kmol for the tube) and weight average molecular weight was 52100 kg/kmol (55300 kg/kmol for the tube). This gives a slightly lower polydispersity index (1.87 for the PFR compared to 2.1 for the tube), indicating a slightly narrower molecular weight distribution. Even in the tube, the mixing is quite fast relative to reaction, leading to almost well-mixed plug flow behavior. If mixing rates decrease, more deviation from plug flow behavior is seen, leading to a broader molecular weight distribution. CFD Simulation - Clean Conditions. The initiator injection port injects initiator at a higher velocity than the surrounding process stream due to the small diameter of the feed nozzle (0.021 in.). This leads to high turbulence levels and intense mixing in the injection’s region. Velocity vectors near the injector’s region are shown in Figure 6. A high velocity is seen at the pipe center (up to 150 ft/s) and a low velocity at the reactor’s

Figure 8. Predicted temperature profile in the first 10 ft of Zone 3 under clean conditions. Solid line ) DynoChem reaction zone temperature, blue diamonds ) default CFD centerline process stream temperatures, line and hollow square ) premixed plug flow reactor temperature predictions, line and crosses ) CFDReaction centerline temperatures, and solid squares ) plant data.

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Figure 9. Process stream temperature profile in Zone 3 predicted by DynoChem under fouled conditions. Squares and solid line ) DynoChem reaction zone temperature under clean conditions, crosses and dashed line ) DynoChem reaction zone temperature under fouled conditions, black squares ) plant data under clean conditions, and blue diamonds ) plant data under fouled conditions, triangles and solid line ) DynoChem reaction zone temperature while defouling, circles ) plant data while defouling.

Figure 10. Monomer conversion profile in Zone 3 predicted by DynoChem under clean and fouled and defouling conditions.

walls. This high velocity is gradually reduced and the velocity profile becomes fully developed some distance downstream of the injector. The predicted process stream temperature profile in the reacting section using CFDReaction is shown in Figure 7. The default CFD simulation does not pick up a hot spot near the injector inlet (left of Figure 7). This is because the default CFD simulation does not account for micromixing or mesomixing. The CFDReaction temperature profile shown in Figure 8 is of a similar shape to the DynoChem predictions. Both simulations predict a hot spot, which is not picked up either with default CFD or a PFR simulation. CFDReaction predicts the hot spot to occur closer to the injector and to have a lower value. DynoChem Simulation - Fouled and Defouling Conditions. Under fouled conditions and while defouling, the process stream inlet temperature for Zone 3 is increased. The resultant temperature profile is shown in Figure 9. The increased process stream inlet tem-

perature has a strong effect on the reaction zone temperature. The magnitude of the hot spot is increased (up to 74 °F above the average process stream temperature for fouled and up to 113 °F above the average process stream temperature for defouling conditions). This results in the hot-spot temperature being higher than the outlet temperature. The outlet temperature is not significantly affected by the increased inlet temperature and is, in fact, lower than the clean conditions. Fouling upstream leads to a large increase in the hotspot temperature, and as less heat is removed, there is an increased probability of ethylene decomposition. The plant data show little difference 60 ft downstream of the injector between clean and fouled conditions. This suggests that either the foulant is removed over the reacting section or that there is less fractional conversion when Zones 2 and 3 are fouled. The predicted fractional conversion profile is shown in Figure 10. The fractional conversion profile shows that, under fouled conditions, the fractional conversion

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Figure 11. Predicted process stream temperature profile in Zone 3 under fouled conditions. Solid line ) DynoChem reaction zone temperature, blue diamonds ) default CFD centerline process stream temperatures, line and hollow square ) premixed plug flow reactor temperature predictions, line and crosses ) CFDReaction centerline temperatures, and solid squares are plant data.

is reduced to 2.41% (from 4.53%). This explains why the outlet temperature is not different between clean and fouled conditions. While defouling, the predicted fractional conversion is further reduced to 0.82%. The number average molecular weight under fouled conditions is lower at 17600 kg/kmol (compared to 26600 kg/kmol under clean conditions). The weight average molecular weight is similar under clean and fouled conditions and is 62500 kg/kmol when fouled (compared to 55300 kg/kmol under clean conditions). The polydispersity is higher under fouled conditions (3.5 compared to 2.1 under clean conditions). Under defouling conditions, ZP increases to 9.0. Fouling has an adverse effect on the performance of the reactor. It decreases fractional conversion, increases the hot-spot temperature, and broadens the molecular weight distribution. CFD Simulation - Fouled Conditions. The CFD simulation under fouled conditions shows similar behavior to the clean conditions. A hot spot is picked up by CFDReaction and not by default CFD. As with the DynoChem simulations, the hot spot has a higher temperature under fouled conditions. The temperature profiles in Zone 3 under fouled conditions are shown in Figure 11. As with the DynoChem runs, monomer fractional conversion is reduced under fouled conditions (from 7% down to 4%) due to an increase in the ratio of termination rate to propagation rate at higher temperatures.

mixing rates in the region near the injector and, hence, reaction happens more quickly, bringing the hot spot closer to the initiator injection point. Design of initiator injectors is key to preventing hot-spot formation and eventually ethylene decomposition reactions. The fractional conversion achieved in a reacting zone downstream of a fouled or defouling cooling zone is lower. The average process stream temperature rise is therefore lower than if the upstream cooling section were clean. In addition, the hot-spot temperature in the reacting section is higher and the polymer molecular weight distribution is broader when the upstream cooling zone is fouled or defouled. To maintain LDPE quality produced in the reactor and to reduce decomposition frequency, it is imperative that one minimize foulant levels, particularly in zones upstream of reacting zones.

Conclusions

Aj ) constant for j reaction in Arrhenius equation, m3/(kg mol s) A• ) initiator free radical concentration, (kg mol)/m3 Ej ) activation energy for j component in Arrhenius equation, J/(kg mol) ITBPO ) TBPO initiator concentration, (kg mol)/m3 ITBPA ) TBPA initiator concentration (kg mol)/m3 IDTBP ) DTBP initiator concentration (kg mol)/m3 kTBPO ) TBPO initiator rate constant, m3/(kg mol s) kTBPA ) TBPA initiator rate constant, m3/(kg mol s) kDTBP ) DTBP initiator rate constant, m3/(kg mol s) kl ) initiator rate constant, m3/(kg mol s) kp ) polymerization rate constant, m3/(kg mol s)

Kinetics for free radical polymerization were included in the CFD model as well as a PC-based model (DynoChem). The kinetics of ethylene decomposition reactions was not included in this phase of the work due to the additional run time required for the CFD simulations. CFDReaction software picks up a hot spot closer to the injector port than does DynoChem. DynoChem calculates the turbulent mixing parameters based on empty pipe turbulence. In reality, the initiator injector generates a jet, which results in higher turbulence than in an empty pipe. This higher turbulence increases the

Acknowledgment Thanks are due to Mike Brown, Tom Srnka, and Mike Hurst, members of the Lyondell-Equistar Process Engineering Management Team, for allowing the publication of this work. Also, Robert Bridges is acknowledged for having provided the financial support for conducting this research project. Dr. Raghu Narayan is thanked for reviewing the paper. Nomenclature

Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1501 ktc ) termination rate constant, m3/(kg mol s) L ) average chain length M ) monomer concentration, (kg mol)/m3 Mn ) molecular weight number average, kg/(kg mol) Mw ) molecular weight weight average, kg/(kg mol) MWm ) monomer molecular weight, kg/(kg mol) MWD ) mass average molecular weight, kg/kmol n ) number of repeat units in polymer molecule NWD ) number average molecular weight, kg/kmol P ) concentration of dead polymer, (kg mol)/m3 Pi ) concentration of life polymer of length “i”, (kg mol)/ m3 Pn ) concentration of life polymer of length “i”, (kg mol)/ m3 R ) ideal gas law constant, (kg m2)/(s2 kg mol K) R1• ) free radical concentration of length “1”, (kg mol)/m3 Ri• ) free radical concentration of length “i”, (kg mol)/m3 Rn• ) free radical concentration of length “n” (kg mol)/m3 Rn+1• ) free radical concentration of length “n+1”, (kg mol)/ m3 Rm• ) free radical concentration of length “m”, (kg mol)/ m3 Pn+m ) dead polymer concentration of length “n+m”, (kg mol)/m3 T ) temperature, K ZP ) polydispersity index (MWD/NWD) ∆HP ) heat of polymerization, J/(kg mol) λ0 ) zero moment of the radical chain length λ1 ) first moment of the radical chain length

λ2 ) second moment of the radical chain length µ0 ) zero moment of the polymer chain length µ1 ) first moment of the polymer chain length

Literature Cited (1) Bokis, C. P.; Ramanathan, S.; Franjione, J.; Buchelli, A.; Call, M. L.; Brown, A. L. Physical Properties, Reactor Modeling, and Polymerization Kinetics in the Low-Density Polyethylene Tubular Reactor Process. Ind. Eng. Chem. Res. 2002, 41, 10171030. (2) Zhang S. X.; Read, N. K.; Ray, W. H. Runaway Phenomena in Low-Density Polyethylene Autoclave Reactors. AIChE J. 1996, 42 (10), 2911-2925. (3) Kiparissides, C.; Verros, G.; Kalfas, G.; Koutoudi, M.; Kantzia, C. A Comprehensive Mathematical Model for a Multizone Tubular High Pressure LDPE Reactor. Chem. Eng. Commun. 1993, 121, 193-217. (4) Tsai, K.; Fox, R. O. PDF Modeling of Turbulent-Mixing Effects on Initiator Efficiency in a Tubular LDPE Reactor. AIChE J. 1996, 42(10), 2926-2940. (5) 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.; Chapter 3 and Chapter 7, Oct 2001. Unpublished results.

Received for review May 14, 2004 Revised manuscript received November 30, 2004 Accepted November 30, 2004 IE040159A