110th Anniversary: Nonideal Mixing Phenomena in High-Pressure

Article Cite This: Ind. Eng. Chem. Res. 2019, 58, 13093−13111

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110th Anniversary: Nonideal Mixing Phenomena in High-Pressure Low-Density Polyethylene Autoclaves: Prediction of Variable Initiator Efficiency and Ethylene Decomposition Prokopis Pladis† and Costas Kiparissides*,†,‡ Chemical Process and Energy Resources Institute, Centre for Research and Technology Hellas and ‡Department of Chemical Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

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†

ABSTRACT: A segregation−backmixing model is developed to simulate the dynamic operation of multifeed high-pressure low-density polyethylene (LDPE) autoclaves, calculate the specific initiator consumption (grams of initiator per kilogram of LDPE), and assess the risk of ethylene decomposition under different nonideal mixing and operating scenarios. To describe the nonideal macro- and micromixing phenomena in a LDPE autoclave, a user-specified multizone model representation of the actual reactor is established. One of the key features of the present model is that the continuously fed initiator into a reaction zone can exhibit two distinct states, namely, a “segregated” or a “molecular” one. Simulation results are presented, showing the effects of macro- and micromixing model parameters on the spatial temperature distribution and specific initiator consumption with respect to the initiator feed concentration to a two-compartment reaction zone. Finally, the operational conditions and process faults that can lead to ethylene decomposition in a reaction zone are investigated. mixed). Thus, a nonideal flow mixing model needs to be employed to approximate the macro- and micromixing phenomena in the reactor.2 The main reason for the nonideal mixing conditions in LDPE autoclaves is the lack of spatial homogeneity in the initiator concentration in a reaction zone. It is important to point out that the characteristic reaction time associated with the initiator decomposition is of the same order (i.e., 0.1 s) with the characteristic micromixing time of the freshly injected initiators into a reaction zone. Spatial homogeneity in the initiator concentration is brought about by mass transfer at two different mixing scales, namely, macroand microscopic. At the former scale, large-scale flow motions (e.g., eddies) result in the so-called macromixing of reactive species. At the microscopic scale, small-scale eddies and molecular diffusion effect mixing at the molecular level. For all practical reasons, macromixing can be associated with residence time distribution (RTD) in a reaction zone, whereas micromixing can be linked with the state of fluid aggregation. It has been shown that due to the spatial variation of initiator concentration in a LDPE autoclave (i.e., nonideal mixing) the specific initiator consumption expressed in grams of initiator consumed per kilogram of polyethylene produced often exhibits a U-shape behavior with respect to polymerization temperature. The observed variation in the specific initiator consumption with respect to polymerization temper-

1. HIGH-PRESSURE ETHYLENE POLYMERIZATION TECHNOLOGY The high-pressure ethylene free-radical polymerization is a manufacturing process of significant economic importance. Two reactor technologies, namely, tubular and autoclave, are currently employed in the production of low-density polyethylene (LDPE). High-pressure LDPE reactors typically operate at high temperatures (160−320 °C) and pressures (1200−3200 atm). Under these conditions, the reaction mixture behaves as a supercritical fluid. At higher temperatures (i.e., above 330 °C) ethylene decomposition can be initiated.1 An autoclave reactor is a constantly mixed vessel made up of two or more reaction zones in series, separated by disks and stirred by a vertical stirrer shaft. Despite the large specific power input to the reacting system (20−100 kW/m3) effected by high agitation rates and specially designed impellers, the zones are not perfectly mixed due to the very fast reaction kinetics and short mean residence times. Ethylene polymerization in autoclaves is practically carried out in an adiabatic way. Cooling of the reaction mixture is effected by the introduction of cold monomer at several side-feed points along the reactor height. The reaction temperature in a zone is controlled by manipulating the corresponding initiator feed rate to the zone. It is important to point out that robust control of temperature profile along the autoclave height is required because the reactor usually operates at an open-loop unstable steady state. In Figure 1, a schematic representation of a highpressure LDPE plant is depicted. The flow behavior in a multizone LDPE autoclave can significantly deviate from ideal flow conditions (i.e., perfectly © 2019 American Chemical Society

Received: Revised: Accepted: Published: 13093

May 7, 2019 June 21, 2019 June 24, 2019 June 24, 2019 DOI: 10.1021/acs.iecr.9b02517 Ind. Eng. Chem. Res. 2019, 58, 13093−13111

Article

Industrial & Engineering Chemistry Research

Figure 1. Schematic representation of a high-pressure LDPE plant.

Figure 2. Effect of macro-/micromixing phenomena on the specific initiator consumption in LDPE autoclaves, Luft et al.:4 (I) tert-butyl per(2ethy1)hexanoate (t-butyl peroctoate), TBPEH; (II) di-isononanoyl peroxide, DTHP; (III) tert-butyl perpivalate, TBPP; (IV) dilauroyl peroxide, LPO; (V) dioctanoyl peroxide, DOP; (VI) dicyclohexyl percarbonate, DCHP; (VII) tert-butyl perneo-decanoate, TBPND; (VIII) tert-butyl per3,5,5-trimethylhexanoate (iso-nonanoate), TBPIN; (IX) tert-butyl perbenzoate, TBPB; (X) ditert-butyl peroxide, DTBP. [Pressure, 1700 bar; mean residence time, 65 s; initiator concentration 40 ppm (molar)]. Adapted with permission from ref 4. Copyright 1977 Taylor & Francis.

ature was experimentally investigated by Van der Molen,3 Luft and co-workers,4−6 and Goto et al.7 Luft and his co-workers4 carried out a comprehensive experimental investigation, using a lab-scale LDPE autoclave, to study the effect of polymerization temperature (e.g., 110−300 °C) on the specific initiator consumption for different organic peroxides (see Figure 2). Most investigators have attributed the observed variable specific initiator consumption with respect to polymerization temperature to the nonideal mixing conditions in LDPE

autoclaves, and a number of macro-/micromixing models have been proposed to quantify this behavior.7−11 The proposed nonideal mixing models describe the lack of spatial homogeneity in the initiator concentration in a reaction zone and can be broadly classified into three categories, namely, (i) fluid-flow models, (ii) fluid−particle (segregation) models, and (iii) fluid-mechanics models. However, as suggested by Smit,12 any nonideal mixing model that explains the slowness in the initiator mixing process, in comparison to the very fast 13094

DOI: 10.1021/acs.iecr.9b02517 Ind. Eng. Chem. Res. 2019, 58, 13093−13111

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Finally, fluid-mechanics models are based on the solution of first-principles Navier−Stokes equations describing the nonideal mixing phenomena in a reactor. They have a sounder theoretical basis but have some difficulties in handling complex polymerization kinetics. Some fluid-mechanics models are based on ultimate microturbulence and molecular diffusion of the fluid element.19 Computational fluid dynamics (CFD) models have been also applied to describe nonideal mixing in reactive systems. In CFD models, the governing flow equations are numerically solved to calculate the complete flow field in a vessel. CFD approaches have been applied to model imperfect mixing in high-pressure ethylene LDPE autoclave and tubular reactors.25−33 Although CFD is a powerful tool for describing nonideal mixing phenomena in a LDPE autoclave, it requires significantly long computational times and careful consideration in handling a comprehensive free-radical ethylene polymerization mechanism coupled with ethylene decomposition, molecular weight developments, and time and spatial variability of reactive mixture transport properties and micromixing parameters. The present paper is organized as follows. In the following section, the ethylene free-radical polymerization mechanism, including ethylene decomposition kinetics, is described, and the leading moment rate functions of the number chain length distribution are derived. Subsequently, the characteristic mixing and reaction times in a LDPE autoclave are defined. On the basis of the calculation of the characteristic mixing and reaction times, a number of important criteria are established regarding the specific initiator consumption in relation to the stability/operability of a LDPE autoclave. Finally, a nonideal mixing model is derived, the so-called segregation−backmixing model to describe the macro- and micromixing phenomena in industrial-scale LDPE autoclaves. In section 4, the simulation results are presented showing the effects of segregation− backmixing model parameters on the spatial temperature distribution, specific initiator consumption with respect to the initiator feed concentration and temperature in a twocompartment zone of a multizonal autoclave. The present model predictions are also compared with the CFD results of Wells and Ray.30 Finally, the operational conditions that can lead to ethylene decomposition in a two-compartment zone in a multizonal industrial autoclave are examined.

polymerization kinetics, can simulate the experimentally measured variability in the specific initiator consumption with respect to polymerization temperature. In the fluid-flow mixing approximation models, the reactor volume is divided into a number of reaction zones with welldefined macro- and micromixing characteristics. These models are simple but realistic representations of the nonideal mixing conditions in an autoclave. They can be further subdivided into compartmental and environmental models. In the former models, the nonideal flow behavior in an autoclave is approximated by a combination of a number of ideal continuous stirred-tank (CST) and/or ideal plug-flow (PF) reactors. These model representations mainly describe the nonideal macromixing flow patterns in an autoclave. However, they can also represent some degree of micromixing if the diffusion of initiator molecules is also considered in the selected ideal-mixing compartments. Several investigators have employed compartmental models to study nonideal mixing phenomena in LDPE autoclaves.2,13−15 Mercx et al.8 developed a parametric macromixing model consisting of a small CSTR followed by a PF reactor to describe the nonideal flow in LDPE autoclaves. Part of the outlet stream from the PF reactor was recycled back to the CSTR. The model was capable of fitting the observed variable specific initiator consumption curve with respect to the polymerization temperature. Environmental models16−18 approximate the nonideal mixing phenomena in a reactor by assuming two different environmental states for reactive species, namely, (i) a state of perfect molecular mixing and (ii) a state of complete segregation. Accordingly, one or more mixing parameters are introduced to quantify the fluid mass fraction corresponding to each environmental state of a reactive species. Zwietering11 developed a plume model to describe the micromixing phenomena in a CSTR. In the plume model, the entering initiator feed stream grows exponentially by means of mixing with the bulk fluid. During this continuous dilution process, the initiator consumption increases due to mixing with monomer. Smit12 utilized the engulfment model of Bourne and Baldyga19,20 to explain the variable initiator consumption in low density polyethylene vessel reactors. Villermaux and his co-workers10,21 considered two different partial segregation models, namely, the shrinking aggregate model (SA) and the interaction by exchange with the mean model (IEM), to explain the variable initiator consumption in LDPE autoclaves. In the SA model, initiator lumps entering the reactor with the feed stream gradually disintegrate while the eroded material is instantaneously mixed with the bulk fluid. In the IEM model, it is assumed that initiator aggregates, having a constant volume, undergoing a reversible mass transfer exchange with the bulk fluid. Fluid−particle models assume the reaction mixture is made of a dispersion of many droplets (e.g., fluid lumps, particles) that interact in pairs by diffusion/coalescence and/or redispersion. The resulting mixing models are actually population balance equations with respect to the droplet properties (e.g., volume, mass, composition, and so on).22−24 In population balance models, two droplets with different compositions can coalescence, be completely mixed, and/or break up into two new droplets with the same composition. By varying the droplet coalescence and breakage rate kernels, the nonideal mixing conditions in a reactor can vary from complete segregation to ideal mixing.

2. NONIDEAL MIXING PHENOMENA IN LDPE AUTOCLAVES 2.1. Free-Radical Ethylene Kinetic Mechanism and Molecular Species Rate Functions. Ethylene can be polymerized via a free-radical mechanism at high temperatures (160−320 °C), and pressures (1200−3200 atm) in the presence of a mixture of chemical initiators (e.g., organic peroxides, hydro peroxides, and azo compounds). A typical free-radical ethylene polymerization mechanism includes the following: (i) an initiation reaction for the formation of primary radicals via the thermal decomposition of chemical initiators and/or thermal self-initiation of monomer, (ii) a polymer chain propagation reaction, (iii) a reaction for transfer of radical reactivity to monomer and/or chain transfer agent, (iv) a reaction for short chain branching formation via an intramolecular (backbiting) mechanism, (v) a transfer to polymer reaction for the formation of long chain branches, (vi) a polymer chain scission reaction and formation of two shorter chains (i.e., a “dead” chain and a “live” one), and (vii) the termination of “live “polymer chains by combination and/or 13095


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Net production rate function of “dead” polymer chains of chain length x:

disproportionation to form dead polymer chains. In particular, the following elementary reactions are considered in the present study:

∞

x−1

∞

∞

1 rDx = (k tmM + k tsS)R x + k tdR x ∑ R x + k tc ∑ R yR y − x 2 y=1 x=1

Chemical initiation: kdi

Ii → 2R*; i = 1, 2, ..., Ni

∞

+ k tpR x ∑ xDx − k tpxDx ∑ R x + kβR x ∑ xDx

Chain initiation:

ij ∞ yz j z + kβ jjj ∑ R x zzz jj zz k χ=1 { x=2

kI

R* + M → R1

Propagation: kp

R x + M → R x+1

x=1 ∞

x=2 ∞

∑

Dy − kβxDx ∑ R x

y=x+1

x=1

(2)

where δ(x) is the Kronecker delta function. On the basis of the above general expressions for the rRx and

Chain transfer to monomer:

rDx rate functions, one can derive a theoretically infinite set of differential equations with respect to the degree of polymerization (i.e., from x = 1 to x = Nmax) to describe the dynamic variation of all polymer chains concentrations in a polymerization reactor. However, instead of solving the resulting large system of differential equations, a lower order system of differential equations is usually obtained by using the method of moments. This method is based on the statistical representation of the “live” and “dead” number chain length distributions (NCLDs) in terms of their respective leading moments. Following the previous developments of Pladis and Kiparissides,33 the single moments of the “live” and “dead” number chain length distributions are defined:

k tm

R x + M ⎯→ ⎯ Dx + R1

Chain transfer to CTA: k ts

R x + S → Dx + R1

Intramolecular transfer (short-chain branching): kb

Rx → Rx Transfer to polymer (long-chain branching): k tp

R x + Dy → Dx + R y

β-Scission of internal radicals: kβ

R x + Dy → Dx + R z + D=y − z

∞

λn =

Termination by combination:

∞

∑ x nR x

;

μn =

x=1

k tc

R x + R y → Dx + y

∑ x nDx x=2

(3)

Accordingly, the general moment rate functions for rλn and rμn will be given by

Termination by disproportionation: k td

R x + R y → Dx + Dy

∞

rλn = ∑ x nrR x

∞

;

rμn = ∑ x nrDx

Let Rx and Dx be the concentrations of “live” and “dead” polymer chains, and let rRx and rDx be the corresponding net rates of production of “live” and “dead” polymer chains, respectively. On the basis of the postulated kinetic mechanism one can derive the following rate functions for rRx and rDx by combining the individual rates of generation and consumptions of “live” and “dead” polymer chains.34

Using the above definitions and the net production rate functions (eqs 1 and 2), one can easily obtain the corresponding moment rate equations for the leading moments (i.e., λn and μn ; n = 0, 1, 2, and 3) of “live” and “dead” number chain length distributions (NCLDs).

Net production rate function of “live” polymer chains of chain length x:

Rate functions for the zero, first, second, and third moment of “live” NCLD:

x=1

∞ | l o o o o rR x = m k R * M + ( k M + k S ) ∑ R δ(x − 1) } x I tm ts o o o o x=1 n ~

x=1

(4)

Nd

rλ0 =

∑ 2fi kdiIi − (k tc + k td)λ02 i=1

∞ ∞ ij ∞ yz j z j z + k tpxDxjj∑ R x zz − k tpR x ∑ xDx − k tcR x ∑ R x j x=1 z x=2 x=1 k {

(5)

+ k p(R x − 1 − R x)M − (k tmM + k tsS)R x

∞

∞

∞

− k tdR x ∑ R x − kβR x ∑ xDx + kβ ∑ R x x=1

x=2

x=1

Nd

rλ1 =

i=1

∞

∑

∑ 2fi kdiIi − (k tmM + k tsS)λ1 + k pMλ0 i1 y + kβ jjj λ 0μ2 − λ1μ1zzz k2 {

− (k tc + k td)λ 0λ1 + k tp(λ 0μ2 − λ1μ1)

Dy

y=x+1

(1) 13096

(6) DOI: 10.1021/acs.iecr.9b02517 Ind. Eng. Chem. Res. 2019, 58, 13093−13111

Article

Industrial & Engineering Chemistry Research Nd

rλ2 =

Table 1. Numerical Values of the Kinetic Rate Constants Used in the Present Study

∑ 2fi kdiIi − (k tmM + k tsS)λ2 + 2k pMλ1 i=1

i1 y 1 1 + kβ jjj λ 0μ3 − λ 0μ2 + λ 0μ1 − λ 2μ1zzz 3 2 6 k { − (k tc + k td)λ 0λ 2 + k tp(λ 0μ3 − λ 2μ1)

(7)

Nd

rλ3 =

∑ 2fi kdiIi − (k tmM + k tsS)λ3 + 3k pMλ2 i=1

i1 y 1 1 + kβ jjj λ 0μ4 − λ 0μ3 + λ 0μ2 − λ3μ1zzz 2 4 k4 { − (k tc + k td)λ 0λ3 + k tp(λ 0μ4 − λ3μ1)

(8)

× × × × × × × × × ×

1.63 1.06 2.00 2.59 2.50 2.50 2.59 3.00 2.54 2.13

10 1016 1016 108 109 109 105 1010 106 108

Ea (kJ/mol)

ΔV (cm3/mol)

126.10 148.79 165.00 37.66 4.184 4.184 37.66 69.04 50.54 50.60

4.9 2.5 10 −19.7 13 13 −19.7 4.41 −16.8 −19.7

(16b)

kdcm3

C2H4 ⎯⎯⎯⎯→ C + CH4 (9)

(16c)

Ethylene decomposition is a highly exothermic reaction (about 125 kJ/mol). Thus, once initiated, it proceeds rapidly, consuming the ethylene and causing large temperature and pressure increases. Ethylene decomposition can be triggered by local temperature hot spots in a reaction zone because of high local polymerization rates, as a result of high local initiator concentrations due to nonideal mixing phenomena. Neuman and Luft36 experimentally investigated the ethylene decomposition and proposed the following ethylene decomposition rate function:

(10)

rμ2 = (k tmM + k tsS)λ 2 + (k tc + k td)λ 0λ 2 + k tcλ12 − k tp(λ 0μ3 − λ 2μ1) + kβ ÄÅ ÉÑ ÅÅ Ñ ÅÅλ μ − λ ijj 2 μ + 1 μ − 1 μ yzzÑÑÑ 0j ÅÅ 2 1 3 2 1zÑ 2 6 {ÑÑÖ ÅÇ k3

kd,TBPEH kd,TBPOA kd,DTBP kp ktc ktd ktm ktp kβ kts

14

kdcm2

rμ1 = (k tmM + k tsS)λ1 + (k tc + k td)λ 0λ1

1 i y − k tp(λ 0μ2 − λ1μ1) + kβ jjjλ1μ1 − λ 0μ2 zzz 2 k {

k0

C2H4 ⎯⎯⎯⎯→ 2C + 2H 2

Rate functions for the zero, first, second, and third moment of “dead” NCLD: i1 y rμ0 = (k tmM + k tsS)λ 0 + jjj k tc + k td zzzλ 02 + kβλ 0μ1 k2 {

kinetic rate constants

i 32.4 yz zz[M] re,dec = 1.1 × 1012 expjjj− k RT {

(11)

(17)

rμ3 = (k tmM + k tsS)λ3 + (k tc + k td)λ 0λ3 + 3k tcλ1λ 2 ÄÅ ÅÅ i3 1 − k tp(λ 0μ4 − λ3μ1) + kβÅÅÅλ3μ1 − λ 0jjj μ4 − μ3 ÅÅÇ 4 2 k É Ñ 1 yÑÑ − μ2 zzzÑÑÑ 4 {ÑÑÖ (12)

where re,dec is the ethylene decomposition rate (mol/(m3 h)), [M] is the ethylene concentration (mol/m3), T is the polymerization temperature (K), and R is the universal gas constant (kcal/mol). Neuman and Luft36 measured the enthalpy of ethylene decomposition in terms of temperature (K) and pressure (atm) and proposed the following equation for ΔHe,dec (kcal/mol):

Finally, the net consumption rates for initiator(s) and monomer will be given by eqs 13 and 14, respectively.

ΔHe,dec = −29.48 − 8.1 × 10−4T + 1.5 × 10−4(P − 500) (18)

Initiator(s) consumption rate(s): rIi = −kdiIi

;

i = 1, ..., Ni

30

Wells and Ray, following the original developments of Zhang et al.,37 proposed a simplified equation for the calculation of the heat released rate, Qe,dec, due to ethylene decomposition in terms of the ethylene decomposition rate, re,dec, and enthalpy, ΔHe,dec. The ethylene decomposition enthalpy, ΔHe,dec, was set equal to 30.2 kcal/mol.

(13)

Monomer consumption rate: rΜ = −k pMλ 0

(14)

In Table 1, the numerical values of the kinetic rate constants used in the present study are reported.30,34 The kinetic rate constants do follow the general Arrhenius equation (eq 15). k=

i E + P ΔV zy zz −jjj a k 0 e jk RT z{

kdcm1

(19)

re,dec = −(1.89k md + k pd1)[M]2 + 0.0714k pd2[M]

(20)

(15)

In Table 2, the numerical values of the three kinetic parameters in eq 19, kmd, kpd1, and kpd2, are reported. 2.2. Characteristic Mixing and Reaction Times. In general, turbulent flows are characterized by two length scales, namely, the integral scale, L, where the inertial subrange begins, and the Kolmogorov length scale, λk, where the inertial subrange ends.38 These two length scales are given by eqs 21 and 22, respectively.

2.1.1. Thermal Decomposition of Ethylene. At high temperatures, above 320 °C, ethylene can undergo thermal decomposition in the absence of an oxidant.35 It is wellestablished that the main ethylene decomposition reactions result in the formation of carbon, hydrogen, and methane. 2C2H4 ⎯⎯⎯⎯→ CH4 + 3C + 2H 2

Q e,dec = ΔHe,dec × re,dec

(16a) 13097


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Industrial & Engineering Chemistry Research Table 2. Ethylene Decomposition Kinetic Rate Constantsa k = k0 e−(E+PΔV)/RT

k0

kmd kpd1 kpd2

4.00 × 10 1.59 × 1020 4.39 × 1020

Ea (kJ/mol)

ΔV (cm3/mol)

271.96 271.96 271.96

−8.0 13.29 −8.0

19

molecular diffusion. Later, this concept was replaced by the engulfment model, which is a more realistic way of treating the breakup of the added reactant to a vessel. Accordingly, they define the following engulfment/micromixing time, te, in terms of engulfment rate, E. te =

ij ν 3 yz λk = jjj zzz kε{

1/2

(21)

0

(22)

where E(k) dk is the turbulent kinetic energy contained in the wavenumber range [k to k + dk], ν is the kinematic viscosity of the fluid, and ε is the turbulent energy dissipation rate. Eddies cascade energy down through smaller and smaller scales until the turbulent energy is dissipated by viscosity to the smallest scales of motion.38 The Kolmogorov length scale is the size of the smallest turbulent eddy. The time required for the dissipation of Kolmogorov sized eddies, tk, is be given by λ 1 ijj ν 3 zyz tk = k = jj zz uλ′ (νε)1/4 k ε {

1/4

iνy = jjj zzz kε{

1/2

(23)

For isotropic homogeneous turbulence, Corrsin determined the time required for the reduction of largest scale concentration fluctuations, LS, through the full range of the inertial convective scales of turbulence to the Kolmogorov scale, λk, and then through the viscous scales to the Batchelor scale, λB, by integrating the scalar concentration and turbulence spectra.39 tm = 2jj zz j ε z k {

1iνy + jjj zzz 2kε{

tr =

1/2

for liquids where Sc ≫ 1

(26)

2.2.1. Importance of Damkoehler Mixing Number in a High-Pressure LDPE Autoclave. In dealing with nonideal mixing phenomena in a LDPE autoclave two quantities are of interest. The first is the blob size of the additive initiator feed stream. The second is the rate of disappearance of the additive initiator feed stream or the inverse local mixing time. For low viscosity liquids and small size reactors (e.g., liter), the local mixing time (tm) can be as short as 0.01 s (i.e., very rapid mixing). However, local mixing times of the order of 0.1 s or longer can typically occur in larger-size reactors due to limitations in mechanical agitation. Estimating the local mixing time at a given point in a reaction zone is not easy and will be strongly affected by the reactor configuration, impeller design and mixing intensity, the reaction conditions (e.g., temperature, solution viscosity, species diffusion coefficient, etc.) as well as the way the reactants are fed into the reactor. However, the value of the characteristic reaction time will determine the extent of a chemical reaction in one characteristic mixing time. The characteristic reaction time for an elementary chemical reaction (i.e., assuming a power law kinetic rate model) will be given by

1/4

1/3 ij Ls2 yz j z

(25)

iεy E = 0.058jjj zzz kν{

∞

3π ∫0 Ε(k)/k dk L= ∞ 4 ∫ E (k ) d k

1 iνy = 17.3jjj zzz Ε kε{

1/2

a Adapted with permission from ref 30. Copyright 2005 American Institute of Chemical Engineers (AIChE).

ln(Sc)

1 , n is the order of reaction kC n − 1

(27)

Accordingly, in a free-radical polymerization, one can define the following characteristic reaction times for initiator decomposition, td, chain propagation, tp, and chain termination, tt:

(24)

LS is the local scale of segregation or the average size of unmixed regions, and ε is the local turbulent energy dissipation rate. The first term in eq 24 arises from the large inertial scales which contain most of the turbulent energy. The second term denotes the time required to reduce a fluid blob from the Kolmogorov length scale to the Batchelor length scale for large values of Sc, where molecular diffusion is much slower than the diffusion of momentum. Baldyga and Bourne40 and Baldyga and Pohorecki41 restated the second term in eq 24 as 2 arcsinh(0.05Sc)(ν /ε)1/2 using a somewhat more rigorous derivation than Corrsin’s. In both cases, this term will be very small most of the time.39 However, for highly viscous reactive systems and/or low values of mass diffusivity of reactive species, the numerical value of Sc number can be significantly larger than 1 (i.e., Sc ≫ 1), thus, the contribution of the second term in eq 24 in the calculation of tm can be notable. Alternative expressions for calculating the local mixing time have been developed by Bourne and co-workers.20,42,43 In their earliest micromixing investigations, it was assumed that the added material did not do anything until the Kolmogorov scale was reached and the subsequent mixing took place by

td =

1 1 1 ; tp = ; tt = kd k p⌊λ 0 ⌋ k t⌊λ 0 ⌋

(28)

where kd, kp, and kt are the respective rate constants for initiator decomposition, chain propagation, and chain termination reactions. In Table 1, the decomposition rate constants for three initiators commonly used in high-pressure free-radical ethylene polymerization as well as the propagation and termination rate constants are reported. [λ0] is the steady-state concentration of “live” polymer chains in a reaction zone given by [λ 0] = (2fkd[I] /kt )1/2

(29)

On the basis of the above definitions (eqs 27 and 28), the dimensionless Damkoehler mixing number is defined as Da = tm/tr (i.e., the ratio of the characteristic mixing time over the characteristic reaction time). Thus, based on the last definition, we can obtain the following Damkoehler mixing numbers for initiator decomposition, polymer chain propagation, and termination reactions: 13098


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Table 3. Initiator Decomposition, Propagation, and Termination Characteristic Reaction and Mixing Times (s) at Different Temperaturesa temperature, °C

td,TBPEH

td,TBPOA

td,DTBP

tp

tt

tm

200 220 240 260 280 300

0.617 0.167 0.050 0.016 0.006 0.002

2.691 0.594 0.147 0.041 0.012 0.004

120.455 21.644 4.446 1.028 0.264 0.075

26.168 18.298 13.157 9.697 7.306 5.615

0.00190 0.00179 0.00168 0.00160 0.00152 0.00145

0.1960 0.1943 0.1929 0.1917 0.1905 0.1895

a td,TBPEH, td,TBPOA, and td,DTBP are the characteristic initiator decomposition characteristic times for the three initiators, tp and tt are the characteristic propagation and termination reaction times, and tm is the Corrsin mixing time (eq 24). The numerical value of the local turbulent energy dissipation rate, ε, can be obtained from CFD calculations for a given reactor design and agitation system. In this study, the value of ε was set equal to 20 W/kg.

Figure 3. Effect of polymerization temperature on the initiator Damkohler mixing number. For the calculation of tm (see eq 24), the value of ε was set equal 20 W/kg.

DaI =

tm = kd,Itm 1/kd,I

DaP =

tm = tmk p⌊λ 0 ⌋ 1/k p⌊λ 0 ⌋

Da t =

tm = tmk t⌊λ 0 ⌋ 1/k t⌊λ 0 ⌋

the numerical value of DaI determines the importance of micromixing in a reaction zone of a multizone autoclave. Thus, if the value of the Damkoehler mixing number (DaI) is less than 1 (i.e., tm < td,I, a slowly decomposing initiator at a given polymerization temperature), then the initiator feed stream can be adequately mixed with the bulk fluid in a reaction zone, resulting in an efficient utilization of initiator (i.e., high specific initiator consumption; see minima in specific initiator consumption curves in Figure 2). However, if the Damkoehler mixing number DaI is larger than 1 (i.e., tm > td,I, a fast decomposing initiator at a given polymerization temperature), then the numerical values of Dap and Dat will determine whether the primary initiator radicals will prematurely selfterminate and/or react with monomer starting a chainpropagation reaction. This means that the ratio Rp/Rt (= kp[M] [λ0])/kt[λ0]2) will determine the extent to which newly formed initiator primary radicals will participate in polymer chain-propagation reactions or undergo premature termination. In the former case, the local polymerization rate and thus the generated polymerization heat (i.e., in the initiator injection zone) can be very high resulting in a high polymerization temperature. In contrast, for low values of (Rp/Rt) ratio, the amount of initiator required to maintain a certain temperature in a reaction zone increases; thus, the specific initiator consump-

(30)

In Table 3, the characteristic reaction and mixing times (s) for the three initiators of Table 1 are reported. It should be pointed that for the calculation of the characteristic mixing time, tm, the Corrsin eq 24 was used. The numerical value of the kinematic viscosity (ν = viscosity/density) will depend on the solution viscosity (i.e., a complex function of temperature, polymer concentration, and MWD of LDPE) and density of the polymerization mixture (i.e., as well a function of temperature, pressure, and polymer content). 2.2.2. Effect of Micromixing Conditions on Specific Initiator Consumption. In Figure 3, the variation of DaI with respect to the polymerization temperature is plotted for the three initiators of interest. For the calculation of the Damkoehler mixing numbers for the three initiators. For a given initiator and known reactor conditions (i.e., polymerization temperature, polymer mass fraction and MWD, etc.), 13099


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Figure 4. (a) Engulfment model. (b) Effect of solution viscosity and energy dissipation rate on engulfment rate, E.

Figure 5. Effect of engulfment parameter E on the evolution of the polymerization temperature in the plume: a: pure initiator feed; b: initiator is mixed with ethylene.

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local hot spots. Subsequently, the two initiator feeding policies are examined using the engulfment model. 2.3.1. Effect of Initiator Concentration in the Feed Stream on Ethylene Decomposition. In Figure 5a the effect of engulfment parameter E on the time evolution of the temperature in the “plume”, T1, is illustrated for a pure initiator feed stream to a reaction zone. That is, the initiator feed is not diluted with ethylene. In this case, the initiator concentration at the feed injection point is very high, while its respective mass flow rate is low. As can be seen from the results of Figure 5, for low values of E a significant increase in the plume temperature is observed. Note that the temperature in the “plume” can increase more than the ethylene decomposition temperature (i.e., ≅330 °C). Figure 5 shows the effect of engulfment parameter E on the evolution of the “plume” temperature for the case of a diluted initiator feed stream, that is, the initiator is fed together with the ethylene into the reaction zone. In this case, the initiator concentration in the feed stream is very low, while the total volumetric ethylene flow rate is high. As can be seen, the temperature in the plume does not exceed the specified bulk temperature in the reaction zone, that is, it exhibits an overdamped dynamic behavior. From the above analysis, it is evident that the introduction of highly concentrated initiator feed streams to an autoclave should be avoided for it can result in local temperature overheating that can trigger the ethylene decomposition reactions.

tion decreases (see specific initiator consumption curves in Figure 2). Subsequently, the operating conditions that may lead to ethylene decomposition in a reaction zone in a multizone autoclave are examined using the engulfment model. 2.3. Engulfment Model. The way in which reactants are mixed in a reactor can have a large influence on the product distribution in a complex chemical reaction system. This mixing process was analyzed in detail by Baldyga and Bourne,19 and in particular, when mixing on the molecular scale is the controlling step (i.e., mixing in the viscousconvective and viscous diffusive parts of the concentration spectrum). Baldyga and Bourne19 proposed a mixing mechanism describing the engulfment, deformation, and diffusion (EDD model) of a reactant stream fed into a reaction zone. This model was later simplified to the “engulfment model” in which the engulfment step is only considered, since in many cases this is the rate determining mixing step. In liquids, micromixing takes place by molecular diffusion, laminar deformation of striations below the Kolmogorov scale, and the mutual engulfment of regions having different compositions leading to growth of the micromixed volume. Under frequently occurring mixing conditions, engulfment is the limiting step of these three mechanisms.20 The formulation of the engulfment model for a single LDPE zone resembles the growing plume model of Zwietering11 (Figure 4). Engulfment of one fluid by another is caused by vorticity and results in the growth of the volume of spots of a substance (inert tracer or reactant) at the expense of the environment according to the following equation: dV1(t ) = E V1(t ) dt

3. SEGREGATION−BACKMIXING MODEL The present model developments were driven by a number of real process requirements including (i) dynamic simulation of industrial LDPE autoclaves using a multizone reactor configuration, (ii) prediction of the varying specific initiator consumption in a zone as a function of the specified zone operating temperature, (iii) prediction of ethylene decomposition onset in an autoclave, and (iv) use of the developed multizone model for process monitoring, process optimization, and control of industrial-scale autoclaves. A great deal of papers have been published on the effect of macro-/micromixing in chemical reactors. This is especially true in the free-radical ethylene polymerization in highpressure multizone LDPE autoclaves in which a mixture of cold monomer and initiators are fed into a number of hot reaction zones. The ethylene−initiator mixture injected into a reaction zone forms a plume near the injection point, in which the initiator concentration is higher than that in the bulk of the reaction zone. In the present study, a nonideal segregation− backmixing model is developed to describe macro- and micromixing phenomena in an industrial-scale LDPE autoclave. To introduce the basic idea behind the proposed segregation−backmixing model, it is assumed that the ethylene−initiator feed stream, after its introduction into a reaction zone, is broken up by the action of stirrer (i.e., turbulent energy dissipation rate) into “lumps of fluid” which are progressively mixed with the bulk polymerization mixture. In particular, a feed stream entering a reaction zone undergoes a number of successive mixing steps including (i) transport through the bulk of the reactor by means of flow circulation, induced by the impeller, and (ii) breakage of feed stream by the action of turbulent whirls into fluid lumps of smaller length scale. However, polymerization is not taking place in the “fluid lumps” yet, since they are still completely segregated from the

(31)

where E is the engulfment rate. The variation of the engulfment rate in terms of the solution viscosity and energy dissipation rate is depicted in Figure 4 (see eq 26). Following the original developments of Smit,12 one can derive the following dynamic mass and energy balances for the monomer, [M1], and initiator, [I1], concentrations as well as the temperature, T1, in the plume. ij 2fk [I ] yz d[M1] = E([M1] − [M 2]) − k p[M1]jjj d 1 zzz j kt z dt k {

(32)

d[I1] = E([I1] − [I 2]) − kd[I1] dt

(33)

ij 2fk [I ] yzji −ΔH zyz dT1 z = E(T1 − T2) − k p[M1]jjj d 1 zzzjjjj j k t zj ρCp zzz dt k {k {

(34)

where [M2], [I2], and T2 denote the steady-state values of the corresponding variables in the bulk of a reaction zone, V2. From the numerical solutions of eqs 26 and 31−34, the polymerization temperature in the plume can be calculated under specific mixing and polymerization conditions. In a multizone autoclave, the initiator(s) stream injected to the top zones of the reactor is commonly mixed with an ethylene stream before entering a reaction zone. Thus, it is in a diluted state. However, in the bottom zone(s) of an autoclave, the initiator is not mixed with ethylene which means that a highly concentrated initiator feed stream is directly injected into a zone. This feeding policy may lead to the appearance of 13101


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Figure 6. Schematic representation of the proposed segregation−backmixing model for a multizone LDPE autoclave.

the characteristic initiator micromixing/erosion time of “fluid lumps”, in terms of the kinematic viscosity of the reaction mixture, ν, initiator diffusion coefficient, D, and local turbulent energy dissipation rate, ε, in a compartment.

bulk reaction mixture and are at relatively lower temperature than that in the bulk. Following this macromixing process, mixing at a molecular scale of the feed fluid lumps with the bulk fluid takes place. Since turbulent velocity fluctuations can reduce the size of fluid lumps to a certain minimum scale (the so-called Kolmogorov length scale), a different mechanism is needed to reduce the lump size even further. It has been postulated that this is done by viscous shearing of adjacent layers of feed and bulk liquid in the energy containing vortices (turbulent whirls having a size of approximately 12 times the Kolmogorov scale). Being deformed into thin lamellae, the initiator feed stream is finally micromixed with the bulk by means of diffusion. Subsequently, “primary” radicals formed via the decomposition of initiator molecules in the bulk can react with ethylene via a chain propagation reaction that results in the formation of long polymer chains with simultaneous release of heat. In Figure 6, the proposed segregation−backmixing model is shown schematically for a multizone autoclave. Q, V1, V2, R (= QR/Q), and δ (= Qb/Q) denote the total volumetric feed rate into a reaction zone, the volume of the feed injection compartment 1, the volume of the bulk compartment 2, the recycle flow ratio between the compartments 1 and 2 (recycle parameter, R), and the flow exchange ratio between two adjacent zones (backmixing parameter, δ), respectively. Initial values of V1, V2, R, and δ parameters in a multizone autoclave configuration can be obtained from CFD simulations. Moreover, fine-tuning of the key model parameters, R and δ, can be carried out based on real-time temperature and initiator feed rate measurements collected from an industrial multizone autoclave. In addition to the above macromixing parameters, the values of the characteristic initiator micromixing parameters tm1 and tm2 in the respective compartments 1 and 2 in a reaction zone should be known. Plasari and Villermaux44 and Villermaux and David45 proposed the following equation for the calculation of

tm = tm ≈ l0 × ν 5/2 × D−2/3 × ε−1/4

(35)

where l0 is the initial size of the initiator “fluid lumps” (e.g., 50−500 μm). The numerical values of ε can be obtained from CFD simulations. The kinematic viscosity ν will depend on the solution viscosity (i.e., a function of temperature, polymer concentration, and polymer MW) and the density of the polymerization mixture (i.e., a function of temperature, pressure, and polymer content). In the proposed segregation−backmixing model, it is assumed that the initiator feed stream entering the compartment V1 is in a segregated state (“fluid lump”), at a concentration [Iseg 0 ]. Subsequently, it undergoes two different mixing processes, namely, a macromixing process with the bulk fluid in compartment V2 and an erosion−micromixing process during which the initiator fluid lumps are mixed, at molecular scale, with the bulk fluid. Accordingly, one can derive the following dynamic mass and energy balances for the initiator(s) (in both segregated, [Iseg i,j ], and molecular state, [Ii,j]), monomer concentration, leading moments of “live” and “dead” NCLDs as well as the polymerization temperature in compartments 1 and 2 in the “j” reaction zone of a multizone autoclave. Compartment 1 in zone “j”, V1,j: V1, j

d[I1,segj ] dt

seg seg seg = Q F, j[I seg 0, j ] + Q R, j[I 2, j ] − Q 1, j[I1, j ] − rI,1, jV1, j

− 13102

[I1,segj ] tm1, j

V1, j

(36) DOI: 10.1021/acs.iecr.9b02517 Ind. Eng. Chem. Res. 2019, 58, 13093−13111

Article

Industrial & Engineering Chemistry Research V1, j

d[I1, j] dt

= Q R, j[I 2, j] − Q 1, j[I1, j] − rI,1, jV1, j +

[I1seg] V1, j tm1

ρ2, j Cp2, jV2, j (37)

V1, j

d[M1, j]

− rM,1, jV1, j V1, j

V1, j

d[λi ,1, j]

d[μi ,1, j ] dt

ρ1, j Cp1, jV1, j

− ρ2, j Cp2, jQ R, j(T2, j − Tref ) − ρ2, j Cp2, jQ 2, j(T2. j − Tref ) − ρ2, j Cp2, jQ b, j(T2, j − Tref ) (39)

+ (( −ΔΗ pol)rM,2, j + ( −ΔΗdec)rdec,2, j)V2, j (47)

= Q R, j[μi ,2, j ] − Q 1, j[μi ,1, j ] + rμi ,1,jV1, j

dT1, j

where QR,j = RjQ2,j and Qb,j = δjQ2, j, j = 1, 2, ..., Nzones and i = 0, 1, 2, and 3. All symbols are defined in the List of Symbols. It should be noted that the values of the kinetic rate constants as well as the viscosity of the polymerization mixture are calculated at the corresponding temperature in each compartment.

(40)

= ρF CpFQ F, j(T0 − Tref )

dt

+ ρ2, j Cp2, jQ R, j(T2, j − Tref ) − ρ1, j Cp1, jQ 1, j(T1, j − Tref )

4. SPECIFIC INITIATOR CONSUMPTION AND ETHYLENE DECOMPOSITION IN A REACTION ZONE It is well-known that the nonideal mixing phenomena in an autoclave do depend on the specific design characteristics of the vertical stirrer shaft (e.g., design, location and number of impellers, etc.), the intensity of agitation (spatial distribution of local turbulent energy dissipation rate), half-lives of initiators, and operating conditions in a reaction zone (i.e., temperature profile, viscosity of the reaction mixture, MWD, polymer mass fraction, etc.). This means that an autoclave operator can change the operating conditions in the reactor but the nonideal mixing in a zone are strongly linked with the specific design characteristics of the stirrer shaft. In the present study, using the segregation−backmixing model, the effects of the above design and operating parameters on the specific initiator consumption (grams of initiator consumed per kilogram of LDPE) as well as on local temperature gradients in a reaction zone were analyzed by varying the key model parameters, namely, the macromixing parameters V1, V2, and R and the micromixing parameters tm1 and tm2. The parameters V1, V2, R, and δ are directly linked with the design characteristics of an autoclave (i.e., number and size of reaction zones, initiator injection points, number and type of impellers in a zone, etc.). The values of the macromixing parameters can be obtained from CFD simulations33 and/or from plant data on initiator feed rate to a zone and local temperatures. This information is strictly proprietary to a company. The values of the micromixing parameters (tm1 and tm2) can be estimated by eqs 24, 25, and 35. The reaction zone is assumed to consist of two compartments V1 and V2 and a recycle stream R. The design and operating conditions used in all simulations are reported in Table 4.30 The numerical values of the kinetic rate constants used in the present simulations are reported in Tables 1 and 2. The ability of the proposed segregation−backmixing model to predict the variable specific initiator consumption (in grams of initiator consumed per kg of LDPE produced) and temperature gradients in a reaction zone is assessed. Present model predictions are directly compared with the CFD simulation results and 100 compartment model of Wells and Ray.30

+ (( −ΔΗ pol)rM,1, j + ( −ΔΗdec)rdec,1, j)V1, j (41)

Compartment 2 in zone “j” (V2,j= Vtot,j − V1,j): V2, j

d[I seg 2, j ]

seg = Q 1, j[I1,segj ] + Q b, j + 1[I seg 2, j + 1] + Q 2, j − 1[I 2, j − 1]

dt

seg seg − Q R, j[I seg 2, j ] − Q 2, j[I 2, j ] − Q b, j[I 2, j ] seg − rI,2, jV2, j −

V2, j

d[I 2, j] dt

[I seg 2, j ] tm2

V2, j

(42)

= Q 1, j[I1, j] + Q b, j + 1[I 2, j + 1] + Q 2, j − 1[I 2, j − 1] − Q R, j[I 2, j] − Q 2, j[I 2, j] − Q b, j[I 2, j] − rI,2, jV2, j +

V2, j

d[M 2, j] dt

[I seg 2 ] V2, j tm2

(43)

= Q 1, j[M1, j] + Q b, j + 1[M 2, j + 1] + Q 2, j − 1[M 2, j − 1] − Q R, j[M 2, j] − Q 2, j[M 2, j] − Q b, j[M 2, j] − rM,2, jV2, j

V2, j

d[λi ,2, j] dt

(44)

= Q 1, j[λi ,1, j] + Q b, j + 1[λi ,2, j + 1] + Q 2, j − 1[λi ,2, j − 1] − Q R, j[λi ,2, j] − Q 2, j[λi ,2, j] − Q b, j[λi ,2, j] + rλi ,2,jV2, j

V2, j

d[μi ,2, j ] dt

= ρ1, j Cp1, jQ 1, j(T1, j − Tref )

+ ρ2, j − 1Cp2, j − 1Q b, j − 1(T2, j − 1 − Tref ) (38)

= Q R, j[λi ,2, j] − Q 1, j[λi ,1, j] + rλi ,1,jV1, j

dt

dt

+ ρ2, j + 1Cp2, j + 1Q b, j + 1(T2, j + 1 − Tref )

= Q F, j[M 0] + Q R, j[M 2, j] − Q 1, j[M1, j]

dt

dT2, j

(45)

= Q 1, j[μi ,1, j ] + Q b, j + 1[μi ,2, j + 1] + Q 2, j − 1[μi ,2, j − 1] − Q R, j[μi ,2, j ] − Q 2, j[μi ,2, j ] − Q b, j[μi ,2, j ] + rμi ,2,jV2, j

(46) 13103


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(i.e., the mean residence time of the fresh initiator feed into compartment 1 decreases), the temperature difference in the two compartments increases (i.e., becomes larger than 20 °C), meaning that significant temperature gradients can appear in the reaction zone. At this point, it is important to mention that one of the key features of the present segregation−backmixing model is that the continuously fed initiator into a reaction zone can exhibit two distinct states, namely, a segregated state [Iseg],in the form of initiator aggregates (“lumps”), and a molecular state in the continuous bulk phase, [I]. Thus, via the proposed micromixing/erosion mechanism, initiator molecules are transferred from an “aggregated” state to the “molecular” one. Initiator molecules in the “aggregated” state can undergo decomposition, but the generated “primary radicals” are selfterminated (i.e., have an almost zero efficiency in initiating new polymer chains). This means that only the initiator molecules in the bulk phase (“molecular state”) can initiate new polymer chains and thus contribute to ethylene polymerization. As a result, the higher the initiator fraction in “aggregated state”, the lower the overall initiator efficiency. The effect of recycle parameter, R, on the steady-state temperatures in the two compartments is shown in Figure 8 for a value of (V1/Vtot) ratio equal to 0.1. It is evident that as the recycle ratio increases the temperature difference in the two compartments decreases due to the improved initiator homogeneity at molecular scale in the two compartments. In contrast, as the recycle ratio decreases, the initiator concentrations in the two compartments are significantly different and thus so are the respective polymerization rates and temperatures in the two compartments of a reaction zone. In Figure 9, the effect of characteristic mixing/erosion times tm1 and tm2 on the steady-state temperatures in the two compartments is illustrated. The characteristic mixing times tm1 and tm2 in Figure 9 were calculated from eq 35 by varying the value of l0 (i.e., the initial size of the initiator fluid “lumps”) from 10 to 500 μm. It is evident that as the characteristic mixing/erosion time increases (i.e., the fraction of initiator in aggregated state increases) the temperatures in both compart-

Table 4. Reactor Design and Operating Parameters operating conditions

value

reactor volume, Vtot pressure, P mean residence time, τ initiator feed concentration recycle parameter, R V1/Vtot initiator efficiency, f 0 dissipation energy rate, W/kg

500 L 2000 bar 32.8 s 2−200 ppm 2−20 0.10 1.0 30

Finally, an analysis is carried out, using the developed segregation−backmixing model, to identify possible process faults that can lead to ethylene decomposition in a reaction zone during the operation of a multizone LDPE autoclave. 4.1. Effect of Macro- and Micromixing Model Parameters. In Figures 7−9, the effects of model parameters V1, V2, R, tm1, and tm2 on the calculated steady-state temperatures in the two compartments of a single reaction zone are shown. Estimates of the compartment volumes V1 and V2 can be obtained either from CFD simulations and/or from plant data on initiator feed rate to a zone and local temperature gradients. The numerical values of tm1 and tm2 in the two compartments were obtained from the application of eq 35 to each compartment. In Figure 7, the effect of the volume fraction of compartment 1, V1/Vtot, on the steady-state temperatures in compartments 1 and 2 is shown for a value of the recycle ratio R = 2. As can be seen as the volume fraction V1/Vtot increases, the temperature difference between the two compartments decreases. The observed decrease in the temperature difference between the two compartments is due to the increased mean residence time of initiator lumps in compartment 1, V1/Q. This results in an increase of the initiator concentration in the bulk phase of compartment 1, [I1], due to the increase in the erosion time of initiator “aggregates”. As a result, both the polymerization rate and temperature in compartment 1 increase. However, as the volume fraction V1/Vtot decreases

Figure 7. Effect of the volume fraction V1/Vtot on the steady-state temperatures in the two compartments of a reaction zone for R = 2 (Tfeed = 420 K, l0 = 100 μm in eq 35). 13104


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Figure 8. Effect of recycle parameter R on the steady-state temperatures in the two compartments for V1 = 0.10 Vtot (Tfeed = 420 K, l0 = 100 μm in eq 35).

Figure 9. Effect of the characteristic mixing times tm1 and tm2 on the steady-state temperatures in the two compartments for R = 10, V1 = 0.1Vtot, Tfeed = 420 K.

is illustrated. The applied operating conditions and the numerical values of all kinetic parameters are reported in the original paper of Wells and Ray.30 In Figure 10, a direct comparison of segregation−backmixing model simulation results (discrete blue solid points) to the original CFD simulations (solid black line) as well as the multicompartment results (i.e., 100 compartments) of Wells and Ray30 is presented. The numerical values of the segregation−backmixing model parameters used in the comparison study were as follows: V1 = 0.1Vtot and R = 10. The characteristic initiator erosion−mixing parameters, tm1 and tm2, were calculated from eq 35. The temperatures calculated by the segregation− backmixing model shown in Figure 10 (blue solid circles) represent the steady-state temperatures in the second compartment, V2. It is apparent that the predictions of the present segregation−backmixing model are in very good agreement with the CFD simulations as well as with the results of 100 compartment model of Wells and Ray.30

ments decrease due to the lower polymerization rate as well as the lower polymerization heat. In this case (i.e., long characteristic mixing times), the specific initiator consumption increases, meaning that a higher initiator feed rate is required to maintain the polymerization rate and temperature in a zone at specified values. High initiator flow rates to a reaction zone due to nonideal mixing (i.e., high specific initiator consumption) can increase the amount of residual initiator from a current zone to the zone below (usually operating at a higher temperature) and thus increase the risk for appearance of a local hot spot. 4.2. Comparison of Segregation−Backmixing Model with the CFD Results of Wells and Ray.30 The simulation potential of the proposed segregation−backmixing model was tested by a direct comparison of present model predictions with the CFD results and 100 compartment model of Wells and Ray.30 In Figure 10, the effect of initiator feed concentration on the steady-state temperature in an autoclave 13105


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Figure 10. Effect of initiator concentration in the feed stream on the final steady-state temperature in a LDPE autoclave as predicted by CFD, the 100 compartment model of Wells and Ray30 and the present segregation−backmixing model (Tfeed = 420 K, l0 = 100 μm in eq 35, V1 = 0.1Vtot, R = 10). Data reprinted with permission from ref 30. Copyright 2005 American Institute of Chemical Engineers (AIChE).

Figure 11. Effect of initiator concentration in the feed stream on the final steady-state temperature in an LDPE autoclave as predicted by CFD, the 100 compartment model of Wells and Ray30 and the present segregation−backmixing model (Tfeed = 360 K, l0 = 40 μm in eq 35, V1 = 0.1Vtot, R = 10). Data reprinted with permission from ref 30. Copyright 2005 American Institute of Chemical Engineers (AIChE).

In Figure 11, the effect of initiator feed concentration on the outlet temperature in the autoclave is illustrated for an ethylene feed temperature of 360 K. It is apparent that there is an excellent agreement of present model predictions with the CFD and 100 compartment model results reported by Wells and Ray.30 At this point, it is important to emphasize that the present model is significantly simpler than the 100 compartments model of Wells and Ray and requires significantly less computational effort than the CFD model. Thus, the segregation−backmixing model can be practically used as an online simulation tool for process monitoring, process optimization and control. In Figure 12 the initiator mass fractions in “segregated” and “molecular” states in the two compartments (V1 = 0.1Vtot and R = 10) are plotted with respect to the initiator concentration in the feed stream. The initiator concentration in segregated state in compartment 1 is higher than that in compartment 2

due to the lower mean residence time of compartment 1. It is apparent that the initiator mass fraction in segregated state in compartment 1 increases as the initiator concentration in the feed stream increases. It should be noted that as the initiator mass fraction in segregated state increases the specific initiator consumption in a zone increases. Figure 13 illustrates the variation of the initiator segregation index (ISI) with respect to the initiator feed concentration. The initiator segregation index denotes the initiator mass fraction in segregated state in the two compartments. The simulation conditions are the same as in Figure 10. The initiator segregation index is a measure of the nonideal mixing conditions in a reaction zone. Thus, under ideal mixing conditions the value of ISI will be equal to 1, while for a completely segregated system the value of ISI will be equal to zero. As can be seen, the value of ISI decreases as the initiator feed concentration increases that results in a lower utilization 13106


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Ray30 shown in Figure 8 of their original paper. They actually showed that the so-called “effective reactor volume fraction” decreased as the initiator concentration in the feed stream increased (i.e., the exit reactor temperature increased). In fact, Wells and Ray clearly demonstrated via CFD simulations that at higher polymerization temperatures the “effective reactor volume fraction” becomes very small. They concluded that lower initiator concentrations in the feed stream resulted in a higher homogeneity of initiator and primary radicals in the autoclave and, thus, a better utilization of the reactor volume for ethylene polymerization with concomitant an improved specific initiator consumption. Finally, Figure 14 shows the specific initiator consumption with respect to the polymerization temperature as calculated

Figure 12. Mass fractions of initiator in segregated and molecular state in the two compartments as predicted by the segregation− backmixing model (Tfeed = 420 K, l0 = 100 μm in eq 35, V1 = 0.1Vtot, R = 10).

of the total amount of initiator fed into the autoclave and concomitant increase of the specific initiator consumption per kg of LDPE polymer. The inset figure depicts the variation of the overall initiator efficiency with respect to the temperature in the second compartment (i.e., for different initiator feed concentrations). In the present study, the overall initiator efficiency is defined as the ratio of the rate of polymer chains initiation by primary radicals over the total initiator decomposition rate in both compartments. The theoretical maximum initiator efficiency obtained under ideal mixing conditions (i.e., absence of macro- and micromixing phenomena) would be equal to 1. However, the present model simulations show that the actual overall initiator efficiency will in general be lower than 1 due to nonideal mixing conditions in LDPE autoclaves. Thus, as the initiator concentration in the feed stream increases, the steady-state temperature increases, while the overall initiator efficiency decreases. It is very interesting to point out that the results of Figure 13 are in full agreement with the results of Wells and

Figure 14. Specific initiator consumption with respect to the polymerization temperature as predicted by the segregation−backmixing model (Tfeed = 360 K, l0 = 40 μm in eq 35, V1 = 0.1Vtot, R = 10).

by the segregation−backmixing (S−B) model. The broken red line denotes the unstable branch of steady-states in the autoclave. The minimum in the specific initiator consumption

Figure 13. Variation of the initiator segregation index with respect to the initiator concentration in the feed stream as predicted by the segregation−backmixing model (Tfeed = 420 K, l0 = 100 μm in eq 35, V1 = 0.1Vtot, R = 10). 13107


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Figure 15. Effect of 10% increase in the initiator feed concentration (injected into a high-temperature polymerization zone) on ethylene decomposition (Tfeed = 420 K, l0 = 100 μm in eq 35, V1 = 0.1Vtot, R = 10).

concentration in the outlet stream leaving an imperfectly mixed zone that subsequently enters the zone below, usually operating at a higher temperature. As a consequence, the risk of a local hot spot in the zone increases.); (iv) the presence of a substantial initiator bypassing stream from an initiator injection zone to the zone below due to inefficient design and location of the impellers and so on. In Figure 15, the effect of a 10% increase in the initiator flow rate to a reaction zone is illustrated for two values of the initiator feed concentration. In particular, at t = 300 s the initiator concentration in the feed stream increases by 10% (i.e., to simulate potential faults in total concentration of radicals). As can be seen, when the reaction zone operates at a higher polymerization temperature (i.e., approximately 285 °C corresponding to a initiator feed concentration of 230 ppm), the temperatures in the two compartments start slowly increasing, and after approximately 2 min, the temperatures in the two compartments exhibit a very sudden increase caused by the highly exothermic ethylene decomposition reactions. However, when the reaction zone operates at a lower temperature (i.e., approximately at 270 °C corresponding to an initiator feed concentration of 100 ppm), the temperatures in the two compartments practically remain unchanged for the same step-change (i.e., a 10% increase) in the initiator feed concentration. Figure 16 depicts the effect of a step-change decrease (i.e., 25%) in the characteristic micromixing/erosion time which affects the “segregated” and “molecular” initiator concentrations in a reaction zone. As previously discussed, the initiator concentration in a “segregated state” does not contribute to the formation of new polymer chains and thus lowers the overall initiator efficiency. In the present study, eq 35 was used to calculate the characteristic micromixing/ erosion times (tm1 and tm2) in the two compartments of a reaction zone. The values of tm1 and tm2 will depend on the kinematic viscosity, ν, of the polymerization mixture (a function of temperature, pressure, polymer concentration, and polymer MW), turbulent energy dissipation rate, ε, initiator mass diffusivity, D, and the initial size of initiator

curve is very close to the minimum value reported by Wells and Ray30 for the TBPOA (tert-butyl peroxyacetate) initiator (blue discrete points). 4.3. Prediction of Ethylene Decomposition in a Reaction Zone of an Autoclave. Ethylene decomposition is known to occur occasionally in LDPE autoclaves. Once ethylene decomposition is initiated both temperature and pressure in the autoclave exhibit very sharp increases, in a matter of seconds, resulting in a reactor runaway.46−50 These runaways cannot be easily predicted during real-time operation of an industrial autoclave; thus, LDPE reactors are commonly fitted with relief valves in order to vent the products of ethylene decomposition and prevent the reactor from exploding. It has been suggested that the appearance of hot spots in an autoclave can initiate ethylene decomposition reactions.49 According to Zhang et al.,37 ethylene decomposition in an autoclave can be triggered by a number of incidents including ethylene feed impurities (reactive and nonreactive), mechanical friction, excess in initiator feed, poor distribution of the fresh initiator feed stream, insufficient micromixing, temperature controller failure, and/or poorly tuned controller, feed temperature disturbances, reactor defouling, and so on. The developed segregation−backmixing model was applied to a typical reaction zone of a multizone autoclave to simulate the effects of some of the process operating faults that may result in local hot spots and possible ethylene decomposition. Thus, the results shown in Figure 15 do actually represent a number of possible operational faults that may cause a sudden increase or decrease of initiator concentration in a reaction zone. These potential faults could include the following: (i) a sudden increase (decrease) of reactive impurities due to reactor purging; (ii) the variations in the initiator feed rate due to poorly tuned controller and/or controller failure; (iii) an increase in the initiator concentration of the incoming stream from the zone above to present zone of interest (High initiator flow rates to a reaction zone can be required in order to maintain a desired temperature due to nonideal mixing phenomena. This can result in an increased initiator 13108


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5. CONCLUSIONS In the present study, a comprehensive mathematical model was developed to simulate the dynamic behavior of multizone, multifeed high pressure ethylene polymerization autoclaves. A dynamic segregation−backmixing model was derived to describe macro- and micromixing phenomena in industrial autoclaves by considering two distinct states of initiator feed, namely, a nonreactive “segregated initiator state” and a reactive “molecular state” in the initiator-ethylene polymerization mixture. The model assumes that a part of initiator feed is in a segregated state that continuously interacts with the rest of the ethylene−initiator reaction medium via an erosion− micromixing process characterized by a characteristic mixing time, tm. Each reaction zone in a multizone autoclave configuration is characterized by three macromixing parameters, namely, the volume of compartment 1, V1, the recycle ratio R, and the backmixing parameter δ. The parameters V1, V2, R, and δ are directly linked with the design characteristics of an autoclave (i.e., number and size of reaction zones, initiator injection points, specific stirrer design, number and type of impellers in a zone, etc.). Estimates of these parameters can be obtained from CFD simulations, cold flow experiments, and/or by fitting model predictions to actual temperature and initiator feed rate measurements to a reaction zone. The initiator micromixing parameters, tm1 and tm2, in the two compartments in a reaction zone can be calculated by available correlations in terms of the kinematic viscosity of the polymerization mixture and local energy dissipation rate (e.g., eq 35). A general free-radical ethylene polymerization mechanism (including a series of ethylene decomposition reactions) was employed to represent the kinetics of ethylene polymerization. It was demonstrated that the proposed segregation−backmixing model can simulate the dynamic behavior of a reaction zone in multiscale LDPE autoclave (polymerization rate, monomer conversion, and presence of significant temperature gradients in a reaction zone) and thus can be employed in the design, simulation, optimization, and control of these reactors. It should be emphasized that the results of the present 2 compartment segregation−backmixing model are in very good agreement with the CFD simulations and 100 compartment model of Wells and Ray30 applied to describe the operation of a 500 L autoclave. Moreover, the present model has been successfully employed to simulate ethylene decomposition incidents in a reaction zone caused by various faults and random changes in operational conditions (e.g., feed impurities and plant disturbances, excess initiator, controller failure, poorly tuned controller, local energy dissipation rate, etc.) and thus can be used as a risk analysis tool of ethylene decomposition in industrial LDPE autoclaves. Finally, it should be noted that the above modeling approach has been applied to industrial multizone LDPE autoclaves to predict its dynamic behavior (e.g., temperature, ethylene conversion and molecular properties (Mn, Mw, LCB, etc.) along the height of a multizone autoclave. Moreover, it can used to analyze potential faults that can lead to reactor runaway and ethylene decomposition. In the latter case, the macro- and micromixing model parameters need to be estimated in real-time in terms of time-varying polymerization conditions so the model can continuously follow the real-time operation of the actual plant.

Figure 16. Effect of a 25% decrease in the characteristic micromixing/ erosion time, tm, on ethylene decomposition (Tfeed = 420 K, l0 = 100 μm in eq 35, V1 = 0.1Vtot, R = 3.6).

“fluid lumps” in the feed stream, l0. According to eq 35, the characteristic times tm1 and tm2 can suddenly decrease (e.g., by 25%) due to a number of process related faults and/or disturbances including (i) a 10% decrease in the kinematic viscosity caused by a decrease in the weight molecular weight by 10% due to an unforeseen increase in the effective concentration of chain transfer agent, (ii) an increase in the local energy dissipation rate (from 20 W/kg near the reactor wall to 60 W/kg near the impeller subzone), and (iii) a 25% decrease in the initial size of initiator “fluid lumps”, and so on. As can be seen when the polymerization temperature in the reaction zone is relatively low (e.g., below 260 °C corresponding to a low initiator concentration in the ethylene−initiator feed stream), a 25% decrease in the characteristic micromixing time does not affect the polymerization temperature in the reaction zone. When the temperature in the reaction zone is high (e.g., above 270 °C) and there are high initiator concentrations in the feed stream, a 25% decrease in value of tm will bring about an increase in the concentration of primary radicals in both compartments with concomitant the increase in the polymerization rates and the appearance of a hot spot that can trigger the ethylene decomposition reactions. Similarly, other operational faults potentially linked with ethylene decomposition can be successfully simulated using the present segregation−backmixing model. The decomposition results shown in the present study clearly demonstrate the effects of macromixing and macromixing parameters on the local temperatures in the two compartments in a selected reaction zone in a multizone autoclave configuration. From the above analysis (see Figures 15 and 16), it is apparent that ethylene decomposition is most likely to occur in middle to bottom reaction zones in a multiscale autoclave, operating at higher temperatures, higher conversions, and higher molecular weights than the upper zones in an autoclave. In addition to the above observations, it is important to mention that in accordance with the results of the engulfment model, a pure initiator feed policy to the lower zones in an autoclave could potentially result in local temperature overheating. Finally, it should be noted that the present simulation results are in line with industrial observations regarding ethylene decomposition incidents. 13109


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ε

AUTHOR INFORMATION

Corresponding Author

λk λn μn

*E-mail: [email protected]. Phone: +30-2310498161. ORCID

Costas Kiparissides: 0000-0003-4649-0001

[λi,1]

Notes

[λi,2]

The authors declare no competing financial interest.

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[μi,1]

LIST OF SYMBOLS Cp Specific heat (kJ/kg·K) D Initiator diffusion coefficient (m2/s) Da Damkoehler number Dx “Dead” polymer chain concentration Ea Activation energy (kJ/mol) E Engulfment parameter (1/s) E(k) Turbulent kinetic energy f Initiator efficiency [I] Initiator concentration (mol/L) [Iseg 0 ] Initiator feed in segregated form (mol/L) [Iseg i,j ] Initiator in segregated form in compartment “i” and zone “j” (mol/L) [Ii, j] Initiator in molecular form in compartment “i” and zone “j” (mol/L) kd Initiator decomposition rate constant (1/s) ki Chain Initiation reaction rate constant (L/mol·s) kp Propagation reaction kinetic rate constant (L/mol·s) kt Termination reaction kinetic rate constant (L/mol·s) ktc Termination by combination reaction kinetic rate constant (L/mol·s) ktd Termination by disproportionation reaction kinetic rate constant (L/mol·s) ktm Transfer to monomer reaction kinetic rate constant (L/ mol·s) ktp Transfer to polymer reaction kinetic rate constant (L/ mol·s) kts Transfer to solvent reaction kinetic rate constant (L/ mol·s) kβ Beta scission reaction kinetic rate constant (L/mol·s) l0 Initial size of the aggregate (m) [Mi] Monomer concentration in compartment “i” (mol/L) P Pressure (bar) Q Volumetric flow (m3/s) r Rate function (mol/L·s) R Recycle parameter Rx “Live” radical chain concentration Sc Schmidt number T1 Temperature of compartment 1 (K) T2 Temperature of compartment 2 (K) tm1 Mixing parameter in compartment 1 (s) tm2 Mixing parameter in compartment 2 (s) te Engulfment characteristic mixing parameter in compartment 2 (s) u′λ Fluctuating velocity (m/s) V1 Volume of compartment 1 (m3) V2 Volume of compartment 2 (m3) Vtot Total volume of reactor (m3)

[μi,2] ν ρ

Subscripts

0 b I p R t

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feed Backflow Initiation Propagation Recycle Termination

REFERENCES

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Greek Letters

δ ΔH ΔV

rate of dissipation of turbulent kinetic energy per unit mass (W/kg) Kolmogorov length scale (m) nth moment of “live” radical chain length distribution nth moment of “dead” polymerl chain length distribution Concentration of the ith “live” moment in compartment 1 (mol/L) Concentration of the ith “live” moment in compartment 2 (mol/L) Concentration of the ith “dead” moment in compartment 1 (mol/L) Concentration of the ith “dead” moment in compartment 2 (mol/L) kinematic viscosity (m2/s) Mixture density (kg/m3)

Backflow mixing parameter Heat of reaction (kJ/mol) Activation volume (cm3/mol) 13110


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