LDPE Production in Tubular Reactors. Comprehensive Model for The

Article Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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LDPE Production in Tubular Reactors: Comprehensive Model for the Prediction of the Joint Molecular Weight-Short (Long) Chain Branching Distributions Maira L. Dietrich,† Claudia Sarmoria,†,‡ Adriana Brandolin,†,‡ and Mariano Asteasuain*,†,‡ †

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Planta Piloto de Ingeniería Química (PLAPIQUI), Universidad Nacional del Sur-CONICET, Camino La Carrindanga km 7, Bahía Blanca 8000, Argentina ‡ Departamento de Ingeniería Química, Universidad Nacional del Sur (UNS), Avenida Alem 1253, Bahía Blanca 8000, Argentina S Supporting Information *

ABSTRACT: Good control on molecular properties, and as a consequence on end-use properties, is very important for lowdensity polyethylene (LDPE) manufacturers. However, the connection between the architecture of polymer chains and the kinetic mechanism and polymerization conditions is still a subject of study. In this work, we present a comprehensive model of the polymerization of ethylene in high-pressure tubular reactors. In addition to the usual predictions of conversion, temperature profiles, and average molecular properties, this model also provides bivariate distributions such as molecular-weight−long-chain branching distribution and molecular-weight−short-chain branching distribution of LDPE produced under different operating conditions. The 2D probability-generating function technique is applied to obtain the bivariate distributions. This is a deterministic technique that allows the calculation of the distributions without any prior assumption of their shape.

1. INTRODUCTION Low-density polyethylene (LDPE) is such a versatile commodity polymer that its current worldwide production is around 20 Mt per year. It is well-known that the polymerization conditions determine the polymer architecture, such as the molecular weight distribution (MWD), the long chain branching distribution (LCBD), and the short chain branching distribution (SCBD). These molecular properties, in turn, determine the end-use properties of the material, such as chemical, mechanical, and rheological properties. LDPE is generally produced in high capacity tubular or autoclave reactors under high pressure and temperatures.1 A typical tubular reactor consists of a pipe with a high length-to-diameter ratio. The operating pressure ranges from 1800 to 2800 bar, and the temperature may reach up to 335 °C at the hottest points. Ethylene monomer, inert species, chain transfer agents, and initiators such as oxygen and/or peroxide mixtures are fed at the reactor entrance and, eventually, at lateral injection points. Since the reaction is highly exothermic, temperature control is achieved using several heating/cooling jackets.2,3 The rigorous operation conditions of the process would make it both expensive and dangerous to try to establish experimentally the relationship between polymerization conditions and the product molecular properties. Therefore, a computational tool able to simulate the reactor output as a function of the polymerization conditions becomes crucial. Since in general © XXXX American Chemical Society

a complete characterization of LDPE requires simultaneous knowledge of all three MWD, LCBD, and SCBD,4−6 an appropriate model should be able to calculate them accurately. Such a model could be used to find the best operating conditions for producing a polymer with specific properties without resorting to experiments. This model-aided exploration, also called simulation-based product development, could be especially attractive for industrial LDPE producers. In spite of its influence on polymer properties, the prediction of the branching distributions of the polymer has only recently been included in modeling and simulation studies, as detailed below. Some overall branching quantities may also be calculated to characterize the polymer, such as the branching density, average segment length, and number and weight-average degree of branching.7,8 Another molecular property directly related with branching architecture is the mean-square radius of gyration ⟨R2g⟩. The ratio of ⟨R2g⟩ of a branched molecule to that of a linear molecule of the same molecular weight is defined as the branching index g.9 This index varies with the type and functionality of the branches, the LCB distribution, and the total number of branches per Received: Revised: Accepted: Published: A

November 16, 2018 February 18, 2019 February 20, 2019 February 20, 2019 DOI: 10.1021/acs.iecr.8b05713 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

a single dimension by computing moments of the number of branches. Even though they did not report the bivariate distribution of molecular weight and degree of branching, they computed the branching index from the calculated branching distribution and compared the model outcome with experimental data. Kim and Iedema7 extended this work to consider LDPE produced in CSTR and tubular reactors. They reported number and weight-average branching and branching density as functions of chain length. Different deterministic models of the production of LDPE in either CSTR or autoclave reactors have been presented by several groups,16−18 aiming at the study of the LCB density curve, the time evolution of the MWD-LCBD, or the branching index in terms of operating conditions. Recently, Neuhaus and co-workers19 and Eckes and Busch20 combined the advantages of probabilistic and deterministic approaches to model the high-pressure polymerization of ethylene in industrial tubular and autoclave reactors. Concentrations, temperatures, and pressures were calculated deterministically, and subsequently the detailed microstructure of individual macromolecules was simulated in an MC algorithm. Several of the previously mentioned modeling studies that account for branching distributions use them to calculate the branching index g and then compare it with experimental data. Those who apply MC methods predict ⟨R2g⟩ and g by reconstructing the 3D spatial configurations of the branched molecules using random-walk molecular simulations.3,12,14 Others use a theoretical graph representation of branched molecules to find the radius of gyration.10,21 Several authors applied a relation between the branching index and the number of branches derived by Zimm and Stockmayer.9 For example, in a polymer sample where both the chain lengths and the number of long chain branches are randomly distributed, the branching index may be expressed in terms of the weight-average number of branch units per molecule. There are conflicting reports on the general shape of the LCB density curve (i.e., number of long chain branches per 1000 carbon atoms (LCB/1000C) vs molecular weight). On one hand, Hutchinson16 found theoretically that in a CSTR it reached a constant value for molecular weights larger than a given threshold, regardless of the presence of scission. He needed considering chains with up to 100 long chain branches to reproduce the theoretical results of Tobita.22 However, those theoretical results do not agree with the experimental trends reported by Axelson and Knapp23 and Iedema and co-workers8 for tubular reactors, which show that the LCB density curve has a maximum. Models by Pladis and Kiparissides,24 Krallis and Kiparissides,17 and Iedema and co-workers8 for CSTR reactors do agree with the experimental trend. In a later work, Kim and Iedema7 found that for a CSTR the branching density was monotonically increasing. For the particular case of the synthesis of LDPE in a tubular reactor, they found that the branching density in the large molecular weight region reaches a plateau, in coincidence with other theoretical works.12,14 In this work, we present a deterministic model of the LDPE production in tubular reactors. This model can predict joint bivariate distributions MWD-LCBD and MWD-SCBD, the branching index g, and the LCB density curve, as well as average molecular properties such as the number- and weight-average molecular weights ( M n , M w ), long and short chain branches per 1000 carbon atoms (LCB/1000C, SCB/1000C), and conversion as a function of the reactor axial distance. To the best of our knowledge, no prior deterministic models of this system

molecule. It is strongly related with the rheological behavior of polymer melts, so it is important to consider it in models that include rheological properties. Several mathematical models have been reported over the years to study the influence of the design and operating conditions on the molecular and end-use properties of LDPE. Most of them do not describe those properties completely because they focus on the prediction of average molecular properties and/or on the complete MWD only. In order to advance in the state of the art in this subject, the present work places special emphasis on the prediction of the joint MWD-LCBD and MWD-SCBD distributions by means of deterministic models. Probabilistic and deterministic approaches for the prediction of bivariate polymer molecular properties have been reported in the literature. For instance, Iedema and Hoefsloot10,11 studied the synthesis of polyethylene with metallocene catalysts and other radical polymerizations in CSTRs using Monte Carlo simulations (MC). They assessed the problem that properties of polymer molecules depend not only on their length and number of branching points but also on their molecular architectures. The latter were characterized in terms of the radius of gyration and seniority/priority distributions, obtained from the predicted bivariate chain length−degree of branching distribution. Meimaroglou and Kiparissides12 presented a stochastic MC algorithm that calculated topological characteristics of LDPE produced in tubular reactors, such as the MWD-LCBD and MWD-SCBD, as well as distributions of the number of long- and short-chain branches per 1000 carbon atoms, average branch length distribution, bivariate branching order-branch MWD, and seniority/ priority segment number fraction distributions. This information was used to calculate some important rheological parameters such as ⟨R2g⟩ and the average branching factor. In other works, this research group used the topological information on the MC model to predict the complex viscosity of LDPE at different temperatures3 and evaluated the effect of different polymer chain structures on the rheological behavior of LDPE.13 Additionally, Meimaroglou and co-workers14 evaluated the impact of the polymerization temperature and the transfer agent concentration on the molecular and topological properties of the LDPE, as well as the effect of the long chain branching content on the radius of gyration and the branching factor. Even though stochastic methods provide highly detailed information about the polymer molecular structure and topological architecture, they are in general computationally demanding. These methods may also have difficulties in the determination of very low concentrations of molecules, such as those in the ends of MWDs.12 The use of multivariate population balance equations (PBEs), derived from the polymerization kinetic mechanism, is an interesting alternative. The PBEs describe the evolution of the concentration of the living radicals and dead polymer chains, characterized by a set of distributed properties. The infinitely large PBE system needs to be treated by any of several methods presented in the literature to reduce the number of model equations. Some of these deterministic methods have been used to predict bivariate distributions in LDPE processes, although not in tubular reactors. For example, Yaghini and Iedema15 proposed a two-dimensional model for the chainlength−branching distribution of LDPE obtained in CSTRs, using the Galerkin-finite element method to solve the PBEs. Iedema and co-workers8 considered bidimensional PBEs that described molecular weight and degree of branching distribution of bimodal LDPE produced in autoclave reactors. They used the pseudo distribution method to reduce the problem to B

DOI: 10.1021/acs.iecr.8b05713 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research Table 1. Kinetic Mechanisms for Long and Short Chain Branching Prediction step peroxide initiation monomer thermal initiation oxygen initiation generation of inert propagation termination by combination thermal degradation chain transfer to monomer chain transfer to polymer chain transfer to transfer agent backbiting

long chain branching mechanism fk kk

Ik ⎯⎯⎯→ 2R1,0 k = 1, 2 km

3M → R1,1 + R1,2 ko

O2 + M → R1,1

fk kk

Ik ⎯⎯⎯→ 2R 0,0 k = 1, 2

(1)

km

3Μ → R 0,1 + R 0,2

(3)

ko

O2 + M → R 0,1

(5)

fo ko

O2 + R n , m ⎯⎯⎯→ X R n,m + M → R n,m+1

R h,m + M → R h,m+1

(9) (11)

R n , m + M ⎯⎯⎯→ Pn , m + R1,1

(15)

R n , m + Pj , y ⎯⎯→ Pn , m + R j + 1, y R n , m + S ⎯→ ⎯ Pn , m + R1,0 k bb

R n , m ⎯→ ⎯ R n,m

(21)

R h , m + R j , y → Ph + j , m + y R h , m ⎯⎯⎯→ Ph , m + R 0,0

k trp

(10)

k tc

k tdt

(13)

k trm

k trs

(8)

kp

R n , m + R j , y → Pn + j − 1, m + y

R n , m ⎯⎯⎯→ Pn , m + R1,0

(6)

O2 + R h , m ⎯⎯⎯→ X

k tc

k tdt

(2) (4)

fo ko

(7)

kp

short chain branching mechanism

(12) (14)

k trm

R h , m + M ⎯⎯⎯→ Ph , m + R 0,1

(16)

k trp

(17)

(19)

R h , m + Pj , y ⎯⎯→ Ph , m + R j , y k trs

R h , m + S ⎯→ ⎯ Ph , m + R 0,0 k bb

R h , m ⎯→ ⎯ R h + 1, m

(18) (20)

(22)

third columns correspond to long and short chain branching descriptions, respectively. Chain transfer to the dead polymer chain (eqs 17 and 18) is responsible for the long chain branching formation. The backbiting reaction (eqs 21 and 22) accounts for the short chain branching. Peroxide (Ik), monomer (M), and oxygen (O2) initiation reactions (eqs 1−6) generate linear radicals with chain lengths of 0, 1, or 2. The propagation reaction (eqs 9 and 10) adds a monomer to the growing radical macromolecules. The macroradicals can be deactivated in several ways, such as by reaction with oxygen (eqs 7 and 8), by thermal degradation (eqs 13 and 14), or by termination by combination (eqs 11 and 12). The chain transfers to monomer (eqs 15 and 16) and to transfer agent (S; eqs 19 and 20) produce lower molecular weight molecules. This mechanism does not include the production of polyradicals or the presence of intramolecular terminal double bond reactions. The set of kinetic constants obtained by Asteasuain and co-workers29 and Brandolin and co-workers31 are used here and are considered to follow an Arrhenius equation with pressure dependence (see Kinetic Parameters section in the Supporting Information file). The configuration of the high-pressure tubular reactor considered in this work allows for two lateral feeds, where the initiator (or a mixture of initiator and monomer) is injected (see Figure 1). As a consequence, two reaction zones are found along the reactor. Several hypotheses and features of the previous works were maintained: plug flow, reaction mixture forming a single supercritical phase, variation of physical and transport properties along the axial distance, piece-wise constant jacket temperature profile, and peroxide and transfer agent mixtures treated, respectively, as a single fictitious species. It was found in our previous work29 that each mixture of peroxides could be treated as a single virtual or fictitious peroxide with the same total molar flow rate without affecting the quality of the results. The use of fictitious peroxide simplified the model because only one kinetic constant needed to be fitted for the mixture. Mixtures were considered different if the particular peroxides or their proportions varied. The same simplification was found to be applicable to mixtures of transfer agents.29 The same

have been reported in the literature with this level of detail. The model is an extension of a previous comprehensive steadystate model.25 The bivariate distributions are predicted using the 2D probability generating functions technique (2D pgf),26−28 without a priori knowledge of the shape of the distributions. It is easy to implement, in spite of the complexities of both the kinetic mechanism and the reactor configuration. The following section provides a description of the 2D pgf reactor model.

2. REACTOR MODEL The fundamental aspects of the model presented here were taken from previous works performed in our group.25,29,30 These previous works assumed a realistic industrial reactor configuration and included detailed correlations for the calculation of the physical and transport properties. It was possible to predict conversion, average molecular weights, and average branching indexes, as well as the complete MWD. The kinetic parameters were found by fitting model predictions with experimental data from an industrial reactor.29,31 Details on the calculation of the above parameters and physical and transport properties are summarized in the Supporting Information. For the present work, we add the calculation of the bivariate distributions MWD-LCBD and MWD-SCBD. The updated mathematical model employs two parallel sets of equations for the prediction of the MWD-LCBD and the MWD-SCBD, respectively. Each set of equations corresponds to the same kinetic mechanism, with different internal coordinates for the macromolecules. Depending on the set of equations, the first internal coordinate keeps track of either the short or the long chain branches. In both sets, the total length of the macromolecules corresponds to the second internal coordinate. Rn,m and Pn,m represent, respectively, living radicals and dead polymer chains with n − 1 long chain branches (the 1 is subtracted because n accounts for both the chain backbone and the long branches) and chain length m. Rh,m and Ph,m are living radicals and dead polymer chains with chain length m and h short chain branches. Table 1 shows the kinetic mechanism expressed in terms of the above conceptual species (eqs 1−22). The second and C


Article

Industrial & Engineering Chemistry Research fictitious transfer agent (S) is employed in all feeds while two fictitious peroxides (I1 and I2) are used in the first and second lateral injections, respectively. In the same work,29 the validity of the constant jacket temperature for each jacket section was assessed. The first set of model equations, shown below, includes the momentum, mass and energy balances and equations for monomer conversion and average molecular properties of the polymer: Global mass balance ρ(x) ν(x) = Fmain +

∫0

x

jij zy jj∑ F (x)zzz dx jj zz j j j z k {

Equation 24 is solved as an ordinary differential equation for the pressure P(x). The coupling with velocity is solved calculating the derivative dv(x)/dx using the approach proposed by Bansal and co-workers,32 according to which a derivative which is not explicitly available, like dv(x)/dx in the reactor model, can be accurately approximated by the derivative of an auxiliary variable (dvaux(x)/dx), where v(x) = vaux(x) + 1 × 10−8

(23)

dvaux(x) dx

Pressure drop (momentum balance)

ij 2f v(x)2 yzz dP(x) dv(x) zz = − ρ(x)jjjjv(x) + f j dx dx D zz k { dT (x) dx 4U (x)(T (x) − Ti(x)) =− + rp(x)(−ΔH ) D

ρ(x) ν(x) C P(x)

+ C̅ p(x)(Tinlet − T (x)) ∑ Fj̅ (x)

dx

= rj(x) +

Fj̅ (x)

j = O2 , M, Ik , S

M w, j

( )(T

dx

(26)

energy balance. Variable Fj(x) is a continuous mass flow of component j per unit cross-sectional area and unit length that is added to the tubular reactor through the reactor wall. The axial profile of Fj(x) is formed by impulses located at the lateral injections, with impulse areas equal to the mass flow of component j at the lateral injection divided by the reactor-crosssectional area. Approximating the impulses as short pulses of length Δx, the following expression results:

(27)

Monomer conversion Conv(x) = 1 −

v(x) CM(x) M w,mon

j j

(31)

where A is the reactor cross-sectional area. Variable Fsidei,j is the mass flow rate of component j at the lateral injection i located at the rector length xsidei and is specified by the operating conditions. Numerical experiments were carried out in a previous work25 to determine the appropriate value of the length interval as Δx = 0.001 m. Equations 32−37 show the reaction rates for small species, namely initiators, monomer, and transfer agent, and for long chain branching content and side groups, which combine with eqs 26 and 27 to give the mass balance equations for these species. Additional model equations include the ones required for the prediction of the bivariate distributions, balance equations for the moments of these distributions, and algebraic equations for the average molecular properties. They are described in the following sections. Initiator mixtures (Ik)

x

FM,main + ∫ FM̅ (x) dx 0

inlet

l Fside i , j o o o xside i < x < xside i + Δx i = 1, 2 o Fj̅ (x) = m AΔx o o o o0 otherwise n

= rj(x)

j = LCB, Me, R n , m, Pn , m , R h , m, Ph , m

( )) ∑ F (x) to the

−T x

the mass balances and C P x

Balances of long chain branches, side groups, polymer and radical chains d(Cj(x) v(x))

(30)

(25)

Mass balances of reactants d(Cj(x) v(x))

=0 x=0

Equation 30 is included in the reactor model. The axial velocity v(x) is available independently from the right-hand side of eq 23 and the density of the reaction mixture. Lateral injections require a special treatment in the formulation of the reactor model. Mathematically, they represent instantaneous additions, at a given axial distance, of mass and energy that cause discontinuities in some of the model variables. This situation was modeled by adding an additional term to the mass and energy balances, Fj(x)/Mw,j to

(24)

Energy balance

j

dvaux(x) dx

(28)

In these equations, x is the axial distance, ρ(x) is the density, v(x) is the axial velocity, P(x) is the reactor pressure, CP(x) is the heat-capacity of the reactor mixture, T(x) is the reactor temperature, U(x) is the global heat-transfer coefficient, Cj(x) is the molar concentration of component j, rj(x) is the generation rate of component j, and rp(x) is the propagation reaction rate. Fmain is the global mass flux at the reactor entrance, FM,main is the monomer mass flux at the reactor entrance, f f is the friction factor, D is the internal diameter, ΔH is the reaction enthalpy, Cp is the heat capacity of lateral injections, Mw,j is the molecular weight of component j, and Mw,mon is the molecular weight of the monomer. Finally, Ti(x) is the temperature of the ith jacket section, Tinlet is the lateral injection temperature, LCB is the molar concentration of long chain branches, and Me is the molar concentration of methyl groups. The friction factor in eq 24 is calculated as follows:

rIk = −kkC Ik(x)

l o 1.18 × 10−2 − 4.66 × 10−4ln(Re) Re < 2.5 × 105 o o o o o ff = o m 7.441 × 10−3 − 1.18033 × 10−4ln(Re) 2.5 × 105 < Re < 106 o o o o o o Re > 106 n 0.00475

(32)

Oxygen ∞

rO2 = −koCO2(x)1.1CM(x) − f0 k 0CO21.1 (x) ∑

∞

∑ CR

(x )

n,m

n=1 m=0

(29)

(33) D


Article

Industrial & Engineering Chemistry Research Transfer agent (S) ∞

rS = −k trsCs(x) ∑

∞

∑ CR

(x )

n,m

(34)

n=1 m=0

Monomer (M) rM = − koCM(x)CO2(x)1.1 − 3kmiCM(x)3 − k pCM(x) ∞

∞

∞

∑ ∑ CR

(x) − k trmCM(x) ∑ n,m

n=1 m=0

∞

∑ CR

(x)

n,m

(35)

n=1 m=0

Methyl groups

yz ij ∞ ∞ rMe = k bbjjjj∑ ∑ C R n,m(x)zzzz z j { kn=1 m=2

Figure 1. Schematic representation of the tubular reactor. Eight different cooling/heating zones are defined by the separate cooling/ heating jackets, each one operating at a constant temperature Tj, j = 1−8. Only one generic lateral feed is shown.

(36)

Long chain branches ∞

rLCB = k trp ∑

∞

∞

∑

C R n,m(x) ∑

n=1 m=0

∞

∑ mCP

(x )

n,m

Dead polymer with chain length m and n − 1 long chain branches (Pn,m)

(37)

n=1 m=1

3. MODELING OF THE COMPLETE BIVARIATE DISTRIBUTIONS If eq 27, with j = Rn,m, Pn,m, Rh,m, and Ph,m, were written out for every possible value of the internal coordinates of the species (total length and number of short and long chains), the bivariate MWD-LCBD and the MWD-SCBD could be obtained in a straightforward manner. Equations 38−41 show the resulting PBEs for the high-pressure tubular reactor under study. Note that eqs 38 and 39 correspond to the long chain branching mechanism and eqs 40 and 41 to the short chain branching mechanism. Note also that the total number of equations is infinite. Even if it were possible to truncate the system at a finite length, its size would still be too large to be tractable. Living radical with chain length m and n − 1 long chain branches (Rn,m) d(C R n,m(x)v(x)) dx

n

d(C Pn,m(x)v(x)) dx

m

1 k tc ∑ ∑ C R (x)C R n−r+1,m−s(x) 2 r = 1 s = 0 r ,s

=

+ k tdtC R n,m(x) + k trsCS(x)C R n,m(x) + k trmCM(x)C R n,m(x) ∞

∞

− k trpmC Pn,m(x) ∑ ∑ C R j ,y(x) + k trpC R n,m(x) j=1 y=0 ∞

∞

∑ ∑ yC P

(x )

j,y

n = 1−∞ ; m = 1−∞ (39)

j=1 y=1

Living radical with chain length m and h short chain branches (Rh,m) d(C R h,m(x)v(x)) dx

= 2f1 k1C I1(x)δh ,0δm ,0

+ 2f2 k 2C I2(x)δh ,0δm ,0 + kmiCM(x)3 δh ,0δm ,1 + kmiCM(x)3

= 2f1 k1C I1(x)δn ,1δm ,0 + 2f2 k 2C I2(x)δn ,1δm ,0

+ kmiCM(x)3 δn ,1δm ,1 + kmiCM(x)3 δn ,1δm ,2

δh ,0δm ,2 − k pCM(x)C R h,m(x) + k pCM(x)C R h,m−1(x)

− k pCM(x)C R n,m(x) + k pCM(x)C R n,m−1(x)(1 − δm ,0)

(1 − δm ,0) − k tdtC R h,m(x) + k tdt ∑ ∑ C R t ,y(x)δh ,0δm ,0

∞

∞

∞

t=0 y=0

− k tdtC R n,m(x) + k tdt ∑ ∑ C R j ,y(x)δn ,1δm ,0 ∞

∞

t=0 y=0 ∞

∞

∑ ∑ yCP

j=1 y=0 ∞ ∞

(x) − fo koCO2(x)1.1C R h,m(x) + koCO2(x)1.1

t ,y

t=0 y=1

− k trpC R n,m(x) ∑ ∑ yC Pj ,y(x) − fo koCO2(x)1.1C R n,m(x)

∞

∞

∞

CM(x)δh ,0δm ,1 − k tcC R h,m(x) ∑ ∑ C R t ,y(x) + k trm

j=1 y=1 ∞

t=0 y=0

+ k 0CO2(x)1.1CM(x)δn ,1δm ,1 − k tcC R n,m(x) ∑ ∑ C R j ,y(x)

∞

∞

CM(x) ∑ ∑ C R t ,y(x)δh ,0δm ,1 + k bbC R h−1,m(x)(1 − δh ,0)

j=1 y=0 ∞

t=0 y=0

+ k trmCM(x) ∑ ∑ C R j ,y(x)δn ,1δm ,1 − k trmCM(x)C R n,m(x)

− k bbC R h,m(x) − k trmCM(x)C R h,m(x) + k trsCS(x)

j=1 y=0 ∞ ∞

∞

∞

∑ ∑ CR

+ k trsCS(x) ∑ ∑ C R j ,y(x)δn ,1δm ,0 − k trsCS(x)C R n,m(x)

(x)δh ,0δm ,0 − k trsCS(x)C R h,m(x)

t ,y

h = 0−∞

t=0 y=0

j=1 y=0

n = 1−∞ ; m = 0−∞

∞

+ k trpmC Ph,m(x) ∑ ∑ C R t ,y(x)(1 − δm ,0) − k trpC R h,m(x)

j=1 y=0 ∞

+ k trpmC Pn−1,m(x) ∑ ∑ C R j ,y(x)(1 − δn ,1)(1 − δm ,0)

∞

∞

; m = 0−∞

(38) E

(40) DOI: 10.1021/acs.iecr.8b05713 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research Dead polymer with chain length m and h short chain branches (Ph,m) d(C Ph,m(x)v(x)) dx

h

Long-chain branches per 1000 carbon atoms 500C LCB(x) 500C LCB(x) LCB (x ) = L = S L S 1000 C (x ) λ 0,1(x) + μ0,1 (x ) λ 0,1(x) + μ0,1

m

1 = k tc ∑ ∑ C R r ,s(x)C R h−r ,m−s(x) 2 r=0 s=0

(46)

It is important to recall that the parallel sets of equations used to build the reactor model differ only in which type of branch (i.e., long or short) is accounted for. Both sets give the same results for the chain length dimension. Therefore, either the moments of the long-chain branching set (superscript “L”) or those of the short-chain branching set (superscript “S”) can be used to calculate the average properties and the concentration of small species, because all the moments involved are of order 0 with respect to the branching content. Bivariate distributions: The pgf values obtained from the solution of eqs S16−S19 (see Supporting Information) are fed to 2D pgf inversion formulas to recover the bivariate distributions. In this work, we used the 2D Papoulis-IFG inversion method,28 as detailed in the Inversion Method section of the Supporting Information. Distributions of branching densities: Besides the bivariate distributions, long and short chain branches per 1000 carbon atoms as a function of the molecular weight are also calculated, as follows:

+ k tdtC R h,m(x) + k trmCM(x)C R h,m(x) − k trpmC Ph,m(x) ∞

∞

∑ ∑ CR

∞

∞

(x) + k trpC R h,m(x) ∑ ∑ yC Pt ,y(x) t ,y

t=0 y=0

t=0 y=1

+ k trsCS(x)C R h,m(x)

h = 0−∞ ; m = 1−∞

(41)

In the present work, we calculate the bivariate MWD-LCBD and the bivariate MWD-SCBD by means of the 2D pgf technique, as applied to the synthesis of LDPE in a tubular reactor. To the best of our knowledge, there are no reports on the use of deterministic approaches for modeling bivariate distributions in LDPE processes in tubular reactors. The pgf method is a general modeling tool that has been applied to a number of polymer systems.33−36 It is a transform technique, easy to implement and with relatively low computational time requirements.27,28 Briefly, the 2D pgf technique consists of two steps. First, the PBEs are transformed to the 2D pgf domain, leading to balance equations for the 2D pgf of the bivariate distribution. These pgf balances are solved instead of the PBEs. Second, the resulting pgf is numerically inverted to recover the bivariate distribution. The pgf balances obtained from the transformation of the PBEs (eqs S16−S19), as well as the necessary double moment equations (eqs S23−S26), are shown in the Supporting Information. Please note that two sets of pgf balances are considered, corresponding to the two parallel sets of mass balance equations. 3.1. Calculation of Molecular Properties. The double index moments, calculated as explained in the Supporting Information, are used to compute the average molecular weights and average branching density as follows: Number-average molecular weight

S S λ 0,1 (x) + μ0,1 (x )

(49) (42)

n w (x , m ) =

Weight-average molecular weight M w (x) = M w,mon = M w,mon

L L λ 0,2 (x) + μ0,2 (x ) L λ 0,1 (x )

+

+

S μ0,1 (x )

M w (x ) M n (x )

=

∞ L ∑n = 1 nd0,0 (x , n , m ) ∞ L ∑n = 1 d0,0(x , n , m)

(50)

MWD-LCBD expressed in weight is the MWD-LCBD expressed in

number fraction. (43)

4. RESULTS AND DISCUSSION This section illustrates the predicting capabilities of the reactor model. The influence of the operating conditions on the polymer molecular structure is shown for two scenarios. Case I corresponds to experimental operating conditions from a real industrial reactor, while case II is a realistic but hypothetical scenario. Table 2 details the operating conditions of these two cases, as well as the calculated conversion and average molecular properties of the polymer obtained in each of them. Case I is a typical operating point involving single feeds of monomer and transfer agent at the reactor entrance, and two side additions of

Dispersity Đ(z) =

∞ L ∑n = 1 nd0,1 (x , n , m) ∞ L ∑n = 1 d0,1(x , n , m)

where dL0,1(x,n,m) is the fraction and dL0,0(x,n,m)

L μ0,1 (x )

S S λ 0,2 (x) + μ0,2 (x ) S λ 0,1 (x )

(48)

Branching index: The branching index is expressed in terms of the weight average number of long branches per molecule nw(x,m):9 ÅÄÅ ÅÅ 1 ij 2 + n (x , m) yz1/2 6 w ÅÅ jj zz g (x , m ) = Å n w (x , m) ÅÅÅÅ 2 jjk n w (x , m) zz{ ÅÇ ÉÑ ÑÑ 1/2 1/2 ÑÑ jij (2 + n w (x , m)) + n w (x , m) zyz ÑÑ z × lnjjj − 1 z ÑÑ 1/2 1/2 z + − (2 n ( x , m )) n ( x , m ) ÑÑÑ w w k { Ö

L L λ 0,0 (x) + μ0,0 (x )

S S λ 0,0 (x) + μ0,0 (x )

(47)

∞

S ∑h = 1 hd0,0 (x , h , m) SCB (x , m) = 500 ∞ S 1000 C m ∑h = 1 d0,0(x , h , m)

L L λ 0,1 (x) + μ0,1 (x )

M n(x) = M w,mon = M w,mon

∞

L ∑ nd0,0 (x , n , m) LCB (x , m) = 500 n =∞0 L 1000 C m ∑n = 0 d0,0(x , n , m)

(44)

Short-chain branches per 1000 carbon atoms 500CMe(x) 500CMe(x) SCB (x ) = L = S L S 1000 C λ 0,1(x) + μ0,1(x) (x ) λ 0,1(x) + μ0,1 (45) F


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Industrial & Engineering Chemistry Research peroxide initiators. For this case, the experimental final conversion and measured average molecular properties31 are also reported in Table 2. Conversion was provided by the industrial Table 2. Operating Conditions of Case I and Case II operating conditions

case I

case II

inlet temperature (°C) inlet pressure (bar) oxygen main feed (kg/s) transfer agent main feed (kg/s) transfer agent first lateral injection (kg/s; Fside 1,S)b transfer agent second lateral injection (kg/s; Fside 2,S)b peroxide first lateral injection (kg/s; Fside 1,I1)b

77 2300 6.9 × 10−5 0.00762

85 2173 8.9 × 10−5 0.064

0

0

0

0.61

0.00102

8.5 × 10−5

peroxide second lateral injection (kg/s; Fside 2,I2)b

1.57 × 10−4

6.9 × 10−5

monomer main feed (kg/s) monomer first lateral injection (kg/s; Fside 1,M)b monomer second lateral injection (kg/s; (Fside 2,M)b location of first lateral injection (x/L) location of second lateral injection (x/L) temperature of water in the jacket zones 1−8 (°C)

11 0

10.2 0

0

0.8

0.12

0.03

0.63

0.62

170−225−170− 170−170−170− 170−170 25.8 (27.8)a

270−150−270− 266−150−150− 159−270 20.5

21602 (23300)a

25670

162958 (172000)a

224149

24.73 (22.34)a

14.79

2.14 (2.40)a

1.37

a

8.73

conversion (%) at the reactor exit M n at the reactor exit

M w at the reactor exit SCB/1000 C at the reactor exit LCB/1000 C at the reactor exit Đ at the reactor exit

7.54 (7.38)

Figure 2. MWDs at the reactor exit for case I and case II.

rate is governed mainly by the peroxide flow rate and the heat exchange with the reactor jacket. In case II, the additional feed of monomer at the second lateral injection is small, less than 10% of the main feed flow rate, and the dilution and decrease of the reactor temperature is small, as can be seen in Figure 3b. Therefore, polymerization rate is not affected significantly. Appropriate jacket temperature values in case II were selected within the feasible operating range, to help achieve the mentioned MWD with a high molecular-weight shoulder. In case II, the jacket temperature is low at the beginning of the first reaction zone, helping to keep a low reactor temperature. Near the ends of both reaction zones in case II, the jacket temperature becomes higher so as to increase the reaction rate to achieve a reasonable conversion. Reaction zones are longer in case II, because longer time is needed to obtain appropriate conversion levels under the lower reaction temperatures. Another particularity of case II is the presence of a significant flow of transfer agent at the second lateral injection. Clearly, this addition of transfer agent helps to generate a population of low-molecular-weight chains, responsible for the peak of the MWD. In addition, the small derivation of monomer to the second side feed accentuates the MWD shoulder. The experimental temperature profile of case I is also shown in Figure 3a. The model follows closely the experimental data. In particular, the hot spots, which are very important for the reactor operation, are accurately predicted. The operating conditions affect the level of chain branching in the synthesized polymer, as may be observed in the characteristics of the materials produced in cases I and II. For the latter case, the lower flow rate of initiator in the first side injection leads to a lower polymerization rate, with a consequent reduction in the heat release. The resulting lower temperature (compare the initial temperature profile of case II (Figure 3b) with respect to case I (Figure 3a)), together with the lower number of active radicals (because of the smaller initiator feed rate) decreases the rate of long- and short-chain branching reactions. Hence, the polymer of case II is less branched (see Table 2). Figure 4 shows the monomer conversion profiles. The reaction zones (increasing conversion) are clearly distinguished. Additionally, Figures 5 and 6 show the pressure and initiators mixtures’ concentration profiles. Figure 7 shows the MWD-LCBD at the reactor exit for cases I and II. This bivariate distribution provides information about

a

Experimental data in parentheses. bAll lateral injections are at the main feed temperature.

reactor operators, while average molecular properties were measured by GPC and infrared spectroscopy as detailed in Brandolin and co-workers.31 It may be observed that the model predicts those quantities very satisfactorily. The polymer that is produced under these conditions has a unimodal MWD (Figure 2). On the other hand, the operating conditions of case II yield a polymer with a high-molecular-weight shoulder (Figure 2). One of the main differences between the operating conditions of cases I and II is that the flow rate of the first lateral feed of initiator is much lower in case II. This helps generating a population of high molecular-weight chains, responsible for the high molecular-weight shoulder of the MWD. The temperature in the reaction zones of case II are lower than in case I (compare Figure 3a and b), something that also contributes to the generation of a high molecular-weight polymer. Both in case I and in case II, the inlet temperatures are typical of industrial operations. Even though they are different, the influence of these temperatures is minimal. As usual in the operation of these reactors, the first portion up to the first peroxide injection acts as a heat exchanger. In this first part of the reactor, the reaction mixture is heated up to the appropriate temperature for the peroxide injection. It is at this point that the reaction actually starts. Afterward, the polymerization G


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Figure 3. (a) Temperature profiles for case I. Solid line, calculated reactor temperature; dashed line, jacket temperature profile (model input); (■) experimental reactor temperature. (b) Temperature profiles for case II. Solid line, calculated reactor temperature; dashed line, jacket temperature profile (model input).

Figure 4. (a) Calculated monomer conversion profile for case I. (b) Calculated monomer conversion profile for case II.

the molecular weight of the LDPE produced in case II is greater than that of case I, as was explained before. Figure 9 shows the bivariate MWD-SCBD of the LDPE at the reactor exit for cases I and II. The majority of the chains have less than 50 short branches, approximately. At low SCB content, the molecular weight distributions at a given branching content shift toward higher molecular weights as the number of branches increases. This is expected since longer chains had larger lifetimes than shorter chains and therefore had more chances to have participated in backbiting reactions. This occurs up to a certain value of SCB (about 50) beyond which the location of the distribution peaks does not change significantly. Figure 10 compares slices of the MWD-SCBD at some representative values of SCB for cases I and II. For the same shortchain branching content, the molecular weight of the LDPE produced in case II is greater than that of case I. Another property of interest for highly branched polymers is the branching density distribution. Figure 11 depicts this

the weight fraction of polymer chains as a function of the molecular weight and long-chain branching content. The bivariate distributions show that, for both case studies, the majority of the polymer chains are linear (0 branches) or lightly branched (1−2 LCBs per chain). Furthermore, the MWD of chains with a given branching content shifts toward higher molecular weights as the number of LCBs increases. This is to be expected because branching formation implies reactivating a dead polymer chain that starts growing again. However, it is interesting to note that this shift to higher molecular weights is more marked at low branching content than at high branching content. In addition, a comparison of the heights of the distributions of Figure 7 shows that the amount of branched chains (i.e., LCB ≥ 1) is smaller in case II, in agreement with its lower LCB/1000 C with respect to case I reported in Table 2. Figure 8 compares slices of the MWD-LCBD at the first few values of LCB for cases I and II. The shift to the right of the curves of case II with respect to those of case I illustrates that H


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Figure 5. (a) Calculated pressure profile for case I. (b) Calculated pressure profile for case II.

Figure 6. (a) Calculated initiator mixtures’ concentration profiles for case I. (b) Calculated initiator mixtures’ concentration profiles for case II.

Figure 7. MWD-LCBD at the reactor exit corresponding to case I (a) and case II (b). I


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Figure 8. Slices of the MWD-LCBD at 0 (a), 1 (b), 2 (c), and 3 (d) long chain branches for cases I and II.

Figure 9. MWD-SCBD at the reactor exit corresponding to case I (a) and case II (b).

property for the LCB and the SCB expressed as branching content every 1000 C atoms vs molecular weight. It is interesting to note that, although the total branching content (LCB

and SCB) increases with molecular weight, the branching densities have a maximum. For both distributions, case I presents a higher branching density. As was mentioned before, J


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Figure 10. Slices of the MWD-SCBD at 0 (a), 12 (b), 24 (c), 32 (d), and 40 (e) short-chain branches for cases I and II.

maximum.9,24 There is less information available in literature about the SCB density curve. Figure 11 shows an initial rapid increase of the SCB density followed by a plateau in a wide range of molecular weights. Meimaroglou and Kiparissides,13 who studied this curve for LDPE produced in tubular reactors, found similar results.

the operating conditions of case II yield a polymer with less branching content than in case I. As was discussed previously, there is some degree of controversy in the literature regarding what the LCB density curve for LDPE should look like. Our results (Figure 11) agree with reported experimental trends that indicate that this curve has a K


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Figure 13. Average number of branching units per molecule nw(m) for case I and case II.

Figure 11. LCB/1000 C and SCB/1000 C vs molecular weight for cases I and II.

As far as we know, the joint distributions have not been presented in previous published deterministic models of the LDPE production in tubular reactors. The present approach does not require any simplifying assumptions regarding the shape of the distributions of live radical and dead polymer chains. The model was tested by applying it to different operating conditions resembling those found in industrial reactors, allowing to evaluate the influence of operating conditions on the properties mentioned above. This model has the potential to be a useful tool for industrial practitioners interested in determining operating conditions suitable for obtaining resins with specific molecular structures that would broaden their range of final applications.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.8b05713. Kinetic parameters, physical and transport properties, pgf balance equations, moment balance equations, and inversion method (PDF)

Figure 12. Branching index for cases I and II.

As commented previously, the branching index is commonly used to reflect the degree of LCB in highly branched polymers. Figure 12 shows this parameter, obtained using eq 49, for the two operating cases. In agreement with the results shown for the average molecular properties (Table 2) and branching densities (Figure 11), case I gives a polymer with a lower branching index, which means that the polymer is more branched. The general shape of the g curves agrees with experimental reports on various LDPEs obtained in tubular and autoclave reactors.37 It also agrees with theoretical trends from Krallis and co-workers,19 obtained with a discretized model for autoclave reactors and from Meimaroglou and Kiparissides13 obtained with an MC model for a tubular reactor. Figure 13 plots nw(m), calculated using eq 50. For both cases, the average number of LCB per molecule increases with molecular weight, due to the fact that larger molecules allow more chain branches in their structure.

■

AUTHOR INFORMATION

Corresponding Author

*Address: Camino La Carrindanga Km 7, Bahía Blanca, Argentina. Phone: 54 - (0)291−4861700. E-mail: [email protected]. ORCID

Mariano Asteasuain: 0000-0003-4599-7310 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS M.L.D., C.S., A.B., and M.A. received financial support from Consejo Nacional de Investigaciones Científicas y Técnicas of Argentina (CONICET) through grant PIP 0653 and from Universidad Nacional del Sur (UNS) through grant PGI 24/ M149. M.L.D. received a doctoral scholarship from CONICET (Res. 3902-2015).

5. CONCLUSIONS This work presents a comprehensive deterministic model of an LDPE tubular reactor able to predict two bivariate distributions: molecular weight−long chain branching distribution (MWD-LCBD) and molecular weight−short chain branching distribution (MWD-SCBD). It may also calculate the average number of short and long branches and the branching index.

■

REFERENCES

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LDPE Production in Tubular Reactors. Comprehensive Model for The

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