Modeling of Suspension Vinyl Chloride Polymerization: From Kinetics to Particle Size Distribution and PVC Grain Morphology

Abstract

A comprehensive multiscale, multiphase modeling approach is developed to describe the dynamic evolution of polymerization rate, average molecular weight, and morphological properties of poly(vinyl chloride) (PVC) produced in batch suspension polymerization reactors. Dynamic evolution of the molecular (molecular weight distribution, long chain branching, short chain branching, terminal double bonds) and morphological (particle size distribution, grain porosity) properties of PVC can be calculated from the numerical solution of the proposed integrated model. In particular, polymer molecular properties are determined by employing a detailed kinetic mechanism that describes the free-radical polymerization of vinyl chloride monomer in both monomer- and polymer-rich phases. The initial monomer droplet size distribution and final polymer particle size distribution depend on the type and concentration of the surface-active agents, the quality of agitation (reactor geometry, impeller type, power input, etc.) and the physical properties (density, viscosity, interfacial tension, etc.) of the continuous and dispersed phases. A dynamic discretized particle population balance equation (PBE) is numerically solved to calculate the dynamic evolution of the particle size distribution of the produced PVC in a batch suspension reactor. Furthermore, the primary particle size distribution inside the polymerizing monomer droplets, which affects the porosity of the final PVC grains, is determined from the solution of a PBE governing the nucleation, growth, and aggregation of primary particles inside the polymerizing monomer droplets. Theoretical model predictions are compared successfully with a comprehensive series of experimental data on polymerization kinetics, particle size distribution, and PVC grain morphology.

List of Symbols

A ₀ :

Reactor outside heat transfer area, m²

A _a :

Reactor heat transfer area to the environment, m²

A _i :

Reactor inside heat transfer area, m²

α _m :

Monomer activity

A _t :

Reactor top heat transfer area, m²

B _i :

Virial coefficient of i component, m³/kmol

B _m :

Virial coefficient of monomer, m³/kmol

B _mw :

Virial coefficient of mixture monomer and water, m³/kmol

B _w :

Virial coefficient of water, m³/kmol

C₁, C₂:

Model parameters

C _pm :

Metal wall heat capacity, kJ/(kg K)

C _pmix :

Mixture heat capacity, kJ/(kg K)

C _PVA :

Concentration of the stabilizer, kg/m³

C _pw :

Water heat capacity, kJ/(kg K)

D _eq :

Jacket equivalent diameter, m

ΔH_r:

Specific reaction enthalpy, kJ/kmol

D _imp :

Impeller diameter, m

D _pq :

Mean particle diameter, m

D _R :

Reactor inside diameter, m

$\widehat{f}$ :

Fugacity, Pa

f _i :

Fractional particle number distribution

f _i,j :

Efficiency of initiator i in the j phase

F _w :

Mass flow rate of the water added to the reaction mixture, kg/h

G :

Particle growth rate due to polymerization in the polymer-rich phase, kg/s

g(u):

Breakage rate of drops of volume u, s⁻¹

h _i :

Heat transfer coefficient of the reaction mixture side, kJ/(m s K)

h _o :

Heat transfer coefficient from the reactor wall to jacket, kJ/(m s K)

I :

Initiator molecule

I ₀ :

Initial initiator concentration, g/kg VCM

K :

Solubility constant for the VCM in the aqueous phase, kg VCM/kg H₂0

k :

Thermal conductivity, kW/K

k(v, u):

Coalescence rate between two drops of volume v and u, m³/s

k _B :

Boltzmann’s constant, m² kg/(s² K)

k _bj :

Intramolecular transfer rate constant in the j phase, s⁻¹

k _di,j :

Decomposition rate constant of initiator i in the j phase, s⁻¹

k _fmj :

Chain transfer to monomer rate constant in the j phase, m³/(kmol s)

k _fpj :

Chain transfer to polymer rate constant in the j phase, m³/(kmol s)

k _I :

Rate constant for initiator decomposition, m³/(kmol s)

k _p1 :

Propagation rate constant in the monomer-rich phase, m³/(kmol min)

k _p2 :

Diffusion-controlled propagation rate constant in the polymer-rich phase, m³/(kmol min)

k _pj :

Propagation rate constant in the j phase, m³/(kmol s)

k _t :

Termination rate constant in the monomer phase, m³/(kmol s)

k _t2 :

Diffusion-controlled termination rate constant in the polymer-rich phase, m³/(kmol min)

k _t20 :

Termination rate constant in the polymer-rich phase, m³/(kmol min)

k _tcj :

Termination by combination rate constant in the j phase, m³/(kmol s)

k _tdj :

Termination by disproportionation rate constant in the j phase, m³/(kmol s)

k _zj :

Inhibition rate constant in the j phase, m³/(kmol s)

L _eq :

Jacket equivalent length, m

L _n :

Number of long chain branches per polymer molecule

M :

Mass of monomer, kg

M ₀ :

Initial mass of monomer, kg

M _n :

Number average molecular weight, kg/kmol

M _w :

Weight average molecular weight, kg/kmol

MW_m:

Molecular weight of monomer, kg/kmol

MW_w:

Molecular weight of water, kg/kmol

MW_x:

Molecular weight of molecular species “x”, kg/kmol

N :

Agitation rate, rpm

n(v, t):

Number density function, m⁻⁶

n₀(v):

Initial drop size distribution of the dispersed phase, m⁻⁶

N _d :

Number of initiators used in the polymerization

N _da :

Number of daughter drops per breakage event

N _sa :

Number of satellite drops per breakage event

N _i :

Particle number distribution

N _We :

Weber number

P :

Total reactor pressure, Pa

P _m :

Monomer partial pressure, Pa

P _m ^sat :

Monomer saturation pressure, Pa

Pr :

Prandtl number

P _w ^sat :

Water saturation pressure, Pa

[P_x]:

“Dead” polymer chains, containing x monomer units, kmol/m³

R :

Ideal gas constant, J/mol/K

r :

Radius of colloidal particles, m

Re :

Reynolds number

R _pm :

Polymerization rates in the monomer-rich phase, kmol/(m³ s)

R _pp :

Polymerization rates in the polymer-rich phase, kmol/(m³ s)

$\left[{R}_x^{\bullet}\right]$ :

“Live” macroradicals, containing x monomer units, kmol/m³

r _λj,j :

“Live” polymer moment rate function, kmol/(m³ s)

r _μj :

“Dead” polymer moment rate function, kmol/(m³ s)

S ₀ :

Nucleation rate of primary particles of volume v₀ in the monomer-rich phase, s⁻¹

S _d :

Number density of SCB per 1,000 monomer units

S _n :

Number of short chain branches per polymer molecule

t :

Time, s

T :

Reactor mixture temperature, K

T ₀ :

Reference temperature, K

T _a :

Ambient temperature, K

T _j :

Reactor’s jacket temperature, K

T _m :

Temperature of the metal wall, K

T _n :

Number of terminal double bonds per polymer molecule

u(u):

Number of droplets formed by the breakage of a drop of volume u

U _a :

Heat transfer coefficient to the reactor environment, kJ/(m s K)

U _t :

Heat transfer coefficient from the reactor top, kJ/(m s K)

V :

Total volume of the polymer particles, m³

v _da :

Volume of daughter drops, m³

v _sa :

Volume of satellite drops, m³

V _f :

Free volume of the mixture in the polymer-rich phase, m³

V _f ^* :

Free volume of the mixture at the critical monomer conversion, m³

V _fm :

Free volume of monomer, m³

V _fp :

Free volume of polymer, m³

V _g :

Volume of gas phase, m³

V _j :

Volume of j-phase, m³

V _m :

Metal wall volume, m³

V _mix :

Reaction mixture volume, m³

V _R :

Total reactor volume, m³

W _ij :

Fuch’s stability ratio

W _w :

Total mass of water loaded in the reactor, kg

W _wα :

Mass of water in the aqueous phase, kg

X :

Monomer conversion

X _c :

Critical monomer conversion

X _f :

Fractional monomer conversion

y _m :

Mole fraction of monomer in the vapor phase

y _w :

Mole fraction of water in the vapor phase

Z :

Inhibitor molecule

1.1 Greek Symbols

β :

Aggregation rate kernel, s⁻¹

β(u, v):

Daughter drop breakage function, accounting for the probability that a drop of volume v is formed via the breakage of a drop of volume u, m⁻³

$\dot{\upgamma}$ :

Mean value of the shear rate, s⁻¹

${\dot{\gamma}}_{eff}$ :

Effective shear rate, s⁻¹

Δv:

Relative droplet velocity, m/s

ε :

Porosity

$\overline{\varepsilon}$ :

Average dissipation rate of turbulent kinetic energy per unit mass, m²/s³

η :

Kinematic viscosity, m²/s

[η]:

Intrinsic viscosity, m³/kg

θ :

Surface coverage

κ ⁻¹ :

Debye length, m

[λ_i,j]:

i-th moment of molecular weight distribution of “live” polymer radicals in the j phase, kmol/m³

λ _b :

Breakage coalescence efficiency

λ _c :

Coalescence efficiency

μ :

Viscosity, kg/(m s)

[μ_k]:

k-th moment of dead polymer chains, kmol/m³

ρ :

Density, kg/m³

ρ _m :

Monomer density, kg/m³

ρ _mix :

Mixture density, kg/m³

ρ _p :

Polymer density, kg/m³

ρ _w :

Water density, kg/m³

σ :

Interfacial tension, kg/s²

σ ₀ :

Standard deviation

σ _da :

Standard deviation of the distribution for daughter drops

σ _sa :

Standard deviation of the distribution for satellite drops

φ :

Volume fraction of the dispersed phase

φ ₁ :

Volume fraction of the monomer-rich phase

φ ₂ :

Volume fraction of the polymer-rich phase

φ _2,C :

Critical value of the polymer volume fraction in the polymer-rich phase

φ _cr :

Polymer volume fraction corresponding to the critical monomer conversion

φ _j :

Volume fraction of the polymer in the j phase

φ _pol :

Volume fraction of the polymer in the dispersed phase

${\widehat{\phi}}_m$ :

Fugacity coefficient of monomer

χ :

Flory–Huggins interaction parameter

ω _b :

Breakage frequency, s⁻¹

ω _c :

Collision frequency, s⁻¹

1.2 Superscripts

g :

Gas phase

m :

Monomer phase

p :

Polymer phase

w :

Aqueous phase