Comprehensive Analysis of Single-Particle Growth in Heterogeneous Olefin Polymerization:  The Random-Pore Polymeric Flow Model

Abstract

In the present study, a comprehensive mathematical model is developed to analyze the effects of initial catalyst size, active site concentration, catalyst morphology (e.g., porosity, extent of prepolymerization, etc.), and hydrodynamic conditions on the growth and overheating of highly active Ziegler−Natta catalyst particles (e.g., fresh or prepolymerized) in gas-phase olefin polymerization. The generalized Stefan−Maxwell diffusion equation for porous solids is combined with the mass balances on the various molecular species (i.e., monomer and “live” and “dead” polymer chains) and the energy conservation equation to predict the temporal−spatial evolution of temperature and monomer concentration, as well as the polymerization rate in a single catalyst/polymer particle. To calculate the equilibrium monomer concentration in the amorphous polymer phase, the Sanchez−Lacombe equation of state is employed. It is shown that the evolution of the catalyst/particle morphology greatly affects the internal and external mass- and heat-transfer resistances in the particle and, thus, its growth rate and overheating. The effect of the hydrodynamic flow conditions on particle overheating is analyzed in detail. It is shown that, depending on the particle size, the concentration of solids in the bulk gas phase, and the dissipation rate of the turbulence kinetic energy of the flow field, the Ranz−Marshall correlation can significantly underestimate the value of the heat-transfer coefficient, resulting in an erroneous overestimation of the particle temperature.

Optimal Catalyst and Cocatalyst Precontacting in Industrial Ethylene Copolymerization Processes

“…In the present study, the external film mass transfer coefficient for the cocatalyst component in a two-phase liquid/solid system was estimated by employing a correlation proposed by Kittilsen et al [24]. Subsequently, the value of the external film mass transfer coefficient, ext , was used to estimate the molecular species concentration at the solid particle surface in a non-well-mixed system following the developments of Kanellopoulos et al [26,27]. According to these studies, the cocatalyst concentration at the catalyst particle surface exhibits exponential dependence with respect to the system energy dissipation rate which in turn depends on the energy input (i.e., stirring speed), solids concentration, and the physical (i.e., density, viscosity) and the transport properties (i.e., components diffusion coefficient in the liquid phase) of the liquid/solid system.…”

“…To assess the external and internal mass transfer limitations of the cocatalyst component during its precontacting with a Z-N spherical catalyst particle, the dynamic sorption model developed by Kanellopoulos et al [26,27] that a porous solid particle of radius is exposed at time = 0 to a liquid environment. Assuming that the initial cocatalyst concentration value at the catalyst particle is equal to zero (i.e., (0, ) = 0), the following dimensionless dynamic mass balance equation accounting for the diffusion of the cocatalyst molecules from the liquid bulk phase to the catalyst particle can be obtained:…”

“…In addition, and are the diffusion coefficient and the concentration of the cocatalyst at time and ,eq is the thermodynamic Journal of Polymers maximum concentration of the cocatalyst in the solid catalyst phase (see Table 1). The external liquid/solid film mass transfer coefficient was calculated using the boundary condition described in (4) [26,27]. In case of poor mixing conditions (i.e., no stirring during catalyst and cocatalyst precontacting), ext attains low values (i.e., 1.8⋅10 −2 1/s), while under intensive mixing conditions, ext reaches order of magnitude higher values (i.e., ext > 1.0 1/s).…”

“…In case of poor mixing conditions (i.e., no stirring during catalyst and cocatalyst precontacting), ext attains low values (i.e., 1.8⋅10 −2 1/s), while under intensive mixing conditions, ext reaches order of magnitude higher values (i.e., ext > 1.0 1/s). It has to be pointed out that the following equation is employed to calculate the value of ext taking into account the energy dissipation rate as well as all the physical and transport properties of the liquid medium [24,26,27]:…”

“…Moreover, the density of the catalyst/cocatalyst mixture is approximately constant during the whole sorption process, since only a relatively small amount of cocatalyst is sorbed into the solid catalyst phase (i.e., smaller than 0.02% w/w). Therefore, the variation of the particle radius,̇( ) = ( ( )/ ), caused by solid particle swelling, is not considered in (1) [26,27]. Moreover, the degree of swelling caused by the sorption of triethylaluminum in the catalyst solid phase was estimated by employing a fundamental thermodynamic model (i.e., Sanchez Lacombe EOS) and it was found to be below 5% in terms of catalyst particle volume change [32,33].…”