Multigrain models

Multigrade batch processing, 20 703 Multigrade oils, 15 225 Multigrain model, of polymer growth, 26 528... 

PCM (polymeric core model), PFM (polymeric flow model), and MGM (multigrain model). 

Figure 10, Propylene concentration profiles in the multigrain model as a function of radius and time for a high-activity catalyst.

Figure 11. Multigrain model predictions for propylene polymerization in a semihatch reactor Q = M /Mn, rate = gm polymer/gm cat.-hr, yield = gm polymer/gm-catalyst.

In the Multigrain model, fractured catalyst microparticles are produced during the polymerization and uniformly dispersed in the polymer each of these particles behaves as a micro Solid core and diffusion within them, as well as in the interstices between them, can take place. In the Polymeric flow model the catalyst microparticles are dispersed in a polymer continuum and move outward in proportion to the volumetric expansion due to polymerization only one value of diffusivity is considered. Both these models predict significant MWD broadening due to mass transfer limitations (Q , 9 for polypropylene in the Polymeric flow model) on the basis of mathematical calculations carried out assuming reasonable values of the kinetic and physical parameters. 

In this way the Multigrain model would take on a precise physical meaning. The microglobular polypropylene surface, obtained with both traditional and high yield catalysts is clearly shown out by electron scanning microscopy. 

Such a different conclusion can be understood by considering the difficulties connected to the experimental determination and the definition of Thiele modulus parameters, such as So and D. According to Chien, S means the catalyst primary particle size with a value of about 10 cm for a-TiCIj instead, in the Multigrain model, Sp seems to correspond to the size of the whole catalyst granule. 

Several models have been proposed to describe intraparticle heat and mass transfer with heterogeneous coordination catalysts [114], but the most commonly accepted is the multigrain model (MGM) [115-126], In the MGM, the... 

Figure 5.13 The multigrain model (MGM). (See insert for the color representation of the figure.)...

The particles in Fig. 30 represent an excellent replica of the catalyst particle distribution through the polymer particle distribution the spherical form of the initial particles is retained and the equivalent circle diameters are enlarged from about 90 pm to about 360 pm after 190 min polymerization time. Their dependence on time indicates a very active catalyst system with a fast copolymerization rate and a fast particle expansimi caused by the loosely agglomerated MgQ2 support and the volume increase of the amorphous copolymer. It seems that this copolymerization system follows the multigrain model. 

More recently, so-caUed multigrain models have been developed (Asua, 2007 Tobita and Yanase, 2007) to describe polymerizations with solid catalysts in a more detailed manner, including a clear link between the micro- and mesoscale. In such models, the polymer growth and the possibility to form radial concentration and temperature gradients are accounted for. The catalyst particle is seen as an agglomeration of macrograins, which in turn are composed of micrograins, as depicted in Fig. 10.17. The catalyst sites are assumed to be present at the surface of the catalyst particle. 

Principle of a multigrain model for the description of radial temperature and concentration gradients in heterogeneous polymerizations with solid catalysts. A macrograin (radius / mac) consists of micrograins (radius Rn,ic with polymer layer, without polymer layer) in concentric layers surrounded... 

Notice that the multigrain model does not deal directly with the initial seconds of particle fragmentation, when the catalyst pores are being filled with polymer chains that start to fragment the catalyst particles. Since many industrial catalysts are in fact pre-polymerized in milder conditions in a separate reactor before being fed to the polymerization reactor, this should not be seen as a limitation of the multigrain model for most industrial applications. Catalyst pre-polymerization in... 

The multigrain model equation for spherical secondary particles is the classic diffusion-reaction equation in a sphere, Eq. (41), where Ms is the monomer concentration in the secondary particle as a function of polymerization time, t, and radial position, rj. 

Finally, is the average volumetric rate of polymerization in the secondary particle at a given radial position. Since, in the multigrain model, the polymerization is assumed to take place only at the surface of the primary particles, this term couples the models for the primary and secondary particles. 

Notice that Eq. (46) does not contain a polymerization reaction term. Because the multigrain model assumes that polymerization takes place at the surface of the catalyst fragment embedded within the primary particle, the reaction term appears as one of the two required boundary conditions [Eq. (48)]. Equation (48) states that the rate of monomer diffusion at the surface of the catalyst fragment, Rc, equals the rate of monomer consumption due to polymerization at rate of R, and Eq. (49) imposes the condition that the concentration at the surface of the primary particle equals the equilibrium concentration of monomer absorbed onto the polymer phase,... 

Tab. 8.2. Temperature profiles in the primary and secondary particles according to the multigrain model.

The multigrain model also includes a set of equations to describe the temperature profiles in the primary and secondary partides. These equations are summarized in Table 8.2. Mathematical models for solving this system of partial differential equations with moving boundaries are involved and have been discussed in the literature [36, 51-60],... 

The equations presented so far for the multigrain model are mass- and energy-balance equations in a spherical catalyst particle used for conventional heterogeneously catalyzed reactions subjected to a moving boundary due to polymer formation. To predict polymer properties such as chain length and chemical composition, these monomer and temperature profiles must be coupled with an additional set of equations that describes polymerization and termination mechanisms... 

Population balances and the method of moments can also be combined with the multigrain model and other polymer particle growth models. In this case, the population balances are defined for each position in the particle to obtain the radial profiles of chain length averages [36, 51-60]. 

The polymerization of olefins with coordination catalysts is performed in a large variety of polymerization processes and reactor configurations that can be classified broadly into solution, gas-phase, or slurry processes. In solution processes, both the catalyst and the polymer are soluble in the reaction medium. These processes are used to produce most of the commercial EPDM rubbers and some polyethylene resins. Solution processes are performed in autoclave, tubular, and loop reactors. In slurry and gas-phase processes, the polymer is formed around heterogeneous catalyst particles in the way described by the multigrain model. Slurry processes can be subdivided into slurry-diluent and slurry-bulk. In slurry-diluent processes, an inert diluent is used to suspend the polymer particles while gaseous (ethylene and propylene) and liquid (higher a-olefins) monomers are fed into the reactor. On the other hand, only liquid monomer is used in the slurry-bulk pro-... 

The term macroparticle diffusion resistance is used as defined by Ray and coworkers (3) in their multigrain model. If there is a very significant resistance, this could be the physical factor limiting rate. At an early stage when polymer particles are small the polymerization rate would be much reduced, later polymerization rate would Increase. 

The grain- or multigrain model consists of separate model equations for characteristic grains or micro-particles and the macro-scale pellet. The characteristic micrograin is described by the following diffusion-reaction partial- differential equation for the product species s ... 

---