Modulus particle effect

This rate depends on particle size (pellet) and in this case spherical particles. If there is no mass transfer limitations (diffusion effects), the effectiveness factor t] = 1. However, if diffusion effects take place this factor decreases significantly in a manner that is dependent on the Thiele modulus. The effectiveness factor varies according to Equation 18.24 ... 

In Figure 2.25 the effectiveness factor as function of the Thiele modulus for different pellet shapes is shown. For small values of the Thiele modulus the effectiveness factor reaches unity in all cases. The reaction rate is controlled by the intrinsic kinetics, and the reactant concentration within the pellet is identical to the concentration at the outer pellet surface. This situation may be observed for low catalyst activity or very small particles as used in fluidized beds or suspension reactors. For large values of the Thiele modulus the dependency of r p approaches an asymptotic solution tjp = micp with w = 1, 2, 3 for a slab, a cylinder, and a sphere, respectively. This situation may occur for very fast reactions or large catalyst particles. The concentration in the center of the catalyst particles approaches zero for rip < 0.2. 

Catalyst particle effectiveness factor E as a function of the Thiele modulus (p for a flat plate and isothermal first-order reaction. 

When Eqns. (9-13) and (9-13a) are used to define the Thiele modulus, the effectiveness factor for an isothermal catalyst particle is almost independent of particle geometry and... 

The intraparticle deactivation of nonisothermal pellets has been analyzed by Ray (21) using a pore mouth poisoning, slab geometry model. Heat flux at the boundary of the active and inactive portions of the particle (Figure 2a) was computed via Bischoff s ( ) asymptotic solution for large Thiele modulus. An effective diffusional modulus for this case can be defined as ... 

Stretching a polymer sample tends to orient chain segments and thereby facilitate crystallization. The incorporation of different polymer chains into small patches of crystallinity is equivalent to additional crosslinking and changes the modulus accordingly. Likewise, the presence of finely subdivided solid particles, such as carbon black in rubber, reinforces the polymer in a way that imitates the effect of crystallites. Spontaneous crystal formation and reinforcement... 

The result is shown in Figure 10, which is a plot of the dimensionless effectiveness factor as a function of the dimensionless Thiele modulus ( ), which is R.(k/Dwhere R is the radius of the catalyst particle and k is the reaction rate constant. The effectiveness factor is defined as the ratio of the rate of the reaction divided by the rate that would be observed in the absence of a mass transport influence. The effectiveness factor would be unity if the catalyst were nonporous. Therefore, the reaction rate is... 

The mass transport influence is easy to diagnose experimentally. One measures the rate at various values of the Thiele modulus the modulus is easily changed by variation of R, the particle size. Cmshing and sieving the particles provide catalyst samples for the experiments. If the rate is independent of the particle size, the effectiveness factor is unity for all of them. If the rate is inversely proportional to particle size, the effectiveness factor is less than unity and

experimental points allow triangulation on the curve of Figure 10 and estimation of Tj and ( ). It is also possible to estimate the effective diffusion coefficient and thereby to estimate Tj and ( ) from a single measurement of the rate (48). 

At the present time it is generally accepted that the toughening effect is associated with the crazing behaviour.Because of the presence of the low-modulus rubber particles most of the loading caused when a polyblend is subject to mechanical stress is taken up by the rigid phase (at least up to the moment of... 

Polyarylate (PAR)-b-PSt and PAR-b-PMMA for compatibiiizers are described 135,39,40). The addition of PAR-b-PSt (1-10 parts) to 100 parts of a blend of PAR-PSt (7w-3w) resulted in improvement of the tensile and flexural modulus (Fig. 4), and PSt dispersed particles were diminished from 1-5 microns to an order that is undetectable by SEM, indicating the excellent, compatibilizing effect of the block copolymer. The alloy thus formed exert the characteristic of PAR, an engineering plastic, as well as easy processability of PSt. Addition of PAR-b-PMMA (3 or 8 parts) to 100 parts of a blend of PAR-polyvinylidenefluoride (PVDF) (7w-3w) resulted in improved microdispersed state of PVDF due to compatibility of PMMA with PVDF, while segregation of PVDF onto the surface was controlled. 

Obviously, the discrepancy between the experimental data [238-241] and predictions of the theory [236,237] can be attributed to the difference of the coefficients of thermal expansion. The polymer exerts pressure on the filler, thereby masking the effect of the strength of adhesion on the modulus. The pressure on the filler may be sufficiently high. In [243] it was found, for example, that in PP, quartz particles experienced a compression force of about 100 MPa after cold drawing of the composite the force reduces to 50 MPa in the direction of drawing but at the same time increases to 300 MPa in the perpendicular direction. 

Under increasing strain the propint volume increases from the voids created around the unbonded solid particles. Nonlinearities in Young s modulus and Poisson s ratio then occur. Francis (Ref 50) shows this effect for a carboxy-terminated polybutadiene composite propellant with 14% binder as in Figure 12. He concludes that nonlinearities in low-temperature properties reduce the predicted stress and strain values upon cooling a solid motor, and therefore a structural analysis that neglects these effects will be conservative. However, when the predictions are extended to a pressurized fiberglas motor case, the nonlinearities in properties produce greater strains than those predicted with linear analysis... 

The relationship between effectiveness factor p and Thiele modulus < >l may be calculated for several other regular shapes of particles, where again the characteristic dimension of the particle is defined as the ratio of its volume to its surface area. It is found that... 

The solution of this equation is in the form of a Bessel function 32. Again, the characteristic length of the cylinder may be defined as the ratio of its volume to its surface area in this case, L = rcJ2. It may be seen in Figure 10.13 that, when the effectiveness factor rj is plotted against the normalised Thiele modulus, the curve for the cylinder lies between the curves for the slab and the sphere. Furthermore, for these three particles, the effectiveness factor is not critically dependent on shape. 

Estimate the Thiele modulus and the effectiveness factor for a reactor in which the catalyst particles are ... 

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