Empirical models, suspensions

Literature data for the suspension polymerization of styrene was selected for the analysi. The data, shown in Table I, Includes conversion, number and weight average molecular weights and initiator loadings (14). The empirical models selected to describe the rate and the instantaneous properties are summarized in Table II. In every case the models were shown to be adequate within the limits of the reported experimental error. The experimental and calculated Instantaneous values are summarized in Figures (1) and (2). The rate constant for the thermal decomposition of benzoyl peroxide was taken as In kd 36.68 137.48/RT kJ/(gmol) (11). 

The Instantaneous values for the initiator efficiencies and the rate constants associated with the suspension polymerization of styrene using benzoyl peroxide have been determined from explicit equations based on the instantaneous polymer properties. The explicit equations for the rate parameters have been derived based on accepted reaction schemes and the standard kinetic assumptions (SSH and LCA). The instantaneous polymer properties have been obtained from the cummulative experimental values by proposing empirical models for the instantaneous properties and then fitting them to the cummulative experimental values. This has circumvented some of the problems associated with differenciating experimental data. The results obtained show that ... 

In general, these formulas hold well for < 0.10, while empirical models using higher-order polynomials in can be used to fit almost any viscosity data. An equation that is often used for emulsions and concentrated suspensions of rigid spheres is that by Krieger and Dougherty ... 

To keep the particles in suspension, the flow should be at least 0.15m/sec faster than either 1) the critical deposition velocity of the coarsest particles, or 2) the laminar/turbulent flow transition velocity. The flow rate should also be kept below approximately 3 m/sec to minimize pipe wear. The critical deposition velocity is the fluid flow rate that will just keep the coarsest particles suspended, and is dependent on the particle diameter, the effective slurry density, and the slurry viscosity. It is best determined experimentally by slurry loop testing, and for typical slurries it will lie in the range from 1 m/s to 4.5 m/sec. Many empirical models exist for estimating the value of the deposition velocity, such as the following relations, which are valid over the ranges of slurry characteristics typical for coal slurries ... 

If we interpret this question as asking whether models exist for the general class of complex/non-Newtonian fluids that are known to provide accurate descriptions of material behavior under general flow conditions, the current answer is that such models do not exist. Currently successful theories are either restricted to very specific, simple flows, especially generalizations of simple shear flow, for which rheological data can be used to develop empirical models, or to very dilute solutions or suspensions for which the microscale dynamics is dominated by the motion deformation of single, isolated macromolecules or particles/drops.24... 

The just-suspended state is defined as the condition where no particle remains on the bottom of the vessel (or upper surface of the liquid) for longer than 1 to 2 s. At just-suspended conditions, all solids are in motion, but their concentration in the vessel is not uniform. There is no solid buildup in comers or behind baffles. This condition is ideal for many mass- and heat-transfer operations, including chemical reactions and dissolution of solids. At jnst-snspended conditions, the slip velocity is high, and this leads to good mass/heat-transfer rates. The precise definition of the just-suspended condition coupled with the ability to observe movement using glass or transparent tank bottoms has enabled consistent data to be collected. These data have helped with the development of reliable, semi-empirical models for predicting the just-suspended speed. Complete suspension refers to nearly complete nniformity. Power requirement for the just-suspended condition is mnch lower than for complete snspension. 

Since, as discussed above, it is impossible to achieve dynamic similarity between laboratory and full scale, the predictive capability of empirical modeling of crystallization is limited. Mathematical modeling also has its shortcomings. Suspension flows in crystallizers are turbulent, two and perhaps even three phase (for boiling crystallizers), the particle size is distributed, and the geometry is complicated with perhaps multiple moving parts (impellers). This is of course beyond the possibility of analytical solution of the equations of motion, so we must turn to computational fluid dynamics (CFD). However, even CFD is not capable of successfully dealing with all of these features. Successful computational models of crystallizers to date are limited to very specific limited problems. 

The application of the reaction rate from the suspension reactor would only be justified if the catalyst in the fixed bed were as completely wetted by liquid as in the stirred autoclave. A semi-empirical model was used to estimate the conversion (Eq. 13-24)... 

For the discrete bubble model described in Section V.C, future work will be focused on implementation of closure equations in the force balance, like empirical relations for bubble-rise velocities and the interaction between bubbles. Clearly, a more refined model for the bubble-bubble interaction, including coalescence and breakup, is required along with a more realistic description of the rheology of fluidized suspensions. Finally, the adapted model should be augmented with a thermal energy balance, and associated closures for the thermophysical properties, to study heat transport in large-scale fluidized beds, such as FCC-regenerators and PE and PP gas-phase polymerization reactors. 

Empirical multiple linear regression models were developed to describe the foam capacity and stability data of Figures 2 and 4 as a function of pH and suspension concentration (Tables III and IV). These statistical analyses and foaming procedures were modeled after data published earlier (23, 24, 29, 30, 31). The multiple values of 0.9601 and 0.9563 for foam capacity and stability, respectively, were very high, indicating that approximately 96% of the variability contributing to both of these functional properties of foam was accounted for by the seven variables used in the equation. 

Table III. Empirical multiple linear regression model describing foaming capacity as a function of pH and suspension concentration.

Cell models constitute a second major class of empirical developments. Among these, only two will be mentioned here as constituting the most successful and widely used. The first, due to Happel (1957,1958), is useful for estimating the effective viscosity and settling velocity of suspensions. Here, the suspension is envisioned as being composed of fictitious identical cells, each containing a single spherical particle of radius a surrounded by a concentric spherical envelope of fluid. The radius b of the cell is chosen to reproduce the suspension s volume fraction

[Pg.21]

In Chapter 2 (Section 2.9) we see how the cluster bonding requirements for the icosahedron, plus two-center and three-center inter-cluster bonds perfectly uses the three available valence electrons and four available valence orbitals in a covalently bonded cluster network. Once one has these advanced bonding models in hand, then the explanation of the B network structure is no more difficult than that of the C diamond structure. One purpose of this text is to provide these advanced models, but for now the solution to the problem remains hidden. Hey, a little suspense always helps the story line. At this empirical stage of the presentation you have learned that the nature of bonding (distribution of electrons) is expressed in geometry. The tricky bit is to interpret the empirical nuclear position in terms of a useful (simplest one that answers the question asked) model for the distribution of valence electrons. 

Subbarao and Basu (1986), Basu and Nag (1987) and Basu (1990) derived the expression of cluster residence time lc on the heat transfer surface based on Subbarao s (1986) cluster model, although the model is not widely accepted. Lu et ai (1990) and Zhang et al. (1987) have also obtained empirical correlations independently for predicting cluster residence time based on their heat transfer experiments. However, because of the lack of available and reliable information about the residence time of clusters at the surface and the fraction of the clusters in solids suspension, a significant discrepancy between the results predicted by the different approaches mentioned earlier has been observed. Besides, it should be pointed out that the major shortcoming in the earlier models is that they all take no account of the heat transfer surface length. 

Table IV shows the overall analysis of variance (ANOVA) and lists some miscellaneous statistics. The ANOVA table breaks down the total sum of squares for the response variable into the portion attributable to the model, Equation 3, and the portion the model does not account for, which is attributed to error. The mean square for error is an estimate of the variance of the residuals — differences between observed values of suspensibility and those predicted by the empirical equation. The F-value provides a method for testing how well the model as a whole — after adjusting for the mean — accounts for the variation in suspensibility. A small value for the significance probability, labelled PR> F and 0.0006 in this case, indicates that the correlation is significant. The R2 (correlation coefficient) value of 0.90S5 indicates that Equation 3 accounts for 91% of the experimental variation in suspensibility. The coefficient of variation (C.V.) is a measure of the amount variation in suspensibility. It is equal to the standard deviation of the response variable (STD DEV) expressed as a percentage of the mean of the response response variable (SUSP MEAN). Since the coefficient of variation is unitless, it is often preferred for estimating the goodness of fit.

Because no general theories exist even for concentrated non-food suspensions of well defined spherical particles (Jeffrey and Acrivos, 1976 Metzner, 1985), approaches to studying the influence of the viscosity of the continuous medium (serum) and the pulp content of PF dispersions, just as for non-food suspensions, have been empirical. In PF dispersions, the two media can be separated by centrifugation and their characteristics studied separately (Mizrahi and Berk, 1970). One model that was proposed for relating the apparent viscosity of food suspensions is (Rao, 1987) ... 

A mathematical model for styrene polymerization, based on free-radical kinetics, accounts for changes in termination coefficient with increasing conversion by an empirical function of viscosity at the polymerization temperature. Solution of the differential equations results in an expression that calculates the weight fraction of polymer of selected chain lengths. Conversions, and number, weight, and Z molecular-weight averages are also predicted as a function of time. The model was tested on peroxide-initiated suspension polymerizations and also on batch and continuous thermally initiated bulk polymerizations. 

C. Parkinson et al. (17) considered the effect of particle size distribution on viscosity. They studied suspensions of polymethylmethacrylate) spheres in Nujol with diameters of 0.1, 0.6, 1.0 and 4.0 microns with different volume fractions and with different particle size combinations to determine the influence of size-distribution on the viscosity. Each particle size gave a certain contribution to the final viscosity based on the volume fraction and the hydrodynamic coefficient obtained from the empirical equation for that particle size. The contributions were expressed in the same form as in Mooney s model, and the viscosity was calculated from the product of each term, n... 

---