Distributions, population density

Experimental MSMPR population density distributions for eooling erystallization are illustrated in Figure 9.4. 

Figure 9.4 Population density distribution of potassium sulphate crystals from continuous cooling crystallization Jones and Mydlarz, 1989)...

Mt suspension, or magma, density, kgm n L, t) population density distribution function, /m" nuclei population density (m m )... 

Population Density Distribution. Integration of Equation 1 yields a semilogarithmic relationship. 

Equations (11) and (12) provide two experimental methods for evaluation K A.. (ss). Experimental agreement confirms the accuracy 85 the population density distribution obtained by GPC. 

N umerical simulations of reactor start-up were programmed, predicting monomer and initiator concentrations, total polymer concentration, weight and number average molecular weights, viscosity and population density distribution dynamics. The following two relationships obtained from steady state observations were utilized in the simulation. 

Gel Permeation Chromatography. The instrument used for GPC analysis was a Waters Associates Model ALC - 201 gel permeation chromatograph equipped with a R401 differential refractometer. For population density determination, polystyrene powder was dissolved in tetrahydrofuran (THF), 75 mg of polystyrene to SO ml THF. Three y -styragel columns of 10, 10, 10 A were used. Effluent flow rate was set at 2.2 ml/min. Total cumulative molar concentration and population density distribution of polymeric species were obtained from the observed chromatogram using the computer program developed by Timm and Rachow (16). 

Steady State Population Density Distributions. Representative experimental population density distri-butions are presented by Figure 1 for two different levels of media viscosity. An excellent degree of theoretical (Equation 8) / experimental correlation is observed. Inasmuch as the slope of population density distribution at a specific degree of polymerization is proportional to the rate of propagation for that size macroanion, propagation rates are also observed to be independent of molecular weight. 

Figure 1, Steady state population density distributions...

Polymerization Dynamics. Relationships presented were utilized for the simulation of monomer concentration, number and weight average molecular weights, and population density distributions for two experimental observations. Experimental values of these variables are in reasonable proximity of calculated values. 

A kinetic study for the polymerization of styrene, initiated with n BuLi, was designed to explore the Trommsdorff effect on rate constants of initiation and propagation and polystyryl anion association. Initiator association, initiation rate and propagation rates are essentially independent of solution viscosity, Polystyryl anion association is dependent on media viscosity. Temperature dependency correlates as an Arrhenius relationship. Observations were restricted to viscosities less than 200 centipoise. Population density distribution analysis indicates that rate constants are also independent of degree of polymerization, which is consistent with Flory s principle of equal reactivity. 

Dynamic simulations for an isothermal, continuous, well-mixed tank reactor start-up were compared to experimental moments of the polymer distribution, reactant concentrations, population density distributions and media viscosity. The model devloped from steady-state data correlates with experimental, transient observations. Initially the reactor was void of initiator and polymer. 

The resin system selected to initiate these studies is a step-growth anhydride cured epoxy. The approach to the kinetic analysis is that which is prevalent in the chemical engineering literature on reactor design and analysis. Numerical simulations of oligomeric population density distributions approximate experimental data during the early stages of the cure. Future research will... 

Equation 12 nay now be expressed in terms of the experimentally measurable population density distribution Pj. 

The cumulative molar concentration of polymeric species P 0 may be evaluated from the population density distribution, Equation 16. The first two rate expressions represent monomeric additions which do not change the molar concentration of polymeric molecules. A rate constant that describes functionality as a separable function of the molecule s degree of polymerization satisfies this constraint. The simplest, realistic function is the linear expression... 

Relationship 23 provides a method for evaluating the parameter "a" that is defined by Equation 2A. The cumulative molar concentration of polymeric species PT0T was numerically evaluated via integration of population density distributions. The contribution of network molecules to the zeroth moment of the distribution is negligible. Results are presented by Figure A and show that... 

Initial numerical simulations of population density dynamics incorporated experimental data of Figures 2 and 4 and Equations 16, 20, 23, and 27 into a Runge-Kutta-Gill integration algorithm (21). The constant k.. was manipulated to obtain an optimum fit, both with respect to sample time and to degree of polymerization. Further modifications were necessary to improve the numerical fit of the population density distribution surface. 

The polymerization sites at the molecule s exterior are likely to experience more fruitful collisions than sites on the interior. To account for such complex phenomena, an additional separable function was introduced to simulate steric hindrance factors at the molecular level. An arbitrary, but continuous, exponential function was selected. The resulting rate constant that yields a good fit of experimental population density distributions is... 

The polystyrene data were collected from a steady state, continuous, well-mixed reactor. The initiator was n-butylli-thlum for data of Figure 2 and was azobisisobutylnitrile for data of Figure 3. Toluene was used as a solvent. The former polymerizatl n y ields an exponential population density distribution ( ), M /M = 1.5 the latter yields a molar distribution defined as th product of degree of polymerization and an exponential ( ), M /M = 2.0. Standards utilized in calibration of both instrumen s ftere polystyrene supplied by Pressure Chemical Company. 

The first moment of the distribution is Pt0T the total, cumulative molar concentration of polymeric material. As the molecular weight of polymeric species increases, branching and crosslinking reactions yield a thermoset resin. Chromatography analysis of epoxy resin extracts confirms the expected population density distribution described by Equation 4, as is shown in Figure 2. Formulations and cure cycles appear in Table II. 

Figure 5. Population Density Distribution of Mixed Product from the Crystallizer.

The first method for AHFs as a function of the population density distribution in urban areas is the one most frequently used and was tested for FUMAPEX NWP models. 

TABLE 4.4 Population Density Distribution Data Obtained from Sieve Analysis in Example 4.3... 

Distribution of particle area as a function of size Distribution of particle mass as a function of size (th moment of the population density distribution as defined by Eq. (4.11)... 

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