Population density distribution steady state

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. 

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...

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 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. 

Figure A3.13.16. Illustration of the level populations (eorresponding to the C-C oseillator states) from various treatments in the model of figure A3.13.15 for C2Hg at a total energy E = (he) 41 000 em and a tlneshold energy = (he) 31 000 em The pomts are mieroeanonieal equilibrium distributions. The erosses result from the solution of the master equation for IVR at steady state and the lines are thennal populations at the temperatures indieated (from [38] quant, is ealeulated with quantum densities of states, elass. with elassieal meehanieal densities.).

Growth and nucleation interact in a crystalliser in which both contribute to the final crystal size distribution (CSD) of the product. The importance of the population balance(37) is widely acknowledged. This is most easily appreciated by reference to the simple, idealised case of a mixed-suspension, mixed-product removal (MSMPR) crystalliser operated continuously in the steady state, where no crystals are present in the feed stream, all crystals are of the same shape, no crystals break down by attrition, and crystal growth rate is independent of crystal size. The crystal size distribution for steady state operation in terms of crystal size d and population density // (number of crystals per unit size per unit volume of the system), derived directly from the population balance over the system(37) is ... 

A population balance can be used to follow the development of a crystal size distribution in batch crystallizer, but both the mathematics and physical phenomena being modeled are more complex than for continuous systems at steady state. The balance often utilizes the population density defined in terms of the total crystallizer volume, rather than on a specific basis n = nVj. Accordingly, the general population balance given by Eq. (51) can be modified for a batch crystallizer to give ... 

Laser-induced fluorescence is a sensitive, spatially resolved technique for the detection and measurement of a variety of flame radicals. In order to obtain accurate number densities from such measurements, the observed excited state population must be related to total species population therefore the population distribution produced by the exciting laser radiation must be accurately predicted. At high laser intensities, the fluorescence signal saturates (1, 2, 3 ) and the population distribution in molecules becomes independent of laser intensity and much less dependent on the quenching atmosphere (4). Even at saturation, however, the steady state distribution is dependent on the ratio of the electronic quenching to rotational relaxation rates (4, 5, 6, 7). When steady state is not established, the distribution is a complicated function of state-to-state transfer rates. 

The population balance equation is employed to describe the temporal and steady-state behavior of the droplet size distribution for physically equilibrated liquid-liquid dispersions undergoing breakage and/or coalescence. These analyses also permit evaluation of the various proposed coalescence and breakage functions described in Sections III,B and C. When the dispersion is spatially homogeneous it becomes convenient to describe particle interaction on a total number basis as opposed to number concentration. To be consistent with the notation employed by previous investigators, the number concentration is replaced as n i,t)d i = NA( i t)dXi, where N is the total number of particles per unit volume of the dispersion, and A(xj t) dXi is the fraction of drops in increment X, to X( + dxi- For spatially homogeneous dispersions such as in a well-mixed vessel, continuous flow of dispersions, no density changes, and isothermal conditions Eq. (102) becomes... 

For an oxide (device) thickness of 0.5 pm and a device population density of 0.8, predicted steady-state wafer temperature distributions are shown in Figs. 18.58a and b. The isotherms of... 

The experiment of Kumar et al (2000) consists of continuously feeding the polydisperse suspension through a vertical column in the well-mixed state and allowing the relative motion of particles to exit at an outlet located at a suitable distance from the point of entry. The relative motion of particles will have established a steady state, spatially uniform distribution of particles with an exit number density that can be measured by a device such as a Coulter counter. The population density, / (z, v) in vertical coordinate z and particle size described by volume v, satisfies the population balance equation... 

Over the years, the FA technique has undergone continuous refinement and development and fotmd a wide variety of applications, e.g., in fundamentals of ion-molecttle reactiorts and in atmospheric and interstellar chemistry [104]. The FA technique enables the generation of high-density, steady state populations of ions and reactive neutral species with well-defined thermal energy distributions. The reaction conditiotts can be carefully controlled among others due to the temporal... 

Because of the assumption of isotropy of medium and source emission, the number of neutrons moving in a given direction per unit solid angle per unit volume will be the same for all directions. Thus for systems at steady state, the energy distribution alone will suffice to describe the neutron population. Consequently, for this calculation, we can use the functions for the neutron density and flux given by (3.49)... 

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