Polydisperse ensemble

It is important to note that this formula gives only an average value of the crystallite size if, for example, we have a polydispersed ensemble of spherical particles with density probability of the diameter sizes N D), the average... 

As already noted in the case of the Scherrer equation, if we have a polydispersed ensemble of spheres the dimensions obtained by Fourier analysis correspond to ... 

The fractal dimension that is determined from Eq. (38) is not in all cases the true fractal dimension of the individual macromolecules. In a polydisperse ensemble one has to take the ensemble average which yields an ensemble fractal dimension df g [7,92] ... 

In the preceding, we assumed implicitly that we were dealing with a mono-disperse system, but if the chains form a (moderately) polydisperse ensemble the same considerations apply. Then, the structure function H(q) is defined on the average. Thus, for Brownian chains, (13.2.98) is replaced by... 

Estimations of characteristic times of drop integration, as described in the previous paragraph, related to the elementary case of monodisperse drop distribution without regard for the forces of hydrodynamic and molecular interactions. We now consider the kinetics of integration of a polydisperse ensemble of drops in view of these forces. 

As earlier, consider ti to be the characteristic coagulation time of a polydisperse ensemble of drops, caused by the mechanism of turbulent diffusion due to the forces of hydrodynamic and molecular interactions. This time should be estimated. For typical values of the flow, Pq = 40 kg m , 2o = 5 x 10 m, Pq = 1.2 X 10 Pa-s, W = 5 X 10 m /m and distribution parameters of = 10 m, k = 3, one obtains 1/ti = 0.257 s. Thus, a twofold increase in drop radius occurs in a time t of 7 s. This time is almost two orders of magnitude higher than for a monodisperse distribution without regard to hydrodynamic and molecular forces. Such a big difference in characteristic times is undoubtedly caused not by taking into account the polydispersivity of the distribution, but as a result of considering the interaction forces. 

Continuous thermodynamics may be applied to all kinds of systems containing polydisperse ensembles. Important examples of such systems are solutions and mixtures of synthetic polymers and copolymers, crude oils and many liquid... 

Fundamentals of continuous thermodynamics as applied to homopolymers characterized by univariate distribution functions have been reviewed extensively [28, 29]. Hence, this chapter will provide the fundamentals in their most general form by considering systems composed of any number of polydisperse ensembles described by multivariate distribution functions and any number of solvents and by referring to the papers [28, 29],... 

A Fortran code for computing the amplitude and phase matrices for a homogeneous, axisymmetric particle in an arbitrary orientation A Fortran code for computing the far-field scattering and absorption characteristics of a polydisperse ensemble of randomly oriented, homogeneous, axisymmetric particles... 

Fet us now consider the 3D equivalent of the aforementioned example an ensemble of uncorrelated homogeneous spheres - with polydispersity, meaning that the observed CLD... 

The technique of stimulated Raman scattering (SRS) has been demonstrated as a practical method for the simultaneous measurement of diameter, number density and constituent material of micrometer-sized droplets. 709 The SRS method is applicable to all Raman active materials and to droplets larger than 8 pm in diameter. Experimental studies were conducted for water and ethanol mono-disperse droplets in the diameter range of 40-90 pm. Results with a single laser pulse and multiple pulses showed that the SRS method can be used to diagnose droplets of mixed liquids and ensembles of polydisperse droplets. 

Computationally, polydispersity is best handled within a grand canonical (GCE) or semi-grand canonical ensemble in which the density distribution p(a) is controlled by a conjugate chemical potential distribution p(cr). Use of such an ensemble is attractive because it allows p(a) to fluctuate as a whole, thereby sampling many different realizations of the disorder and hence reducing finite-size effects. Within such a framework, the case of variable polydispersity is considerably easier to tackle than fixed polydispersity The phase behavior is simply obtained as a function of the width of the prescribed p(cr) distribution. Perhaps for this reason, most simulation studies of phase behavior in polydisperse systems have focused on the variable case [90, 101-103]. 

For ensembles of particles the light extinction is additive, as discussed previously. However, in certain cases simplifying assumptions can be made. For example, suppose there is a polydisperse aerosol having n d) particles of diameter d per unit volume. From Eq. 16.11... 

The remaining chapters in this book are organized as follows. Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. There, the mathematical definition of a number-density function (NDF) formulated in terms of different choices for the internal coordinates is described, followed by an introduction to population-balance equations (PBE) in their various forms. Chapter 2 concludes with a short discussion on the differences between the moment-transport equations associated with the PBE and those arising due to ensemble averaging in turbulence theory. This difference is very important, and the reader should keep in mind that at the mesoscale level the microscale turbulence appears in the form of correlations for fluid drag, mass transfer, etc., and thus the mesoscale models can have non-turbulent solutions even when the microscale flow is turbulent (i.e. turbulent wakes behind individual particles). Thus, when dealing with turbulence models for mesoscale flows, a separate ensemble-averaging procedure must be applied to the moment-transport equations of the PBE (or to the PBE itself). In this book, we are primarily... 

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