Size distribution function equation

The size distribution of micellar aggregates Ng/F is plotted against the aggregation number, g, for an amphiphile with an octyl hydrocarbon tail and for a 2 X 1G4 cal A2t molsb (Figure 1). Equation 19 leads to wjt = 3.88. For < crit, the size distribution is a monotonic decreasing function of g. At = CT t, the size distribution function has ah inflection point. At > mt, the size distribution function has two extrema. It can be seen that if increases both the number and the average size of the micellar aggregates increase. 

There is some analogy between the critical concentration as defined here and the critical temperature predicted by the van der Waals equation of state, since each of them separates two kinds of behavior of the size distribution function and pressure-volume relationship, respectively. 

The introduction of this correction makes the algebra leading to the equations for gCTn and c,a more complex. For this case, the value of eJjt can be computed only by numerical methods. Figure 2 represents the size distribution function for 4 = 3 A and a 8 X 104 cal A2/mol for an amphiphile with an octyl hydrocarbon tail. 

A number of analytical solutions have been developed since that of von Smoluchowski, all of which contain some assumptions and constraints. Friedlander [33] and Swift and Friedlander [34] developed an approach relaxing the above constraint of an initially monodisperse suspension. Using a continuous particle size distribution function, a nonlinear partial integro-differential equation (with no known solution) results from Eq. (5). Friedlander [35] demonstrated the utility of a similarity transformation for representation of experimental particle size distributions. Swift and Friedlander [34] employed this transformation to reduce the partial integro-differential equation to a total integro-differential equation, and dem-... 

In this paper we present a new characterisation method for porous carbonaceous materials. It is based on a theoretical treatment of adsorption isotherms measured in wide temperature (303 to 383 K) and pressure ranges (0 to 10000 kPa) and for different adsorbates (N2, CH4, Ar, C3H8 and n-C4Hio). The theoretical treatment relies on the Integral Adsorption Equation concept. We developed a local adsorption isotherm model based on the extension of the Redlich-Kwong equation of state to surface phenomena and we improved it to take into account the multilayer formation. The pore size distribution fimction is assumed to be a bi-modal gaussian. By a minimisation procedure, it is possible to determine the bi-modal pore size distribution function witch can be used for purely characterisation purposes or to predict adsorption isotherms. 

The methods depend on the theoretical treatment which is used. A majority of them are based on the Generalised Adsorption Isotherm (GAI) also called the Integral Adsorption Equation (LAE). The more recent approaches use the Monte Carlo simulations or the density functional theory to calculate the local adsorption isotherm. The analytical form of the pore size distribution function (PSD) is not a priori assumed. It is determined using the regularization method [1,2,3]. Older methods use the Dubinin-Radushkevich or the Dubinin-Astakhov models as kernel with a gaussian or a gamma-type function for the pore size distribution. In some cases, the generalised adsorption equation can be solved analytically and the parameters of the PSD appear directly in the isotherm equation [4,5,6]. Other methods which do not rely on the GAI concept are sometimes used the MP and the Horvath-Kawazoe (H-K) methods are the most well known [7,8]. 

Kamack offered the following solution to equation (8.12). If Q is plotted as a function of v, = rJSy with f = o9-t as parameter, a family of curves is obtained whose shape depends on the particle size distribution function. The boundary conditions are that Q - 1 when f = 0 for all r, (i.e. the suspension is initially homogeneous) and = 0 for r, = S when f>0 (i.e. the surface region is particle free as soon as the centrifuge bowl spins). Hence all the curves, except for f>0, pass through the point Q = 0, S, and they will all be asymptotic to the line / = 0, which has the equation Q = - Furthermore, from equation (8.12), the areas under the curves are each equal to F(r/., ). 

Figure 10. Representative particle-size distribution function (PSDF) for the storms fit with a cubic regression equation.

The asymmetrical cell size distribution functions may be described by the following equation... 

Fiiedlander (1960), Hunt (1980), Filella and Buffle (1993), and others have analyzed the effect of colloid agglomeration by coagulation and particle removal by settling on the shape of the particle size distribution function as expressed by equation 4. The predictions of model calculations are often consistent with the range of values of /3 observed in aquatic systems. 

We can conclude the following from an Inspection of Figures 20, 21 and 22. Equation 32 gives an accurate pore size distribution function for the porous polymeric membrane prepared by the microphase separation method. The mean radius Increases and the pore size distribution broadens with S. and Pr. The reduced pore distribution N(r)S vs. r/S curve is Independent of S. but dependent on Pr. The effect of Pr on N(r) Is more remarkable than that of S. The reduced pore size distribution curves widen with an Increase In Pr. 

If the particles are sufficiently large, it is permissible to pass from the discrete to the continuous particle size distribution and particle current. The basic starting equation for the analysis of the behavior of the stable aerosol in a batch reactor is the continuity relationship (10.33) for the continuous size distribution function. In terms of nj and dp, this can be written... 

The General Dynamic Equation for the Particle Size Distribution Function. ... 

At the beginning of the chapter it is shown that the usual models for coagulation and nucleation presented in Chapters 7 and 10 arc special cases of a more general theory for very small particles. An approximate criterion is given for determining whether nucleation or coagulation is rate controlling at the molecular level. The continuous form of the GDE is then used to derive balance equations for several moments of the size distribution function. 

Let the distribution function for particles in a volume range between v and vand the area range between a and a + da at time t be n(u, a, /). The Smoluchowski equation for the continuous size distribution function (Chapter7) becomes (Koch and Friedlander, 1990)... 

Chapter 11 THE GENERAL DYNAMIC EQUATION FOR THE PARTICLE SIZE DISTRIBUTION FUNCTION 306... 

One of my goals in writing the book was to introduce the use of the equation for the dynamics of the particle size distribution function at the level of advanced undergraduate and introductory graduate instruction. This equation is relatively new in applied science, but has many applications in air and water pollution and the atmospheric sciences. 

For a general dimension d, the cluster size distribution function n(R, i) is defined such that n(R, x)dR equals the number of clusters per unit volume with a radius between R and R + dR. Assuming no nucleation of new clusters and no coalescence, n(R, i) satisfies a continuity equation... 

Although (13.82) is a rigorous representation of the system, it is impractical to deal with discrete equations because of the enormous range of k. Thus it is customary to replace the discrete number concentration Nk(t) (cm-3) by the continuous size distribution function n(v,t) (pm-3 cm-3), where v = kv, with w, being the volume associated with a monomer. Thus n(v, t)dv is defined as the number of particles per cubic centimeter having volumes in the range from to t) + dv. If we let wo = g v be the volume of the smallest stable particle, then (13.82) becomes in the limit of a continuous distribution of sizes... 

Although (12.101) is a rigorous representation of the system, it is impractical to deal with discrete equations because of the enormous range of k. Thus it is customary to replace the discrete number concentration N it) (cm ) by the continuous size distribution function... 

Let us consider the continuous condensation equation, (12.8), written in terms of particle mass m as the size variable, so that n(m, t) is the size distribution function based on particle mass, and / ,(m, t) is the rate of change of a mass of a particle of mass m due to condensation ... 

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