Moments of the size distribution

When the more general Mie scattering theory is applied the approach adopted in deriving the previous fomulae cannot he used. One is, however, able to derive an analytical expression for the moments of the size distribution within the detector cell. They are given as ... 

Thus, we plot M(x,t)IMi vs xls(t). As noted earlier, the cluster size distribution and the first moment of the size distribution are averaged over the entire journal bearing. As indicated by the behavior of P in Fig. 39b, the cluster size distribution becomes self-similar when the average size is about 10 particles per cluster. 

Figure 6 shows the size distributions for the samples taken from one of the runs, presented as the cumulative number oversize per ml of slurry. From the lateral shift of the size distributions, the growth rate can be determined. Figure 7 shows values of growth rate, G, supersaturation, s, and crystal content determined during the run. As a material balance check, the crystal contents were evaluated from direct measurements, from solution analyses and from the moments of the size distribution. The agreement was satisfactory. No evidence of size dependent growth or size dispersion was observed. 

To illustrate the application of Eq. (88) we will derive equations needed to calculate the second moment of molecular size distribution. The r-th moment of the size distribution which, taking into account Eq. (84), is defined thus ... 

The moments of the size distribution function are useful parameters. These have the form ... 

In this case, an eigth order approximation was used in order to include some of the higher moments of the distribution. KnU,m) represents the weight for the nth moment of the size distribution. The values for Kn are given in Appendix II. A similar expression can be obtained in terms of the D32 average if Equation 24 is applied to Q(aav,m) in equation 19. 

Several parameters can be used to express the mean size of the milk fat globules. These parameters are derived from the so-called moments of the size distribution function the th moment of the distribution function (Sn) is equal to ... 

With respect to the moments of the size distribution, it is common to define the number-average degree of polymerization Xw and the corresponding weight average X ... 

To evaluate a it is necessary to solve for the SPD, which depends on Df. Values of a for the free molecule regime vary little with Df in the range 2 to 3 as shown in Table 8.1. along with the 1/0/ moment of the size distribution function. Mi/Of For Df 3, ctsa function of the size of the primary particle, ctpo. 

In many cases of practical interest condensation takes place through both pathways discussed above, homogeneous and heterogeneous. These are very complex systems that in general must be analyzed on an ad hoc basis. In this section, an approximate set of equations is derived incorporating both processes in terms of the moments of the size distribution of the stable aerosol, dp > d (Friedlander, 1983). 

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. 

Figure 11.4 Evolution of the moments of the size distribution function for the aerosols shown in Fig. 11.3, The peak in the number distribution probably results when formation by homogeneous nudeaiiun is balanced by coagulation. Total aerosol volume increases with time as gas-tO partide conversion takes place. Total. surface area, A, increases at first and then approaches an approximately constant value, due probably toa balance between growth and coagulation (Husarand Whitby, 1973). The results should be compared with Pig. 11.2.

To compute the results of Equation 14.45 applied to Equation 14.44, we define the ith moment of the size distribution... 

The polymer species in the liquid phase span from L to jVmax, where L represents the minimum number of units for species in the liquid phase and Amax is the corresponding maximum, thereby limiting the mass MWD to 99.9%. Monomers and oligomers up to L-l units are in the gas phase. Frenklach (1985) and Bockhorn (1999b) assume the following definition of statistical moments of the size distribution of the polymer species ... 

Sum of Growth Rate Distributions. Berglund and de Jong (1990) devised a method of data treatment that allows use of the entire size distribution. It was previously shown that the moments of the growth rate distribution can be related to the moments of the size distribution, thus. 

The moments of the size distribution, and can be used to determine the relative extent of Ag condensation and particle coagulation on the growth mechanism. By measuring the arithmetic mean radius, r =Y rJN > cube-mean... 

Since is the average radius defined as the first moment of the size distribution f(R,t), the condition u=i must be satisfied. From Equation 6 with n=2, together with Equation 1 and the mass-conservation condition (Eq. 2), we obtain... 

In this equation, the average droplet radius is effectively a ratio of moments of the size distribution, Eq. (21), i.e.. 

Figure 4.11 Variation of the mean aggregate size ( n > number of elementary units) with the simulation time (Monte Carlo steps) at 0 = 0.05. The mean aggregate size is defined as (n ) = Mi/Mo, where A/, = is the zth-order moment of the size distribution [69].

---