Size distribution function particle diameter equation

In practice, when one measures the size distributions of aerosols using techniques discussed in Chapter 11, one normally measures one parameter, for example, number or mass, as a function of size. For example, impactor data usually give the mass of particles by size interval. From such data, one can obtain the geometric mass mean diameter (which applies only to the mass distribution), and crg, which, as discussed, is the same for all types of log-normal distributions for this one sample. Given the geometric mass mean diameter (/) ,) in this case and crg, an important question is whether the other types of mean diameters (i.e., number, surface, and volume) can be determined from these data or if separate experimental measurements are required. The answer is that these other types of mean diameters can indeed be calculated for smooth spheres whose density is independent of diameter. The conversions are carried out using equations developed for fine-particle technology in 1929 by Hatch and Choate. 

Specifically, the calculations had as their goal the computation of (dZi/dz) as a function of particle size for clouds of different heights at various altitudes and sampling times, including the parameters applicable to the samples analyzed. A detailed exposition of the theory and its limitations is presented in the Appendix. The values of (dzjbz) are divided point by point into the measured size distribution—i.e., f(a,z,t)— to arrive at the size distribution at stabilization time—i.e.y f[a,z(a,z,t),0], according to Equation 2. An additional output of the calculations are the cutoff diameters (smallest and largest diameters) in the samples. 

The solution of the above equation in order to obtain W(y) requires an appropriate form of the spreading function and the numerical values of its parameters. Furthermore, to convert W(y) into a size distribution requires a relationship between the mean retention volume y and the particle diameter D (i.e., a calibration curve). 

As suggested earlier, both sedimentation and flow FFF (in steric or hyperlayer modes) can be used to characterize particles in the size range of chromatographic silica. However, the two FFF approaches are more complementary than they are redundant. This is once again (as found also in the normal mode) a consequence of the different force laws that control particle behavior in the two systems. The driving force in flow FFF depends only on the Stokes diameter ds as shown in equation 8. Thus for spherical silica particles, the diameter and diameter distribution, and nothing more, is characterized by flow FFF. However, the force acting on particles in a sedimentation FFF channel is a function of both particle diameter d and the density difference Ap as shown by equation 7. Measurement by sedimentation FFF alone thus yields a mix of diameter and density information. However, if d can be established independently by other means (such as flow FFF or microscopy), then Ap can be obtained... 

Physically, the characteristics of the partial differential equation give us the trajectories in the time-diameter coordinate system of the various particle sizes. If, for a moment, we think of the particle population not as a continuous mathematical function but as groups of particles of specific diameters, the characteristic equations describe quantitatively the evolution of the sizes of these particles with time. The characteristic curves in Figure 12.2 are not the complete solution of the p.d.e., but they still convey information about the evolution of the size distribution for this case. 

Very early on, Aitken (1923) showed that most particles in the atmosphere are smaller than 0.1 pm diameter and that their concentrations vary from some hundreds per cm over the ocean to millions per cm in urban areas. Junge (1955,1963,1972) measured the atmospheric aerosol number size distribution and concentration in urban and non-urban areas as functions of altitude and site. He established the standard form for plotting size distribution data log of AN/ADp versus logD, where N = number and Dp = particle diameter. He observed that this plot was a straight line that could be described by the equation AN/ADp = AD, where A and k were constants. He also noted that in the range from 0.1 to 10.0 pm particle diameter, k was approximately equal to 4.0. This distribution mode was widely known as the Junge distribution or the power law distribution. 

In Equation (3.71), Q3 represents the mass summation curve ranging from 0 to 1, d is the particle diameter. The parameters of the distribution function are characteristic particle size d, which can be simply read off from the summation curve at a value of 0.632, and the parameter of homogeneity n, which requires linearization. 

Table VIII summarizes the physical processes that affect the evolution of aerosol in a unit volxame of atmosphere. To develop the general dynamic.equation governing aerosol behavior let us assxime that the aerosol is composed of liquid droplets of M chemical species. We let c denote the concentration of species i in a droplet, i = 1,2,..., M, and Dp denote the diameter of the particle. We then define n(Dp, c, ..., cjyj,r,t) as the size-composition distribution function, such that n dDp dc. .. dcj is the n umber of particles per unit vol ume of atmosphere at location r at time t of diameter Dp to Dp + dDp and of composition c to spegies ...

Substituting all the possible combinations of characteristics, i.e. values of p and q, info equation 1.10 gives rise to a number of differenf definitions of the mean size of a distribution. At minimum fluidization the drag force acting on a particle due to the flow of fluidizing gas over the particle is balanced by the net weight of fhe particle. The former is a function of surface area and the latter is proportional to particle volume. Consequently the surface-volume mean diameter, with p = 2 and = 3, is the most appropriate particle size to use in expressions for minimum fluidizing velocity. It is defined by equafion 1.11... 

Figure 71 shows some of the results as a function of the mean particle size Xq and Xi Xq is the surface equivalent diameter and Xi is the maximum of the diameter distribution density (see Section 3.1.1.1). The diagonal lines represent the maximally transferable tensile strength calculated according to equation (4) using a = 6 and a = 8. 

Particulate products, such as those from comminution, crystallization, precipitation etc., are distinguished by distributions of the state characteristics of the system, which are not only function of time and space but also some properties of states themselves known as internal variables. Internal variables could include size and shape if particles are formed or diameter for liquid droplets. The mathematical description encompassing internal co-ordinate inevitably results in an integro-partial differential equation called the population balance which has to be solved along with mass and energy balances to describe such processes. 

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