Volume or mass distribution

Volume or Mass Distribution v(D), the percentage of particles with a certain volume or mass, respectively, plotted against the equivalent diameter. 

Since atmospheric aerosols comprise particles with a wide range of sizes, it is often convenient to use mathematical models to describe the atmospheric aerosol distribution (Seinfeld and Pandis, 1998). A series of mathematical models have been proposed, of which the lognormal distribution has been the most used in atmospheric applications (Seinfeld and Pandis, 1998 Horvath, 2000). Useful discussions of the various aerosol size distribution models are provided by Seinfeld and Pandis (1998) and Jaenicke (1998). In general, atmospheric aerosols size distributions are shown graphically in terms of the volume (or mass) distributions, surface area distributions, or number distributions as a function of particle size (Jaenicke, 1998). 

Qi(x) Qx, Cumulative volume or mass distribution Cumulative volume distribution till class i — — Z(x) Electrical mobility of particle size x c G... 

If the quantity measure is number, Qfx) is called a cumulative number distribution. If it is length, area, volume, or mass, then the corresponding length [Qfx ], area [Qf,x ], volume, or mass distributions are formed [Qs(x), mass and volume are related by the specific density p. The index r in this notation represents the quantity measure (ISO 9276, Representation of Results—Part 1 Graphical Representation). The choice of the quantity measured is of decisive importance for the appearance of the PSD, which changes significantly when the dimension r is changed. As, e.g., one 100- lm particle has the same volume as 1000 10- lm particles or lOVl-pm particles, a number distribution is always dominated by and biased to the fine fractions of the sample while a volume distribution is dominated by and biased to the coarse. 

Standard quartz is officially designed as DQ12 < 5 (un (Robock 1973). The maximum of the particle number distribution is ca. 0.8 pm that of the surface distribution ca. 1pm while the maximum of the volume or mass distribution is ca. 1.3 pm. The upper size of the quartz particles is between 5 and 6 pm. The specific surface found by the BET-method is 7.4 mVg. 

Geometric standard deviation (GSD) is a measure of the distribution of particle sizes which can be used for log-normal volume (or mass) distributions as function of the diameter ... 

All currently marketed inhaler devices produce polydisperse aerosols of which the individual particles have different sizes. Therefore, they cannot be characterised by a single diameter. In fact, for most solid aerosols from dry powder inhalers the particles may have different shapes too, which is the reason to characterise them with aerodynamic diameters. To be able to express polydisperse aerosols with a single parameter, the median aerodynamic diameter (MAD) was introduced. When the aerodynamic size range which covers the population of particles in the aerosol is divided into different classes and the volume or mass fraction within each size class is expressed as function of the class mean diameter, a volume or mass distribution as function of the aerodynamic diameter is obtained. This volume or mass frequency distribution can be transferred into a cumulative percent distribution of which the 50 % value corresponds with the volume or mass median aerodynamic diameter (VMAD or MMAD). This is the diameter indicating that 50 % of the total aerosol volume or mass is in larger, and 50 % is in smaller particles. When particles of all sizes in the aerosol have the same density, which is mostly the case, then VMAD equals MMAD. 

Means of different distributions can be equivalent. For example, as is shown below, the arithmetic mean of a surface distribution is equivalent (numerically equal to) the harmonic mean of a volume (or mass) distribution ... 

In Figure 1.2 (NRC, 1979), the normalised frequency plots of number, surface and volume or mass distributions are presented. In this figure, the apparent area under the curves is proportional to the number, surface area and volume or mass in a given size range. Most particles, i.e. the number distribution, are of approximately 0.01 pm diameter. The number of particles decreases sharply with increasing size. Most of the surface area is provided by particles averaging 0.2 pm diameter. The volume or mass distribution is bimodal one mode is around 0.3 pm diameter, the other about 10.0 pm diameter. The mass of fine particles of size smaller than 2.0 pm is almost equal to the mass of coarse particles of size larger than 2.0 pm. Atmospheric aerosol size distributions consist basically of three separate modes ... 

Depending on their source there may be from one to three distinct maxima in the surface and volume or mass distributions. The activity size distribution of a radionuclide-associated aerosol particle is a surface distribution (Papastefanou and Bondietti, 1987). 

Normally the SMD is calculated with the help of the statistical moments because a measured DSD based on the surface area is not common. Then xi can be calculated with the (-l)-moment of the volume or mass distribution Q3, see (20.2). If a DSD... 

Pattemators may comprise an array of tubes or concentric circular vessels to coUect Hquid droplets at specified axial and radial distances. Depending on the pattemator, various uniformity indexes can be defined using the accumulated relative values between the normalized flow rate over a certain sector or circular region and a reference value that represents a perfectly uniform distribution. For example, using an eight-sector pie-shaped coUector, the reference value for a perfectly uniform spray would be 12.5%. The uniformity index (28) could then be expressed as foUows, where is the normalized volume or mass flow rate percentage in each 45-degree sector. 

The significance of this novel attempt lies in the inclusion of both the additional particle co-ordinate and in a mechanism of particle disruption by primary particle attrition in the population balance. This formulation permits prediction of secondary particle characteristics, e.g. specific surface area expressed as surface area per unit volume or mass of crystal solid (i.e. m /m or m /kg). It can also account for the formation of bimodal particle size distributions, as are observed in many precipitation processes, for which special forms of size-dependent aggregation kernels have been proposed previously. 

If a 1-g soil sample is extracted with 10 mL of extractant, then the component extracted is evenly distributed throughout the lOmL. This means the final result will need to be multiplied by 10 because the component was diluted 1 10 (this assumes an extractant density of lg/mL). This then is related back to the volume or mass of soil in the original sample, or it may be directly related back to the field. It may also be necessary to apply other conversion or correction factors, such as the percent water present in the original soil sample, depending on the procedure used. 

The volume of distribution of a drug molecule is, as described previously, a theoretical number that assumes the drug is at equal concentration in the tissue and in the circulation and represents what volume (or mass) of tissue is required to give that concentration. Volume of distribution, therefore, provides a term that partially reflects tissue affinity. However, it is important to remember that affinity may vary between different tissues and a moderate volume of distribution may reflect moderate concentrations in many tissues or high concentrations in a few. For an illustration of... 

Note that this result can also be found by using Eq. 2.11. With a lognormal distribution, the volume or mass median diameter will always be greater than the surface median diameter which will in turn be greater than the number median diameter. 

The mass is related to the cumulative (percent less than) size distribution by volume (or mass), F iL), as follows ... 

Here and below, the radius distribution of voids, /(r), is assumed to be defined so that f r) dr is the number of voids with the radius from r to r + dr per unit volume (or mass) of the sample. The radius distribution of necks, first derived by Wall and Brown 14) and were then employed by Kheifets and Neimark (15-17). 

The moments have physical meaning. The zeroth order moment (iq is the total number of particles per unit volume (or mass, depending on the basis). The first-order moment fii is the total length of the particles per unit volume, with the particles lined up along the characteristic length. The second-order moment is proportional to the total surface area, and the third-order moment is proportional to the total volume. Many physical characteristics of the particles such as the number-mean crystal size, weight-mean crystal size, the variance of the distribution function, and the coefficient of variation also can be represented in terms of the lower order moments of the distribution. 

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