Hatch-Choate equations

Special attention must be paid to the interpretation of particle size data presented in terms of either weight or number of particles. Particle weight data may be more useful in sedimentation studies, whereas number data are of particular value in surface-related phenomena such as dissolution. Values on the basis of number can be collected by a counting technique such as microscopy, while values based on weight are usually obtained by sedimentation or sieving methods. Conversion of the estimates from a number distribution to a weight distribution, or vice versa, is also possible using adequate mathematical approaches, e.g., the Hatch-Choate equations. 

Table 2 Hatch-Choate Equations for Conversion of Diameters...

TABLE 9.3 Values of the Constant b in the Hatch - Choate Equations for Converting the Count Geometric Mean Diameter to Mass, Surface, or Volume Diameters... 

Roller obtained good agreement with an extensive amount of experimental data. However, the application of his equation is dependent on straight-line plots between log w/ /d.and 1/d in order that a and b may be determined. Roller s equations appear to have no distinct advantage over the Hatch-Choate equations. 

The terms av and t s are called volume and surface shape-factors, respectively. Some interesting derivations to which the Hatch-Choate equations may be applied are deducible from these relationships. Let N be the number of particles per unit-weight of material having a density p. Then... 

In its original form the Hatch-Choate equation for conversion of number to mass is given by... 

Using the Hatch-Choate equation for surface median diameter gives... 

Using the Hatch-Choate equation, compute the mass median diameter from the information developed in Prob. 5. If the aerosol contains 1 million particles per cubic foot and the particle density is 1 g/cm3, find the aerosol concentration in micrograms per cubic meter. 

Show that the Hatch-Choate equations are just special cases of the general equation for lognormal distributions, Eq. 2.11. 

Hatch-Choate equation, 27—29 Haze, definition of, 3 Heat (see Temperature)... 

Equation 2.11 is a more general form of a well-known relationship used for converting particle number measurements to mass measurements and vice versa known as the Hatch-Choate equation (Drinker and Hatch, 1954). 

An advantage of the log normal distribution is that multiplication becomes addition and exponential terms become multiplicative. As a result of the properties of logarithms, the geometric standard deviation is the same for the number-length, number-surface, number-volume, etc. distributions. This fact allows one to calculate the relationships between different averages for the log normal distribution, and these equations are called the Hatch-Choate equations. For example, the number length diameter can be expressed as ... 

Dallavale defined a new shape factor that is modified from the correction factor. The Dallavalle shape factor can be useful for a log normal distribution because the shape of the size-hfcquency (density distribution) curve can be taken into account when combining Martin s correction factor with the Hatch-Choate equation (1,34,37). Applying the Hatch-Choate equation of dy, and from Table 6 to in Equation (79) and Sw in Equation (81) yields. 

Appendix 3 Derivation of the Hatch-Choate Equations / 105 Problems / 108... 

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