One-Moment Exponent Averages

The best known of these one-moment exponent averages is what is known as the viscosity-average molar mass  

According to Equation (8-42), averages from two moments are also possible. The order ofthe moments [that is, (/ + 1) in the numerator and (q — 1) in the denominator] must be combined with the exponent 1/pin such a manner that the total expression has the same physical units as the property. Since the physical units on both sides of the equation must be the same, the so-called exponent rule can be directly obtained from Equation (8-42), 

The exponent rule is especially significant in combination with the rule that states that the relationships between two variables can always, at least over a limited range of values, be written as an exponential relationship. It has been found empirically that over wide ranges of the molar mass the following relationships between the molecular weight and the sedimentation coefficient 5, the diffusion coefficient D, or the intrinsic viscosity [17] are valid  

The molar mass can be obtained from any pair of the three quantities (for derivation, see Chapter 9) 

The quantities KsOy Ks, and Kd are accessible through independent measurements and are independent of the molar mass. They are consequently called physical constants. On the other hand, A sd, A sr, and A dv are model constants, since they are based on certain assumptions. If, for example, the frictional coefficients from sedimentation and diffusion are of equal magnitude (see Chapter 9), XhttiAso = 1. The model constants can, of course, influence the numerical value of the molar mass, but they have no effect on the composition of the average from the various individual molecular species contributions. Consequently, model constants can always be assumed to have a value of unity until evidence to the contrary is obtained. 

According to these equations, the following must always hold  

8 Molecular Weights, Molecular- Weight Distributions 

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The MOM is also applicable for bivariate PBEs, as explained by Hulburt Katz (1964). In this case the moments of the NDF are defined as wjk = / n d, where and are the two internal coordinates and k = k, k2) is the exponent vector containing the order of the moment with respect to the two internal coordinates. Moments can have a particular physical meaning and, as for the univariate case, mo,o is the total particle number density. Likewise, the ratio of different moments can be used to characterize integral properties of the population, for example mi,o/mo,o is the number-average value of the first internal coordinate whereas mo,i/mo,o is the number-average value of the second one 2- For details and examples on the relationship between bivariate moments and measurable quantities, readers are referred, for example, to the work of Rosner et al. (2003). As for the univariate case, one has to solve the evolution equation for the moments. 

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