Velocity mass-weighted average

Within a fluid packet, the net directed fluid velocity V is a mass-weighted average of the individual molecular velocities ... 

The summation in equation (14) is extended over all N distinct chemical components. Thus v[ is the mass-weighted average velocity of the fluid mixture, and Vf is said to denote the diffusion velocity of species K. Since... 

In order to show that the model of independent, coexistent continua represents correctly a real mixture of gases composed of different chemical species, we must compare the results obtained from this model with those of the kinetic theory of nonuniform gas mixtures (see Appendix D). Quantities such as the density p, the mass-weighted average velocity v j, and the body force fj have obviously analogous meanings in both the kinetic theory and the coexistent-continua model. On the other hand, the precise kinetic-theory meaning of terms such as the stress tensor, the absolute internal energy per unit mass and the heat-flux vector qf is not immediately apparent. In view of the known success of continuum theory for one-com-... 

In each of the above relations, the property for the mixture is equal to the sum, over all species, of the corresponding property for the components plus various diffusion terms. The diffusion terms arise because the reference coordinate system for species K is taken to move with velocity v j Vf which is the mass-weighted average velocity for molecules in species K alone the reference coordinate system for the mixture on the other hand is taken to move with velocity v j, which is the mass-weigh ted average velocity for all molecules in the mixture. The latter is a natural coordinate system in that it is consistent with equations (19) and (34). 

The mass weighted average velocity is not determined in the vast majority of cases and, for example, in the case of cross correlation the velocity measured in the above hypothetical example would be vo. Whilst the example is merely illustrative it does demonstrate that the flow patterns within the pipeline may have a major effect on the performance of the measuring instrument. 

In the subsequent temporal decomposition of the scalar variables is as a main rule performed in accordance with (3.323), while the velocity is decomposed using the approach for mass-weighted variables (3.325). After time averaging the mass flux (3.350) of phase k, we obtain ... 

After introducing a temporal decomposition of the mass-weighted velocity variable and time averaging the volume averaged term in (3.370), we get ... 

In this section we derive the diffusion mixture model equations starting from the time averaged multi-fluid model expressed in terms of phase- and mass weighted variables [112]. The relative movement of the individual phases is given in terms of diffusion velocities. 

This approach makes the velocities in the population balance different from the mass- or phase weighted average velocities obtained solving the two-fluid model. This discrepancy is an argument for the formulation of a mass density population balance instead of a number balance to achieve a consistently integrated population balance within the two-fluid modeling framework. 

In order to separate the average of products into products of average, weighted averaged values are commonly introduced. The phasic - and mass averages have been defined by (3.277) and (3.278), respectively. Hence, the instantaneous velocity is decomposed into a weighted mean component and a fluctuation component in accordance with (3.279). 

Earlier in section 4.4 the weight-average velocities u were introduced (see (4.23)), as well as the mass fluxes J relative to the average velocity (see (4.26)). In view of these equations, (5.41) may be rewritten as... 

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