Average intrinsic

Besides the phase average, it is also important to introduce the intrinsic average, which is defined by... 

Hence, the phase average can be related to the intrinsic average by the volume fraction of phase k, c, as... 

Usually, the intrinsic average reflects the real physical property or quantity such as density and velocity, while the phase average gives a pseudoproperty or quantity based on the selection of control volume. Phase averages are used to construct the continuum of each phase to which Eulerian description can be applied. 

Note that volume averaging can only be applied to quantities per unit volume, such as density, momentum per unit volume, and energy per unit volume. The intrinsic average of the velocity of phase k may thus be defined as... 

Furthermore, the intrinsic average of velocity correlation may be approximated by a product of intrinsic averages of velocities as... 

The approximation in the preceding equation is to circumvent any conflict with Eq. (5.100), which defines the intrinsic average of velocity. This approximate relation becomes an equivalent one under the isothermal condition of phase k with the expansion work estimated by... 

Assuming the material density or the intrinsic averaged density of phase k to be constant (as is true for the solids and almost true for the fluid when the effect of temperature variation on the material density can be neglected), we have... 

The relationship between regional average and intrinsic average is... 

To simplify the problem further, we adopt a concept similar to that for the intrinsic averaged velocity and intrinsic velocity tensor as in Eqs. (5.129) and (5.130). Thus, Eqs. (6.95) and (6.96), respectively, become... 

For subsequent discussions, the symbol () is removed with an understanding that the velocities and densities involved are intrinsically averaged. 

The dimensionless intrinsic average of the total mass density having been expanded in a Taylor series... 

Intrinsic averaging corresponds to calculating density, concentration, or other parameter directly obtainable from experiments. Comparison of (A-7) and (A-8) shows that... 

The average of the spatial deviation V fc is computed applying the intrinsic average operator (3.118) on (3.208). The obvious result is ... 

If i( is a constant, then < /f )f = /<. The intrinsic average is useful in analysis of multiphase flow and in dealing with the energy equation. 

When dealing with porous catalysts, one generally does not work with the intrinsic average transport equation given by... 

In this notation for the volume averages, a Greek subscript is used to identify the particular phase under consideration, while a Greek superscript is used to identify an intrinsic average. Since the intrinsic and superficial averages differ by a factor of Sy, it is essential to make use of a notation that clearly distinguishes between the two averages. 

Here we have used the intrinsic average concentration since this is most closely related to the concentration in the fluid phase, and we have used the superficial average velocity since this is the quantity that normally appears in Darcy s law (Whitaker, 1999) or the Forchheimer equation (Whitaker, 1996). Use of equation 1.87 in equation 1.84 leads to... 

Dressing surfactant density Magnetic core density Effective relaxation time of magnetization Brownian relaxation time Neelian relaxation time Hydrodynamic volume fraction of the surfactant-dressed nanoparticles Suspension volume fraction of magnetic material embedded in the nanoparticles y-phase-side values of the variable eg on K-phase-side values of the variable eg on Spatial fluctuation of the bounded piecewise continuous scalar field cp around the intrinsic average in u... 

A certain fraction of these positrons, however, may enter the bound state of the positronium (Ps), by combining with an electron. Ps can be formed in two ground states either in the triplet or ortho state, with parallel spin orientation and an intrinsic average annihilation lifetime of 1.4 x 10" s, or in the singlet or para state with antiparallel spin orientation, which has an intrinsic average lifetime of 1.25 x 10"l s (Fig. 3). 

The average of the spatial deviation is computed applying the intrinsic average... 

Data from a single molecule are 100% pure there is no intrinsic averaging or collective effect hidden in the data. But how do we know that we have a single molecule, and especially the correct one ... 

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