Averages phase average

Note that we have also introduced the Reynolds-average, phase-average density (p). Applying the Favre average to Eq. (157) yields a closed expression for the mass balance as follows ... 

Any one of these is optional, fh e l un li lg phase is ofleri thought of as first an equilibration phase and then an averaging phase, although HyperChem does n ot specifically dlstin gulsh these two. 

Thermodynamic average. An average over all points in phase space at a single time. 

Note that the intrinsic phase average and the overall phase average are related by the poros ity or volume fraction of the phases by [423]... 

The average molar flux across some domain in terms of the intrinsic phase average concentration or the phase average concentration is [427]... 

For the case of a three-phase problem, where the solute is accessible to the a, (3, and y phases, Whitaker [427] finds the overall average phase concentration for the case of local mass equilibrium given by... 

Limitation to the term with / = 1 (whieh dominates for sufficiently small h and the same phase averaging as in appendix D leads to [24,25]... 

Figure. 1 Structural parameters variations of the evaporated and sol-gel samples for the main crystalline phases average grain dimension (a), unit cell mass modifications Am (b), lattice microstrain factor (c), lattice profile anisotropy factor AO (d).

Figure 20. The phase-averaged mean decay time as a function of frequency of the driving signal for different values of noise intensity, kT = 0.3,0.1,0.05, A — 1. Solid hnes represent results of computer simulation, and dashed lines represent adiabatic approximation (6.15).

Phase-averaged values of 4 in a plane midway between two baffles of a stirred tank have been plotted in Fig. 1 (from Hartmann et al., 2004a) for two different SGS models (Smagorinsky and Voke, respectively) in LES carried out in a LB approach. The highest values, i.e., the strongest deviations from isotropy, occur in the impeller zone, in the boundary layers along wall and bottom of the tank, and at the separation points at the vessel wall from which the anisotropy is advected into the bulk flow. In the recirculation loops, the turbulent flow is more or less isotropic. 

Fig. 1. Phase-averaged plots of the anisotropy distanced in a plane midway between two baffles in a stirred vessel provided with a Rushton turbine, as obtained by means of LES, with two different SGS models (a) the Smagorinsky model (b) the Voke model. Reproduced with permission from Hartmann et al. (2004a).

Note that these expressions (Eqs. 156 and 157) appear deceptively simple (i.e., as if the problem can be reduced to modeling a variable-density, single-phase flow) because we have hidden the difficult terms in the definition of some new symbols First, the phase-average stress a is defined by... 

The phase average (or ensemble average) of a dynamical quantity A(q,p) is defined as... 

Yet such an approach would be hampered by self-cancellation due to phase averaging of the NMR signal by integration over r. A solution would then be the use a surface coil whose receptivity is confined to a limited number of propagation wavelengths, i.e., for low wave frequencies and/or high velocities. 

When pellet-diffusion controls the exchange rate, the rate is often expressed in terms of a hypothetical solid-film coefficient kp and a contrived driving force (C — C s) where C is the concentration of the resin phase in equilibrium with C, and C s is the concentration of resin phase, averaged over the pellet. 

The quality of the measured phases ean be ehaiacterised by the averaged phase error (phase residual ( )Res) of s mimetry-related refleetions ... 

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