The Volume Averaging Procedure

The condition under which the volume averaging operators can be applied handling the scale disparity in a suitable manner can be expressed as [235, 156, 89, 205, 206]  

It is noted that the requirement of proper separation of scales represents the main drawback of the volume averaging method. The constitutive equations used generally depend strongly on this assumption which is hardly ever fulfilled performing simulations of laboratory, pilot and industrial scale reactor units. 

The volume averaging operator which can be applied to any scalar, vector or tensor valued property function ipk associated with phase k, is defined by  

It is emphasized that tph)v is the average of the quantity 4 in phase k over the entire averaging volume V. This averaging operator is sometimes called the extensive or superficial volume average of ip for phase k. 

Mathematicians may say that this approach leads to more elegant mathematics, since the phase indicator function might be imagined as a sensor which enables local instantaneous phase identification. However, this somewhat mathematical formulation may also impede physical understanding for an inexperienced reaction engineer having a more practical background. 

The Xk is the phase indicator function which equals 1 in phase k and zero elsewhere  

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Proceeding in the same way as explained presenting the volume averaging procedure, we obtain the following result... 

A similar procedure can be applied to the energy transport terms. When the volume-averaged energy equation is expressed in terms of the internal energy and time averaging... 

In the case of two immiscible fluids, a characteristic phase indicator function, Xj, may be defined that is equal to 1 In one of the phases and 0 in the other phase. Then Xi and nj are related by nidi = VXi analogue to the relations used In standard volume averaging procedures [54] [164]. An averaged representation of this relation may be given as f mSida = —Vai... 

As distinct from the time and volume averaging procedures, the application of the ensemble averaging operator is not restricted by any space- and time scales. The ensemble averaging is thus considered the fundamental method [58, 59]. 

The time after volume averaging procedure can be applied under a unified set of conditions denoting the sum of the two sub-sets of requirements formulated in sects 3.4.1 and 3.4.2 for the pure volume and time averaging procedures to handle the scale disparity in a proper manner. 

For multiphase flows perturbed by the presence of particles to obtain a turbulence like behavior the local instantaneous velocity of the continuous phase can for example be decomposed adopting the Reynolds averaging procedure (i.e., other methods including time-, volume-, ensemble-, and Favre averaging have been used as well) and expressed as Vg = Vg- - < v >g, where v(. is the fluctuating component of the continuous phase velocity. Introducing the peculiar velocity for the dispersed phase this relation can be re-arranged as ... 

The 2D steady-state two-fluid model presented in this section is based on the early work by Torvik and Svendsen [22], Svendsen et al [21], and Jakobsen [12], The two-fluid model was derived based on the time-after volume averaging procedure, described in sect 3.4.4. 

The volume-averaging technique has been developed by Whitaker (7), Slattery (8), Bear (33), and Lundgren (9). Assuming that the solid matrix of the porous medium is incompressible, immobile, and not supported by the fluid, equations 17 and 18 can be subjected to a volumeaveraging procedure in an REV and reduce to the following equations (32) ... 

In mineral processing, in addition to size, the particle density and possibly the shape may also vary. Using the measured variation of —dpf/dz) with z, the procedure of Di Felice et al. (1987), just described, can also be applied to determine the changing composition with bed level of two spherical particle species differing in both diameter and density, by means of an additional equation for the volume-average density,... 

In a continuous stirred tank separator (CSTS) employed in particulate systems, there are no spatial gradients inside the separator therefore we need to integrate either equation (6.2.51b) or equation (6.2.51c) over the total volume V of the separator. To that end, we may employ the type of relations (6.2.4b-d) utilized in volume-averaging procedures. For example, the second term, V- Upn rp)j, on the left-hand side of (6.2.51b) will lead to... 

The nonlinear master curve for dilute emulsions has furthermore been extended to take polydisperse droplet distributions into account. Modelling procedures analogous to the ones presented above were used for a variety of distributions [36], finding that the same universal curve can be used for polydisperse systems if the volume averaged droplet radius is used in Eq. 13. 

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