Volume averaging

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Eulerian averaging is the most important and widely-used method of averaging, because it is consistent with the control volume analysis that we used to develop the governing equations in the preceding section. It is also applicable to the most common techniques of experimental observations. Eulerian averaging is based on time-space description of physical phenomena. In the Eulerian description, changes in the various dependent variables, such as velocity, temperature, and pressure, are expressed as functions of time and space coordinates, which are considered to be independent variables. One can average these independent variables over both space and time. The integral operations associated with these averages smooth out the local spatial or instant variations of the properties within the domain of integration.

For a generalized function Φ = Φ(x,y,z,t), the most widely-used Eulerian averaging includes time averaging and volumetric averaging. The Eulerian time average is obtained by averaging the flow properties over a certain period of time, t, at a fixed point in the reference frame, i.e.,

\bar \Phi  = \frac{1}{{\Delta t}}\int_{\Delta t} {\Phi (x,y,z,t)dt}     \qquad \qquad(1)

for this equation, the time period Δt is chosen so that it is larger than the largest time scale of the local properties’ fluctuation, yet small enough in comparison to the process macroscopic time scale of the process. During this time period, different phases can flow through the fixed point. Eulerian time averaging is particularly useful for a turbulent multiphase flow as well as for the dispersed phase systems (Faghri and Zhang, 2006).

Eulerian volumetric averaging is usually performed over a volume element, ΔV, around a point (x,y,z) in the flow. For a multiphase system that includes Π different phases, the total volume equals the summation of the individual phase volumes, i.e.,

\Delta V = \sum\limits_{k = 1}^\Pi  {\Delta {V_k}}     \qquad \qquad(2)

The volume fraction of the kth phase, {\varepsilon _k}, is defined as the ratio of the elemental volume of the kth phase to the total elemental volume for all phases, i.e.,

 {\varepsilon _k} = \frac{{\Delta {V_k}}}{{\Delta V}}    \qquad \qquad(3)

The volume fraction of all phases must sum to unity:

\sum\limits_{k = 1}^\Pi  {{\varepsilon _k} = 1}      \qquad \qquad(4)

Eulerian volume averaging is expressed as

\left\langle \Phi  \right\rangle  = \frac{1}{{\Delta V}}\sum\limits_{k = 1}^\Pi  {\int_{\Delta {V_k}} {{\Phi _k}(x,y,z,t)dV} }     \qquad \qquad(5)

where the volume element Δ must be much smaller than the total volume of the multiphase system so that the average can provide a local value of Φ in the flow field. The volume element ΔV must also be large enough to yield a stationary average. Since the volume element includes different phases, information about the spatial variation of Φ for each individual phase is lost and \left\langle \Phi  \right\rangle represents the average for all phases.

For any variable or property that is associated with a particular phase, Φk, the phase-average value of any variable or property for that phase is obtained with the following equations

Intrinsic phase average:

 {\left\langle {{\Phi _k}} \right\rangle ^k} = \frac{1}{{\Delta {V_k}}}\int_{\Delta {V_k}} {{\Phi _k}dV}     \qquad \qquad(6)

Extrinsic phase average:

\left\langle {{\Phi _k}} \right\rangle  = \frac{1}{{\Delta V}}\int_{\Delta {V_k}} {{\Phi _k}dV}     \qquad \qquad(7)

Intrinsic means that it forms to the inherent part of a phase and is independent of other phases in the volume element. In contrast, extrinsic means it is a property that depends on the phase’s relationship with other phases in the volume element.

While the intrinsic phase average is taken over only the volume of the kth phase in eq. (6), the extrinsic phase average for a particular phase is taken over an entire elemental volume in eq. (7). These two phase-averages are related by

\left\langle {{\Phi _k}} \right\rangle  = {\varepsilon _k}{\left\langle {{\Phi _k}} \right\rangle ^k}     \qquad \qquad(8)

The intrinsic and extrinsic phase averages defined in eqs. (6) and (7) are related to the volume average defined in eq. (5) by

 \left\langle \Phi  \right\rangle  = \sum\limits_{k = 1}^\Pi  {\left\langle {{\Phi _k}} \right\rangle }  = \sum\limits_{k = 1}^\Pi  {{\varepsilon _k}{{\left\langle {{\Phi _k}} \right\rangle }^k}}    \qquad \qquad(9)

The deviation from a respective intrinsic phase-average value is

	{\hat \Phi _k} \equiv {\Phi _k} - {\left\langle {{\Phi _k}} \right\rangle ^k}     \qquad \qquad(10)

When the products of two variables are phase-averaged, the following relations are needed:

	{\left\langle {{\Phi _k}{\Psi _k}} \right\rangle ^k} = {\left\langle {{\Phi _k}} \right\rangle ^k}{\left\langle {{\Psi _k}} \right\rangle ^k} + {\left\langle {{{\hat \Phi }_k}{{\hat \Psi }_k}} \right\rangle ^k}     \qquad \qquad(11)
\left\langle {{\Phi _k}{\Psi _k}} \right\rangle  = {\varepsilon _k}{\left\langle {{\Phi _k}} \right\rangle ^k}{\left\langle {{\Psi _k}} \right\rangle ^k} + \left\langle {{{\hat \Phi }_k}{{\hat \Psi }_k}} \right\rangle     \qquad \qquad(12)
Control volume for volume averaging
Control volume for volume averaging.

In order to obtain the volume-averaged governing equations, the volume average of the partial derivative with respect to time and gradient must be obtained. For a control volume ΔV shown in figure on the right, the volume averaging of the partial derivative with respect to time is obtained by the following general transport theorem:

 \left\langle {\frac{{\partial {\Omega _k}}}{{\partial t}}} \right\rangle  = \frac{{\partial \left\langle {{\Omega _k}} \right\rangle }}{{\partial t}} - \frac{1}{{\Delta V}}\int_{{A_k}} {{\Omega _k}{{\mathbf{V}}_I} \cdot {{\mathbf{n}}_k}d{A_k}}    \qquad \qquad(13)

where Ak is the interfacial area surrounding the kth phase within control volume ΔV, ΔVk is the volume occupied by the kth phase in the control volume and ΔV, {{\mathbf{V}}_I} is the interfacial velocity, and nk is the unit normal vector at the interface directed outward from phase k (see figure).

The volume average of the gradient is

\left\langle {\nabla {\Omega _k}} \right\rangle  = \nabla \left\langle {{\Omega _k}} \right\rangle  + \frac{1}{{\Delta V}}\int_{{A_k}} {{\Omega _k}{{\mathbf{n}}_k}d{A_k}}     \qquad \qquad(14)

and the volume average of a divergence is

	\left\langle {\nabla  \cdot {\Omega _k}} \right\rangle  = \nabla  \cdot \left\langle {{\Omega _k}} \right\rangle  + \frac{1}{{\Delta V}}\int_{{A_k}} {{\Omega _k} \cdot {{\mathbf{n}}_k}d{A_k}}     \qquad \qquad(15)

The general quantity Ωk in eqs. (13) and (14) can be a scalar, vector, or tensor of the second order. It can be a vector or tensor of the second order in eq. (15).

The formulation of macroscopic equations for multiphase systems can be classified into two groups: (1) the multi-fluid model, and (2) the homogeneous model, also known as the mixture or diffuse model. If the averaging is performed for each individual phase within a multiphase control volume, as shown in eqs. (6) and (7), one obtains the multifluid model, in which Π sets of averaged conservation equations – each set includes continuity, momentum and energy equations – describe the flow of a Π − phase system. The equations will also include source terms that account for the transfer of momentum, energy, and mass between phases. If only two phases are present, the multifluid model is referred to as the two-fluid model. However, if spatial averaging is performed over both phases simultaneously within a multiphase control volume, as indicated in eq. (5), the homogeneous model is obtained; in this case the mixture of a two-phase fluid would be considered a whole. The governing equations for the homogeneous model comprise a single set of equations including continuity, momentum, and energy equations, with one additional diffusion equation to account for the concentration change due to interphase mass transfer by phase change. Continuity, momentum, and energy equations for the mixture model can be obtained by adding together the governing equations for the multifluid models; a diffusion model must be developed to account for mass transfer between phases.

References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

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