# Averaging approaches

(Difference between revisions)
 Revision as of 14:28, 28 June 2010 (view source)← Older edit Current revision as of 01:43, 13 July 2010 (view source) (14 intermediate revisions not shown) Line 4: Line 4: ==Volume Averaging== ==Volume Averaging== + 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. + ''See Main Article'' [[Volume averaging|Volume Averaging]] ''See Main Article'' [[Volume averaging|Volume Averaging]] - ==Area Averaging== + ==Lagrangian Averaging== - ''See Main Article'' [[Area averaging|Area Averaging]] + Lagrangian averaging is directly related to the Lagrangian description of a system, which requires tracking the motion of each individual fluid particle. Therefore, Lagrangian averaging is a very useful tool when the dynamics of individual particles are of interest. To obtain Lagrangian time averaging, it is necessary to follow a specific particle and observe its behavior for a certain time interval. Then, the behavior of this particle is averaged over the time interval. + + ''See Main Article'' [[Lagrangian Averaging]] ==Boltzmann Statistical Averaging== ==Boltzmann Statistical Averaging== + When the collective mechanics of a large number of particles is of interest, molecular statistical averaging may be employed. This relies on the concept of particle number density, which is the number of particles per unit volume. For a system with a large number of particles, the behavior of each individual particle is random because random collisions occur. To describe the behavior of each particle, it is necessary to track the motion resulting from each collision – an impractical and often unnecessary task. Although the behavior of each particle is random, the collection of particles may demonstrate some statistical behaviors that are different from those of the individual particles. When the number of molecules involved in the averaging process is large enough, the statistical average value becomes independent of the number of molecules involved. The statistical average value of the microscopic properties for a large number of molecules is related to the macroscopic properties of the system. + ''See Main Article'' [[Boltzmann statistical averaging|Boltzmann Statistical Averaging]] ''See Main Article'' [[Boltzmann statistical averaging|Boltzmann Statistical Averaging]] + + + The objectives of the various averaging methods are twofold: (1) to define the average properties for the multiphase system and correlate the experimental data, and (2) to obtain solvable governing equations that can be used to predict the macroscopic properties of the multiphase system. This chapter will address the application of averaging methods to the governing equations. + Based on the physical concepts used to formulate multiphase transport phenomena, the averaging methods can be classified into three major groups: (1) ''Eulerian averaging, (2) Lagrangian averaging'', and (3) ''Molecular statistical averaging''. These averaging techniques are briefly reviewed below. + + + ===2.4.1.1 Eulerian Averaging=== + + 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 $\Phi =\Phi (x,y,z,t)$, 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, $\Delta$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 $\Delta t$ 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. 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 [[#References|(Faghri and Zhang, 2006)]]. + + ''Eulerian volumetric averaging'' is usually performed over a volume element, $\Delta V$, around a point ($x,y,z$) in the flow. For a multiphase system that includes $\Pi$ 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 ${k^{th}}$ phase, ${{\varepsilon }_{k}}$, is defined as the ratio of the elemental volume of the ${k^{th}}$ 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 $\Delta V$ must be much smaller than the total volume of the multiphase system so that the average can provide a local value of $\Phi$ in the flow field. The volume element $\Delta V$ must also be large enough to yield a stationary average. Since the volume element includes different phases, information about the spatial variation of $\Phi$ 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, ${{\Phi }_{k}}$, 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 ${k^{th}}$ 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: + + +
${{\left\langle {{\Phi }_{k}}{{\Psi }_{k}} \right\rangle }^{k}}={{\left\langle {{\Phi }_{k}} \right\rangle }^{k}}{{\left\langle {{\Psi }_{k}} \right\rangle }^{k}}+{{\left\langle {{{\hat{\Phi }}}_{k}}{{{\hat{\Psi }}}_{k}} \right\rangle }^{k}}\qquad \qquad(11)$
+ + + +
$\left\langle {{\Phi }_{k}}{{\Psi }_{k}} \right\rangle ={{\varepsilon }_{k}}{{\left\langle {{\Phi }_{k}} \right\rangle }^{k}}{{\left\langle {{\Psi }_{k}} \right\rangle }^{k}}+\left\langle {{{\hat{\Phi }}}_{k}}{{{\hat{\Psi }}}_{k}} \right\rangle \qquad \qquad(12)$
+ + + [[Image:chapter2_(1).gif |thumb|400 px|alt=Control volume for volume averaging|Figure 1: 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 $\Delta V$ shown in Fig. 1, 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 ${{A}_{k}}$ is the is the interfacial area surrounding the ${k^{th}}$ phase within control volume $\Delta V$, $\Delta {{V}_{k}}$ is the volume occupied by the ${k^{th}}$ phase in the control volume and $\Delta V$, ${{\mathbf{V}}_{I}}$ is the interfacial velocity, and ${n_k}$ is the unit normal vector at the interface directed outward from phase k (see Fig. 1). + + 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 ${{\Omega }_{k}}$ 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'' (Section 2.4.2), and (2) the ''homogeneous model'' (Section 2.4.3), 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 $\Pi$ sets of averaged conservation equations – each set includes continuity, momentum and energy equations – describe the flow of a $\Pi -$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. In this section, it is assumed for the sake of simplicity that the reference frame is stationary. + + ===2.4.1.2 Lagrangian Averaging=== + + Lagrangian averaging is directly related to the Lagrangian description of a system, which requires tracking the motion of each individual fluid particle. Therefore, Lagrangian averaging is a very useful tool when the dynamics of individual particles are of interest. To obtain Lagrangian time averaging, it is necessary to follow a specific particle and observe its behavior for a certain time interval. Then, the behavior of this particle is averaged over the time interval. + + For a generalized function $\Phi =\Phi (X,Y,Z,t)$, X, Y, and Z are material coordinates moving with the particle, and X, Y, Z are functions of the spatial coordinates $x, y, z$, and time t, i.e., + +
$\begin{matrix} + X=X(x,y,z,t), & Y=Y(x,y,z,t), & Z=Z(x,y,z,t) \\ + \end{matrix}$
+ + + The most widely used Lagrangian averaging is ''time averaging'', where the time average of the function $\Phi$ in time interval of $\Delta t$ is + + +
$\bar{\Phi }=\frac{1}{\Delta t}\int_{\Delta t}{\Phi (X,Y,Z,t)dt}\qquad \qquad(16)$
+ + + + Lagrangian time averaging is performed for a distinct particle moving in the field; therefore, X, Y, and Z in the time interval $\Delta t$ are not fixed in space. This focus on specific particles moving in space and time distinguishes Lagrangian averaging from Eulerian time averaging, which treats a fixed point in space relative to the reference frame. An example from daily experience will serve to illustrate this difference. In order to monitor traffic on the highway, the speed of all cars passing a point can be measured and averaged over a certain time interval – a case of Eulerian averaging. To catch an individual speeder, the police must follow the vehicle of interest to measure its speed as it moves in space over a certain time interval – a case of Lagrangian averaging. + + ===2.4.1.3 Molecular Statistical Averaging=== + + When the collective mechanics of a large number of particles is of interest, molecular statistical averaging may be employed. This relies on the concept of particle number density, which is the number of particles per unit volume. For a system with a large number of particles, the behavior of each individual particle is random because random collisions occur. To describe the behavior of each particle, it is necessary to track the motion resulting from each collision – an impractical and often unnecessary task. Although the behavior of each particle is random, the collection of particles may demonstrate some statistical behaviors that are different from those of the individual particles. When the number of molecules involved in the averaging process is large enough, the statistical average value becomes independent of the number of molecules involved. The statistical average value of the microscopic properties for a large number of molecules is related to the macroscopic properties of the system. For example, temperature is a statistical measure of the kinetic energy of the individual molecules, and the pressure of a gas in a container is the result of many molecules’ collisions with the wall. For some engineering problems, the macroscopic properties of the fluid as well as the microscopic properties are required for design or analysis. + + Most numerical codes are based on the Navier-Stokes equations, which treats a fluid as a continuous field. It is well known that a fluid is made of a discrete number of particles or molecules. Since the number of molecules is extremely large (Avogadro’s number = 6.022×1023 atoms/mole) for almost all practically sized systems, it may never be computationally viable to track each particle and its interactions with other particles. The number of molecules in a given region and the molecular interaction are described through the fluid’s density and transport coefficients (i.e., viscosity) in the continuous model. Modeling the individual molecules for a small system over a small period of time has been achieved by molecular dynamic simulations (MDS). The computational requirements needed in these simulations can be greatly reduced if the degrees of freedom of the system are reduced.  Also, instead of considering individual molecules, groups of molecules can be considered.  The degrees of freedom can be reduced by restricting the movement of the molecules to a lattice. A lattice is simply a predefined direction in which a molecule can move. + + From this standpoint the independent variables are space, velocity and time, while the dependent variable is a molecular distribution function for species $i$, ${{f}_{i}}\left( \mathbf{x},\mathbf{c},t \right)$. The Boltzmann equation relates the distribution function at $\left( \mathbf{x},\mathbf{c},t \right)$ to the distribution function at  $(\mathbf{x}+\Delta \mathbf{x},$ $\mathbf{c}+\Delta \mathbf{c},$ $t+\Delta t).$ + + The location in space is x, and the particle velocity is c. It is important to note that the particle velocity is directly related to the mass average velocity, V, that is used throughout this book.  This distribution function can be related to the Navier-Stokes equations as well as other transport equations; these relationships give insight to the origin of transport coefficients such as viscosity. A detailed presentation of Boltzmann statistical averaging including the discussion of Lattice Boltzmann model for both single and multiphase systems can be found in [[#References|Faghri and Zhang (2006)]]. + ==References== ==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. ==Further Reading== ==Further Reading== ==External Links== ==External Links==

## Current revision as of 01:43, 13 July 2010

The objectives of the various averaging methods are twofold: (1) to define the average properties for the multiphase system and correlate the experimental data, and (2) to obtain solvable governing equations that can be used to predict the macroscopic properties of the multiphase system. This chapter will address the application of averaging methods to the governing equations.

Based on the physical concepts used to formulate multiphase transport phenomena, the averaging methods can be classified into three major groups: (1) Eulerian averaging, (2) Lagrangian averaging, and (3) Molecular statistical averaging.

## Volume Averaging

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.

See Main Article Volume Averaging

## Lagrangian Averaging

Lagrangian averaging is directly related to the Lagrangian description of a system, which requires tracking the motion of each individual fluid particle. Therefore, Lagrangian averaging is a very useful tool when the dynamics of individual particles are of interest. To obtain Lagrangian time averaging, it is necessary to follow a specific particle and observe its behavior for a certain time interval. Then, the behavior of this particle is averaged over the time interval.

See Main Article Lagrangian Averaging

## Boltzmann Statistical Averaging

When the collective mechanics of a large number of particles is of interest, molecular statistical averaging may be employed. This relies on the concept of particle number density, which is the number of particles per unit volume. For a system with a large number of particles, the behavior of each individual particle is random because random collisions occur. To describe the behavior of each particle, it is necessary to track the motion resulting from each collision – an impractical and often unnecessary task. Although the behavior of each particle is random, the collection of particles may demonstrate some statistical behaviors that are different from those of the individual particles. When the number of molecules involved in the averaging process is large enough, the statistical average value becomes independent of the number of molecules involved. The statistical average value of the microscopic properties for a large number of molecules is related to the macroscopic properties of the system.

See Main Article Boltzmann Statistical Averaging

The objectives of the various averaging methods are twofold: (1) to define the average properties for the multiphase system and correlate the experimental data, and (2) to obtain solvable governing equations that can be used to predict the macroscopic properties of the multiphase system. This chapter will address the application of averaging methods to the governing equations. Based on the physical concepts used to formulate multiphase transport phenomena, the averaging methods can be classified into three major groups: (1) Eulerian averaging, (2) Lagrangian averaging, and (3) Molecular statistical averaging. These averaging techniques are briefly reviewed below.

### 2.4.1.1 Eulerian Averaging

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, Δt, 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. 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 ΔV 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:

${{\left\langle {{\Phi }_{k}}{{\Psi }_{k}} \right\rangle }^{k}}={{\left\langle {{\Phi }_{k}} \right\rangle }^{k}}{{\left\langle {{\Psi }_{k}} \right\rangle }^{k}}+{{\left\langle {{{\hat{\Phi }}}_{k}}{{{\hat{\Psi }}}_{k}} \right\rangle }^{k}}\qquad \qquad(11)$

$\left\langle {{\Phi }_{k}}{{\Psi }_{k}} \right\rangle ={{\varepsilon }_{k}}{{\left\langle {{\Phi }_{k}} \right\rangle }^{k}}{{\left\langle {{\Psi }_{k}} \right\rangle }^{k}}+\left\langle {{{\hat{\Phi }}}_{k}}{{{\hat{\Psi }}}_{k}} \right\rangle \qquad \qquad(12)$

Figure 1: 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 Fig. 1, 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 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 Fig. 1).

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 (Section 2.4.2), and (2) the homogeneous model (Section 2.4.3), 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. In this section, it is assumed for the sake of simplicity that the reference frame is stationary.

### 2.4.1.2 Lagrangian Averaging

Lagrangian averaging is directly related to the Lagrangian description of a system, which requires tracking the motion of each individual fluid particle. Therefore, Lagrangian averaging is a very useful tool when the dynamics of individual particles are of interest. To obtain Lagrangian time averaging, it is necessary to follow a specific particle and observe its behavior for a certain time interval. Then, the behavior of this particle is averaged over the time interval.

For a generalized function Φ = Φ(X,Y,Z,t), X, Y, and Z are material coordinates moving with the particle, and X, Y, Z are functions of the spatial coordinates x,y,z, and time t, i.e.,

$\begin{matrix} X=X(x,y,z,t), & Y=Y(x,y,z,t), & Z=Z(x,y,z,t) \\ \end{matrix}$

The most widely used Lagrangian averaging is time averaging, where the time average of the function Φ in time interval of Δt is

$\bar{\Phi }=\frac{1}{\Delta t}\int_{\Delta t}{\Phi (X,Y,Z,t)dt}\qquad \qquad(16)$

Lagrangian time averaging is performed for a distinct particle moving in the field; therefore, X, Y, and Z in the time interval Δt are not fixed in space. This focus on specific particles moving in space and time distinguishes Lagrangian averaging from Eulerian time averaging, which treats a fixed point in space relative to the reference frame. An example from daily experience will serve to illustrate this difference. In order to monitor traffic on the highway, the speed of all cars passing a point can be measured and averaged over a certain time interval – a case of Eulerian averaging. To catch an individual speeder, the police must follow the vehicle of interest to measure its speed as it moves in space over a certain time interval – a case of Lagrangian averaging.

### 2.4.1.3 Molecular Statistical Averaging

When the collective mechanics of a large number of particles is of interest, molecular statistical averaging may be employed. This relies on the concept of particle number density, which is the number of particles per unit volume. For a system with a large number of particles, the behavior of each individual particle is random because random collisions occur. To describe the behavior of each particle, it is necessary to track the motion resulting from each collision – an impractical and often unnecessary task. Although the behavior of each particle is random, the collection of particles may demonstrate some statistical behaviors that are different from those of the individual particles. When the number of molecules involved in the averaging process is large enough, the statistical average value becomes independent of the number of molecules involved. The statistical average value of the microscopic properties for a large number of molecules is related to the macroscopic properties of the system. For example, temperature is a statistical measure of the kinetic energy of the individual molecules, and the pressure of a gas in a container is the result of many molecules’ collisions with the wall. For some engineering problems, the macroscopic properties of the fluid as well as the microscopic properties are required for design or analysis.

Most numerical codes are based on the Navier-Stokes equations, which treats a fluid as a continuous field. It is well known that a fluid is made of a discrete number of particles or molecules. Since the number of molecules is extremely large (Avogadro’s number = 6.022×1023 atoms/mole) for almost all practically sized systems, it may never be computationally viable to track each particle and its interactions with other particles. The number of molecules in a given region and the molecular interaction are described through the fluid’s density and transport coefficients (i.e., viscosity) in the continuous model. Modeling the individual molecules for a small system over a small period of time has been achieved by molecular dynamic simulations (MDS). The computational requirements needed in these simulations can be greatly reduced if the degrees of freedom of the system are reduced. Also, instead of considering individual molecules, groups of molecules can be considered. The degrees of freedom can be reduced by restricting the movement of the molecules to a lattice. A lattice is simply a predefined direction in which a molecule can move.

From this standpoint the independent variables are space, velocity and time, while the dependent variable is a molecular distribution function for species i, ${{f}_{i}}\left( \mathbf{x},\mathbf{c},t \right)$. The Boltzmann equation relates the distribution function at $\left( \mathbf{x},\mathbf{c},t \right)$ to the distribution function at $(\mathbf{x}+\Delta \mathbf{x},$ $\mathbf{c}+\Delta \mathbf{c},$ t + Δt).

The location in space is x, and the particle velocity is c. It is important to note that the particle velocity is directly related to the mass average velocity, V, that is used throughout this book. This distribution function can be related to the Navier-Stokes equations as well as other transport equations; these relationships give insight to the origin of transport coefficients such as viscosity. A detailed presentation of Boltzmann statistical averaging including the discussion of Lattice Boltzmann model for both single and multiphase systems can be found in Faghri and Zhang (2006).

## 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.