Averaging approaches

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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.,

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

(2.311)

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.,

$\Delta V=\sum\limits_{k=1}^{\Pi }{\Delta {{V}_{k}}}\qquad \qquad( )$

(2.312)

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.,

${{\varepsilon }_{k}}=\frac{\Delta {{V}_{k}}}{\Delta V}\qquad \qquad( )$
	 (2.313)


The volume fraction of all phases must sum to unity:

$\sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}=1}\qquad \qquad( )$

(2.314)

Eulerian volume averaging is expressed as

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

(2.315)

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:

${{\left\langle {{\Phi }_{k}} \right\rangle }^{k}}=\frac{1}{\Delta {{V}_{k}}}\int_{\Delta {{V}_{k}}}{{{\Phi }_{k}}dV}\qquad \qquad( )$

(2.316)

Extrinsic phase average:

$\left\langle {{\Phi }_{k}} \right\rangle =\frac{1}{\Delta V}\int_{\Delta {{V}_{k}}}{{{\Phi }_{k}}dV}\qquad \qquad( )$

(2.317)

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. (2.316), the extrinsic phase average for a particular phase is taken over an entire elemental volume in eq. (2.317). These two phase-averages are related by

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

(2.318)

The intrinsic and extrinsic phase averages defined in eqs. (2.316) and (2.317) are related to the volume average defined in eq. (2.315) by

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

(2.319)

The deviation from a respective intrinsic phase-average value is

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

(2.320)

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

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

(2.321)

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

(2.322)

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. 2.15, the volume averaging of the partial derivative with respect to time is obtained by the following general transport theorem:

$\left\langle \frac{\partial {{\Omega }_{k}}}{\partial t} \right\rangle =\frac{\partial \left\langle {{\Omega }_{k}} \right\rangle }{\partial t}-\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\Omega }_{k}}{{\mathbf{V}}_{I}}\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}\qquad \qquad( )$

(2.323)

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. 2.15).

The volume average of the gradient is

$\left\langle \nabla {{\Omega }_{k}} \right\rangle =\nabla \left\langle {{\Omega }_{k}} \right\rangle +\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\Omega }_{k}}{{\mathbf{n}}_{k}}d{{A}_{k}}}\qquad \qquad( )$

(2.324)

and the volume average of a divergence is

$\left\langle \nabla \cdot {{\Omega }_{k}} \right\rangle =\nabla \cdot \left\langle {{\Omega }_{k}} \right\rangle +\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\Omega }_{k}}\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}\qquad \qquad( )$

(2.325)

The general quantity Ωk in eqs. (2.323) and (2.324) can be a scalar, vector, or tensor of the second order. It can be a vector or tensor of the second order in eq. (2.325).

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. (2.316) and (2.317), 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. (2.315), 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

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

(2.326)

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).

2.4.2 Volume-Averaged Multi-Fluid Models

If spatial averaging is performed for each individual phase within a multiphase control volume, the multi-fluid model is obtained. Additional source terms are needed in these equations to account for the interaction between phases.

Continuity Equation

The volume average of the continuity equation for the kth phase is obtained by taking extrinsic phase averaging from eq. (2.51)

$\left\langle \frac{\partial {{\rho }_{k}}}{\partial t} \right\rangle +\left\langle \nabla \cdot {{\rho }_{k}}{{\mathbf{V}}_{k}} \right\rangle =0\qquad \qquad( )$

(2.327)

where the two terms on the left-hand side can be obtained by using eqs. (2.323) and (2.325), i.e.,

$\left\langle \frac{\partial {{\rho }_{k}}}{\partial t} \right\rangle =\frac{\partial \left\langle {{\rho }_{k}} \right\rangle }{\partial t}-\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\rho }_{k}}{{\mathbf{V}}_{I}}\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}$

$\left\langle \nabla \cdot {{\rho }_{k}}{{\mathbf{V}}_{k}} \right\rangle =\nabla \cdot \left\langle {{\rho }_{k}}{{\mathbf{V}}_{k}} \right\rangle +\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\rho }_{k}}{{\mathbf{V}}_{k}}\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}$

Substituting the above expressions into eq. (2.327), the volume-averaged continuity equation becomes

$\frac{\partial \left\langle {{\rho }_{k}} \right\rangle }{\partial t}+\nabla \cdot \left\langle {{\rho }_{k}}{{\mathbf{V}}_{k}} \right\rangle =-\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\rho }_{k}}({{\mathbf{V}}_{k}}-{{\mathbf{V}}_{I}})\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}\qquad \qquad( )$

(2.328)

The right-hand side of eq. (2.328) represents mass transfer per unit volume from all other phases to the kth phase due to phase change; it can be rewritten as

$-\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\rho }_{k}}({{\mathbf{V}}_{k}}-{{\mathbf{V}}_{I}})\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}=\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk}}}\qquad \qquad( )$

(2.329)

where ${{{\dot{m}}'''}_{jk}}$ represents mass transfer per unit volume from the Failed to parse (syntax error): {j^{th}

to the kth phase due to phase change. The value of  ${{{\dot{m}}'''}_{jk}}$ depends on the phase change process that takes place in the multiphase system, and the conservation of mass requires that ${{{\dot{m}}'''}_{jk}}=-{{{\dot{m}}'''}_{kj}}$.


The extrinsic phase-averaged density, $\left\langle {{\rho }_{k}} \right\rangle$, is related to the intrinsic phase-averaged density, ${{\left\langle {{\rho }_{k}} \right\rangle }^{k}}$, by

$\left\langle {{\rho }_{k}} \right\rangle ={{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}\qquad \qquad( )$

(2.330)

Furthermore, the intrinsic phase-averaged density is equal to the density ρk.

Substituting eqs. (2.329) and (2.330) into eq. (2.328), and considering eq. (2.322), the continuity equation for the kth phase becomes

$\frac{\partial }{\partial t}\left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}} \right)+\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}+\left\langle {{{\hat{\rho }}}_{k}}{{{\mathbf{\hat{V}}}}_{k}} \right\rangle \right)=\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk}}}\qquad \qquad( )$

(2.331)

The dispersive term in eq. (2.331), $\left\langle {{{\hat{\rho }}}_{k}}{{{\mathbf{\hat{V}}}}_{k}} \right\rangle$, is generally small compared with ${{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}$; it is assumed that it can be neglected. The continuity equation for the kth phase becomes

$\frac{\partial }{\partial t}\left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}} \right)+\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} \right)=\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk}}}\qquad \qquad( )$

(2.332)

Momentum Equation

The extrinsic phase-averaged momentum equation for the kth phase can be obtained by performing extrinsic phase-averaging on the momentum equation (2.65):

$\left\langle \frac{\partial ({{\rho }_{k}}{{\mathbf{V}}_{k}})}{\partial t} \right\rangle +\left\langle \nabla \cdot ({{\rho }_{k}}{{\mathbf{V}}_{k}}{{\mathbf{V}}_{k}}) \right\rangle =\left\langle \nabla \cdot {{{\mathbf{{\tau }'}}}_{k}} \right\rangle +\left\langle {{\rho }_{k}}{{\mathbf{X}}_{k}} \right\rangle \qquad \qquad( )$

(2.333)

where the body force per unit mass is assumed to be the same for different species for sake of simplicity. After evaluating each term in eq. (2.333), the multi-fluid volume-averaged momentum equation becomes(Faghri and Zhang, 2006)

\begin{align} & \frac{\partial }{\partial t}\left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} \right)+\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}}{{\mathbf{V}}_{k}} \right\rangle }^{k}} \right) \\ & =\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{{\mathbf{{\tau }'}}}_{k}} \right\rangle }^{k}} \right)+{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\mathbf{X}}_{k}}+\sum\limits_{j=1(j\ne k)}^{\Pi }{\left( \left\langle {{\mathbf{F}}_{jk}} \right\rangle +\left\langle {{{{\dot{m}}'''}}_{jk}} \right\rangle {{\left\langle {{\mathbf{V}}_{k,I}} \right\rangle }^{k}} \right)} \\ \end{align}\qquad \qquad( )

(2.334)

where ${{\left\langle {{\mathbf{V}}_{k,I}} \right\rangle }^{k}}$ is intrinsic phase-averaged velocity of the kth phase at the interface. The difference between two adjacent phases results solely from the density difference between the two phases. $\left\langle {{\mathbf{F}}_{jk}} \right\rangle$ is an interactive force between the jth and the kth phase, and depends on the friction, pressure, and cohesion between different phases. Newton’s third law requires that the interactive forces satisfy

$\left\langle {{\mathbf{F}}_{jk}} \right\rangle =-\left\langle {{\mathbf{F}}_{kj}} \right\rangle \qquad \qquad( )$

(2.335)

The interactive force can be determined by

$\left\langle {{\mathbf{F}}_{jk}} \right\rangle ={{K}_{jk}}\left( {{\left\langle {{\mathbf{V}}_{j}} \right\rangle }^{j}}-{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} \right)\qquad \qquad( )$

(2.336)

where Kjk is the momentum exchange coefficient between phases j and k. Determining the momentum exchange coefficient is a very challenging task because interphase momentum exchange depends on the structure of the interfaces. If a secondary phase j is dispersed in the primary phase k, as is the case with the dispersed phase system summarized in Table 1.8, one can assume that the secondary phase is spherical in shape and an appropriate empirical correlation can be used to obtain the momentum exchange coefficient.

Since liquid-vapor flow is widely used in various applications, we will use liquid-vapor flow as an example to explain the determination of the momentum exchange coefficient. If liquid is the primary phase and vapor is the secondary phase, the vapor phase is dispersed in the liquid as vapor bubbles. If vapor is the primary phase and liquid is the secondary phase, the liquid phase is dispersed in the vapor as liquid droplets. Boysan (1990) suggested that the momentum exchange coefficient could be estimated by

${{K}_{jk}}=\frac{3}{4}{{C}_{D}}\frac{{{\varepsilon }_{j}}\left\langle {{\rho }_{k}} \right\rangle }{{{d}_{j}}}\left| {{\left\langle {{\mathbf{V}}_{j}} \right\rangle }^{j}}-{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} \right|\qquad \qquad( )$

(2.337)

where phase k is the primary phase, and phase j is the secondary phase, and dj is the diameter of vapor bubbles or liquid droplets of the secondary phase j. CD is the drag coefficient based on the relative Reynolds number, which obtained by the following empirical correlations:

{{C}_{D}}=\left\{ \begin{align} & \frac{24}{\operatorname{Re}}(1+0.15{{\operatorname{Re}}^{0.687}}) \\ & 0.44 \\ \end{align} \right.\begin{matrix} {} & \begin{align} & \operatorname{Re}\le 1000 \\ & \operatorname{Re}>1000 \\ \end{align} \\ \end{matrix}\qquad \qquad( )

(2.338)

where

$\operatorname{Re}=\frac{\left\langle {{\rho }_{k}} \right\rangle \left| {{\left\langle {{\mathbf{V}}_{j}} \right\rangle }^{j}}-{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} \right|{{d}_{j}}}{{{\mu }_{k}}}\qquad \qquad( )$

(2.339)

Energy Equation

The extrinsic phase-average of the energy equation, (2.92), is

\begin{align} & \left\langle \frac{\partial ({{\rho }_{k}}{{h}_{k}})}{\partial t} \right\rangle +\left\langle \nabla \cdot {{\rho }_{k}}{{\mathbf{V}}_{k}}{{h}_{k}} \right\rangle \\ & =-\left\langle \nabla \cdot {{{\mathbf{{q}''}}}_{k}}_{k} \right\rangle +\left\langle {{{{q}'''}}_{k}} \right\rangle +\left\langle \frac{\partial {{p}_{k}}}{\partial t} \right\rangle +\left\langle {{\mathbf{V}}_{k}}\cdot \nabla {{p}_{k}} \right\rangle +\left\langle \nabla {{\mathbf{V}}_{k}}:\mathbf{\tau }{}_{k} \right\rangle \\ \end{align}\qquad \qquad( )

(2.340)

which can be used to obtain the volume-averaged energy equation and the result is (Faghri and Zhang, 2006)

\begin{align} & \frac{\partial }{\partial t}\left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}} \right)+\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}}{{h}_{k}} \right\rangle }^{k}} \right)=-\nabla \cdot \left\langle {{{\mathbf{{q}''}}}_{k}} \right\rangle +\left\langle {{{{q}'''}}_{k}} \right\rangle \\ & +{{\varepsilon }_{k}}\frac{D{{\left\langle {{p}_{k}} \right\rangle }^{k}}}{Dt}+\nabla \left\langle {{\mathbf{V}}_{k}} \right\rangle :\left\langle {{\mathbf{\tau }}_{k}} \right\rangle +\sum\limits_{j=1(j\ne k)}^{\Pi }{\left[ \left\langle {{{{q}'''}}_{jk}} \right\rangle +{{{{\dot{m}}'''}}_{jk}}{{\left\langle {{h}_{k,I}} \right\rangle }^{k}} \right]} \\ \end{align}\qquad \qquad( )

(2.341)

where $\left\langle {{{{q}'''}}_{jk}} \right\rangle$ is the intensity of heat exchange between phase j and k. It can be obtained by using Newton’s law of cooling:

$\left\langle {{{{q}'''}}_{jk}} \right\rangle =\frac{{{h}_{c}}\Delta {{A}_{j}}\left( {{\left\langle {{T}_{j}} \right\rangle }^{j}}-{{\left\langle {{T}_{k}} \right\rangle }^{k}} \right)}{\Delta {{V}_{j}}}\qquad \qquad( )$

(2.342)

where hc is the convective heat transfer coefficient, ΔAj is the area of the interface between phases j and k, and ΔVj is the volume of the secondary phase in the volume element. Like the momentum exchange coefficient, the interphase heat transfer also depends on the structure of the interfaces. If a secondary phase j is dispersed in the primary phase k – as in the dispersed phase summarized in Table 1.8– the following empirical correlation recommended is widely used:

$Nu=2+(0.4{{\operatorname{Re}}^{1/2}}+0.06{{\operatorname{Re}}^{2/3}})\Pr _{k}^{0.4}{{\left( \frac{{{\mu }_{k}}}{{{\mu }_{k,s}}} \right)}^{1/4}}\qquad \qquad( )$

(2.343)

where the Reynolds number, Re, is obtained by eq. (2.339), the Nusselt number is defined as

$Nu=\frac{{{h}_{c}}{{d}_{j}}}{{{k}_{k}}}\qquad \qquad( )$

(2.344)

and all thermal properties of the primary phase are evaluated at Tk except μk,s, which is evaluated at Tj. Equation (2.343) is valid for $3.5<\operatorname{Re}<7.6\times {{10}^{4}}$ and $0.71<{{\Pr }_{k}}<380$, which covers a wide variety of problems.

If the secondary phase is liquid and the primary phase is vapor (gas), eq. (2.343) can be simplified to

$Nu=2+0.6{{\operatorname{Re}}^{1/2}}\Pr _{k}^{1/3}\qquad \qquad( )$

(2.345)

Species

If the fluid undergoing phase change involves multiple components, it is also necessary to write the equation for conservation of the species mass in the kth phase. The extrinsic phase-average of conservation of species mass can be obtained by

$\left\langle \frac{\partial {{\rho }_{k,i}}}{\partial t} \right\rangle +\left\langle \nabla \cdot {{\rho }_{k,i}}{{\mathbf{V}}_{k}} \right\rangle =-\left\langle \nabla \cdot {{\mathbf{J}}_{k,i}} \right\rangle +\left\langle {{{{\dot{m}}'''}}_{k,i}} \right\rangle \qquad \qquad( )$

(2.346)

where each term can be obtained by

$\left\langle \frac{\partial {{\rho }_{k,i}}}{\partial t} \right\rangle =\frac{\partial \left\langle {{\rho }_{k,i}} \right\rangle }{\partial t}-\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\rho }_{k,i}}{{\mathbf{V}}_{I}}\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}$

$\left\langle \nabla \cdot {{\rho }_{k,i}}{{\mathbf{V}}_{k}} \right\rangle =\nabla \cdot \left\langle {{\rho }_{k,i}}{{\mathbf{V}}_{k}} \right\rangle +\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\rho }_{k,i}}{{\mathbf{V}}_{k}}\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}$

$\left\langle \nabla \cdot {{\mathbf{J}}_{k,i}} \right\rangle =\nabla \cdot \left\langle {{\mathbf{J}}_{k,i}} \right\rangle +\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\mathbf{J}}_{k,i}}\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}$

Substituting the above expression into eq. (2.346), one obtains

\begin{align} & \frac{\partial \left\langle {{\rho }_{k,i}} \right\rangle }{\partial t}+\nabla \cdot \left\langle {{\rho }_{k,i}}{{\mathbf{V}}_{k}} \right\rangle =-\nabla \cdot \left\langle {{\mathbf{J}}_{k,i}} \right\rangle +\left\langle {{{{\dot{m}}'''}}_{k,i}} \right\rangle \\ & -\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\rho }_{k,i}}({{\mathbf{V}}_{k}}-{{\mathbf{V}}_{I}})\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}+\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\mathbf{J}}_{k,i}}\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}} \\ \end{align}\qquad \qquad( )

(2.347)

where the third and fourth terms in the right-hand side of eq. (2.347) represent mass source (or sink) of the ith component in the kth phase due to phase change from other phases to the kth phase, as well as mass transfer at the interface due to diffusion, respectively, i.e.,

$-\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\rho }_{k,i}}({{\mathbf{V}}_{k}}-{{\mathbf{V}}_{I}})\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}+\frac{1}{\Delta V}\int_{{{A}_{k}}}{{{\mathbf{J}}_{k,i}}\cdot {{\mathbf{n}}_{k}}d{{A}_{k}}}=\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk,i}}}\qquad \qquad( )$

(2.348)

where ${{{\dot{m}}'''}_{jk,i}}$ represents the mass source (or sink) of the ith component in phase k due to phase change from phase j to phase k, as well as diffusive mass transfer at the interface between phases j and k.

Substituting eq. (2.348) into eq. (2.347) and considering eq. (2.322), the volume-averaged species equation becomes

$\frac{\partial \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k,i}} \right\rangle }^{k}} \right)}{\partial t}+\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k,i}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} \right)=-\nabla \cdot \left\langle {{\mathbf{J}}_{k,i}} \right\rangle +\left\langle {{{{\dot{m}}'''}}_{k,i}} \right\rangle +\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk,i}}}\qquad \qquad( )$
(2.349)

where the three terms on the right-hand side represent the effects of mass diffusion, species source/sink due to chemical reaction, and phase change at the interfaces.

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.