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FIGURE 10.15 Simulated results at typical r-z plane for the case of a pitched blade turbine.

FIGURE 10.29 Contours of gas hold-up on horizontal plane located at a distance of BW/3 from impeller center plane (impeller rotation is counter-clockwise). Ten uniform contours between 0 and 0.1; Red: 0.1; Blue: 0.

algorithm discussed in the previous chapter. It has three specific components developed for the simulation of multiphase flows, namely, derivation of pressure equations, a partial elimination algorithm (PEA) to handle tight coupling between momentum equations of two phases, and reformulation of continuity equations to calculate phase volume fractions to ensure that the sum of the volume fractions is unity.

Let us consider these three aspects in reverse order. Before we proceed, it must be emphasized that the usual interpolation rules, calculation of mean (harmonic/algebraic) and procedures to obtain finite volume discretized equations discussed in the previous chapter are applicable to multiphase flows as well. Applying the general methodology of the finite volume technique to two fluid flows, a discretized equation for any general variable <p can be written:

nb nb

The coefficients a consist of all the inflow contributions (convective as well as diffusive) while the coefficients b consist of all the outflow contributions. In the absence of any source or sink, the mass conservation equation dictates that the sum of inflow contributions is equal to the sum of outflow contributions. In the presence of linearized source terms, one can write,

nb nb

If the flows are unsteady, the terms containing aP0 can be added on both sides of Eq. (7.10) (refer to Section 6.4). It must be noted that for multiphase flows, the inflow and outflow terms require considerations of interpolations of phase volume fractions in addition to the usual interpolations of velocity and the coefficient of diffusive transport. The source term linearization practices discussed in the previous chapter are also applicable to multiphase flows. It is useful to recognize that special sources for multiphase flows, for example, an interphase mass transfer, is often constituted of terms having similar significance to the a and b terms. Such discretized equations can be formulated for each variable at each computational cell. The issues related to the phase continuity equation, momentum equations and the pressure correction equation are discussed below.

(a) Phase continuity equation: To calculate a phase volume fraction, it is necessary to solve the phase continuity equation. There is one such equation for every phase. It is, however, useful to solve equations for all phases except one. The volume fraction of the remaining phase can then be deduced from the knowledge that the sum of volume fractions of all phases at any point (computational cell) is unity. The discretized phase continuity equation for each phase can be written in the form of Eq. (7.10). However, the as and bs appearing in this equation must be defined without the phase volume fractions, since phase volume fraction replaces the general variable <p in this case. Because of this, Eq. (7.11) relating inflow and outflow contributions is no longer valid for discretized phase continuity equations. This may lead to difficulties with iterative solutions of phase continuity equations. Before convergence, the values of as and bs may not be in proper balance. Direct solution of discretized equations may generate non-physical values of phase volume fractions. It is, therefore, useful to build 'traps' into the solution procedure to catch and avoid non-physical values.

To illustrate the two possible ways of avoiding non-physical values of phase volume fractions, let us consider a dispersed two-phase flow. The volume fraction of two phases can be obtained by solving the following equations:

where the superscript indicates either a continuous (C) phase or a dispersed phase (D). F consists of all the inflow contributions including the sources at any grid node P and a contains all the coefficients of the outflow contributions. As mentioned earlier, straightforward solution of Eq. (7.10) may lead to non-physical values of volume fractions (less than zero or greater than one). One of the ways to avoid this is to build a constraint on the sum of two volume fractions within the solution procedure. This requires formulation of a discretized equation for the volume fraction of continuous phase and combining the two discretized equations to obtain:

This formulation ensures that the sum of the two volume fractions is unity. This method is very simple to implement when the volume fraction equation is solved using a point-by-point method, and is recommended by Spalding (1980). For more implicit calculations of volume fractions, further algebraic manipulations are necessary to accommodate Eq. (7.13) within the implemented numerical technique.

The other alternative is to subtract the two discretized continuity equations to obtain:

It is also possible to subtract the original continuity equations before discretization and then apply the usual discretization procedures to derive the corresponding algebraic equations. This alternative has a special advantage when used with the pressure correction equations derived from the overall continuity equations (sum of two individual phase continuity equations). This combination then ensures that, simultaneously, both phase continuity equations are satisfied. Pressure correction enforces D + C = 0, while the volume fraction calculation enforces D - C = 0, leading to satisfaction of D = 0 and C = 0 (D and C denote dispersed phase and continuous phase continuity equations, respectively). This option is, however, restricted to two-phase flow simulations. The earlier option (Eq. (7.13)) suggested by Spalding (1980) can be extended to simulations of any number of phases in a straightforward manner.

Once the possibility of non-physical values of volume fraction is eliminated, solving phase continuity equations does not exhibit any other peculiarities, and the methods discussed in the previous chapter can be applied. One more point that must be mentioned while discussing the solution of phase continuity equations is of numerical or false diffusion. Numerical diffusion or false diffusion is not specific to multiphase flows and is related to any fixed-grid numerical solution procedure. However, it becomes very important in simulating multiphase flows. For example, suppose that in a field of uniform velocity, a 'front' exists across which phase volume fraction exhibits a discontinuity. In the absence of diffusion, such a front will move within the fluid without losing its discontinuous character. Numerical computations discussed here, however, would lead to smearing of the discontinuous front. Extra precautions are, therefore, necessary to recognize and control the extent of numerical diffusion when simulating multiphase flows.

(b) Momentum equations: Momentum equations differ from general transport equations because of the pressure gradient terms in the source terms. It is necessary to estimate the pressure field before solving the momentum equations. Of course this is necessary even for the simulation of single-phase flows and is discussed in the previous chapter. As discussed there, several ways of treating pressure-velocity coupling have been developed. The most widely applied are the SIMPLE family of algorithms, in which the pressure field is obtained either by solving the pressure correction equation (SIMPLE) or by solving directly the pressure equation (SIMPLER). Similar algorithms can be developed for multiphase flows. However, the presence of more phases widens the possible choices for deriving the discretized pressure correction or pressure equations. These issues are discussed later. At this stage, it may be assumed that a suitable pressure field is available to solve the multiphase momentum equations.

The other specific feature of multiphase momentum equations is the term containing interphase momentum transport (Eq. (4.16)). The interphase momentum transport terms invariably contain the velocities of all interacting phases at that grid node. Typically, the discretized momentum equation for two-phase flows (for the node P) can be written:

Superscripts indicate phase index. All coefficients appearing in this equation can be derived by following the standard methods discussed earlier. The coefficient fi12 represents all the relevant interphase interaction terms, which cause slow convergence, as mentioned earlier. Large interaction coefficients 'tie together' the velocities of two phases. Therefore, any iterative procedure involving one variable at a time proceeds very slowly. It is necessary to manipulate the discretized equations to eliminate the presence of the velocity of the other phase from the discretized momentum equations. This can easily be done by using an equation for Up analogous to Eq. (7.15). The modified equation can be written:

[aPaP + P12 (aP + aP ) ]uP = (aP + P12) (E aib Ui + SC) + P12 (E a2b Ub + SC )

nb nb

This equation implicitly accounts for the interaction between the velocities of two phases and therefore, enhances convergence rate. When the interphase interaction coefficient is zero, the above equation reduces to that for single-phase flows. It is useful to note here that the coefficient of pressure gradient term also becomes modified by the presence of the second phase. These modified coefficients should be used when deriving the discretized pressure correction or pressure equations. These equations are now discussed below.

(c) Pressure correction equation: Following the procedures discussed in Section 6.3.2, it is possible to derive the pressure correction equation for multiphase flows. For this purpose, any one of the phase continuity equations can, in principle, be used. In general, it has been found that use of an overall continuity equation is more suitable. The overall continuity equation is obtained by adding the continuity equations of all phases. Since the phases present may have large difference in densities, it is often advantageous to normalize the phase continuity equations by phase-specific reference densities. Practical experience indicates that the density of each phase can be used as a reference density of that phase. This means that pressure correction equations or pressure equations are derived on the basis of overall volumetric (since the phase continuity equation is normalized by the density of that phase) continuity equations. It may be noted that in the overall continuity equation, the transient term as well as diffusion and mass source terms are absent. This is because the sum of volume fractions of all phases is unity and by definition, the sum of diffusion terms is zero. The sum of all interphase mass source terms is zero, because even though mass may be transferred from one phase to another, the net mass source must be zero.

Usual interpolation rules and definitions of velocity and pressure corrections, similar to single-phase flows (Eq. (6.29)), can be used to derive a pressure correction equation from the discretized form of the overall continuity (normalized) equation. The momentum equation for multiphase flows (Eq. (7.16)) can also be written in the form of Eq. (6.28) for single-phase flows. Again, following the approximation of SIMPLE, one can write an equation for velocity correction in terms of pressure correction, p:

A similar equation can be derived for velocity corrections for all components and for all phases. The above expression neglects velocity corrections from the right-hand side, following the single-phase flow practice. It is useful to note that in handling pressure-velocity coupling, the coefficients of the pressure gradient term appearing in Eq. (7.17) need not be very precise. The skilled numerical analyst can often enhance convergence by changing their magnitudes selectively or by further approximating them to reduce the computational demands. What is, however, important is that the coefficients have the right sign and that they properly reflect the relative sensitivities of the various phases to changes of pressure. Straightforward application of the procedure discussed here, does achieve this. Substitution of corrected velocities in the discretized overall continuity equation after using appropriate expressions of 8p' results in a pressure correction equation in the form of Eq. (6.32). The mass imbalance term in Eq. (6.32) will be replaced by an overall (normalized) imbalance term.

Having established the required discretized equations, the overall solution procedure for simulations of multiphase flows is as follows:

• Solve the volume fraction equations for all but one phase. The methods for avoiding non-physical values of volume fractions should be employed.

• Solve the momentum equations for all phases based on the guessed pressure field (or that obtained from solution of the pressure correction equation during the previous iteration). The partial-elimination algorithm (PEA) should be used to handle the tight coupling between velocities of various phases.

• The pressure correction equation is solved based on the normalized overall continuity equation.

• The velocity and pressure fields are corrected based on the pressure correction field (for the SIMPLE algorithm).

• This cycle of adjustments is repeated several times until the errors remaining in all equations are acceptably small.

If the SIMPLER algorithm is to be used, corresponding pressure equations need to be solved following the procedures discussed in the previous chapter. The usual solution methods employed for single-phase flow simulations will be equally applicable to solving discretized equations of multiphase flows and, therefore, need not be discussed here again. When there are additional transport equations such as enthalpy or species equations (in addition to the momentum equations), PEA should be used to tackle the tight coupling between enthalpies and concentrations in various phases. When simultaneous mass and energy transfers are taking place, special manipulations may be applied to retain the applicability of PEA.

It is often useful to simulate multiphase flows by employing the techniques of unsteady flows. This provides an added under-relaxation, which is often necessary for multiphase flows with large dispersed phase volume fraction. There are several physical and numerical parameters which influence predicted results and the convergence behavior of the computational model. It is not possible to discuss all these issues here. Just to illustrate the possible results of Eulerian-Eulerian simulations, an example considered in the previous section is simulated here using the Eulerian-Eulerian approach, which allows simulation with higher gas flow rates. Therefore, the gas mass flow rate through the sparger was specified as ten times higher than that considered for the case simulated using a Eulerian-Lagrangian approach (0.1 kg s-1). In these simulations, the top surface of the cavity was considered as a wall when solving the liquid phase and as an outlet while solving the gas phase. Details of the implementation of such a boundary condition are discussed in Chapter 11. The predicted results are shown in Fig. 7.18. It can be seen that strong circulatory flow generated by the impeller leads to the accumulation of gas near the eyes of circulation (Fig. 7.18c). The approach can be used to simulate complex multiphase flows as illustrated in Chapters 9

to 14. The convergence behavior of multiphase flow simulations depends on several factors including selected algorithm, extent of PEA, applied under-relaxation and initial guess. In most cases, the convergence behavior of multiphase flows cannot be discussed independently of the solver employed (and programs). Some general comments are made when discussing applications of the Eulerian-Eulerian approach to simulate complex multiphase flows in later chapters (Chapters 10 to 14).

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