## 23.5.3 Conservation Equations

The general conservation equations from which the equations solved by FLUENT are derived are presented in this section, followed by the solved equations themselves.

Equations in General Form

Conservation of Mass

The continuity equation for phase is

 (23.5-4)

where is the velocity of phase and characterizes the mass transfer from the to phase, and characterizes the mass transfer from phase to phase , and you are able to specify these mechanisms separately.

By default, the source term on the right-hand side of Equation  23.5-4 is zero, but you can specify a constant or user-defined mass source for each phase. A similar term appears in the momentum and enthalpy equations. See Section  23.7 for more information on the modeling of mass transfer in FLUENT's general multiphase models.

Conservation of Momentum

The momentum balance for phase yields

 (23.5-5)

where is the phase stress-strain tensor

 (23.5-6)

Here and are the shear and bulk viscosity of phase , is an external body force, is a lift force, is a virtual mass force, is an interaction force between phases, and is the pressure shared by all phases.

is the interphase velocity, defined as follows. If (i.e., phase mass is being transferred to phase ), ; if (i.e., phase mass is being transferred to phase ), . Likewise, if then , if then .

Equation  23.5-5 must be closed with appropriate expressions for the interphase force . This force depends on the friction, pressure, cohesion, and other effects, and is subject to the conditions that and .

FLUENT uses a simple interaction term of the following form:

 (23.5-7)

where ( ) is the interphase momentum exchange coefficient (described in Section  23.5.4).

Lift Forces

For multiphase flows, FLUENT can include the effect of lift forces on the secondary phase particles (or droplets or bubbles). These lift forces act on a particle mainly due to velocity gradients in the primary-phase flow field. The lift force will be more significant for larger particles, but the FLUENT model assumes that the particle diameter is much smaller than the interparticle spacing. Thus, the inclusion of lift forces is not appropriate for closely packed particles or for very small particles.

The lift force acting on a secondary phase in a primary phase is computed from [ 90]

 (23.5-8)

The lift force will be added to the right-hand side of the momentum equation for both phases ( ).

In most cases, the lift force is insignificant compared to the drag force, so there is no reason to include this extra term. If the lift force is significant (e.g., if the phases separate quickly), it may be appropriate to include this term. By default, is not included. The lift force and lift coefficient can be specified for each pair of phases, if desired.

 It is important that if you include the lift force in your calculation, you need not include it everywhere in the computational domain since it is computationally expensive to converge. For example, in the wall boundary layer for turbulent bubbly flows in channels, the lift force is significant when the slip velocity is large in the vicinity of high strain rates for the primary phase.

Virtual Mass Force

For multiphase flows, FLUENT includes the "virtual mass effect'' that occurs when a secondary phase accelerates relative to the primary phase . The inertia of the primary-phase mass encountered by the accelerating particles (or droplets or bubbles) exerts a "virtual mass force'' on the particles [ 90]:

 (23.5-9)

The term denotes the phase material time derivative of the form

 (23.5-10)

The virtual mass force will be added to the right-hand side of the momentum equation for both phases ( ).

The virtual mass effect is significant when the secondary phase density is much smaller than the primary phase density (e.g., for a transient bubble column). By default, is not included.

Conservation of Energy

To describe the conservation of energy in Eulerian multiphase applications, a separate enthalpy equation can be written for each phase:

 (23.5-11)

where is the specific enthalpy of the phase, is the heat flux, is a source term that includes sources of enthalpy (e.g., due to chemical reaction or radiation), is the intensity of heat exchange between the and phases, and is the interphase enthalpy (e.g., the enthalpy of the vapor at the temperature of the droplets, in the case of evaporation). The heat exchange between phases must comply with the local balance conditions and .

Equations Solved by FLUENT

The equations for fluid-fluid and granular multiphase flows, as solved by FLUENT, are presented here for the general case of an -phase flow.

Continuity Equation

The volume fraction of each phase is calculated from a continuity equation:

 (23.5-12)

where is the phase reference density, or the volume averaged density of the phase in the solution domain.

The solution of this equation for each secondary phase, along with the condition that the volume fractions sum to one (given by Equation  23.5-2), allows for the calculation of the primary-phase volume fraction. This treatment is common to fluid-fluid and granular flows.

Fluid-Fluid Momentum Equations

The conservation of momentum for a fluid phase is

 (23.5-13)

Here is the acceleration due to gravity and , , , and are as defined for Equation  23.5-5.

Fluid-Solid Momentum Equations

Following the work of [ 7, 52, 81, 121, 199, 222, 268, 364], FLUENT uses a multi-fluid granular model to describe the flow behavior of a fluid-solid mixture. The solid-phase stresses are derived by making an analogy between the random particle motion arising from particle-particle collisions and the thermal motion of molecules in a gas, taking into account the inelasticity of the granular phase. As is the case for a gas, the intensity of the particle velocity fluctuations determines the stresses, viscosity, and pressure of the solid phase. The kinetic energy associated with the particle velocity fluctuations is represented by a "pseudothermal'' or granular temperature which is proportional to the mean square of the random motion of particles.

The conservation of momentum for the fluid phases is similar to Equation  23.5-13, and that for the solid phase is

 (23.5-14)

where is the solids pressure, is the momentum exchange coefficient between fluid or solid phase and solid phase , is the total number of phases, and , , and are as defined for Equation  23.5-5.

Conservation of Energy

The equation solved by FLUENT for the conservation of energy is Equation  23.5-11.

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Next: 23.5.4 Interphase Exchange Coefficients