The general conservation equations from which the equations solved by Ansys Fluent are derived are presented in this section, followed by the solved equations themselves.
The continuity equation for phase 
is
 | (14–193) |
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 14–193
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 Modeling Mass Transfer in Multiphase Flows
for more information
on the modeling of mass transfer in Ansys Fluent’s general multiphase
models.
The momentum balance for phase 
yields
 | (14–194) |
where 
is the 
phase stress-strain tensor
 | (14–195) |
Here 
and 
are the shear and bulk viscosity of phase 
, 
is an external body force, 
is a lift force (described in Lift Force),
is a wall lubrication force (described in Wall Lubrication Force), 
is a virtual mass force, and 
is a turbulent dispersion force (in the case of turbulent flows only).
is an interaction force between phases, and 
is the pressure shared by all phases.
is the interphase velocity, defined as follows. If 
(that is, phase 
mass is being transferred to phase 
), 
; if 
(that is, phase 
mass is being transferred to phase 
), 
. Likewise, if 
then 
, if 
then 
.
Equation 14–194 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 
.
Ansys Fluent uses a simple interaction term of the following form:
 | (14–196) |
where 
(
) is the interphase momentum exchange coefficient (described in Interphase Exchange Coefficients), and 
and 
are the phase velocities. Note that Equation 14–196
represents the mean interphase momentum exchange and does not include any contribution due to
turbulence. The turbulent interphase momentum exchange is modeled with the turbulent dispersion
force term, 
in Equation 14–194, as described in Turbulent Dispersion Force.
To describe the conservation of energy in Eulerian multiphase applications, a separate
enthalpy equation for the 
phase can be written as:
 | (14–197) |
where 
is the effective conductivity, 
is a source term that includes sources of energy (for example, due to
chemical reaction or radiation), 
is the intensity of heat exchange between the 
and 
phases. The heat exchange between phases must comply with the local balance
conditions 
and 
. 
is the interphase enthalpy (for example, the enthalpy of the vapor at the
temperature of the droplets, in the case of evaporation), 
is enthalpy of species 
in phase 
, and 
is diffusive flux of species 
in phase 
.
In the above equation the energy and enthalpy of phase are defined as follows; the enthalpy 
is defined for ideal gas as:
 | (14–198) |
and for an incompressible material includes a contribution from pressure work
 | (14–199) |
The sensible enthalpy of species 
is the part of enthalpy that includes only changes in the enthalpy due to
specific heat
 | (14–200) |
The internal energy 
is defined uniformly for compressible and incompressible materials as
 | (14–201) |
In the above formulas 
and 
are the gauge and operating pressure, respectively. Such definitions of
enthalpy and internal energy accomodate an incompressible ideal gas in common formulation
 | (14–202) |
The equations for fluid-fluid and granular multiphase flows, as solved by Ansys Fluent, are
presented here for the general case of an 
-phase flow.
The volume fraction of each phase is calculated from a continuity equation:
 | (14–203) |
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 14–191), allows for the calculation of the primary-phase volume fraction. This treatment is common to fluid-fluid and granular flows.
The conservation of momentum for a fluid phase 
is
 | (14–204) |
Here 
is the acceleration due to gravity and 
, 
, 
, 
, 
, and 
are as defined for Equation 14–194.
Following the work of [14] , [99] , [142] , [205] , [347] , [397] , [485] , [640] , Ansys 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 14–204
, and that for the
solid phase
is
 | (14–205) |
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 defined in the same manner as the analogous terms in Equation 14–194.
The equation solved by Ansys Fluent for the conservation of energy is Equation 14–197.