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8. MODELLING CONTINUA
8.6. SELECTING CONTINUUM AND ITS PROPERTIES
macroscopic level. The most common is the Euler–Euler modelling approach which requires the
solution of volume fraction equations for fluid phases. For a small control volume V , the volume
occupied by a fluid phase k, Vk , is Vk = αkV , where αk is a volume fraction. The volume fraction
is a function of space and time. As the volume of a phase can not be shared by another phase,
the sum of all volume fractions is equal to one, ∑k αk = 1. The mass fraction of the k–th phase,
ck = mk /m, m = ∑k mk , is related to the volume fraction through the expression:
ck =
mk
ρkVk
ρk
=
= αk
,
m
ρm V
ρm
(8.44)
where ρk = mk /Vk is the phase "thermodynamic" density, and ρm is the multiphase mixture density:
m
(8.45)
ρm = = ∑ mk /V = ∑ mk αk /Vk = ∑ ρk αk
V
k
k
k
Depending on the selected Euler–Euler model, each fluid phase can have its own flow field (the
full Eulerian–Eulerian model) or it can share some common fields (VOF model for immiscible
fluids and the mixture model). The properties that govern the transport equations of VOF and
mixture models therefore depend on the properties of constituent fluid phases. In case of N ph fluid
phases, the volume–fraction mean properties φm are calculated in a similar way as the density,
Equation (8.45):
N ph
φm =
∑ αk φk ,
(8.46)
k=1
where φk denotes the property value of the constituent phase. Note that for the multicomponent
phase the property φk should be calculated according to Equation (8.39).
8.6
Selecting Continuum and Its Properties
It follows from the previous sections that the physical properties can be associated with species,
phases and with the mixture of phases. The association depends on the type of fluid flow:
Single–phase, single–component fluid. The phase is the same entity as the species (i.e. pure
substance) and the phase properties appearing in the transport equations are required.
Single–phase, multicomponent fluid. Some species properties are required to solve mass fraction equations and others to calculate the properties of the multicomponent fluid phase,
see Equation (8.39). The latter are needed for the fluid phase transport equations.
Multiphase, multicomponent fluid. The properties required depend on the multiphase flow
modelling:
– Full Euler–Euler model: Phase properties are required. For the multicomponent
phases, the properties can be derived from constituent species properties, Equation
(8.39)
– Mixture and Volume of Fluid (VOF) model: Properties of the multiphase mixture
must be defined, see Equation (8.46), from properties of constituent phases. For the
multicomponent phases, the properties can be calculated from constituent species
properties, Equation (8.39).
©Ricardo Software, December 2009
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