The Euler–Euler Model Equations
Mass Balance
The following continuity relations hold for the continuous and dispersed phases (Ref. 3):
(6-51)
(6-52)
Here ϕ (dimensionless) denotes the phase volume fraction, ρ (SI unit: kg/m3) is the density, u (SI unit: m/s) the velocity of each phase, and mdc is the mass transfer rate from the dispersed to the continuous phase (SI unit: kg/(m3·s)). The subscripts c and d denote quantities relating to the continuous and the dispersed and phase, respectively. The following relation between the volume fractions must hold
Both phases are considered to be incompressible, in which case Equation 6-51 and Equation 6-52 can be simplified as:
(6-53)
(6-54)
If Equation 6-54 and Equation 6-53 are added together, a continuity equation for the mixture is obtained:
(6-55)
In order to control the mass balance of the two phases, the Euler–Euler Model interfaces solves Equation 6-54 together with Equation 6-55. Equation 6-54 is used to compute the volume fraction of the dispersed phase, and Equation 6-55 is used to compute the mixture pressure.
Momentum Balance
The momentum equations for the continuous and dispersed phases, using the nonconservative forms (see for example Ref. 3 or Ref. 9), are:
(6-56)
(6-57)
Here p (SI unit: Pa) is the mixture pressure, which is assumed to be equal for the two phases. In the momentum equations the viscous stress tensor for each phase is denoted by τ (SI unit: Pa), g (SI unit: m/s2) is the vector of gravitational acceleration, Fm (SI unit: N/m3) is the interphase momentum transfer term (that is, the volume force exerted on each phase by the other phase), F (SI unit: N/m3) is any other volume force term, and uint is the interphase velocity (SI unit: m/s).
In these equations, the influence of surface tension in the case of liquid phases has been neglected, and the potential size distribution of the dispersed particles or droplets is not considered.
For fluid-solid mixtures where ρd >> ρc, Equation 6-57 is modified in the manner of Enwald (Ref. 5)
(6-58)
where ps (SI unit: Pa) is the solid pressure.
The fluid phases in the above equations are assumed to be Newtonian and the viscous stress tensors are defined as
where and (SI unit: Pa·s) are the dynamic viscosity model of the respective phase. Observe how the viscous terms in equation Equation 6-56 and Equation 6-57 appear with their volume fractions outside the divergence operators. The equations can also be derived so that the terms read ∇ ⋅ (ϕcτc) and ∇ ⋅ (ϕdτd) instead (see for example Ref. 4 or Ref. 5). It depends on exactly how the derivation is carried out. formulation in Equation 6-56 and Equation 6-57 are so that the momentum equation can be divided by ϕc and ϕd respectively as described below.
In order to avoid singular solutions when the volume fractions tend to zero, the governing equations above are divided by the corresponding volume fraction. The implemented momentum equation for the continuous phase is:
The implemented momentum equations for the dispersed phase in the case of dispersed liquid droplets or bubbles is:
(6-59)
and in the case of dispersed solid particles:
(6-60)
Viscosity Models
The Newtonian viscosities of interpenetrating media are not readily available. Instead empirical and analytical models for the dynamic viscosity of the two-phase mixture have been developed by various researchers, usually as a function of the dispersed volume fraction. Using an expression for the mixture viscosity, the default values for the dynamic viscosities of the two interpenetrating phases are taken to be:
(6-61)
A simple mixture viscosity covering the entire range of particle concentrations is the Krieger type model (Ref. 5):
(6-62)
Here ϕd,max is the maximum packing limit, by default 0.62 for solid particles. Equation 6-62 can be applied when μc << μd. An extension of Equation 6-62 can be applied for liquid droplets/bubbles:
(6-63)
For liquid droplets/bubbles the default value of ϕd,max is 1.
For fluid-fluid systems, the simple relation
(6-64)
can sometimes be applied or the viscosity for each individual phase can be applied to the respective momentum equation.
An alternative to mixture viscosity for dispersed solid particles, that is particles in gas flow, is to model the particle interface with a solid pressure instead (see Solid Pressure). A small viscosity is however necessary for numerical robustness and Gidaspow (Ref. 6) suggested to use
(6-65)
This is consistent with the requirement that when ϕd → 0 (Ref. 9), but in practice it can suffice with any small enough, nonzero value.
Interphase Momentum Transfer
In all the equations, Fm denotes the interphase momentum transfer, that is the force imposed on one phase by the other phase. Considering a particle, droplet, or bubble in a fluid flow, it is affected by a number of forces, for example, the drag force, the added mass force, the Basset force, and the lift force. The most important force is usually the drag force, especially in fluids with a high concentration of dispersed solids, and hence this is the predefined force included in the Euler–Euler model. The drag force added to the momentum equation is defined as:
(6-66)
where β is a drag force coefficient and the slip velocity is defined as
The drag force coefficient, β, is for solid particles often written as (Ref. 2)
(6-67)
For fluid-fluid system, Equation 6-67 is regularized to read (Ref. 12)
(6-68)
which is consistent with a vanishing β if ϕc → 0. All drag force models except Gidaspow are based on Equation 6-67 for solid particles and on Equation 6-68 for bubbles and droplets.
Gidaspow
For dense particle flows with a high concentration of the dispersed phase — for example, in fluidized bed models — the Gidaspow model (Ref. 6) for the drag coefficient can be used. It combines the Wen and Yu (Ref. 7) fluidized state expression:
For ϕc > 0.8
(6-69)
with the Ergun (Ref. 8) packed bed expression:
For ϕc < 0.8
(6-70)
In the above equations, dd (SI unit: m) is the dispersed particle diameter, and Cd is the drag coefficient for a single dispersed particle. The drag coefficient is in general a function of the particle Reynolds number
No universally valid expression for the drag coefficient exists. Using the implemented Gidaspow model, Cd is computed using the Schiller–Naumann relation.
Schiller–Naumann
The Schiller–Naumann model describes the drag coefficient in Equation 6-67 and Equation 6-68 for a single rigid sphere (see for example Ref. 3)
(6-71)
It is valid for particle Reynolds numbers up to some critical Reynolds number, approximately equal to 2.5·105 where drag crisis occurs.
The Schiller–Naumann model is appropriate for diluted flows since the correlation in Equation 6-71 is based on a single particle and does not take particle-particle interaction into account.
The Schiller–Naumann model can be applied to small fluid particles since they do not deform significantly.
Hadamard–Rybczynski
The Hadamard–Rybczynski drag law is valid for dilute flows with particle Reynolds number less than one for particles, bubbles, and droplets and is defined as:
for bubbles and droplets and as
for solid particles (the Stokes limit). For very small gas bubbles, the drag coefficient is observed to be closer to the solid-particle value. This is believed to be caused by surface-active impurities collecting on the bubble surface.
Ishii–Zuber
The Ishii–Zuber model (Ref. 10) can for solid particles be regarded as a generalization of the Schiller–Naumann model. For solid particles it can be formulated as
(6-72)
where
(6-73)
and Rem is a Reynolds number based on the mixture viscosity
(6-74)
The Ishii–Zuber model is formulated with the assumption that μm is described by a Krieger type model (Equation 6-62 for solids or Equation 6-63 for droplets and bubbles). The dependence on μm means that the Ishii–Zuber model is valid also for dense suspensions.
For bubbles and droplets, Ishii and Zuber identify three regimes. At low Reynolds numbers, bubbles and droplets behave in the same way as solid particles and the drag coefficient can be described by
(6-75)
When the Reynolds number becomes higher, bubbles and droplets start to deform in such a way that the drag coefficient increases linearly with diameter but is independent of viscosity. Ishii and Zuber refer to this as the distorted particle regime and suggest the following correlation for this regime:
(6-76)
where Eo is the Eötvös number
(6-77)
where in turn g is the gravitational constant and σ is the surface tension coefficient. The Eötvös number relates surface tension forces to gravitational forces. The second, “distorted” regime ends when hydrodynamic instabilities prevents the drag from increasing further and the distorted regime is capped by the following value:
(6-78)
The complete model for bubbles and droplets can hence be written:
(6-79)
Haider–Levenspiel
The Haider–Levenspiel model is applicable to dilute flows with nonspherical solid particles. It models the drag coefficient according to
where A, B, C, and D are empirical correlations of the particle sphericity (see Slip Velocity Models for further details).
Tomiyama and Others
Tomiyama and others (Ref. 11) provided three correlations for diluted bubbles and droplets. The correlation for purified fluids read
(6-80)
Fluid pure enough for Equation 6-80 to be valid is typically only achievable in laboratory environments. Even small amounts of surface-active impurities can collect at the droplet/bubble interface and cause a surface tension gradient which resists surface movement. The correlation for slightly contaminated fluids therefore prescribe a slightly higher drag coefficient:
(6-81)
This correlation is the default one and applies to droplets and bubbles in fluids of contamination levels comparable to ordinary tap water.
When the contamination level is high, the impurities effectively prevent all motion within the droplet or bubble and the following correlation is suggested:
(6-82)
Solid Pressure
For fluid-solid mixtures where ρd >> ρc, for example solid particles in gas flow, a model for the solid pressure, ps in Equation 6-60, is needed. The solid pressure models the particle interaction due to collisions and friction between the particles. The solid pressure model implemented uses a gradient diffusion based assumption:
where the empirical function G can be thought of as a modulus of elasticity or powder modulus. A common form for G is (Ref. 5)
(6-83)
The available predefined models (all defined in Ref. 5) are those of Gidaspow and Ettehadieh,
(6-84)
Ettehadieh,
(6-85)
and Gidaspow,
(6-86)
Observe that the solid pressure term theoretically replaces the stress term in the dispersed phase momentum equation. A small viscosity is however necessary for numerical robustness. The Gidaspow formulation or a small constant value is appropriate. The viscosity for the continuous phase, , can be taken as the single phase value, that is .
Notes on the Implementation
There are several equations with potentially singular terms and expressions such as the term
in Equation 6-59. Also, ϕd is a degree of freedom and can therefore, during computations, obtain unphysical values in small areas. This can, in turn, lead to unphysical values of material properties such as viscosity and density. To avoid these problems, the implementation uses the following regularizations:
1k, k = c,d is replaced by1k,pos where
The residue volume fractions are per default set to 104 but can be changed in the Advanced Settings section of the interface settings.
Note however that the continuity equation, Equation 6-55, uses ϕk without regularization in order to guarantee mass conservation.