T h e o r y f o r t h e E u l e r - E u l e r M o d e l , L a m i n a r F l o w I n t e r f a c e

The Euler-Euler Model, Laminar Flow Interface is based on averaging the Navier-Stokes equations phases over volumes, which are small compared to the computational domain, but large compared to the dispersed phase particles, droplets or bubbles. The present phases, the continuous and the dispersed phase, are assumed to behave as two continuous and interpenetrating fluids. The equations solved for corresponds to two separate sets of Navier-Stokes equations.

In this section:

•The Euler-Euler Model Equations

•References for the Euler-Euler Model, Laminar Flow Interface

The Euler-Euler Model Equations

M A S S B A L A N C E

Assuming that the mass transfer between the two phases is zero, the following continuity equation holds for the continuous and dispersed phase respectively (Ref. 3):

Here denotes the phase volume fraction (unit less), the density (SI unit: kg/m3), and u the velocity (SI unit: m/s) of each phase. The subscripts c and d denote quantities relating to the dispersed and the continuous phase, respectively.

It is assumed that the resulting fluid is a mixture of the two phases, implying that:

c = 1 – d

Both phases are considered incompressible, in which case Equation 9-1 and Equation 9-2 can be simplified as:

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If Equation 9-3 and Equation 9-4 are added together, a continuity equation for the mixture is reached:

In order to control the mass balance of the two phases, the Euler-Euler Model interface solves Equation 9-3 together with Equation 9-5. Equation 9-3 is used to compute the volume fraction of the dispersed phase, and Equation 9-5 the mixture pressure.

M O M E N T U M B A L A N C E

The momentum balance equations for a continuous and a dispersed phase, using the non-conservative forms of Ishii (Ref. 4), are respectively:

d d t ud + ud ud = – d p + d d + d dg + Fm,d + dFd (9-7)

Here p is the mixture pressure (SI unit: Pa), which is assumed to be equal for the two phases. In the momentum equations the viscous stress tensor (SI unit: Pa) of each phase is denoted by , g is the vector of gravitational acceleration (SI unit: m/s2), Fm is the interphase momentum transfer term, that is the volume force exserted on the phase by the other phase (SI unit: N/m3), and F (SI unit: N/m3) is any other volume force term present.

In these equations, the influence of a surface tension in the case of the liquid phases has been neglected, and the potential size distribution of the dispersed particles or droplets is not considered.

For fluid-solid mixtures, Equation 9-7 is modified in the manner of Enwald (Ref. 5)

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where ps is the solid pressure (SI unit: Pa).

The fluid phases in the above equations are assumed to be Newtonian and the viscous stress tensors are defined as

where is the dynamic viscosity (SI unit: Pa·s) of the respective phase.

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 equations for the continuous phase are:

Fm,d

d

D I S P E R S E D P H A S E V I S C O S I T Y

The Newtonian viscosity of a dispersed phase alone is rarely known. 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 Euler-Euler Model, Laminar Flow interface retrieves the dispersed phase viscosity by using the following linear relationship proposed by Enwald et al. (Ref. 5):

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A simple mixture viscosity covering the entire range of particle concentrations is the Krieger type model (Ref. 5):

Here d,max is the maximum packing limit, by default 0.62. Equation 9-12 is derived under the assumption that c « d .

I N T E R P H A S E M O M E N T U M T R A N S F E R

In all the equations, F denotes the interphase momentum transfer, that is the force imposed on one phase by the other phase. Observing a particle, droplet, or bubble in a fluid flow, it is affected by a number of forces, for example, the drag force, added mass force, Basset force, and lift forces. 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 momentum equations is defined as:

where is a drag force coefficient and the slip velocity is defined as

uslip = ud – uc

The drag force on the dispersed phase is equal to the one continuous

phase, but is working in the opposite direction.

Note

Dense Flows

For dense flows with high concentration of 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 c > 0.8

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with the Ergun (Ref. 8) packed bed expression:

For c < 0.8

In the above equations dd is the dispersed particle diameter (SI unit: m) and Cdrag 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, Cdrag is computed using the Schiller-Naumann relation

Dilute Flows

For dilute flows the drag force coefficient can be modeled as:

In this case the drag coefficient can be either computed from the Schiller-Naumann model in Equation 9-16, or using the Hadamard-Rybczynski drag law. The Hadamard-Rybczynski drag law is valid for particle Reynolds number less than one for both particles, bubbles, and droplets and defined as:

S O L I D P R E S S U R E

For fluid-solid mixtures, a model of the solid pressure, ps in Equation 9-10, is needed. The solid pressure models the particle interaction due to collisions and friction

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between the solid particles. The solids pressure model implemented uses a gradient diffusion based assumption in the manner of:

ps = –G c c

where the empirical function G follows the form

described in Ref. 5. The predefined models available corresponds to that of Gidaspow and Ettehadieh,

N O T E S O N I M P L E M E N T A T I O N

There are several equations with potentially singular terms and expressions, for example the term

F

-----m,d------

d

in Equation 9-9. Also, d is a degree of freedom, and can therefore during computations obtain nonphysical values in small areas. This can in turn lead to non-physical values of material properties such as viscosity and density. To avoid these problems, the following regularizations have been introduced in the implementation.

•1 k, k c,d is exchanged for 1 k,pos where

k pos = 0.001 + 0.999 flc1hs k – 5e–4 5e–4

•k k is evaluated as log k pos .

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