### 5.5.2. Interphase Drag for the Particle Model

For spherical particles, the coefficients may be derived analytically. The area of a single particle projected in the flow direction, , and the volume of a single particle are given by:

| (5–37)

where is the mean diameter. The number of particles per unit volume, , is given by:

| (5–38)

The drag exerted by a single particle on the continuous phase is:

| (5–39)

Hence, the total drag per unit volume on the continuous phase is:

| (5–40)

Comparing with the Momentum Equations for phase , where the drag force per unit volume is:

| (5–41)

you get:

| (5–42)

which can be written as:

| (5–43)

This is the form implemented in CFX.

The following section describes drag correlations specific to dispersed multiphase flow.

#### 5.5.2.1. Sparsely Distributed Solid Particles

At low particle Reynolds numbers (the viscous regime), the drag coefficient for flow past spherical particles may be computed analytically. The result is Stokes’ law:

| (5–44)

For particle Reynolds numbers, such as Equation 5–9, that are sufficiently large for inertial effects to dominate viscous effects (the inertial or Newton’s regime), the drag coefficient becomes independent of Reynolds number:

| (5–45)

In the transitional region between the viscous and inertial regimes, for spherical particles, both viscous and inertial effects are important. Hence, the drag coefficient is a complex function of Reynolds number, which must be determined from experiment.

This has been done in detail for spherical particles. Several empirical correlations are available. The one available in CFX is due to Schiller and Naumann (1933) [6]. It can be written as follows:

##### 5.5.2.1.1. Schiller Naumann Drag Model

| (5–46)

CFX modifies this to ensure the correct limiting behavior in the inertial regime by taking:

| (5–47)

#### 5.5.2.2. Densely Distributed Solid Particles

##### 5.5.2.2.1. Densely Distributed Solid Particles: Wen Yu Drag Model

| (5–48)

Note that this has the same functional form as the Schiller Naumann correlation, with a modified particle Reynolds number, and a power law correction, both functions of the continuous phase volume fraction .

You may also change the Volume Fraction Correction Exponent from its default value of -1.65, if you want.

Note: Although the Wen Yu drag law implemented in ANSYS CFX follows the implementation by Gidaspow [18] and its subsequent wide use, this implementation of the drag law is, in fact, quite different from that given in the original Wen and Yu paper [181].

##### 5.5.2.2.2. Densely Distributed Solid Particles: Gidaspow Drag Model

| (5–49)

This uses the Wen Yu correlation for low solid volume fractions , and switches to Ergun’s law for flow in a porous medium for larger solid volume fractions.

Note that this is discontinuous at the cross-over volume fraction. In order to avoid subsequent numerical difficulties, CFX modifies the original Gidaspow model by linearly interpolating between the Wen Yu and Ergun correlations over the range .

You may also change the Volume Fraction Correction Exponent of the Wen Yu part of the correlation from its default value of -1.65, if you want.

#### 5.5.2.3. Sparsely Distributed Fluid Particles (Drops and Bubbles)

At sufficiently small particle Reynolds numbers (the viscous regime), fluid particles behave in the same manner as solid spherical particles. Hence, the drag coefficient is well approximated by the Schiller-Naumann correlation described above.

At larger particle Reynolds numbers, the inertial or distorted particle regime, surface tension effects become important. The fluid particles become, at first, approximately ellipsoidal in shape, and finally, spherical cap shaped.

In the spherical cap regime, the drag coefficient is well approximated by:

| (5–50)

Several correlations are available for the distorted particle regime. CFX uses the Ishii Zuber [19] and Grace [35] correlations.

##### 5.5.2.3.1. Sparsely Distributed Fluid Particles: Ishii-Zuber Drag Model

In the distorted particle regime, the drag coefficient is approximately constant, independent of Reynolds number, but dependent on particle shape through the dimensionless group known as the Eotvos number, which measures the ratio between gravitational and surface tension forces:

| (5–51)

Here, is the density difference between the phases, is the gravitational acceleration, and is the surface tension coefficient.

The Ishii-Zuber correlation gives:

| (5–52)

In this case, CFX automatically takes into account the spherical particle and spherical cap limits by setting:

| (5–53)

The Ishii Zuber Model also automatically takes into account dense fluid particle effects. For details, see Densely Distributed Fluid Particles.

##### 5.5.2.3.2. Sparsely Distributed Fluid Particles: Grace Drag Model

The Grace drag model is formulated for flow past a single bubble. Here the drag coefficient in the distorted particle regime is given by:

| (5–54)

where the terminal velocity is given by:

| (5–55)

where:

| (5–56)

and:

| (5–57)

| (5–58)

is the molecular viscosity of water at 25°C and 1 bar.

In this case, CFX automatically takes into account the spherical particle and spherical cap limits by setting:

| (5–59)

#### 5.5.2.4. Densely Distributed Fluid Particles

##### 5.5.2.4.1. Densely Distributed Fluid Particles: Ishii-Zuber Drag Model

The Ishii Zuber [19] drag laws automatically take into account dense particle effects. This is done in different ways for different flow regimes.

In the viscous regime, where fluid particles may be assumed to be approximately spherical, the Schiller Naumann correlation is modified using a mixture Reynolds number based on a mixture viscosity.

##### 5.5.2.4.2. Densely Distributed Fluid Particles: Dense Spherical Particle Regime (Ishii Zuber)

| (5–60)

Here, is the user-defined Maximum Packing value. This is defaulted to unity for a dispersed fluid phase.

In the distorted particle regime, the Ishii Zuber modification takes the form of a multiplying factor to the single particle drag coefficient.

##### 5.5.2.4.3. Densely Distributed Fluid Particles: Dense Distorted Particle Regime (Ishii Zuber)

| (5–61)

##### 5.5.2.4.4. Densely Distributed Fluid Particles: Dense Spherical Cap Regime (Ishii Zuber)

| (5–62)

The Ishii Zuber correlation, as implemented in CFX, automatically selects flow regime as follows:

##### 5.5.2.4.5. Densely Distributed Fluid Particles: Automatic Regime Selection (Ishii Zuber)

| (5–63)

##### 5.5.2.4.6. Densely Distributed Fluid Particles: Grace Drag Model

The Grace [35] drag model is formulated for flow past a single bubble. For details, see Sparsely Distributed Fluid Particles (Drops and Bubbles).

For high bubble volume fractions, it may be modified using a simple power law correction:

| (5–64)

Here, is the single bubble Grace drag coefficient. Advice on setting the exponent value for the power law correction is available in Densely Distributed Fluid Particles: Grace Drag Model in the CFX-Solver Modeling Guide.

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