Granular Stress

Particle-particle interactions can be significant in solid particle flows. The particle-particle interaction forces are considered as internal forces within phases.

Simcenter STAR-CCM+ provides two models to account for the solid particle-particle interaction forces. These models apply exclusively to the dispersed phase.

Solid Pressure Force

A compaction, or 'solid pressure', force is provided as one model of particle-particle interactions within a solid particle flow.

It is implemented as the following internal force term in the momentum equation for a solid particle phase:

Figure 1. EQUATION_DISPLAY

{(F_{i n t})}_{i} = - \frac{α_{i}}{{Σ_{j = 1}}^{M} α_{p, j}} [e^{A (α_{p, \max} - \sum_{j = 1}^{M} α_{p, j})}] \nabla (\sum_{j = 1}^{M} α_{p, j})

(2348)

where:

$α_{p, m a x}$ is the maximum packing limit

$α_{p, j}$ is the volume fraction of the jth particle phase

$M$ is the number of particle phases

$A$ is the model constant, set to -600.0 by default

When activated during a multiphase flow simulation, the solid pressure force comes into action only when the solid volume fraction in any part of the domain approaches $α_{p, m a x}$ . The force acts to resist the formation of unphysical solid packing fractions by increasing exponentially as the solid volume fraction approaches the maximum packing limit. This model is useful to study a multiphase flow containing solids without needing to consider the effects of solid stress due to friction and/or collision.

The maximum packing limit is set to 0.624 by default, representative of rigid spherical solid particles.

Granular Pressure

The granular pressure model provides a way to estimate the stresses in the granular medium. The model is applicable to gas-solid flows and is typically used for fluidized bed applications.

Fluidized bed reactors are widely used in combustion, catalytic cracking and various other chemical and metallurgical processes. The particle motion can be classified into two regimes - kinetic and frictional. In the kinetic regime, the particle motion is governed by collisions between particles. By analogy with gas kinetic theory, and after assuming distribution and collision properties for the particles, a kinetic theory of granular pressures can be derived for single particle flows [466], [512].

In the frictional regime, where the particle motion is contact-dominated, empirical formulations that are borrowed from soil mechanics are employed. It is possible to work with multiple particle phases (that is, particles of different sizes) when using the Granular Pressure Model. Schaeffer, [542] described the stresses in the frictional regime that is based on a simplified version of the critical state theory. Johnson and Jackson, [483] also developed constitutive relations for stresses in the frictional regime.

A full stress tensor is then obtained that describes the particle-particle interaction terms in the solid phase momentum equation:

Figure 2. EQUATION_DISPLAY

{(F_{int})}_{p} = \nabla\cdot {\bar{S}}_{p}

(2349)

where the granular stress tensor is defined as:

Figure 3. EQUATION_DISPLAY

{\bar{S}}_{p} = [- P_{p} + (ξ_{p} - \frac{2}{3} μ_{p}) \nabla\cdot v_{p}] I

(2350)

where:

$P_{p}$ is the solid pressure.
$μ_{p}$ is the effective granular viscosity.
$I$ is the identity tensor.
The subscript p,is used to denote the particle phase.

$ξ_{p}$ is the bulk viscosity. If you activate only the frictional regime, $ξ_{p}$ is set to 0. If you activate only the kinetic regime, $ξ_{p}$ is computed using Eqn. (2381). If you activate both regimes, $ξ_{p}$ is computed based on the frictional model you select.

Kinetic Regime

The kinetic regime is characterized by particle volume fractions below the maximum packing limit, $α_{p, m a x}$ . In the kinetic regime, the particle motion is governed by collisions between particles. The granular temperature, $θ_{p}$ , is a measure of the fluctuating kinetic energy of particles:

Figure 4. EQUATION_DISPLAY

θ_{p} = \frac{1}{3} 〈 {v′}_{p i} \cdot {v′}_{p i} 〉

(2351)

where ${v′}_{p i}$ is the random fluctuating velocity of the particle phase and $〈〉$ denotes ensemble averaging.

The probability of particle collisions is characterized by the particle radial distribution function, $g_{0}$ . The radial distribution function represents the spatial distribution of the particles, and therefore their proximity, and can be related to the dispersed-phase volume fraction. The form in [451] has been implemented:

Figure 5. EQUATION_DISPLAY

g_{0} = \frac{3}{5} {[1 - {(\frac{α_{p}}{α_{p, m a x}})}^{1 / 3}]}^{- 1}

(2352)

This function becomes undefined when $α_{p}$ approaches the maximum packing limit, $α_{p, m a x}$ . In this limit, Eqn. (2352) is supplemented by the equation that is proposed in [563] for $α_{p} > α_{c r i t}$ :

Figure 6. EQUATION_DISPLAY

g_{0} = 1.08 \times 10^{3} + 1.08 \times 10^{6} (α_{p} - α_{c r i t}) + 1.08 \times 10^{9} {(α_{p} - α_{c r i t})}^{2} + 1.08 \times 10^{12} {(α_{p} - α_{c r i t})}^{3}

(2353)

where:

Figure 7. EQUATION_DISPLAY

α_{c r i t} = α_{p, m a x} {(1 - \frac{0.6}{1.08 \times 10^{3}})}^{3}

(2354)

The energy that is dissipated in particle collisions is characterized by the coefficient of restitution, e. You can specify e (by default e = 0.9).

Mixture Maximum Packing Fraction

Yu and Standish [575] proposed an empirical correlation to estimate a maximum packing limit for a mixture of multiple particles. There is no limit to the number of particles in the mixture for this correlation.

The cumulative particle volume fraction, $α_{p, s u m}$ , for a mixture of $M$ particles is:

Figure 8. EQUATION_DISPLAY

α_{p, s u m} = \sum_{n = 1}^{M} α_{n}

(2355)

The mixture maximum packing fraction is:

Figure 9. EQUATION_DISPLAY

α_{p, m a x}^{m} = m i n {\frac{α_{i, m a x}}{1 - \sum_{j = 1}^{i} (1 - \frac{α_{i, m a x}}{p_{i j}}) (\frac{c x_{i}}{X_{i j}}) - \sum_{j = i + 1}^{M} (1 - \frac{α_{i, m a x}}{p_{i j}}) (\frac{c x_{i}}{X_{i j}})}}

(2356)

where $i$ =1, 2, ... , $M$ are the particle phases in the simulation arranged in order from coarsest to finest.

Figure 10. EQUATION_DISPLAY

c x_{i} = \frac{α_{i}}{α_{p, s u m}}

(2357)

Figure 11. EQUATION_DISPLAY

X_{i j} = {\begin{matrix} \frac{1 - r_{i j}^{2}}{2 - α_{i, m a x}} & j < i \\ 1 - \frac{1 - r_{i j}^{2}}{2 - α_{i, m a x}} & j \geq i \end{matrix}

(2358)

Figure 12. EQUATION_DISPLAY

r_{i j} = {\begin{matrix} \frac{d_{p, i}}{d_{p, j}} & j \leq i \\ \frac{d_{p, j}}{d_{p, i}} & j > i \end{matrix}

(2359)

Figure 13. EQUATION_DISPLAY

p_{i j} = {\begin{matrix} α_{i, m a x} + α_{i, m a x} (1 - α_{i, m a x}) (1 - 2.35 r_{i j} + 1.35 r_{i j}^{2}) & r_{i j} \leq 0.741 \\ α_{i, m a x} & r_{i j} > 0.741 \end{matrix}

(2360)

Frictional Regime

The following models are available in the frictional regime:

Modified Johnson
Schaeffer

Modified Johnson Frictional Model

The Modified Johnson [483] model gets activated in regions where the cumulative particle volume fraction, $α_{p, s u m}$ , exceeds the specified minimum frictional volume fraction packing limit, $α_{p, m i n}^{f}$ . The solid pressure in this regime is given by the frictional solid pressure, $P_{p}^{f}$ :

Figure 14. EQUATION_DISPLAY

P_{p}^{f} = {\begin{matrix} \frac{α_{p}}{α_{p, s u m}} F r (\frac{{(α_{p, s u m} - α_{p, m i n}^{f})}^{r}}{{(α_{p, m a x}^{m} - α_{p, s u m})}^{s}}) & α_{p, s u m} > α_{p, m i n}^{f} \\ 0 & α_{p, s u m} \leq α_{p, m i n}^{f} \end{matrix}

(2361)

where:

$α_{p, m a x}^{m}$ is the mixture maximum packing fraction

$F r$ , $r$ , and $s$ are user-specified empirical constants

The effective granular viscosity in this regime, $μ_{p}^{f}$ , is given by the frictional viscosity:

Figure 15. EQUATION_DISPLAY

μ_{p}^{f} = {\begin{matrix} m i n (\frac{P_{p}^{f} \sin ϕ}{\sqrt{2 D_{p} D_{p} + \frac{θ_{p}}{d_{p}^{2}}}}, μ_{p, m a x}) & α_{p, s u m} > α_{p, m i n}^{f} \\ 0 & α_{p, s u m} \leq α_{p, m i n}^{f} \end{matrix}

(2362)

where:

$ϕ$ is the angle of internal friction. The recommended value is 28.5 $°$ when using the Johnson model.
$D_{p}$ is the strain rate tensor for the particle phase.
$μ_{p, m a x}$ is the user-defined Maximum Solid Viscosity limit applied to $μ_{p}$ .

The term involving granular temperature, $θ_{p}$ is ignored when the kinetic regime is not activated.

In the frictional regime, the bulk viscosity $ξ_{p}$ is set to 0 where the cumulative particle volume fraction exceeds the minimum frictional volume fraction $α_{p, m i n}^{f}$ , and is computed using Eqn. (2381) everywhere else.

Schaeffer Frictional Model

The Schaeffer [542] model gets activated in regions where the cumulative particle volume fraction, $α_{p, s u m}$ , exceeds the specified mixture maximum packing fraction, $α_{p, m a x}^{m}$ . The solid pressure in this regime is given by the frictional solid pressure, $P_{p}^{f}$ :

Figure 16. EQUATION_DISPLAY

P_{p}^{f} = {\begin{matrix} \frac{α_{p}}{α_{p, s u m}} (10^{25} {(α_{p, s u m} - α_{p, m a x}^{m})}^{10}) & α_{p, s u m} > α_{p, m a x}^{m} \\ 0 & α_{p, s u m} \leq α_{p, m a x}^{m} \end{matrix}

(2363)

The effective granular viscosity in this regime, $μ_{p}^{f}$ , is given by the frictional viscosity:

Figure 17. EQUATION_DISPLAY

μ_{p}^{f} = {\begin{matrix} m i n (\frac{P_{p}^{f} \sin ϕ}{\sqrt{4 I_{2 D}}}, μ_{p, m a x}) & α_{p, s u m} > α_{p, m a x}^{m} \\ 0 & α_{p, s u m} \leq α_{p, m a x}^{m} \end{matrix}

(2364)

where:

$ϕ$ is the angle of internal friction and is set to 25^o by default.
$μ_{p, m a x}$ is the user-defined Maximum Solid Viscosity limit applied to $μ_{p}$ .
$I_{2 D}$ is the second invariant of the deviator of the strain rate tensor:

Figure 18. EQUATION_DISPLAY

I_{2 D} = \frac{1}{6} [{(D_{p 11} - D_{p 22})}^{2} + {(D_{p 22} - D_{p 33})}^{2} + {(D_{p 33} - D_{p 11})}^{2}] + D_{p 12}^{2} + D_{p 23}^{2} + D_{p 31}^{2}

(2365)

In the frictional regime, where both regime and frictional are specified, the bulk viscosity, $ξ_{p}$ , is set to 0 above the mixture minimum packing fraction, $α_{p, m a x}^{m}$ , and is computed using Eqn. (2381) everywhere else.