Many industrially important gas-solid systems often include slow and dense solid flows, which is dominated by sustained frictional contacts between the particles. For simulations of these dense flows, in addition to the kinetic and collisional stresses (i.e., kinetic theory), the model should account for the frictional stresses as well, which could be calculated based on soil mechanics principles.

Makkawi and Ocone (2005) have reviewed the modeling approaches to include the frictional effects. The most common approach to consider the effect of frictional stresses is the kinetic frictional model based on the addition of stress from the two limiting regimes at a critical solid volume fraction (e_{cr}) (Johnson and Jackson 1987; Syamlal et al. 1993; Ocone et al. 1993)

where

where

or switching between the two limiting regimes (Laux 1998; Makkawi and Ocone 2005) at a critical solid volume fraction as

This approach is based on the assumptions of Savage (1998) that consider the solid stress comes from the kinetic, collisional, and frictional contributions in an additive manner, where the frictional contributions appear only at higher solid volume fractions (i.e., greater than 0.5). Although, this approach lacks a strong physical justification and the hypothetical assumption of the critical solid volume fraction remains without experimental proof (Makkawi and Ocone 2005), the theory has been shown to capture the qualitative features of slow dense solid flows (Srivastava and Sundaresan 2003).

Most of the reported frictional stress models in the literature (Makkawi and Ocone 2005) and CFD codes such as Ansys and MFIX are based on the critical state theory of soil mechanics, where the shear stress тf. is described in terms of a frictional viscosity based on the work of Schaeffer (1987). Under a normal stress, a well-compacted granular material will shear only when the shear stress attains a critical magnitude. This is described by a Mohr-Coulomb law based on the laws of sliding friction. However, the Mohr-Coulomb law does not provide any information on how the granular material deforms and flows; rather, it describes the onset of yielding (Dartevelle 2003).

Schaeffer (1987) derived the following expression for the frictional stress, by assuming the system to be perfectly rigid-plastic, incompressible, non-cohesive, Coulomb powder with a yield surface of von Mises type, and the eigenvectors of the strain rate and stress tensors are parallel, as

or equivalently

where ф is the angle of internal friction, II_{2}D is the second invariant of the deviatoric stress tensor S, and p_{c} is the critical state pressure. According to Srivastava and Sundaresan (2003), p_{c} increases monotonically with e and is expected to become very large (i.e., diverge) as e approaches random close packing e_{max}. Various expressions have been proposed for the functional dependence of pc on e in the literature (Srivastava and Sundaresan 2003; Atkinson and Bransby 1978; Schaeffer 1987; Tardos 1997; Johnson and Jackson 1987; Savage 1998; Prakash and Rao 1988; Syamlal et al. 1993).

Johnson and Jackson (1987) proposed a critical state solid frictional pressure that allows for a slight compressibility with very limited particle concentration change (Makkawi and Ocone 2005). The Johnson and Jackson correlation for frictional pressure can be written as

where Fr is a coefficient with different values reported in the literature from 0.05 to 5 (Johnson and Jackson 1987; O’Brien et al. 2010). The coefficient Fr was modified by others (Nikolopoulos et al. 2013; Abbasi 2013) assuming to be a function of the volume fraction as Fr = 0.1e_{s} while limiting the solid volume fraction to values less than 0.629 to prevent divergence. Note that p_{c} is the critical state frictional pressure and many studies assumed the critical state frictional pressure is equal to solid frictional pressure pf (Nikolopoulos et al. 2012; Abbasi and Arastoopour 2011; Tsuo and Gidaspow 1990), although clearly it is not an accurate assumption.

Srivastava and Sundaresan (2003) modified the Schaeffer expression for the frictional stress and also the Johnson and Jackson (1987) expression for the frictional pressure (see Eq. 2.108), to approximately account for strain rate fluctuations and slow relaxation of the assembly to the yield surface following Savage’s (1998) argument of existence of fluctuations in the strain rate, even in purely quasistatic flow. The standard deviation a of the fluctuations is related to the granular

temperature of the powder в (taken from the rapid granular flow regime) and the

1

particle diameter d_{p}, as a = bf-, where b is a constant of order unity.

Laux (1998) suggested a correlation of the following form:

where pf is calculated based on the following Srivastava and Sundaresan (2003) equation:

The angle of repose and, in turn, frictional forces for solid particles are significantly affected by the value of the compactness factor (n). The value for compactness factor (n) may be determined by comparing the experimental angle of repose with the calculated values for solid packing (Ghadirian 2016). Based on the above equation, if the granular material dilates as it deforms, V.u_{s} > 0, then pf < p_{c} if it compacts as it deforms, V.u_{s} > 0, then pf > p_{c} and when it deforms at constant volume, V .u_{s} = 0, which is the critical state, pf = p_{c}. This behavior is in line with the experimental measurements (see Das 1997). The value of n (the compactness factor) is different in the dilation and compaction parts of the system. Srivastava and Sundaresan (2003) suggested a value of sincp for the dilation branch, to

ensure that the granular assembly is not required to sustain tensile stress on the yield surface. They also pointed out that n for the compaction branch can be any value marginally greater than one. They suggested that a value of 1.03 be used when no additional information is available. This value is measured for glass beads by Jyotsna and Rao (1997). It is worth mentioning that decreasing the value of n in the compactness branch will cause more deviation from the critical state frictional pressure. In other words, n may be an indicator for nonlinearity of the т-a relation.

With such a formulation, numerical singularity is avoided in regions where S:S is zero as long as the granular temperature в is nonzero. If, however, the physical system does contain regions where both S:S and в are zero (e.g., in a bin discharge problem, the stagnant shoulders at the bottom corners of the bin), the model will fail (Srivastava and Sundaresan 2003). However, their model captures four qualitative behaviors of a dense granular flow: (1) the height-independent rate of discharge of particles from a bin, (2) the dilation of particle assembly near the exit orifice, (3) the significant effect of the interstitial air on the discharge behavior of fine particles, and (4) the occurrence of pressure deficit above the orifice. In addition, in a bubbling fluidized bed, the model captured the significant effect of frictional stresses on the bubble shape.

Tardos et al. (2003) proposed another approach for intermediate granular flows, which smoothly merges the slow-intermediate regime with the rapid dilute flow (i.e., nonadditive approach). The continuous function may be written as (Nikolopoulos et al. 2012)

The basic assumption of this model is that the stress during particle flow is not constant but fluctuates around an average value. The model is restricted to the simple geometry of the Couette device and to an incompressible material:

where p_{s} is the solid pressure and is calculated from the kinetic theory. ф is the angle of internal friction, and K is a parameter defined in terms of the average strain rate

(S) and its standard deviation, a, such that K = 2) .As concluded by Makkawi

and Ocone (2005), by comparing various approaches, the superiority of these approaches is a matter of open debate and subject to further experimental validations.

Up to this point, all of the discussed approaches were based on the von Mises/ Mohr-Coulomb law that, as mentioned earlier, does not provide any information on how the granular material deforms and flows. Although these models can properly simulate dilatancy, they do not capture consolidation (Dartevelle 2003). Therefore, the standard von Mises/Coulomb yield criterion cannot model the effect of compressibility phenomena (i.e., changes of bulk density) occurring in the returning system of circulating fluidized bed units and subsequently results in severe underestimation of the exerted frictional viscous forces (Nikolopoulos et al. 2013).

To overcome the shortcomings of the von Mises/Coulomb yield criterion, several modifications are proposed in the literature (Dartevelle 2003), such as the Gray yield criterion (Gray et al. 1991). The Gray yield criterion can be written as

It should be mentioned that the Gray criterion reduces to the von Mises/Coulomb criterion if (p_{p}— AV • л_{s} — a) = 0 and a = p_{p} (in other words, these two yield criteria are the same if and only if V • л_{s} = 0, which is the case only at the critical state of the soil mechanics where solids deform without volume changes). The Gray yield criterion is based on the approximation that normal stresses in the particulate phase are caused not only by the pressure but also by the viscous normal stresses.

According to Dartevelle (2003), the plastic potential theory combined with the critical state approach can successfully describe the phenomenon of dilatancy, consolidation, and independence between the rate-of-strain tensor and the stress tensor. Using this approach assuming a slightly compressible, dry, non-cohesive, and perfectly rigid-plastic system, the expression for the frictional viscosity may be written

and solid-phase bulk viscosity as

Equation (2.113) reduces to Eq. (2.105) if, V ,u_{s} = 0, corresponds to the critical state of soil mechanics and linear т-a relation. A detailed discussion and derivation of the models can be found in Dartevelle (2003).

Nikolopoulos et al. (2012)) have pointed out that the numerical results indicate that the values calculated by the Laux (Eq. 2.107) and Dartevelle (Eq. 2.112) expressions are of the same order of magnitude for values of solid volume fractions lower than 0.5. However, the Laux expression predicts higher solid frictional viscosity compared to the Dartevelle model for solid volume fractions higher than 0.55. Nikolopoulos et al. (2012) also showed that the results of simulations using the Dartevelle (2003) model are less accurate compared to the results of simulations using the Laux model in calculating the angle of repose.