# Conservation and Constitutive Equations for Fluid-Particle Flow Systems

## Introduction

A major computational advance in the calculation of multiphase flow regimes made in the 1980s was the use of computational codes based on the Navier-Stokes equation for solving the governing equations presented in Chap. 1.

The basic momentum balances for the fluid and particulate phases are as follows: Fluid momentum balance

Solid momentum balance

where p is density, e is volume fraction, *t* is time, v is velocity vector, *P* is pressure, *g* is gravity acceleration, т is stress tensor, and *в* is the interface momentum exchange coefficient.

The summation of volume fractions for all phases is equal to one

In order to close the conservation equations for the momentum, one needs to calculate the stress tensors and consequently the solid-phase viscosity. Two types of models were used to close the coupled Navier-Stokes equations for both the fluid and disperse particles. The first group of models requires an empirical input of particulate viscosity and gradient of the disperse pressure. The second group of models is based on the kinetic theory of granular flow. These models compute the particulate viscosity and the gradient of the solid pressure as a function of the

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H. Arastoopour et al., *Computational Transport Phenomena of Fluid-Particle Systems, *Mechanical Engineering Series, DOI 10.1007/978-3-319-45490-0_2

granular temperature (Arastoopour 2001; Gidaspow 1994; Gidaspow and Jiradilok 2009). The granular temperature is a measure of the random particle kinetic energy per unit mass. It is produced due to “viscous-type dissipation” and consumed due to inelastic collisions. The random granular temperature equation for the particle phase can be expressed as

Accumulation + net outflow = production + conduction — dissipation

+ granular energy exchange between phases

where в is granular temperature (which is defined as the mean of the squares of particle velocity fluctuation), *k _{s}* is granular conductivity,

*у*is the collisional energy dissipation, and

gs is the granular energy exchange between phases which is defined as

gs *= —3fi _{gs}e* for laminar flows (Gidaspow et al. 1991) and

gs *= ft _{gs }{y_{s}j2kffb@ — 2kf)* for disperse turbulent flows (Sinclair and Mallo 1998). In the latter case,

*Kf*is the turbulent kinetic energy of the fluid and

*ft*is the gas-solid exchange coefficient.

_{gs}The stress tensor for each phase is given by a Newtonian-type viscous approximation, as

Particle pressure, *P _{s};* shear viscosity,

*/л*; and bulk viscosity, ?

_{s}_{s}, are expressed as a function of granular temperature based on the kinetic theory model (Gidaspow 1994).

The constitutive equation for the shear viscosity consists of three sources, namely, kinetic, collision, and friction, which could be written either in an additive manner or a continuous form as (see Sect. 2.7 for more details)

The first two parts are calculated based on the kinetic theory. In dense granular flows, in addition to the kinetic and collisional stresses (described by the kinetic theory), the model should account for the frictional stresses as well, which is dominant in flow regimes denser than the bubbling regime. The frictional behavior of granular matters is discussed in this chapter based on soil mechanics principles (see Sect. 2.7).

This chapter starts with the derivation of the conservation equation for mass, momentum, and granular temperature based on the kinetic theory approach for uniform and multi-type particles.