Multiphase Continuum for Gas-Solids Reacting Flows

Multiphase Continuum Formulation for GasSolids Reacting Flows

Madhava Syamlal & Sreekanth Pannala

DOI: 10.4018/978-1-61520-651-3.ch001

Gas-solids reactors, which are critical components in many energy and chemical conversion processes. There are many examples: coal gasifiers that react coal with oxygen and steam to produce synthesis gas (syngas)—a mixture of hydrogen and carbon monoxide; circulating fluidized-bed combustors that burn coal to generate heat and electric power; or fluid catalytic cracking (FCC) risers that crack heavy oil with the help of hot catalyst particles, producing light hydrocarbons such as gasoline.

In multiphase devices, the particles collide, shear, and interact; the particles and gas exchange momentum and interact with the device boundaries; the particles and gas exchange heat and mass; and heterogeneous and homogeneous chemical reactions occur at greatly different scales.

1. Direct numerical simulation (DNS) method, which fully resolves the flow around individual particles by solving Navier-Stokes equations and tracks the particle motion by solving Newton’s equations of motion. This method is the cheapest in modeling effort and the most expensive computationally. The size of the system as well as the physics that can be described by this method is limited.

2. A (computationally) less expensive approach is the lattice-Boltzmann method (LBM), which resolves the flow around particles by solving lattice-Boltzmann equations and tracks the particle motion by solving Newton’s equations of motion.

3. Much computational expense can be avoided by not resolving the flow field around the particles, which leads to the discrete element method (DEM)but a price in modeling effort needs to be paid for not resolving the flow field around the particles in terms of developing constitutive relations for the gas-solids drag. The DEM approach quite effectively accounts for the transfer of momentum between colliding particles in fleeting contact or sliding particles in enduring contact and for the effect of particle size and shape.

a. Much of the computational time required in DEM simulations is for particle contact detection and integration through the contacts.

b. The computational effort for contact detection can be reduced by probabilistic detection of the collisions between sampled particles (rather than all the individual particles) as in direct simulation Monte Carlo. or altogether avoided by obtaining the collisional stresses from an Eulerian grid as in Multiphase particle in cell methods (MPPIC) or by not tracking individual particles and treating their collective motion as that of a fluid. When the equations of motion of discrete particles are averaged, the resulting continuum-solids phase co-locates with the fluid phase, leading to an interpenetrating continuum model (also called a two-fluid model or an Eulerian-Eulerian model)

  1. Gas-SOlids continuum MOdel

Two approaches can be used to arrive at the multiphase flow equations: an averaging approach and a mixture theory approach.

a. In the averaging approach, the equations are derived by space, time, or ensemble averaging of the local, instantaneous balances for each of the phases.

b. In the mixture theory approach, the multiphase flow equations are postulated, and restrictions on the constitutive relations are derived from general principles of continuum mechanics

Both approaches yield a set of balance equations for mass, momentum, and energy

 1.1 Mixture theory

Work by Truesdell (1957). In that paper, Truesdell assigned motion, density, body force term, partial stress tensor, partial internal energy, partial heat flux, and partial heat supply density to each constituent of the mixture. Bowen (1967) pointed out that some of the incorporations of partial stress tensors, partial heat fluxes, and partial heat supplies lead to the special case of ideal mixtures and made further generalizations to the theory of thermodynamics and mechanics of mixtures without invoking the strong assumption that the total quantity is the sum of the partials. Passman (1977) suggested another approach to multiphase mixtures where one can retain the general notion of the continuum but attribute a more complex structure for each particle to account for the discrete nature of the interactions