FLOW SOLVER FOR GAS-PARTICULATE FLOWS

Research shows that the maximum value of the steady-state velocity achieved by the particle phase within the domain decreases, while that of the gas phase increases with the increase in the Stokes number of the particles. 126 7.5 Comparison of the time-averaged volume fraction along the radial direction 126 7.6 Schematic representation of the channel used to investigate the effects of particle diameter.

Classification of two phase flows

The subject of two-phase flow is very important in a number of engineering systems such as power systems (such as nuclear reactors), heat transfer systems (such as heat exchangers, spray cooling towers, dryers), process systems (such as chemical reactors, fluidized beds), transportation systems (such as air lift pump, pipeline transportation of gas and oil mixtures, powdered solid particles) and many other applications for their optimal design and safe operation. Separated two-phase flows are the flows in which the two phases are separated by a clear interface and do not mix with each other.

Figure 1.1: Separated flow, (a) film flow, (b) annular flow, (c) jet flow

Study of two phase flow problems

Homogeneous equilibrium model

Both the phases are assumed to have the same velocity, or in other words, the slip between the phases is neglected. This model is mathematically simpler and is generally applied in the solution of flows such as bubbling flow of water in air or flow of steam at high pressure.

Drift flux model

Two-fluid model

Gas-particle flow

Basic terminology related to gas-particle flows

Volume fraction (α)
Phase coupling
Stokes number (St)
Particle-wall specularity coefficient (φ)
Particle-particle restitution coefficient (e)
Packing limit (α s,max )
Minimum volume fraction (α s,min )

Depending on the roughness of the wall, there is always some loss of particle momentum due to the interactions with the wall. In dense gas-particle flow scenarios, this is defined as the volume fraction above which the frictional stresses become dominant in the overall hydrodynamics of the flow.

Concept of fluidization

Due to the higher kinetic energy of the gas, a greater number of particles can be trapped from the bed and the entrained particles circulate through the outer loop back into the bed. Annular Fluidized Beds: In these types of beds, there is a large nozzle in the middle of the bed that introduces the gas at high velocity, and additional fluidized gas is introduced through an annular ring of nozzles that facilitates intense mixing of the gas and solids. in the lower part of the chamber.

Solution methods

Eulerian-Eulerian approach

In the Eulerian-Eulerian approach, it is assumed that the properties of the dispersed particle phase are continuous like those of a liquid. The main problem is the modeling of surface boundary conditions for the particle phase.

Eulerian-Lagrangian approach

The particle phase is solved using fluid-like equations and thus it becomes relatively easy to implement, solve and interpret the particle phase along with the fluid-phase equations [19]. 13] established a set of Eulerian formulation with generalized wall boundary conditions and developed a particle wall collision model to better represent the particle wall momentum transfer.

Review of literature

Eulerian-Eulerian approach
Gas-particle flows considering two-way coupling
Particle-particle and particle-wall interactions in channel flows 16
Gas-particle flows in fluidized beds

The various physical properties of the particle phase have important effects on the flow hydrodynamics of gas particle streams. Much research has been conducted on the simulation of gas particle flows in fluidized beds.

Motivation and objectives

In this study, the effects of varying φ and e on the recirculation length will be investigated for dispersed gas particle flows with sudden expansion. This study will address the effects of particle diameter variation on fluidization behavior in swirling gas-solid fluidized beds.

Thesis overview

3αgµg(∇ ·ug)I (2.4) where µg is the dynamic viscosity of the gas phase and I is the unit tensor. In the Euler-Euler approach, the governing equations for the particle phase are written in a similar way to the governing equations for the gas phase.

Kinetic theory for granular flows model

Algebraic form of θ equation

This algebraic form of θ is derived by assuming that θ is distributed locally and neglecting the convection and diffusion terms and keeping only the dissipative and source terms of the corresponding PDE, given in Eq. 2.12,tr(Ds) is the trace of the viscous stress tensor of the solid phase velocity and the coefficients k1 ,k2 , k3 and k4 are calculated as follows.

Calculation of solid pressure and radial distribution function . 29

The solid phase shear viscosity (µs) arises due to translational (kinetic) motion and collisional interactions between the solid particles. Several empirical models are available for calculating the effective solid phase shear viscosity (µs), which are described below.

Use of frictional stress model

Like fixed kinetic stresses, the fixed frictional stresses also consist of frictional shear stress and normal frictional stress (or frictional pressure) [88]. These fixed frictional stresses are added to the fixed kinetic stresses (psand µs) calculated using KTGF when the solid volume fraction exceeds a critical minimum value αs,min.

Interphase interaction forces

Magnus Force: The Magnus force is the lift force that develops due to the rotation of particles. In the case of two-phase currents, it arises due to the relative motion of the two phases.

Boundary conditions

Therefore, the boundary conditions for the particle velocity at the wall vary according to the value of φ. For the volume fraction αs and αg of the phases, a homogeneous Neumann boundary condition is specified at the walls.

Numerical methodology

So to calculate the values at the planes of the control volumes, if and when necessary, interpolation. technology must be used. To calculate the corresponding value of φ at the plane center f of plane “abc”, an inverse volume-weighted linear interpolation method is used which can be written as. 2.49) where ∆Vco and ∆Vnb indicate the volumes of the cells with centers co and nb, respectively.

Figure 2.1: Different types of cell. (a) tetrahedron, (b) prism, (c) hexahedron, (d) pyramid, (e) quadrilateral, (f) triangle

Discretization procedure

Discretization of the continuity equation

The terms in the continuity equation (equation 2.1 and equation 2.6) are discretized for each cell as given below. Ffαf (2.53) uf is the velocity defined at the center f of the surface, Sf is the surface vector representing the area of this surface.

Discretization of the momentum equation

On the other hand, the surface flux f using CDS can be calculated by inverse volume-weighted interpolation as shown below. 2.59), where ∆Vco is the volume of the owner cell co and ∆Vnb is the volume of the corresponding neighboring cell nb. Where psf is the solid pressure at the center of the face and Sf is the face area vector f.

Figure 2.3: Pictorial representation of diffusive flux calculation.

Discretized form of the governing equations

This term exists only in the particle phase momentum equation and the Green Gauss reconstruction is used to discretize it as follows. Similarly, the solution form of the particle phase momentum equation can be written as. ρsαsn+1g)co∆Vco (2.71) where the term Fsfn+1 is the volume flux and Fdsn+1uf is the diffuse flux for the particle phase at a given plane f.

Momentum interpolation technique

Derivation of the pressure equation

Solution algorithm

This solution algorithm is implemented to develop the native finite volume flow solver for gas particle flow and the different stages of development of the solver can be listed as follows. These six stages represent the chronological sequence of development, validation and application of the flow solver and form the core of the present thesis.

Closure

Particles released at a very low velocity into a uniform fluid

In this test case, the dispersed phase particles are considered to enter a wall-free gas flow domain at low velocity as shown in Fig. It can be observed that for both meshes the particles with a higher diameter must travel a greater distance downstream of the domain before a constant velocity is achieved.

Particles falling freely into a quiescent gas medium

The former corresponds to the density of n-heptane particles and the latter is close to the density of glass beads. These two test cases together convincingly demonstrate that the implementation of one-way coupling in the proposed flow solver is indeed correct and that the solver can be used to simulate dilute laminar gas particle flows quite accurately.

Figure 3.4: Particle phase velocity variation along the flow direction for freely falling particles for a diameter of (a) 400 µm, (b) 200 µm compared with [102].

Validation for two-way coupling

The graphs show that the agreement is excellent, which validates the developed solution for gas-particle flows with two-way coupling.

Validation for three-way coupling

Settling suspension under the effect of gravity

The instantaneous contours of the volume fraction of the particle phase for each of the three grids are shown in Figs. For each of the three grids, similar changes in particle phase volume fraction contours can be observed with time.

Figure 3.13: The schematic diagram of the domain of settling suspension.

Closure

A detailed study is also carried out to find out the effects of 'inlet slip' (or difference of phase velocities at inlet) on the flow physics within a horizontal channel. In order to investigate the effects of particle diameter and inlet particle phase volume fraction on the hydrodynamics of dispersed laminar gas particle flow, a vertical channel with height (H) 0.4 m and width (D) of 0.155 m is considered as shown in Fig. .

Figure 4.1: Schematic of the vertical channel used to study effects of particle diam- diam-eter and inlet particle volume fraction.

Study of effects of particle diameter

Effects on steady state particle velocity profiles

This can be attributed to the fact that the motion of particles with a larger diameter (or larger Stokes number) is more affected by their inertia. Because of this large inertial and gravitational force compared to the interfacial drag forces for larger diameter particles, the steady-state particle velocity at a given channel section decreases as the particle diameter increases for a given particle inlet volume fraction.

Table 4.2: Maximum particle velocity inside the channel for particles of different sizes at constant inlet volume fraction of 0.01

Effects on steady state gas velocity profiles

4.4 (a) and (b) show the contours of the stable gas and the particle velocity in the channel, respectively, for a particle density of 1000 kg/m3 and a particle diameter of 500 µm.

Study of effects of inlet particle phase volume fraction

Effects on steady state gas velocity profiles

It can be seen that as the particle phase volume fraction increases from 0.0001 to 0.01, the steady state velocity achieved by the gas phase in the midplane of the channel decreases. This in turn reduces the constant maximum velocity of the gas phase at any section of the channel.

Effects on steady state particle velocity profiles

The maximum values of particle phase velocity obtained inside the channel for different inlet volume fractions are also shown in Table 4.5.

Figure 4.6: Variation of steady state particle velocity profiles at midplane of the vertical channel with inlet particle volume fraction for particle diameter 300 µm be observed with increase in particle phase volume fraction

Study of effect of inlet slip between the phases

When gas velocity is greater than particle velocity at inlet

As a result, the steady-state gas velocity in each section for this case is less than the case where there is the same inlet velocity for both phases. But since the steady state velocity of the gas phase in each section in case (ii) is lower than that of case (i), so the volume fraction of the gas phase must increase in order to maintain the velocity of flow of the same mass in both cases.

Figure 4.8: Comparison of plots of steady state (a) gas velocity,(b) gas volume fraction and (c) particle volume fraction profiles in the midplane of the channel for (i) u g =u s =1 m/s; (ii) u g =1 m/s, u s =0.5 m/s

When particle velocity is greater than gas velocity at inlet

Closure

Particle-wall interactions (quantified by reflection coefficient, φ) and particle-particle interactions (quantified by coefficient of restitution, e) significantly influence the overall flow characteristics of dispersed gas-particle flows. In this chapter, a study was carried out using the developed solver to investigate the effects of φ and e on the overall hydrodynamics of dispersed gas particle flows through horizontal channels.

Figure 5.1: Schematic of horizontal channel used to study the effects of specularity coefficient and particle-particle restitution coefficient.

Grid independence study

Effects of variation of specularity coefficient

Effect on particle velocity profiles

Effect on gas velocity profiles

Effect on particle phase volume fraction profiles

The maximum value of the particle volume fraction at the mid-plane of the channel is found to be achieved for φ=0.8; to a certain value ofe. At the central or core region of the channel, which is far away from the wall, the steady state volume fractions for all the three values of φ are almost equal and equal to the inlet value of particle volume fraction.

Effects of variation of particle-particle restitution coefficient

Effect on particle velocity profiles

As a result, the maximum value of the steady-state particle velocity can be reached at any channel section for e=0.99. For e=0.8, the loss of momentum due to particle collisions is the largest of the three values of e, so the steady-state particle velocity at any section of the channel is the smallest for e=0.8.

Effect on gas velocity profiles

As a result, the maximum gas velocity within the channel is obtained for e=0.99 and it decreases as the value of e decreases. Table 5.3 shows that for a given value of φ, the gas velocity obtained within the domain is maximum when the value of e is maximum, and minimum when the value of e is minimum.

Effect on particle phase volume fraction profiles

A point worth noting is that the length of the region where the steady state particle phase volume fraction is constant and equal to the inlet value is minimum when e=0.99 and it increases as the value of ede decreases .

Comparison of results for different combinations of φ-e pairs

Comparison of particle phase velocity profiles

But the velocity near the center line of the channel is maximum for φ=0.8 for both values of e.

Figure 5.9: Comparison of plots of steady state particle velocity profiles for (a) φ=0, e=0.99; (b) φ=0.8, e=0.99; (c) φ=0, e=0.8 and (d) φ=0.8, e=0.8.

Comparison of gas phase velocity profiles

Comparison of particle phase volume fraction profiles

Effect of drag models

Study of wall shear stress distribution of individual phases

Particle phase

Gas phase

As already discussed in Sections 5.3.1 and 5.3.2, the particle velocity near the wall is maximum for φ=0 and decreases as the value of φ increases. So the value of ug,co becomes maximum for φ=0 and as a result, the value of gradient ∂u∂yg at the wall also becomes maximum.

Closure

It is found that for a particular value of φ, the recirculation lengths tend to decrease with increase in the value of e when other parameters are held invariable. At a particular value ofφ, for all other parameters being invariant, the length of the recirculation zone tends to increase with decreasing value ofe.