2.1 Mass balances

The mass of phase i in a volume element V moving with phase i velocity is as follows:

where ε_i is the volume fraction of phase i, and ρ_i is defined as the density of phase i. This volume fraction is the only new variable in multiphase flow. Traditionally, this volume fraction, ε_i, was derived by volume averaging. For multicomponent systems, ε_i is 1, because it is assumed that molecules of component i occupy the same space at the same time. Such an approximation cannot be made in multiphase flow.

The balance on mass m_i moving with the velocity v_i is

where m_i′ is the rate of production of phase i; d/dtⁱ is the substantial derivative moving with phase i velocity.

The Reynolds Transport Theorem gives the continuity equation for phase i:

Conservation of mass requires that

The volume fraction of phase i was defined as follows:

where ∑i=1nεi=1

For incompressible fluid with no phase change, the continuity equation becomes

where i=1, 2, …n phases

Such an equation (6) does not appear in conventional transport phenomena theory.

2.2 Momentum balances

The rate of change of momentum of phase i moving with the velocity v_i equals the forces acting on the system and can be mathematically expressed by the following equation:

where force f_i is given by

The first term represents the surface forces acting on the differential area da; the second term is the external forces; the third term is the interaction forces between phases, and the last term is the force due to the phase change.

The stress tensor T_i for phase i is composed of nine components and can be written as follows:

Figure 2 represents the stress tensor acting on surfaces of volume of phase i.

Figure 2:

Stress tensor acting on surfaces of volume of phase i.

The Reynolds Transport Theorem is applied to the left side of equation (7) and divergence theorem (∯Tida=∭∇⋅TidV) to the first term of equation (8). The result is the momentum balance for phase i:

Syamlal et al. (2016) used a different treatment of momentum transfer in their MFIX code.

Using the continuity equation (3), the momentum balance for phase i moving with phase i velocity becomes

where ρiεidvidti is acceleration of phase i; ∇·T_i is the momentum in flow due to the surface forces; ρ_iε_iF_i is the body force, and p_i is the interaction force between phases.

where ∑i=1n phasepi=0

The interaction forces between phases (p_i) account for the drag between the phases. Note that moving with the velocity v_i, there is no force due to the phase change, because the constant mass in the balance (7) moved with the velocity v_i.

Hence, our form of phase change momentum is correct. This representation was verified experimentally and computationally for detonation of solid particles, such as TNT, in which the solid particles are rapidly converted into gases (Aldis and Gidaspow 1990, Pape and Gidaspow 1998).

The simplest expression for the stress is

Each phase i has its own pressure P_i similar to Euler equation for single-phase flow.

2.3 Energy balance

The energy balance for an open system with phase change can be written as follows (Gidaspow 1994):

where U_i is the internal energy per unit mass of phase i; Q_i is the heat per unit mass of phase i flowing into the system and the only work done by the system is the mechanical work due to the volume changes (P_i dV_i). In the open system with phase change there is inflow due to phase change, consisting of the energy inflow (U_iⁿ) and pressure P_iⁿ. The energy dissipation by the system is D_iss.

The rate of heat transfer is related to the flux, q_i by the following relations:

In Equation (21) the heat flux q_i need not be multiplied by the volume fraction of phase i.

The Reynolds Transport Theorem applied to the energy balance (19) produces energy equation for phase i.

where Diss=∭u′idV,u_i^t internal energy per unit of volume, h_iⁿ is the enthalpy of phase i flowing into the system; q_i is the rate of heat transfer by conduction.

In the above equation there appears a strange term, the work due to the volume fraction changes of phase i, not found in single-phase flow.

3.1 Frequency distributions

The frequency distribution of velocities of particles, f, is a function of position, r and the instantaneous velocity, c, as well as time, t.

The six coordinates, the position, r, and the velocity, c, are sufficient to determine the location of a particle, since Newton’s second law has six integration constants.

The number of particles per unit volume, n, is the integral over the velocity space, c.

The mean values of a quantity, ϕ, such as mass, momentum, energy and stress is defined to be in the usual way as

Hence, the hydrodynamic velocity, v, is the integral over all the velocity space, as shown below:

3.2 Peculiar velocity and transport

The transport of a quantity, ϕ, such as heat, must be invariant under a change of frame. Hence, it cannot be a function of the velocity, c. Otherwise, it will have different values in different frames of reference. But c−v is independent of the frame of reference. Hence, we define the difference between the instantaneous (c) and the hydrodynamic velocities (v) as

In kinetic theory (Chapman and Cowling 1961), this difference is called the peculiar velocity. Its mean is 0, as shown below, since the mean of c is v:

This property is the same as that of the turbulent velocity, defined as the instantaneous minus the time average velocity.

The flux vector of ϕ is defined as n <Cϕ (C)>. For example, if ϕ=E, the internal energy, then the conduction flux, q, becomes q=n <EC>.

Since momentum is the mass, m times the velocity, C, the kinetic stress tensor, P_k, is as follows:

since the bulk density, ρ=nm. Table 1 shows the components of the stress tensor.

For a Maxwellian distribution, the kinetic viscosity is 0. Particle viscosity is nonzero due to collisions and due to a non-Maxwellian distribution.

The hydrostatic pressure, p is the mean of the sum of the normal components of the stress tensor p:

3.3 Granular temperature and the equation of state

In kinetic theory of gases the thermal temperature, T is defined as the average of the random kinetic energy, with the conversion factor of the Boltzmann constant from joules to degrees kelvin, as shown below.

where the subscript of the peculiar velocity C indicates the component of C in the x, y and z directions, respectively. The Boltzmann constant has the value

The ideal gas law constant equals the very small Boltzmann constant, due to the small mass of the molecule, m times the large value of the Avogadro’s number, 6.023×10²³, the number of molecules per mole. Converting from joules to calories gives the gas law constant of 1.987 cal/g mole ° K.

Elimination of the squares of the peculiar velocities in

and in the definition of temperature, Equation (34), gives the ideal gas law equation of state,

where N is the number of moles, V is the volume, and R is the gas constant.

The granular temperature is defined as the random kinetic energy of the particles without the conversion of joules to calories. Equation (34) suggests that it can be defined in two ways: similarly to Equation (34) or as kinetic energy per unit mass. Let θ be the granular temperature, the random kinetic energy per unit mass. Then

in three dimensions. In two dimensions, we would have only two random velocities, and we would divide <C²> by 2. In one dimension we have only one random velocity, and the granular temperature is then simply the variance of the measured instantaneous velocities. However, its behavior is not the same as the three-dimensional granular temperature in Equation (38) (Strumendo et al. 2005). The units of the granular temperature are (m/s)².

These units are convenient. The alternate definition with mass multiplying θ is not that convenient for a single particle size mixture. It may, however, be useful for a mixture of particles (Gidaspow 1994). For a gaseous mixture of molecules, where there is no dissipation of energy, mass times velocity square of molecules is the same, since there is only one temperature.

The equation of state for particles is obtained by eliminating <C²> between Equations (36) and (38). This gives

where the subscript s was added to emphasize that it is the solids pressure. But nm is the bulk density. In terms of the volume fraction of solids, ε_s and the solids density, ρ_s, the ideal equation of state for particles becomes as follows:

The more complete equation of state for particles containing the collisional contribution has been verified experimentally by Gidaspow and Huilin (1998). Its comparison to theory is discussed in the experimental chapter.

3.4 FCC equation of state

The experimental granular temperature for flow of FCC particles is presented in Figure 3. As the particle concentration increases, the granular temperature (turbulent kinetic energy of particles) increases, similar to the rise of thermal temperature upon compression of a gas. The decrease of granular temperature in the collisional regime is due to the decrease of the mean free path, which becomes 0 in the packed state. The Sandia National Laboratory (Bhusarapu et al. 2006) measured the granular temperatures in a large riser with a splash plate using a radioactive particle tracer technique for flow of 150 μm glass beads at velocities of 4.5–7.7 m/s and fluxes of 102–145 kg/m² s. Their granular temperatures were of the order of magnitude of those shown in Figure 3.

Figure 3:

Granular temperature for 75 μm FCC in CFB. Granular temperature=2/3 σ_x²+1/3 σ_y² =2/3 turbulent kinetic energy. Reprinted from Gidaspow et al. (2004) with permission from ©Elsevier.

The particulate pressure measured with a specially designed transducer (Gidaspow and Huilin 1998) is shown in Figure 4. To construct a complete equation of state for FCC particles, we have used the statistical mechanics of liquid theory and our CCD camera system to determine radial distribution functions as a function of solids volume fraction.

Figure 4:

Time-average solid pressure for FCC particles. Reprinted from Gidaspow et al. (2004) with permission from ©Elsevier.

Figure 5 presents the equation of state. It shows that the kinetic theory of granular flow is valid for flow of FCC particles below about 5%. For dilute flow, we have an analogy of the ideal gas law: The ratio of solids pressure to bulk density multiplied by the granular temperature is 1, as a limit. For volume fractions above about 5%, the standard granular flow theory had to be corrected for a cohesive pressure, obtained from measurements of the radial distribution functions and granular temperature, using a modified Boltzmann relation, as explained by Gidaspow and Huilin (1998).

Figure 5:

Equation of state for 75 μm FCC particles determined in IIT CFB. Dimensionless solid pressure: Psρsεsθ=1+2(1+e)εsgo−(0.73εs+8.957εs2). Reprinted from Gidaspow and Huilin (1998) with permission from ©Wiley.

3.5 Particle and molecular velocities

To estimate the average molecular velocity, the concept of thermal temperature (Equation 34) is used. Thus, Equation (34) is multiplied by the Avogadro’s number, A, resulting in the following expression:

where mA (kg/molecule×molecules/mole) represents the molecular mass (M), and k_BA is the gas law constant (R).

Based on Equation (41), the equation for molecular velocity (C²) can be obtained as a function of thermal temperature:

where R=8314 J/mol K; 3R=158

The molecular velocity can be expressed in terms of molecular mass and temperature as follows:

For gases as CO₂ and H₂ at 273 K, the velocity obtained by Equation (43) gives 393 m/s and 1845 m/s. The critical or the sonic velocity is evaluated at a constant entropy and hence is about 20% higher due to the ratio of specific heats at constant temperature to constant volume in the square root relation in Equation (43).

For particles, the average velocity, the hydrodynamic velocity, cannot be 0, since an energy input is required to keep the particles in motion due their inelasticity. So Equation (38) shows that the granular temperature is of the order of the hydrodynamic velocity square. Thus, for fluidization of small particles in a bubbling bed the velocity is of the order of cm/s, while for fluidization in risers it is of the order of m/s. Due to dissipation of energy, the granular temperatures are not inversely proportional to the square root of masses, as shown for gases in Equation (43), although they are smaller for large particles in a binary mixture of small and large particles (Gidaspow et al. 2004).

3.6 Maxwellian distribution

The Maxwellian distribution using the granular temperature can be shown to be as follows (Gidaspow and Jiradilok 2009):

This is the expression found in the kinetic theory of gases (Chapman and Cowling 1961) with k_B/m=1. By comparing Equation (44) with that originally reported by Chapman and Cowling (1961), it can be noted that the thermal temperature was replaced by granular temperature, which is expressed in the units of velocity square.

3.7 Restitution coefficients

Figure 6 shows typical restitution coefficients based on data reported in the book by Johnson (1985). During impact of particles, the work of deformation can be expressed in terms of the elastic pressure P_e and the plastic pressure P_p and the deformation volume by means of the usual relation, as

Figure 6:

Restitution coefficients for various materials.

This work equals the relative velocity of impact square times half the mass. However, the relative kinetic energy equals only the integral of the elastic pressure. Hence, the restitution coefficient e can be expressed as

where v is the relative velocity before impact and v′ is the rebound velocity. Equation (46) suggests that for low velocities, where the plastic deformation is small, the restitution coefficient will be nearly 1. It also clearly shows that the restitution coefficient is a function of the material properties, as well as the dynamic properties associated with plastic flow. Indeed, data summarized by Johnson (1985) show that for hard materials the restitution coefficients are nearly 1 for impact velocities of 0.1 m/s and less (see Figure 6).

3.8 Frequency of binary collisions

Based on Chapter 16 in Chapman and Cowling (1961), the classical binary frequency of collisions corrected for the dense packing effect (g₀) is obtained. A collisional pair distribution function, f⁽²⁾, is introduced, which is analogous to the single frequency distribution given by Equation (26):

The probability of finding a pair of particles in the volume dr₁dr₂ centered on points r₁, r₂ and having velocities within the ranges of c₁ and c₁+dc₁, and c₂ and c₂+dc₂ are defined as follows:

Figure 7 shows the geometry of collisions and the spherical coordinates and solid angle. It is that given by Savage and Jeffrey (1981) with a generalization to two rigid spheres of unequal diameters. Equation (48) is integrated over the volume of collision cylinder as fully described in Gidaspow (1994) to obtain the collision frequency:

Figure 7:

Geometry of a collision of two spheres of diameter, d₁ and d₂.

For dilute gases (m₀/m₁=2 and k_B/m₂=1), Equation (49) is similar to that obtained by classical kinetic theory of gases proposed by Chapman and Cowling’s (1961), which can be written as follows:

where

and

Analyzing Equations (49) and (50), the difference between the granular and the thermal temperature can be demonstrated. In Equation (50), the Boltzmann constant k_B serves to assign the scale of the temperature, whereas in Equation (49) g₀ becomes infinite at maximum packing. However, at this point, the granular temperature is expected to approach 0, leaving N₁₂ undefined.

3.9 Mean free path

In the kinetic theory of gases described by Chapman and Cowling (1961), molecules are considered as rigid spheres and exert no intermolecular forces. In addition, the collisions between the molecules are assumed to be completely elastic. Based on this theory, the concepts of transport coefficients such as viscosity and diffusion coefficient can be obtained.

The mean time between successive collisions, called the collision time (τ), is obtained as shown below for dilute flow:

where g₀=1

Hence,

The mean free path of particles can be understood as the distance between successive collisions and can be expressed as the product of the average velocity and the collision time:

where

Substituting Equation (54) into Equation (55), the mean free path becomes independent of θ:

Using the relation

the mean free equation can be defined in terms of diameter and bed porosity as

3.10 Elementary treatment of transport coefficients

3.10.2 Viscosity

To obtain the viscosity of the particulate phase, let the momentum flux be

(67)Q=ρv〈v〉(Momentum/Volume×Average of oscillating velocity)

Then for a constant density ρ, the change in momentum flux is

(68)ΔQ=lρ〈v〉dvdx

The viscosity for fully developed incompressible flow is defined by

(69)Shear=μdvdx

The momentum transport ΔQ equals the force per unit area

(70)ΔQ=Shear

Therefore, the viscosity assumes the form

(71)μ=lρ〈v〉

From Equation (62) it can be seen that

(72)μ=ρD

as in the kinetic theory of gases. Therefore, using Equation (63), a simple formula for the collisional viscosity is

(73)μ=(13π)ρpdpθ

Kinematic viscosity=Mean free path×Fluctuating velocity

For smooth rigid spherical molecules of diameter d_p (Chapman and Cowling 1961)

(74)μl=516dp2(kBmTπ)1/2

where m=ρpπ6dp3

To convert T to granular temperature θ, let kBm=1. Thus,

(75)DILUTE  μs=5π96ρpdpθ1/2

Multiplying and dividing Equation (75) by ε_s and using the definition of bulk density (ρ_p.ε_s), the following physical interpretation of viscosity can be obtained:

(76)μs=5π96⋅(ρpεs)⋅(dpεs)⋅θ1/2

Viscosity=Constant×Bulk density×Mean free path×Oscillation velocity

For critical flow experiments (Gidaspow 1994), it can be found that θ≅1 m/s. Considering critical flow, for 100 μm particles with 1500 kg/m³ specific density, the viscosity estimated by Equation (76) results in 1.384×10⁻³ kg/m s≅15 cP (15 centipoises). Such a value is 15 times that of water at 20°C temperature. It is close to the measurements (Gidaspow and Huilin 1996). The collisional viscosity includes the effect of particle concentration.

Figure 8 shows the measured viscosity in the Illinois Institute of Technology (IIT) riser for transport of FCC particles. In the figure the dilute viscosity is that given by Equation (75). This dilute viscosity expression as well as the kinetic and the collisional contributions are those derived in Chapman and Cowling’s (1961) book for molecules. The dilute expression for viscosity agrees with that derived here using the elementary kinetic theory, except for the constant, as is evident by comparing Equations (73) and (75). The derivation of the collisional and the kinetic terms are given in Gidaspow’s (1994) book. In Figure 8 the circles are the measurements obtained using the CCD camera method, where the viscosity was obtained by substituting the measured granular temperature and the solids volume fractions into the equations for viscosity obtained from kinetic theory. Figure 8 shows that these viscosities agree with the other measurements obtained earlier by Miller and Gidaspow (1992) where the triangles are for data obtained using the shear pressure drop method, and the squares are for the data using Brookfield viscometer (Miller and Gidaspow 1992).

Figure 8:

FCC powder viscosity measured in three different ways.

4.1 Experimental

To test the validity and the accuracy of the kinetic theory model, a two-story riser was built at IIT with a splash plate on top of the riser to obtain symmetry (Gidaspow and Jiradilok 2009). Figure 9 shows the IIT CFB with splash plate and measurement equipment. The γ-ray source was used to measure the particle concentration. The particle velocities were measured using kinetic theory based particle image velocity (PIV).

Figure 9:

IIT circulating fluidized bed with splash plate.

4.2 Kinetic theory based PIV

Figure 10 illustrates our improved PIV method of obtaining instantaneous velocities for a binary mixture of glass beads. This technique was recently described fully for flow of 530 μm glass beads by Tartan and Gidaspow (2004). In the CFB regime, a curtain of solids at the wall of the pipe restricts the use of laser-Doppler velocity meters for obtaining velocities in the core of the riser, vertical pipe. Hence, a probe, shown in Figure 10, was used. Figure 11 shows the typical steak images captured by CCD camera. The velocity is the length of the streak divided by exposure time. The order of the colors on the rotating transparency establishes the direction. The study of Tartan and Gidaspow (2004) was generalized to a mixture of two sizes of particles. Large particles form thicker streaks than the small particles. To obtain radial profiles, a probe was inserted into the riser. The size of the probe was varied to establish an optimum balance between its hydrodynamic interference and sufficient number of streaks, 10–30, in a picture to obtain meaningful statistics of velocity averages and their variances. Figure 12 shows typical instantaneous axial and radial velocities measured by particle image velocity meter in the riser.

Figure 10:

Particle image velocity measurements system with probe. Reprinted from Tartan and Gidaspow (2004) with permission from ©Wiley.

Figure 11:

Schematically typical streak images captured by CCD camera.

Figure 12:

Typical axial (A) and radial (B) particle velocities. Reprinted from Tartan and Gidaspow (2004) with permission from ©Wiley.

The hydrodynamic velocity v was calculated from measurement of the instantaneous velocity c as follows:

The kinetic stresses are calculated as follows:

where n is the total number of streaks in each frame and C_i=c_i−v_i. We see that the particle stresses, 〈CzCz¯〉, in the direction of flow are much larger than the tangential and the radial stresses, similar to turbulent flow of gases in a pipe (Schlichting 1960, Kim et al. 1987), but are an order of magnitude larger. The orders of magnitude of larger particle weight for fluidization are at expense of an order of magnitude larger pressure drop.

The particle Reynolds stresses are calculated from hydrodynamic velocity v as follows:

where v_i was calculated from Equation (86)

The laminar granular temperature is the average of normal stresses shown below.

4.3 Core-annular flow regime explanation

Recently, Benyahia et al. (2007) studied the ability of multiphase continuum models to predict the core-annulus flow. Figure 13 shows that the time-averaged particle velocity was parabolic and that the particle concentration was uniform in the center of the 7.6 cm tube and high at the wall. Such a concentration distribution is known as the core-annular flow.

Figure 13:

Developed time average particulate phase axial velocity for 530 μm glass beads in the IIT two-story riser (U_g=4.9 m/s; W_s=14.2 kg/m²·s). Reprinted from Gidaspow et al. (2004) with permission from ©Elsevier.

The granular temperate energy balance is similar to the energy balance (22) discussed earlier and is shown below:

For steady state and fully developed flow, the production of granular temperature equals the conduction of granular energy and inelastic dissipation:

The production of granular temperature due to oscillation of particles reduces itself to μ(∂v∂r)2. In transport phenomena texts, this term corresponds to production of heat due to viscous dissipation. In a suspension the particle collisions can be assumed to be elastic. During collisions, the fluid between the particles has to be pushed out, requiring a large force. Therefore, the dissipation of granular energy (γ) is 0.

In cylindrical coordinates, the balance of granular energy then becomes

The measured velocity distribution in our riser as shown in Figure 13 is

Similar parabolic distributions have been reported by Berruti et al. (1995).

Substituting Equation (93) into Equation (92), the balance of granular temperature can be expressed by

Integrating Equation (94) with zero wall granular temperature, the solution of the above equation is as follows:

As reported by Gidaspow (1994), Figure 14 shows that measured granular temperatures can be approximated by the fourth order parabolic equation obtained by solving the granular temperature derived above.

Figure 14:

A comparison of measured granular temperature in the IIT two-story riser to the analytical solution. Reprinted from Gidaspow et al. (2004) with permission from ©Elsevier.

For dilute flow, the granular pressure shown in Figure 5 can be approximated by the kinetic term only:

This assumption corresponds to the use of ideal gas law for molecules, where the particles are far apart from each other.

In developed flow, the radial variation of this pressure is approximately 0. This approximation allows us to obtain the very simple expression for the particle volume fraction distribution shown below:

where μ_s is the particle viscosity, κ the granular conductivity and v̅² is the average particle velocity.

The above Equation (97) explains the core-annular particle distribution in Figure 13. This expression is not valid at the tube wall. The kinetic theory shows that the core annular regime is independent of the pipe radius, not believed to be so by the oil industry, until they finally made the measurements (Mohammad-Reza 2002).

With a few simplifications (Gidaspow and Chandra 2014), this equation can give the ratio of the number of particles per unit volume (n) to its inlet into the system.

where n=6εsπdp3;

These simplifications are as follows: (i) from kinetic theory, for dilute flow μsκs≅415 and (ii) the inlet granular temperature can be approximated by θinlet=μsκsv¯2 since μsκs≅1.

Equation (98) describes the platelets distribution for blood flow in a fully developed regime (Gidaspow and Chandra 2014).

4.4 Turbulent granular temperature

There are two granular temperatures. A laminar granular temperature that is computed using the granular temperature equation in the CFD codes, such as FLUENT (Syamlal et al. 1993), MFIX (Syamlal et al. 1993) or the IIT code (Gidaspow and Jiradilok 2009) shown in Table 2, and a turbulent granular temperature computed from the normal Reynolds stresses per unit bulk density. For dilute riser flow, the granular kinetic theory agrees well with CFB experiments, as shown by Tartan and Gidaspow (2004).

Unfortunately, for the commercially useful dense flow, the turbulent granular temperatures exceed the laminar granular temperatures, as shown in Table 2. In Table 2 the values of solids volume fractions and granular temperatures for flow of Geldart D particles are from the paper of Kashyap et al. (2011) done for UOP for high solids flux in the IIT riser. The flow was in a solids slugging regime. The computations were done using the standard drag model, as described in Gidaspow’s 1994 book. The values for Geldart A particles are for the riser of Wei et al. (1998), with computations done using a correction for the drag, derived using the energy minimization principle invented by Li and Kwauk (1994).

Figure 15 shows the dimensionless turbulent granular temperature in the dense bubbling bed, called “bubble-like” in Jung et al. (2005). It is represented by solid circles and is almost an order of magnitude larger than the laminar or particle granular temperature. For the dilute risers, the turbulent granular temperatures for both 156 and 530 μm particles were smaller than the laminar granular temperatures which agree with the theoretical analytical solution for the granular temperature equation for elastic particles shown in Figure 14.

Figure 15:

Measured dimensionless laminar and turbulent granular temperatures in the IIT two-story riser in dilute flow and in rectangular bubbling beds. Reprinted from Gidaspow et al. (2004) with permission from ©Elsevier.

4.5 Flow regime computation

An excellent review of flow regimes before wide use of CFD was given by Berruti et al. (1995). One-dimensional two-phase models of three decades ago required a specification of measured flow regimes (Lyczkowski et al. 1982). The two- and three-dimensional models used today can successfully compute these flow regimes. Tables 3–5 summarize our kinetic theory model.

Figure 16 shows batch fluidization flow regimes for nanoparticles and Geldart A (aerated), B (bubbling) and C (cohesive) particles. The batch fluidization flow of 10 nm Tullanox nanoparticles occurs without bubbles due to the formation of clusters (Jung and Gidaspow 2002). Geldart C particles are small cohesive particles, which fluidize with the formation of small bubbles. Group A particles form aerated beds with small bubbles, whereas Group B particles fluidize with the formation of large bubbles. Details can be found in Gelderbloom et al. (2003). As the gas velociy is increased, the particles are blown out of the bed.

Figure 16:

A summary of basic batch fluidization flow regimes computed at Illinois Institute of Technology. Reprinted from Gidaspow et al. (2004) with permission from ©Elsevier.

To obtain continuos particle flow, the particles are fed into the bed through one or more jets. Figure 17 summarizes our computed fluidization flow regimes. Turbulent fluidization is the name given to the flow regime in which there exists a dense phase at the bottom of the bed and a dilute particle phase on the top of the bed. The volume fractions of the particles in the bottom and top sections of the bed were estimated by Matsen (2000) using the drift flux model, knowing the gas and particles flow rates. At IIT we had seen a sharp interphase between the dilute and dense portions of the bed. Our code describes the axial experimental measurements of solid volume fractions of Wei et al. (1998). In the fast fluidization regime, large clusters are formed, which descend near the wall. Dense suspension flow is formed at high gas velocities and high solid fluxes. In this regime there is a core-annular flow. For large particles (d_p=1093 μm) and very high velocities, slugging fluidization was observed and computed. In a reactor, high velocities are needed to obtain high production rates. But if the velocity is too high, we will be in the pneumatic transport flow regime with a low catalyst concentration.

Figure 17:

A summary of continuous basic fluidization flow regimes computed at Illinois Institute of Technology. Reprinted from Gidaspow et al. (2004) with permission from ©Elsevier.

4.6 Heat transfer

To design better fluidized bed reactors for new processes, such as for CO₂ capture from flue gases or for the production of pure silicon for solar collectors, wall to bed heat transfer must be improved. The values of the wall to bed heat transfer coefficients limit the size of the fluidized bed reactors. Since the temperatures inside such reactors are nearly constant, the wall to bed heat transfer coefficients are simply the thermal conductivity divided by the small boundary layer thickness. To obtain a high thermal conductivity, the turbulent granular temperature must be as high as possible. Since the turbulent heat flux=ρkcpkεkT′v′¯, we must also have the solids volume fraction as high as possible. Our kinetic theory based CFD code (Gidaspow and Jiradilok 2009, Chaiwang et al. 2014) computes the thermal temperature and its standard deviations. The standard deviation is the highest near the walls, in the thermal boundary layer, in agreement with measurements done in the IIT CFB. The average standard deviations of the temperature and velocities are 0.75° and 0.15 m/s, respectively. Then the heat transfer coefficient obtained by this method and from an overall energy balance is 0.39kwm2−K. If we increase the standard deviation of velocity from 0.15 m/s to 1 m/s, the heat transfer coefficient becomes 2.6, still well below boiling heat transfer coefficients but much larger than the usual CFB heat transfer coefficients. This example demonstrates the potential of CFD to improve fluidized bed reactors.

4.7 Mass transfer

For half a century it was known that the mass transfer coefficients in fluidized beds when expressed in conventional dimensional way as the Sherwood number, where Sherwood number=mass transfer coefficient×particle diameter/gas diffusivity, are orders of magnitude below the theoretical value of 2 for diffusion to a sphere in stagnant fluid. Typical 1970 data obtained in C. Y. Wen’s laboratory in West Virginia (Kato et al. 1970) and our CFD calculation is shown below in Figure 18.

Figure 18:

Sherwood number as a function of riser height.

In Figure 18 the behavior of the Sherwood number is similar to that of the diffusion in a channel in fully developed laminar flow, called the Graetz problem, except for the much lower Sherwood number. There the Sherwood number is high in the diffusion boundary layer and reaches a constant value of about 2 for large heights. For the fluidized bed riser in Figure 18, our explanation (Chalermsinsuwan et al. 2008a,b, Kashyap and Gidaspow 2012a,b) is the formation of clusters, as shown for the fast fluidization regime in Figure 17.

If we define the Sherwood number based on the cluster size rather than particle diameter, the low Sherwood number in Figure 18 becomes close to the theoretical value of two, since the ratio of cluster size to particle diameter can be as much as 1000. This theory suggests that we reduce the cluster size to eliminate potential mass transfer limitation which may require much taller reactors than those computed from reaction rate data.