Many chemical and biochemical reactions are heterogeneous in nature, involving all three phases of solid, liquid and gas. Filtration, sedimentation and dispersion of solids in a liquid involve two phases. One phase is fluid, and the other phase is solid particles. The solid particles may be stationary as in fixed beds, or it may be moving as in fluidised beds or mechanically stirred vessels. Gas–liquid contactors involve the gas phase in the form of bubbles dispersed in liquid or liquid dispersed as droplets in gases. Many processes are carried out in which all three phases (solid, liquid and gas) are present (e.g. slurry bubble column, three-phase fluidised beds and slurry reactors). In bioprocesses, microorganisms or enzymes supported on solids may be present as the solid phase in bioreactors. Some multiphase contactors are presented in Figure 5.1.
The laws of conservations presented in Chapter 4 are applicable in a single phase only. This does not account for a mass transfer across a boundary between two phases, since the mechanisms of mass transfer across boundaries are different from those of convection or diffusion. Mass transfer across the interface is governed by an equilibrium relationship.
The equation of continuity for individual chemical species is applicable in each individual phase. For this purpose, each dispersed unit should be considered an individual phase since each of these has an interface with a continuous phase.
Phenomena such as heat transfer between heat transfer surface and the dis-persed medium involve a wall of the vessel or an outer surface of the tubes that is used to supply or remove heat. The interface does not offer any resis-tance to heat transfer. Mass transfer in electrochemical systems also involves a surface, either of the vessel or of the immersed body.
The movement of individual phases (e.g. settling of particles and bubble velocity in bubble columns) is governed by the forces of buoyancy, gravity and drag force acting on the individual dispersed phase.
Various applications of laws of conservation (discussed in Chapter 4) and simple laws (discussed in Chapter 3) to model processes in multiphase systems in the absence of chemical reactions are presented in this chapter.
5.1 Consideration of a Continuous-Phase Axial Solid Profile in a Slurry Bubble Column
Let us take a case in which the region of interest is away from the transfer surface, which may be the interface of the phases or an external surface such as a wall or immersed objects. The multiphase system may be considered in such cases as a continuous phase.
A number of reactions require the solids to be in suspension form. If the bubbling of gas is used to keep the solids in suspension, the reactor is known as a ‘slurry bubble column’ (SBC). These columns find uses in F-T synthe-sis, coal liquefaction, hydrogenation of oil, wastewater treatment and many other chemical and biochemical reactions.
The SBC consists of a vertical column filled with slurry, in which the gas is bubbled through a suitably designed distributor into the column. It may be operated in batch or in continuous modes. In the fluidised beds, the solids are uniformly distributed in the column. But in SBC, the solids are not uniformly distributed. An axial distribution of solids is observed in SBC. It has been modelled by a one-dimensional sedimentation–dispersion model.
Gas is introduced through a gas distributor placed at the bottom of the SBC.
As the gas bubbles move up, the slurry also moves up with it. The solids in the slurry remain suspended due to the local liquid velocities. The frequently changing velocity of the fluid resulting in back-mixing may be modelled, considering the solid dispersion coefficient analogous to the diffusion and
Liquid
Gas (a) Fixed bed
Liquid Liquid
(d) Emulsion column Gas
(c) Bubble column Liquid
or gas (b) Fluidised bed
FIGURE 5.1
Various multi-phase systems: (a) fixed bed, (b) fluidised bed, (c) slurry bubble column and (d) emulsion column.
131 Multiphase Systems without Reaction
eddy diffusion. Since the particles tend to settle, another term takes this into account. The model based on this concept is known as the sedimentation–
dispersion model. It is based on the assumption of ‘continuum’ (i.e. the slurry is treated as a fluid whose properties may be estimated from the properties of solid and fluid). This assumption is useful in cases in which the region near the interface is not important.
Let us assume that complete radial mixing exists in the column (i.e. the concentration of solids is the same at all radial positions at a given height).
This assumption results in a one-dimensional (1D) model. Writing material balance around a differential strip at a position z from the bottom gives (Figure 5.2). of solids due to solid dispersion
rate of solids due to settling
rate of solids due to slurry flow
= +
+
(5.1)
With appropriate signs, substituting the various terms as shown in Figure 5.2, the following equation was obtained (Suganama and Yamanishi 1966; Farkas and Leblond 1969; Yamanaka et al. 1970; Smith and Ruether 1985):
t C
A sedimentation–dispersion model for an axial solids concentration profile in a slurry bubble column.
Assuming gas holdup, εg; solid dispersion coefficient, Es; and hindered settling velocity, Uc, to be independent of z, Cova (1966) obtained the follow-ing equation to describe the solid distribution in an SBC:
C
Equation 5.3 is applicable for a co-current operation. For counter-current operations, the superficial velocity of the slurry, Usl, may be used. Smith and Ruether (1985) represented settling velocity in terms of slip velocity as Us εl. In batch mode operation (Usl = 0), the solids concentration does not change with time. Therefore, Equation 5.3 reduces to
E C
The general solution of Equation 5.4 is
Cs=A Be+ (−U z Ec /s) (5.5)
The following are the boundary conditions:
At z = 0: Cs=Cs0
Integration constants, A and B, were obtained and substituted in Equation 5.5.
C
The following boundary equation can also be used (Farkas and Leblond 1969):
At z = 0: Cs =Cs0
At z→∞: Cs→ ∞
The solution is the same as given in Equation 5.6, although the above boundary condition is not valid as the column extends up to z = L only and there is no slurry beyond that.
Equation 5.6 is exponential in nature and represents the axial distribu-tion of solids in an SBC when operated in batch mode. The concentradistribu-tion is maximal at the bottom and decreases as the axial distance from the bottom increases. It has the least value at the top of the column.
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Equation 5.6 can be expressed as the concentration profile in terms of the ratio of the solids concentration to the average solids concentration, Cs, as
Although two constants, Uc and Es, appear in the model, in fact there is only one constant, the ratio Uc:Es (Imafuku et al. 1968; Murray and Fan 1989; Zhang 2002). This ratio can be determined by plotting ln(Cs) versus z.
The slope of the line is Uc–Es, and the intercept on the y axis is Cs0.
Several investigators used the free settling velocity of solids as Uc (Cova 1966; Imafuku et al. 1968; Farkas and Leblond 1969; Kojima and Asano 1981;
Kojima et al. 1986). Correlations for Uc are also available. Cova (1966) has taken the solid dispersion coefficient equal to the liquid dispersion coefficient (i.e. the values for Es for particles were taken as equal to those of liquid, El).
A few correlations for Uc and Es are presented in Table 5.1. The Peclet num-ber, Pep1, was based on Ugc, the critical velocity (i.e. the minimum gas veloc-ity to suspend the solids). Fr is the Froude number.
5.2 Single Interface: The Wetted Wall Column
A wetted wall column is used to study the mass transfer coefficient at the gas–liquid interface due to a well-defined interfacial area and velocity pro-file in the liquid. The liquid falls down over the column wall under the influence of gravity. The gas flows through the core of the tube. There is only one interface between the gas and liquid (Figure 5.3). Other equipment TABLE 5.1
A Few Correlations for Hindered Settling Velocity, Uc, and Dispersion Coefficient for the Solids, Es
Investigators Uc Es
Imafuku et al.
using the concept of falling film (e.g. as a falling film evaporator or falling film absorbers) behaves similar to a wetted wall column. The liquid may flow inside or outside of the tube. In all of this equipment, the interfacial area, a, is equal to πD. The transfer rates can be estimated from experimen-tal data or by using an appropriate model due to the well-defined interfa-cial area.
For incompressible Newtonian fluid, the equation of change for momen-tum and equation of continuity give
d v
dx22z = ρg (5.8)
The boundary conditions are as follows:
At x = δ: νz = 0 At x = 0: dv
dxz =0
The solution of Equation 5.8 gives the velocity profile
v g x
2 1
z
2 2
=ρ δ
− δ (5.9)
Assuming that the liquid film is very thin in comparison to the column diameter, D, the average fluid velocity, vz, can be obtained as
v
v dx dy dx dy
g
z 3
z D
D0 0
0 0
∫
2∫
∫
=
∫
= ρ δδ π
δ
π (5.10)
Wall
Gas Liquid
x z
x z
δ
FIGURE 5.3
Mass transfer in a wetted wet column.
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And liquid film thickness, δ, is given by Q
where Q is the volumetric flow rate. The average time of contact, tc , can be written as
Equations 5.8 through 5.12 are valid in all vertical column walls, irrespec-tive of the geometry, in light of the assumed thin film thickness. For mass transfer, the equation of continuity for chemical species, after substituting the velocity profile from Equation 5.9 into it, gives us
g x C
The boundary conditions are as follows:
At z = 0: CA = 0
The solution to Equation 5.13 can be obtained numerically. Many models are based on the assumption of short tubes, which means that the length of the tube is so short that the concentration of A does not penetrate the liquid film even at the lower end. It is mathematically equivalent to impose the con-dition that x 1
δ << . For short tubes, Equation 5.13 is reduced to
g C
The solution of Equation 5.14 is (Bird et al. 1960) C
The total mass transfer rate of A at the interface is
∫
In cases of short contact time, the region in which the solution is required is near the surface, where the velocity is almost constant. The time of contact can be written as t z
vz
c = , and hence t z vz
∂ =∂ . Equation 5.14 is converted to Equation 4.69. The solution for constant concentration at the wall (boundary condition of the first kind) is given by Equation 4.70.
If the surface concentration is not constant and the gas phase also offers resistance to mass transfer, then the mass transfer coefficient in the gas should also be taken into account. Zhi and Kai (2009) studied the absorption of CO2 by monoethanolamine (MEA) in a wetted wall column. Unsteady-state diffu-sion of CO2 into the liquid film, accompanied by a pseudo-first-order revers-ible chemical reaction, was assumed. Equation 4.69 is modified as
C
The boundary conditions are as follows:
CA = Ceq at t = 0 for all z CA = CA,0 at z = 0 for all t CA = CA0 at z = 0 for t→∞
The solution of Equation 5.17 is
C C C C x k
The mass transfer rate is given by
N D dC
All of these results look elegant, but in practice it is very difficult to main-tain a well-defined laminar flow at the wall. Often, a wavy surface is encoun-tered which not only increases the interfacial area but also enhances the mass transfer rate due to induced flow near the interface.
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5.3 Stationary Dispersed-Phase Systems (Gas–Solid Systems) Packed beds are used as fluid solid contactors and are used as adsorbers, absorbers, distillation columns etc. The solid particles, which may be inert, are packed in a column. The fluid, which may be gas or liquid, enters from one end of the column and leaves from the other end. The packed beds are also used as reactors, where the solids act either as catalysts or as a catalyst support. In all of these applications, the particles remain stationary, and it is the fluid phase which is moving. It is not only a formidable task to model the performance of a packed bed, considering the detailed flow around each particle, but also a not very fruitful exercise because the particles are ran-domly packed in the bed.
The processes in packed beds, and in general for all multiphase systems, can be divided into two major classes based on the region of interest. In many cases, the processes around the particles are not very important and may be neglected in simple models. But if catalysts are actively involved, then the region of interest is in the neighbourhood of the particles. This classifica-tion is valid only for simple models. In rigorous models, processes in both regions are considered. Heat transfer between the wall and packed bed is now being considered to illustrate some of the modelling concepts.
5.3.1 Heat Transfer in Packed Beds
Let us understand the modelling problems in packed beds. One can recall the complexities presented in Chapter 1. Wall-to-bed heat transfer takes place through two mechanisms. The heat transfer is mainly between the wall and the fluid flowing through the interstitial space near the wall. The mechanism of heat transfer is convection. This term is called ‘fluid convection’. The fluid conduction may be very small compared to the convective transport, except at a low Reynolds number. The particles, which are adjacent to the wall and are in its contact, are heated (or cooled) by the fluid present around it. Due to the thermal conductivity of the particle being higher than that of the gas, the heat transfer between the particle and gas takes place at a higher rate than in the fluid convection. Let us call this ‘particle convection’ (Figure 5.4a).
Heat transfer models may be classified as ‘one-temperature models’ and
‘two-temperature models’ (Figure 5.4b). The one-temperature models are frequently called ‘pseudo-homogeneous models’. In these models, the solids and fluids are assumed to be a continuum phase. The laws of conservation are written as if only one phase is present. The steady-state model for heat transfer considered steady-state conduction to a homogeneous phase having properties estimated from the properties of solids, fluids and fluid velocity.
The simplest model for heat transfer in packed beds assumes that the heat transfer by conduction is negligible as compared to the convective heat
transfer in the axial direction. This assumption is valid only at high fluid velocity. The bed temperature at any cross-section is constant except in very thin regions adjacent to the wall. This condition also is possible at high fluid velocity. The laws of conservation of thermal energy give
U C dT
dz h T T
g g pgρ =
(
W−)
(5.20)Equation 5.20 is an ordinary differential equation which can be solved analytically as well as numerically by using the fourth-order Runga–Kutta method. For constant wall temperature and a constant heat transfer coeffi-cient, the solution of Equation 5.20 is
T T
The solution of Equation 5.20 for constant heat flux, q, is
T T h T T
The wall temperature, TW, is a function of the axial position, z.
The pseudo-homogeneous models considering radial thermal diffusivity for gas flowing in packed beds may be written as
ρ ∂
Mechanisms of heat transfer in fluidised beds: (a) fluid convection and particle convection and (b) one-temperature and two-temperature models.
139 Multiphase Systems without Reaction
If axial thermal diffusivity is also considered, then the model equations will be
The solution of Equation 5.24 is T T
If a gas–liquid mixture is flowing through the packed bed, then the term on the left-hand side is modified accordingly:
U C U C T
The boundary conditions are the following:
At the centre of the column (i.e. at r = 0): T r 0
∂
∂ = Far away from the entrance (i.e. as z→∞): T → TW
One more boundary condition is required. For constant wall temperature, the boundary conditions of the first kind will be used:
At r = 0, T = TW
The solution of Equation 5.26 is given as (Moreira et al. 2006)
T T
If the boundary condition of the third type is used:
At r = 0: k T
r h T T
r
(
r R= W)
− ∂
∂ = −
T T
These models are one-temperature models. Since the fluid and solids are considered one phase, no heat transfer between these two phases is written.
In the two-temperature models, fluid and solids are considered to be at different temperatures. The heat transfer between the fluid and solids can also be taken into account. This model is applicable at a low fluid velocity at which the rate of heat transfer between the fluid and solid is small, thus caus-ing a temperature difference between the two phases. The fluids and solids are considered separate phases.
A two-temperature model for 2D packed beds consisting of spheres was proposed by Laguerre et al. (2006). The position of each particle was specified by subscripts i and j in the horizontal and vertical directions, respectively.
The temperatures of air and particle are Ta and Tp(i,j), respectively. Equating the accumulation of heat in the particle as equal to the heat transferred to the air and the surrounding sphere, the temperature of the particle is described by the following equation:
C d dT
Only four spheres were considered for the 2D cubic arrangement. For the variables, the heat transfer coefficient between the air and jth particle, hj, and the conduction between two spheres, C, the following correlations were used:
The temperature of the air is given as
C T
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The unsteady-state term is the ‘thermal inertia of air’. It was neglected in comparison to the thermal inertia of the solids. For spherical particles, the surface per unit volume, S
p d
p
= π. These partial differential equations were solved by discretising with Δz = Δr = dp. After discretising, the following equation is obtained: Equations 5.29 through 5.32 were solved numerically.
The assumption of continuum results in a simple model providing only average axial variations of temperature. A 2D model describes the radial temperature profile. A two-temperature model considering the temperature difference between the fluid and particles is much more complex, and the use of empirical equations cannot be avoided. The model equations are to be solved numerically after discretising the equations.
5.3.2 Mass Transfer in Packed Beds
The equation of continuity presented in Chapter 4 is based on the assump-tion that the diffusivity or thermal conductivity is the same in all direc-tions. However, many industrial operations are carried under turbulent flow conditions in which the diffusivity or thermal conductivity in axial and radial directions may be different. In cases of mass transfer in a packed bed under turbulent flow conditions, the effective diffusivity is taken to be Deff = DAB + εd. Let us assume that the packed bed behaves as a continuous single fluid (i.e. the assumption of continuum is considered). In reality, the bed is not homogeneous and is known as pseudo-homogeneous. The con-centration profile may be obtained by simplifying the equation of continuity for the species in cylindrical coordinates, if the diffusivity is isotropic, as follows:
Due to different values of eddy diffusivity in radial and axial directions, the effective diffusivity will also be different. The equation of continuity for the species will be
v C
In terms of the dimensionless numbers, Equation 5.34 takes the following form:
Note that if N is the ratio of the column diameter to the particle diameter then DT =Ndp. The following boundary conditions were used by Dixon et al.
(2003):
The analytical solutions for Equation 5.35:
J y
143 Multiphase Systems without Reaction
The heat and mass transfer in packed beds are generally based on the assumption of continuum. This assumption simplifies the equations to an extent that analytical solutions are possible for different types of bound-ary conditions. Since the detained temperature or concentration distribu-tion around the particles is not of interest, therefore, these soludistribu-tions are adequate.
The heat and mass transfer in packed beds are generally based on the assumption of continuum. This assumption simplifies the equations to an extent that analytical solutions are possible for different types of bound-ary conditions. Since the detained temperature or concentration distribu-tion around the particles is not of interest, therefore, these soludistribu-tions are adequate.