2. Theoretical background
2.3 Crystallization in fluidized bed
2.3.4 Force balance model for evaluation of particle position
Ch. 2 Theoretical background
Theoretical background Ch. 2
suspension, and applications of these equations to the prediction of system behavior in steady state. A starting point for the examination of the mechanism of the fluidization process, which involves the suspension of a large number of solid particles in an upward flowing fluid, is the simple case of the single particle with fixed diameter. The fluidization process of the particle is essentially based on force balance considerations. Herby the forces acting on a single particle are depicted in Fig. 2.15
Fig. 2.15 Single particle in a suspension with important forces acting on it.
As seen from figure 2.15, important forces acting on a fluidized particle with a diameter L are the fluid-particle interaction force fp and the particle effective force fe. Under equilibrium conditions, they are equal and the particle position can be quantified:
fp + fe = 0
The fluid-particle interaction force fp can be represented as the buoyancy force, fb, and the total drag force, fd, in the following form:
fp = fb + fd =πL3
6 gρf+ 3πufLµf Fluid flow
(superficial velocity, uf)
f
pf
e(2.20)
(2.21) L
particle segregation velocity, up
Ch. 2 Theoretical background
where L is the particle diameter, g is the standard Earth gravity, uf represents fluid superficial velocity; ρf and µf are fluid density and fluid viscosity, respectively.
The description of the particle effective force, fe, includes the gravity force acting on the particle and the buoyancy force, fb:
fe = fg+ fb = −πL3
6 (ρp − ρf)g
From both equations it can be seen that in equilibrium conditions the drag force equals the gravity force but with opposite sign, i.e. fd = - fg.
If a low fluid flow rate is considered around a sphere, where the fluid streamlines follow its contours (the so called creeping flow regime, or laminar flow), then the empirical relation for the drag force can be expressed dimensionless by introducing the drag coefficient, CD, which is [Gibilaro2001]:
CD = fd
(ρfuf2/2)(πL2/4)= 24µf
ufρfL = 24 Rep
where Rep is the Reynolds number. It is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions [Gibilaro2001].
At equilibrium, the drag force can be rewritten in order to include the particle segregation velocity, up:
CD =4 3
gL up2
(ρp − ρf) ρf
From equations 2.23 and 2.24 can be seen that CD is a function of Rep and hence of up. So equation 2.24 can be solved for up, which for the laminar flow regime has the following form:
(2.22)
(2.23)
(2.24)
(2.25)
Theoretical background Ch. 2
up =(ρp − ρf)gL2 18µf
The above expression for the segregation (or terminal) velocity of a single particle can be expressed in dimensionless form, thereby introducing the Archimedes number, Ar, which can be used in further characterization of the fluidized state:
Ar =ρf(ρp − ρf)gL3 µf2
Consider a defined volume with height H and a cross-sectional area A, representing a fluidized bed of uniform particles. In order to consider the forces acting, one must take into account the total energy content of the fluid entering and leaving, as well as the dissipation of this energy brought by the individual particles. Thus, the hydrodynamics of such fluid includes the so called unrecoverable pressure loss due to particle-fluid frictional iterations, which rises with the bed height. Moreover, the fluid superficial velocity, uf, depends on the fraction of the bed cross-section, A, free from particles, which is the void fraction, ε, available for flow. Combining all these factors and including tortuosity and the friction factor, for the unrecoverable pressure loss can be written [Gibilaro2001]:
∆P = (18
Rep+ 0.33)Hρfuf2
L (1 − ε)ε−4.8
where, the power value of –4.8 will be explained later. This convenient relation between the particle drag and the unrecoverable pressure loss is a revised version of the known Ergun equation [Ergun1949]. It is applicable over the full expansion range, 1 > ε ≥ 0.4. In contrast, the standard Ergun equation is applicable only for normal packed beds with ε ≈ 0.4 and deviates with increase of the void fraction.
Richardson and Zaki have done an extensive experimental investigation of the fluidized bed behavior [Richardson1954]. Their large amount of observations can be described by the following empirical equation:
(2.26)
(2.27)
Ch. 2 Theoretical background
The parameter nRZ was found to correlate with the particle Reynolds number. A more convenient relation enables nRZ to be evaluated from the Archimedes number:
4.8 − nRZ
nRZ− 2.4= 0.043Ar0.57
nRZ =4.8 + 0.1032Ar0.57 1 + 0.043Ar0.57
Under viscous conditions (Ar << 1), the value for nRZ is 4.8 and under inertial flow conditions (Ar >> 1) nRZ = 2.4, thus the border conditions for nRZ are set.
Concluding this section, the primary forces acting on a fluidized particle can be identified as the gravity, buoyancy and the drag forces. Thus, for single particle suspension an effective weight, We, of a particle can be defined as the net effect of gravity and buoyancy in correlation with particle diameter:
We = fg+ fb = −πL3
6 (ρp − ρf)gε
This relation shows the effective weight of an average fluidized particle under equilibrium conditions is proportional to the void fraction.
For the complete drag force adopted for all flow regimes can be written:
Fd =πL3
6 (ρp − ρf)g (uf up)
n4.8RZ
ε−3.8
The consistency of the equations for We and Fd can be confirmed at equilibrium conditions (where both are equal), thus, yielding the Richardson-Zaki relation:
We+ Fd = 0
−πL3
6 (ρp − ρf)gε +πL3
6 (ρp− ρf)g (uf up)
n4.8RZ
ε−3.8 = 0 After truncating the equal terms, equation 2.34 can be reduced to:
(2.31)
(2.32)
(2.33)
(2.34) (2.29)
(2.30)
Theoretical background Ch. 2
ε4.8 = (uf up)
n4.8RZ
From equation 2.35 it is easily seen, that solving for uf after unifying the power values gives the Richarson-Zaki equation 2.28.
The application of equation 2.28 is only valid at steady state operation of the fluidization process, thus allowing the estimation of a position of particles with certain diameters, depending on the experimental process parameters and crystallizer geometry.
In order to take into account the difference between the assumed spherical particles and real crystal shapes, a sphericity parameter, Ѱ, can be introduced. It was defined by Wadell as the ratio of the surface area of a sphere, which has the same volume as the particle, to the surface area of the particle [Wadell1935]:
Ѱ = 3√π(6Vp)2 Ap
where Vp and Ap are the particle volume and particle area respectively. Hereby, the calculated particle diameter, L, has to be multiplied by the sphericity parameter, Ѱ, to achieve the predicted mean particle size, d50, to be compared with corresponding experimental values of component specific measured CSD i.e. d50= L. Ψ.
From equation 2.36 can be seen, that Ѱ is dimensionless and can have values between 0 and 1, meaning that particle forms similar to a sphere will have sphericity parameter closer to 1 and particles with long, needlelike forms will have a sphericity parameter closer to 0.
This simplified balance force model will be applied in chapter 4.4.1.4 to predict the steady state position of crystals in the fluidized bed crystallizer and to compare calculated data with experimental measurements. Since the model comprises only the segregation of a single particle in the fluidized bed, an extension to the model is introduced in chapter 4.1, where the crystallization dynamics and size distributions are is considered in addition.
(2.35)
(2.36)
Ch. 2 Theoretical background
Main goal of this thesis is to extend essentially the knowledge for the crystallization process in fluidized bed by conducting systematic experimental work in a specially designed experimental setup for selected challenging separation problems, which will be described in the next chapter.