Richardson-Zaki equation

This result follows from the Richardson-Zaki equation. In their original work, Richardson and Zaki (1954) studied batch sedimentation, in particular the settling of coarse solid particles through a liquid in a vertical cylinder with a closed bottom. Richardson and Zaki found that the settling speed uc of the equal-sized particles in the concentrated suspension was related to the terminal settling speed u, of a single particle in a large expanse of liquid by the equation... 

For a particular bubble column, u, = 0.5 m/s and n (in the Richardson-Zaki equation) may be approximated as 2. Determine the value of the void fraction a for each of the following conditions ... 

Particulate fluidization, where the fluidizing medium is usually a liquid, is characterised by a smooth expansion of fhe bed. Liquid-solid fluidized beds are used in continuous crystallisers, as bioreactors in which immobilised enzyme beads are fluidized by the reactant solution and in physical operations such as the washing and preparation of vegetables. The empirical Richardson-Zaki equation (Richardson... 

The Richardson-Zaki equation has been found to agree with experimental data over a wide range of condifions. Equally, if is possible fo use a pressure drop-velocity relationship such as Ergun to determine minimum fluidization velocity, just as for gas-solid fluidizafion. An alternative expression, which has the merit of simplicify, is fhaf of Riba ef al. (1978)... 

If the particle concentration in a suspension is high, then the particles do not sediment independently, but are influenced by the motions of surrounding particles (hindered settling). Hindered settling behaviour can be described by applying a correction factor to the Stokes Law terminal settling velocity (dx/dt from Eq. 2.20) to obtain a hindered settling velocity. Several equations have been advanced to describe the correction [69]. The Richardson-Zaki equation for this is ... 

In liquid-solid systems, and for gas-solid systems of small particles, increasing the fluidising velocity beyond wmf gives rise to a uniform expansion of the bed. The homogeneous expansion of a gas-solid system is generally described using the Richardson-Zaki equation 13... 

The Richardson Zaki equation (Equation 37) is used in fluidisation to describe the homogeneous expansion. It relates the superficial gas velocity to the fluid bed voidage and particle terminal fall velocity ... 

Expansion behavior according to the Richardson-Zaki equation provides reassurance that the particles behave properly. The next factor to determine is dispersion. A pulse of a tracer is passed through the bed and monitored after leaving the expanded bed. The dispersion is calculated by comparing the shape of the elution profile with that of the original plug of liquid. Often acetone or phenol is used as a tracer and is monitored using UV. 

In the preceding analysis, the velocity-hold-up relationship was expressed in terms of the Richardson-Zaki equation. Alternatively, the relationship can be derived from the Ergun (1952) equation. This latter approach was used by Molerus (1993). 

In the case of solid-liquid fluidized beds, the setthng velocity of the particle is less than the terminal settling velocity. The hindered settling velocity is given by the well-known Richardson-Zaki equation,... 

Bed expansion in particulate fluidization may be described by the Richardson-Zaki equation [38]. 

Defining this point as s, gives = [w/( +l)] so that the Richardson-Zaki equation takes the form ... 

For liquid fluidization, the Richardson-Zaki equation as given below can be used to describe the bed expansion ... 

Resin manufacturers often express the fluidization properties of thejr resins in terms of the percentage expension of the bed related to its pseked depth. This expansion cannot be calculatad directly from the above equation since a range of bead sizes is present. Typical data are shown in Fig. 13.3-1, taken from a Rohm and Haas pamphlet. These curves can be correlated by an equation giving the expension as a fouction of the 1.5 power of the flow rale. An equation of almost identical form can be derived from the Richardson-Zaki equation by expressing the expansion in cerma of the void fraction and using appropriate values for n. 

Particulate and Aggregative Fluidization. When the fluid phase is liquid, the difference in the densities of fluid and solid is not very large, and the particle size is small, the bed is fluidized homogeneously with an apparent uniform bed structure as the fluid velocity exceeds the minimum fluidization velocity. The fluid passes through the interstitial spaces between the fluidizing particles without forming solids-free bubbles or voids. This state of fluidization characterizes particulate fluidization. In particulate fluidization, the bed voidage can be related to the superficial fluid velocity by the Richardson-Zaki equation. The particulate fluidization occurs when the Froude number at the minimum fluidization is less than 0.13 [9],... 

The expansion of a fluidized bed of resin is an important variable in determining how much resin will be held in a vessel. This property of a resin is measured very readily in a simple laboratory experiment. For uniform bead size, correlation of data can be done with the Richardson-Zaki equation expressed in the form... 

This relationship, known as the Richardson-Zaki equation, has been modified in order to provide a better correlation of the experimental data [14]. 

The relation shown in eqn (4.4) appears to have been first observed by Lewis el al. (1949), but is now universally known as the Richardson-Zaki equation after the authors of an extensive experimental investigation into its applicability (Richardson and Zaki, 1954a, 1954b Figure 4.2). 

This form is identical to the Richardson-Zaki equation, eqn (4.4). It therefore relates the parameter n in that empirical relation to the ratio of the void fraction and fluid flux exponents in the expression for unrecoverable pressure loss, eqn (4.8) n = —b/a. Note that under both viscous and inertial flow conditions (a=l, = 4.8 and a = 2, =2.4, respectively), the void fraction exponent b assumes the value of —4.8. This unexpected coincidence will now be put to effective use. 

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