Fractional phase holdup

The actual volume of each phase in element AV is that of the total volume of the element, multiplied by the respective fractional phase holdup. Hence considering the direction of solute transfer to occur from the aqueous or feed phase into the organic or solvent phase, the mass balance equations become ... 

Note that the above formulation includes allowance for the fractional phase holdup volumes, hL and ho, the phase flow rates, L and G, the diffusion coefficients Dl and Dq, and the overall mass transfer capacity coefficient Klx a, all to vary with position along the extractor. 

Hydrodynamic parameters that are required for trickle bed design and analysis include bed void fraction, phase holdups (gas, liquid, and solid), wetting efficiency (fraction of catalyst wetted by liquid), volumetric gas-liquid mass-transfer coefficient, liquid-solid mass-transfer coefficient (for the wetted part of the catalyst particle surface), gas-solid... 

Eddy dispersion coefficient Inverse residence time Inverse dispersion time Fractional phase holdup Superficial phase velocity... 

For sound process design, we need values of numerous design parameters such as fractional phase holdups, pressure drop, dispersion coefficients (the extent of axial mixing) of all the compounds, heat and mass transfer coefficients across a variety of fluid-fluid and fluid-solid interfaces depending on the type of multiphase system, type of reactor, and the rate-controlling steps. To clarify the scope of the case studies selected, their salient features are next listed. 

Z. 5-25-Y, large huhhles = AA = 0.42 (NG..) Wi dy > 0.25 cm Dr luterfacial area 6 fig volume dy [E] Use with arithmetic concentration difference, ffg = fractional gas holdup, volume gas/total volume. For large huhhles, k is independent of bubble size aud independent of agitation or liquid velocity. Resistance is entirely in liquid phase for most gas-liquid mass transfer. [79][91] p. 452 [109] p. 119 [114] p. 249... 

Ross (R2) measured liquid-phase holdup and residence-time distribution by a tracer-pulse technique. Experiments were carried out for cocurrent flow in model columns of 2- and 4-in. diameter with air and water as fluid media, as well as in pilot-scale and industrial-scale reactors of 2-in. and 6.5-ft diameters used for the catalytic hydrogenation of petroleum fractions. The columns were packed with commercial cylindrical catalyst pellets of -in. diameter and length. The liquid holdup was from 40 to 50% of total bed volume for nominal liquid velocities from 8 to 200 ft/hr in the model reactors, from 26 to 32% of volume for nominal liquid velocities from 6 to 10.5 ft/hr in the pilot unit, and from 20 to 27 % for nominal liquid velocities from 27.9 to 68.6 ft/hr in the industrial unit. In that work, a few sets of results of residence-time distribution experiments are reported in graphical form, as tracer-response curves. 

An evaluation of the retardation effects of surfactants on the steady velocity of a single drop (or bubble) under the influence of gravity has been made by Levich (L3) and extended recently by Newman (Nl). A further generalization to the domain of flow around an ensemble of many drops or bubbles in the presence of surfactants has been completed most recently by Waslo and Gal-Or (Wl). The terminal velocity of the ensemble is expressed in terms of the dispersed-phase holdup fraction and reduces to Levich s solution for a single particle when approaches zero. The basic theoretical principles governing these retardation effects will be demonstrated here for the case of a single drop or bubble. Thermodynamically, this is a case where coupling effects between the diffusion of surfactants (first-order tensorial transfer) and viscous flow (second-order tensorial transfer) takes place. Subject to the Curie principle, it demonstrates that this retardation effect occurs on a nonisotropic interface. Therefore, it is necessary to express the concentration of surfactants T, as it varies from point to point on the interface, in terms of the coordinates of the interface, i.e.,... 

The dispersed-phase holdup fraction is, for example, responsible for many important interactions. These are indicated by the dashed lines of Fig. 1, which show the main interrelationships that govern the capacity of a given dispersion. Some of these interrelationships, such as the effects of residence-time... 

In each subreactor, the dispersed-phase holdup fraction should be the same as the overall fraction thus... 

Under changing flow conditions, it can be important to include some consideration of the hydrodynamic changes within the column (Fig. 3.53), as manifested by changes in the fractional dispersed phase holdup, h , and the phase flow rates, Ln and G . which, under dynamic conditions, can vary from stage to stage. Such variations can have a considerable effect on the overall dynamic characteristics of an extraction column, since variations in hn also... 

The fractional dispersed phase holdup, h, is normally correlated on the basis of a characteristic velocity equation, which is based on the concept of a slip velocity for the drops, VgUp, which then can be related to the free rise velocity of single drops, using some correctional functional dependence on holdup, f(h). 

P = plug flow, M = mixed flow, eD = fractional dispersed phase holdup, td = residence time of the dispersed phase (Doraiswamy Sharma, 1984). 

Hence, the gas phase conversion for a pseudo-first order reaction in the churn turbulent regime can be calculated if the fractional gas holdups, rise velocities, and interfacial areas for the two bubble classes as well as the physicochemical data are known. 

In the above equations, AG, AL, and As are the gas-phase, liquid-phase and calalyst-surface concentrations of the reacting species, ACi is the average gas-phase concentration at the reactor inlet, Z is the axial distance from the reactor inlet, L is the total length of the reactor, m = H/RgT, where H is the Henry s law constant (cm3 atm g-mol" ), Rg is the universal gas constant, and T is the temperature of the reactor. UG is the mean gas velocity, Us is the mean settling velocity of the particles, t is the time, k is the first-order rate constant, W is the catalyst loading, zc and ZP are the axial dispersion coefficients for the gas and solid phases, respectively. Following the studies of Imafuku et al.19 and Kato et al.,21 the axial dispersion coefficient for the liquid phase was assumed to be the same as that for the solid phase, w is the concentration of the particles and hG the fractional gas holdup. Other parameters have the same meaning as described earlier. 

The above equations assume that the liquid-phase reactant C, the product of the reaction, and the solvent are nonvolatile. The effective interfacial area for mass transfer (nL) and the fractional gas holdup (ii0o) arc independent of the position of the column. The Peclet number takes into account any variations of concentration and velocity in the radial direction. We assume that Peclet numbers for both species A and C in the liquid phase are equal. For constant, 4 , Eq. (4-73) assumes that the gas-phase concentration of species A remains essentially constant throughout the reactor. This assumption is reasonable in many instances. If the gas-phase concentration does vary, a mass balance for species A in the gas phase is needed. If the gas phase is assumed to move in plug flow, a relevant equation would be... 

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