Fractional approach to equilibrium

Solutions are provided for external mass-transfer control, intraparticle diffusion control, and mixed resistances for the case of constant Vf and F0 in = FVi out = 0. The results are in terms of the fractional approach to equilibrium F = (ht — hf)/(nT — nf), where hf and are the initial and ultimate solute concentrations in the adsorbent. The solution concentration is related to the amount adsorbed by the material balance - (hi - nf )M,Ay. 

For a finite fluid volume (A90 > 0), the fractional approach to equilibrium is given by ... 

When the sorbent is initially free from solute, Equation 37 can be solved analytically (73) to give the ratio of the mass sorbed at time t to the mass sorbed at equlibrium (i.e., the fractional approach to equilibrium). The mathematical solution depends on the mass fraction ultimately sorbed from the aqueous phase (F), and is most conveniently presented in terms of t, a dimensionless time parameter given by... 

Figure 7. The fractional approach to equilibrium (after 73). Fractional uptake = mass of solute sorbed at equilibrium/mass of solute added to system fractional approach to equilibrium = mass of solute sorbed at time r/mass of solute sorbed at time .

Since the concentration within the particle varies with time, instantaneous mass transfer rates are difficult to measure. Experimental data are frequently presented in terms of the fractional approach to equilibrium ... 

If the fluids are stagnant (i.e., Pe = Pep == 0), the concentration profiles display angular symmetry and the fractional approach to equilibrium is a function only of H, p/, and x or Xp. The corresponding solution for F (K3, PI) is shown in Figs. 3.12-3.14, for a wide range of values of these parameters. Fluid motion always increases F for given Xp, so that these solutions give a lower limit for the fractional approach to equilibrium. 

Fig. 3.12 Variation of fractional approach to equilibrium F with dimensionless time, ip = for a sphere in stagnant surroundings with H =. ...

Fig. 3.15 Variation of fractional approach to equilibrium with dimensionless time for spheres in creeping flow with negligible external resistance.

Fig. 3.19 Variation of fractional approach to equilibrium with time for rigid spheres with negligible internal resistance in creeping flow at high Pe.

Mass transfer rates from drops are obtained by measuring the concentration change in either or both of the phases after passage of one or more drops through a reservoir of the continuous phase. This method yields the average transfer rate over the time of drop rise or fall, but not instantaneous values. For measurements of the resistance external to the drop this is no drawback, because this resistance is nearly constant, but the resistance within the drop frequently varies with time. The fractional approach to equilibrium, F, is calculated from the compositions and is then related to the product of the overall mass transfer coefficient and the surface area ... 

Fig. 7.16 Fractional approach to equilibrium for circulating and oscillating drops in gases. Data of Garner and Lane (G4).

If a fluid particle oscillates violently enough to mix its contents in each oscillation cycle, the average internal resistance is constant if the driving force is based upon the mixed mean concentration within the drop. The fractional approach to equilibrium is then given by Eq. (7-40) or (7-41). 

Consider the gel collapse in more detail. As hydrogen ions diffuse into the gel, they will rapidly react to form neutral carboxylic acid groups. Thus, a nonionic shell of collapsing gel will develop around a still-swollen ionized core. The diffusion of ions occurs freely in this nonionic shell, so that the collapse is limited by the Fickian diffusion of water out of the gel. We have confirmed this Fickian behavior by measuring the collapse of cylindrical gel samples of differing radii and ionic compositions [5]. The data for the fractional approach to equilibrium fall on a single curve against [Dt/R2]1/2, where D is the diffusion coefficient and R is the initial radius of the gel cylinder. 

Fig. 7. Design chart for estimation of average-flow ratio in absorption (4). R7 = LM / GM at gas outlet = GM /LM at gas inlet y2 = mole fraction in outlet gas y1 = mole fraction in inlet gas Ray effective averageLM /GM / = y /y = fractional approach to equilibrium.

Ionic self-diffusion coefficient, mVs [Eqs. (16-73), (16-74)] Fractional approach to equilibrium Volumetric flow rate, mVs Enthalpy, J/mol ... 

For a binary system, under conditions of small mass transfer fluxes, the unsteady-state diffusion equations may be solved to give the fractional approach to equilibrium F defined by (see Clift et al., 1978)... 

When the size of the bubble (or droplet) exceeds a certain limit the dispersed phase may begin to circulate or oscillate. The Kronig-Brink model for circulation within the dispersed phase gives the following expression for the fractional approach to equilibrium... 

The fractional approach to equilibrium in a multicomponent system is given by the n - 1 dimensional matrix analog of Eq. 9.4.2. 

The fractional approach to equilibrium in the small bubbles is given by Eq. 12.1.64 and is computed as... 

Figure 7 shows that the rate of adsorption, expressed as the time dependence of the fractional approach to equilibrium, decreased with increased partial pressure of SO2, which is rather unexpected. 

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