Comparable Diffusion Phase Resistances

Let us consider a transient solute concentration field in a liquid outside and inside a spherical drop of radius a moving at a constant velocity U in an infinite fluid medium. We assume that the fluid velocity fields for the continuous and disperse phases are determined by the Hadamard-Rybczynski solution [177, 420], obtained for low Reynolds numbers. The concentration far from the drop is maintained constant and equal to C,. At the initial time f = 0, the concentration outside the drop is everywhere uniform and is equal to C inside the drop, it is also uniform, but is equal to Co- 

The following boundary conditions hold at the drop surface  

In view of the discussion above, we conclude that at the final stage of the process the concentration on the interface is constant by virtue of the first boundary condition in (4.13.2) and is equal to 

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Figure 2-4 Typical concentration profiles of instantaneous reaction between the gas A and the reactant C, based on film theory, ids Diffusion controlled - slow reaction, (fcl kinetically controlled-slow reaction, (c) gas-film-controlled desorption - fast reaction, 0 liquid-film-controlled desorption-fast reaction, (e) liquid-film-controlled absorption -instantaneous reaction between A and C, (/) gas-film-controlled absorption-instantaneous reaction between A and C, (g) concentration profiles for A, B, and C for instantaneous reaction between A and C-both gas- and liquid-phase resistances are comparable.1 2...

Internal circulation occurs inside the bubbles, and the resistance to diffusion in the gas phase is negligible compared to that in the liquid phase. 

To transfer ozone to a water phase very often fine bubble columns or stirred or agitated tank reactors are applied. The bubbles in these reactors are small. The value of m for ozone is relatively high. Furthermore the diffusion coefficient of ozone in the gas phase is very high compared with the diffusion coefficient of ozone in the water phase. These three reasons lead to the conclusion that in bubble columns and stirred tank reactors the resistance to mass transfer in the gas phase can be neglected. Equation (20) can than be simplified to ... 

Supercritical fluids possess favorable physical properties that result in good behavior for mass transfer of solutes in a column. Some important physical properties of liquids, gases, and supercritical fluids are compared in Table 4.1 [49]. It can be seen that solute diffusion coefficients are greater in a supercritical fluid than in a liquid phase. When compared to HPLC, higher analyte diffusivity leads to lower mass transfer resistance, which results in sharper peaks. Higher diffusivity also results in higher optimum linear velocities, since the optimum linear velocity for a packed column is proportional to the diffusion coefficient of the mobile phase for liquid-like fluids [50, 51]. 

The three contributions to dispersion are also shown as separate curves in figure 1. It is seen that the major contribution to dispersion at the optimum velocity, where the value of (H) is a minimum, is the multipath effect. Only at much lower velocities does the longitudinal diffusion effect become significant. Conversely, the mobile phase velocity must be increased to about 0.2 cm/sec before the resistance to mass transfer begins to become relatively significant compared to that of the multipath effect. 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

Useful simplifications are often made in Equation 2.2. We will use gas-liquid contact as an example, and assume gas-filled homogeneous membrane of high porosity, thin wall, and low tortuosity. Since diffusion in gas phase is generally of three orders of magnitude faster than in liquid phase, one can show that and ka are quite high in this case compared to ki, and so the controlling resistance to mass transfer is in the liquid phase. This means A total is essentially the same as If is constant within the contactor the total mass transfer rate in Equation 2.4 can be approximated for the entire contactor as... 

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