Diffusion, eddy unsteady-state

Penetration theory (Higbie, 1935)assumes that turbulent eddies travel from the bulk of the phase to the interface where they remain for a constant exposure time te. The solute is assumed to penetrate into a given eddy during its stay at the interface by a process of unsteady-state molecular diffusion. This model predicts that the mass-transfer coefficient is directly proportional to the square root of molecular diffusivity... 

In the five chapters that make up Part II (Chapters 7-11) we consider the estimation of rates of mass and energy transport in multicomponent systems. Multicomponent mass transfer coefficients are defined in Chapter 1, Chapter 8 develops the multicomponent film model, Chapter 9 describes unsteady-state diffusion models, and Chapter 10 considers models based on turbulent eddy diffusion. Chapter 11 shows how the additional complication of simultaneous mass and energy transfer may be handled. 

This relationship, derived for molecular diffusion in a stagnant film, is assumed to hold reasonably well for unsteady-state diffusion or a combination of molecular and eddy diffusion. Sometimes the mass-transfer coefficient for one-way transfer is denoted by or /c, and the coefficients then follow the same relationship as the fluxes in Eq. (21.37) ... 

A theory which incorporates some of the principles of both the two-film theory and the penetration theory has been proposed by TOOR and Marchello The whole of the resistance to transfer is regarded as lying within a laminar film at the interface, as in the two-film theory, but the mass transfer is regarded as an unsteady state process. It is assumed that fresh surface is formed at intervals from fluid which is brought from the bulk of the fluid to the interface by the action of the eddy currents. Mass transfer then takes place as in the penetration theory, except that the resistance is confined to the finite film, and material which traverses the film is immediately completely mixed with the bulk of the fluid. For short times of exposure, when none of the diffusing material has reached the far side of the layer, the process is identical to that postulated in the penetration theory. For prolonged periods of exposure when a steady concentration gradient has developed, conditions are similar to those considered in the two-film theory. 

Figure 7 Determination of backmixing (eddy diffusivity) by steady state method (A), showing devices for tracer injection, sampling electrodes and arrangement of sampling points. Unsteady state arrangement is shown in B.

In Equation 6.18, denotes the depth of penetration by an eddy in the neighboring mass transfer boundary layer. Harriott s analysis of his data indicated that both slip velocity and transient effects are important in determining the overall mass transfer process. Although Harriott could not provide actual slip velocity and unsteady-state mass transfer data, his conclusions based on effects of diffusivity and particle density on in stirred tanks are quite logical. In his conclusion, Harriott argued that the slip velocity theory is relatively easy to use for quantitative predictions as compared to the modified penetration theory. 

Initially, the concentiaJUon of dissobteilgas in the eddy is internally the eddy is considered to be stagnant. When the eddy is expoited to the. gas.aL,the surface, the concentration in the Jiq.ui(lat the gas-liquid surface is Cj which may be taken as the eqnilibrium solubility of the gas in the liquid. During the timeuti, the hquid particle is subject to unsteady-state.diffusion or penetration of solute in the z direction, and, as an approximation, Eq. (2.18) may be applied ... 

Aging of the transfer coefficients was also noticed by Baird and Davidson (Bl) in experiments on absorption of very large CO2 bubbles in tap water. As suggested earlier by Garner (G12), the unsteady value of the transfer coefficient can probably be attributed to slow saturation of the turbulent wake under the spherical-cup bubble, where transfer is by eddy diffusion rather than by convection. With bubbles less than 2.5 cm diam, time-independent transfer coefficients were obtained. This was attributed (Bl) to more rapid bubble circulation and renewal of the wake, which permit a steady state to set in faster behind the smaller bubbles it is consistent both with the dependence of the steady-state time on the Peclet number (B6), which for spherical bubbles yields oc R, and with the relationship 0 oc suggested by Levich (L8). However, due to the irregular hydrodynamic... 

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