Stagnant-film boundary layer model

Methane is also lost from surface waters by air-sea exchange. If the surface concentration of methane exceeds its equilibrium concentration, there will be a net flux to the atmosphere. The empirical relationship commonly used to quantify the transfer flux (F) is the stagnant-film boundary layer model (28, 29). 

According to the assumptions in Section 6.2.1, the liquid phase concentration changes only in axial direction and is constant in a cross section. Therefore, mass transfer between liquid and solid phase is not defined by a local concentration gradient around the particles. Instead, a general mass transfer resistance is postulated. A common method describes the (external) mass transfer mmt i as a linear function of the concentration difference between the concentration in the bulk phase and on the adsorbent surface, which are separated by a film of stagnant liquid (boundary layer). This so-called linear driving force model (LDF model) has proven to be sufficient in... 

The process of equilibration of the atmosphere with the ocean is called gas exchange. Several models are available however, the simplest model for most practical problems is the one-layer stagnant boundary layer model (Fig. 9-18). This model assumes that a well-mixed atmosphere and a weU-mixed surface ocean are separated by a film on the liquid side of the air-water interface through which gas transport is controlled by molecular diffusion. [A similar layer exists on the air side of the interface that can be neglected for most gases. SO2 is a notable exception (Liss and Slater, 1974).]... 

Four of the simplest and best known of the theories of mass transfer from flowing streams are (1) the stagnant-film model, (2) the penetration model, (3) the surface-renewal model, and (4) the turbulent boundary-layer model... 

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

This can be further integrated from the wall to the boundary layer thickness y = 8, where the component is at the bulk concentration Cj,. Substituting / = - o and k = D/o, the mass-transfer coefficient yields the stagnant film model [Brian, Desalination by Reverse Osmosis, Merten (ed.), M.I.T. Press, Cambridge, Mass., 1966, pp. 161-292] ... 

Early investigators assumed that this so-called diffusion layer was stagnant (Nernst-Whitman model), and that the concentration profile of the reacting ion was linear, with the film thickness <5N chosen to give the actual concentration gradient at the electrode. In reality, however, the thin diffusion layer is not stagnant, and the fictitious t5N is always smaller than the real mass-transfer boundary-layer thickness (Fig. 2). However, since the actual concentration profile tapers off gradually to the bulk value of the concentration, the well-defined Nernst diffusion layer thickness has retained a certain convenience in practical calculations. 

The mechanism of dissolution was proposed by Nernst (1904) using a film-model theory. Under the influence of non-reactive chemical forces, a solid particle immersed in a liquid experiences two consecutive processes. The first of these is solvation of the solid at the solid-liquid interface, which causes the formation of a thin stagnant layer of saturated solution around the particle. The second step in the dissolution process consists of diffusion of dissolved molecules from this boundary layer into the bulk fluid. In principle, one may control the dissolution through manipulation of the saturated solution at the surface. For example, one might generate a thin layer of saturated solution at the solid surface by a surface reaction with a high energy barrier (Mooney et al., 1981), but this application is not commonly employed in pharmaceutical applications. 

In the previous sections, stagnant films were assumed to exist on each side of the interface, and the normal mass transfer coefficients were assumed proportional to the first power of the molecular diffusivity. In many mass transfer operations, the rate of transfer varies with only a fractional power of the diffusivity because of flow in the boundary layer or because of the short lifetime of surface elements. The penetration theory is a model for short contact times that has often been applied to mass transfer from bubbles, drops, or moving liquid films. The equations for unsteady-state diffusion show that the concentration profile near a newly created interface becomes less steep with time, and the average coefficient varies with the square root of (D/t) [4] ... 

The solution of such an equation for an actual membrane device for ultrafiltration is difficult to obtain (see Zeman and Zydney (1996) for background information). One therefore usually falls back on the stagnant film model for determining the relation between the solvent flux and the concentration profile (see result (6.3.142b)). To use this result, we need to estimate the mass-transfer coefficient kit = Dit/dt), for the protein/macromolecule. One can focus on the entrance region of the concentration boundary layer, assume to be constant for a dilute solution, V = V, Vj, = 0 in the thin boundary layer, v = y ,y (where is the wall shear rate of magnitude AVz/Ay ) and obtain the result known as the Leveque solution at any location z in terms of the Sherwood number ... 

Similar to the widely adopted model of air-water exchange (Schwarzenbach et ah, 2003), the mass transfer from the snow pack-air interface to the bulk atmosphere is usually interpreted as occurring by molecular diffusion across a thin stagnant boundary layer film. Because the thickness of this boundary layer is unknown, some approaches (Wania, 1997 Daly and Wania, 2004) simply treat few as a constant. In particular, it is assumed that febi adopts values similar to a typical MTC for the boundary layer above soil (e.g., 0.14cms or 5mh Mackay and Stiver, 1991). In a more realistic approach, Hansen et al. (2006) allow for the influence of variable wind speed on febi, which they expressed as ... 

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