Diffusion boundary layer total

Different situations are summarized in Fig. 20.12. When radial diffusion boundary layers totally overlap, i.e. when the diffusion hemisphere is larger than the mean... 

Reduction of particle size increases the total specific surface area exposed to the solvent, allowing a greater number of particles to dissolve more rapidly. Furthermore, smaller particles have a small diffusion boundary layer, allowing faster transport of dissolved material from the particle surface [58]. These effects become extremely important when dealing with poorly water-soluble drugs, where dissolution is the rate-limiting step in absorption. There are numerous examples where reduction of particle size in such drugs leads to a faster dissolution rate [59-61], In some cases, these in vitro results have been shown to correlate with improved absorption in vivo [62-64]. 

This is the classic equation relating distance and time for species in a totally quiescent medium where transport is controlled by molecular diffusion. The simple formula, sometimes referred to as Einstein s equation, is veiy useful for order of magnitude calculations determining the distance a substance will travel by molecular diffusion. Eor example, if one wanted to know if a chemical species with rate constant k would react significantly during transport across a molecular diffusion boundary layer of thickness Ax one would compare the time calculated from the above equation with the reciprocal of the reaction rate constant, k (see below). 

In the diffusion boundary layer approximation with allowance for the corrections (with respect to the Reynolds number) to the potential flow past the bubble, one can obtain the following two-term expansion of the dimensionless total flux I ... 

The fnndamental difference between both approaches is caused by the different transport mechanisms involved (Fig. 12.3). Whereas only a diffusion-controlled transport of material can be measured by applying microelectrodes in the diffusion boundary layer (DBL) and in the pore water (cf. Section 5.2.3), enrichment and/or depletion in the overlying bottom water deliver rates of total exchange. 

If we consider the solute profiles of Fig. 4.9(a), we can see that the total amount of solute incorporated into the diffusion boundary layer increases as the growth rate decreases and vice versa. Thus if a... 

Mooney et al. [70] investigated the effect of pH on the solubility and dissolution of ionizable drugs based on a film model with total component material balances for reactive species, proposed by Olander. McNamara and Amidon [71] developed a convective diffusion model that included the effects of ionization at the solid-liquid surface and irreversible reaction of the dissolved species in the hydrodynamic boundary layer. Jinno et al. [72], and Kasim et al. [73] investigated the combined effects of pH and surfactants on the dissolution of the ionizable, poorly water-soluble BCS Class II weak acid NSAIDs piroxicam and ketoprofen, respectively. 

Sampling rates for the case of total boundary layer-control can be expected to be nearly independent of temperature, since both the diffusion coefficients in air, and the kinematic viscosity of air are only weak functions of temperature (Shoeib and Harner, 2002). This leaves the air-flow velocity as the major factor that can be responsible for the seasonal differences among sampling rates observed by Ockenden et al. (1998). The absence of large R differences between indoor and outdoor exposures may be indicative of membrane-control, but it may also reflect the efficient damping of high flow velocities by the deployment devices used for SPMD air exposures (Ockenden et al., 2001). 

Although Rs values of high Ks compounds derived from Eq. 3.68 may have been partly influenced by particle sampling, it is unlikely that the equation can accurately predict the summed vapor plus particulate phase concentrations, because transport rates through the boundary layer and through the membrane are different for the vapor-phase fraction and the particle-bound fraction, due to differences in effective diffusion coefficients between molecules and small particles. In addition, it will be difficult to define universally applicable calibration curves for the sampling rate of total (particle -I- vapor) atmospheric contaminants. At this stage of development, results obtained with SPMDs for particle-associated compounds provides valuable information on source identification and temporal... 

We consider a spherical particle aggregate with radius r0 surrounded by a concentric boundary layer of thickness 8 (Fig. 19.16). Transport into the aggregate is described by the linear approximation of the radial diffusion model. Thus, the total flux from the particle to the fluid is given by Eq. 19-85 ... 

In this equation, if the rate of diffusion is faster than that of the catalytic reaction at the surface (ko kc), the Arrhenius plot of rr gives the apparent activation energy Ec of kc. This is the reaction-controlled condition. On the other hand, if the rate of the catalytic reaction is faster than that of diffusion (kc 2> kid, the Arrhenius plot of rr gives the characteristics of temperature dependence of ko. This is the diffusion-controlled condition. Under diffusion-controlled conditions, the transferred reactant decreases at once at the surface (Cs = 0) because of the fast catalytic reaction rate. The gas flow along the catalyst surface forms a boundary layer above the surface, and gas molecules diffuse due to the concentration gradient inside the layer in the thickness direction. As the total reaction... 

The leaf resistance is in series with that of the boundary layer, Thus the total resistance for the flow of water vapor from the site of evaporation to the turbulent air surrounding a leaf (r al) is 4- (see Fig. 8-5). We now must face the complication created by the two leaf surfaces representing parallel pathways for the diffusion of water vapor from the interior of a leaf to the surrounding turbulent air. We will represent the leaf resistance for the pathway through the upper (adaxial) surface by rj afu and that for the lower (abaxial) surface by r af. Each of these resistances is in series with that of an air boundary layer—and for the upper and the lower leaf surfaces, respectively. The parallel arrangement of the pathways through the upper and the lower surfaces of a leaf leads to the following total resistance for the diffusion of water vapor from a leaf ... 

Next we note that gj, f is in series with a boundary layer conductance, g , and that the two sides of a leaf act as parallel conductances for water vapor diffusing from the interior of a leaf. We therefore obtain the following expression for the total conductance of a leaf with air boundary layers on each side ... 

Changes in the thickness of the air boundary layers adjacent to a leaf have a greater influence on the flux of water vapor than on the flux of C02. For instance, the total resistance for water vapor diffusion can equal 4-C + rJJ (Eq. 8.16), whereas (see Eq. 8.19) is usually only... 

Adamson (51) proposed a model for W/0 microemulsion formation in terms of a balance between Laplace pressure associated with the interfacial tension at the oil/water interface and the Donnan Osmotic pressure due to the total higher ionic concentration in the interior of aqueous droplets in oil phase. The microemulsion phase can exist in equilibrium with an essentially non-colloidal aqueous second phase provided there is an added electrolyte distributed between droplet s aqueous interior and the external aqueous medium. Both aqueous media contain some alcohol and the total ionic concentration inside the aqueous droplet exceeds that in the external aqueous phase. This model was further modified (52) for W/0 microemulsions to allow for the diffuse double layer in the interior of aqueous droplets. Levine and Robinson (52) proposed a relation governing the equilibrium of the droplet for 1-1 electrolyte, which was based on a balance between the surface tension of the film at the boundary in its charged state and the Maxwell electrostatic stress associated with the electric field in the internal diffuse double layer. 

As the molar flux of each of the two components is independent of the position coordinate y, the total molar flux N/A and likewise the quotient Nx/N are also independent of y. Therefore (4.59) can be integrated easily. The integration extends from the condensate surface (index I) to the vapour space (index G). The thickness of the vapour boundary layer will be 5. We assume constant values for the pressure and temperature. Under the assumption that the gas phase exhibits ideal behaviour, the diffusion coefficient and the molar concentration c = N/V = p/(Rm T) are likewise independent of the coordinate y. The integration yields... 

One can see that the interaction of the diffusion wake of the first drop with the boundary layer of the second drop is more intensive than for the case of a solid particle. In addition, the total mass exchange of the second drop with the liquid is less than half that of the first. 

The turbulent boundary layer model accounts for the transfer of a solute molecule A from a turbulent stream to a fixed surface. Eddy diffusion is rapid in the turbulent stream and molecular diffusion is relatively insignificant. It is supposed that the turbulence is damped out in the immediate vicinity of the surface. In the intermediate neighborhood between the turbulent stream and the fixed surface, it is supposed that transport is by both molecular and eddy diffusion which take place in parallel. The total rate of transfer (moles of A transferred per unit time per unit area) is given by an extended form of Fick s law... 

Although the total heat flux at the surface in a binary gas is composed of the sum of a conduction term and a diffusion term, the results of analyses are expressed solely in terms of the heat conduction term. The reason is that this term is equal to the heat gained by the coolant in passing from its reservoir to the surface in contact with the boundary layer. This simple heat balance is... 

Geometric Optics Results with Emission. When the temperature of a semitransparent layer is large, emission of radiation becomes significant, and the problem of radiative transfer becomes more complex. The change in refractive index at each interface causes total internal reflection of radiation in the medium with higher refractive index at the boundary. This effect must be treated in the RTE at the boundary of the medium, and diffuse boundary conditions are no longer correct for the exact solution of this type of problem. Various approaches have been attempted. 

In equation 6.85, ([J (x, r)] nL) is the total liquid-solid particles interface, averaged, molar diffusive flux of component f. Note that Ag - = Asr +Asjf The second term of the right-hand side of equation 6.85 is 0 because at the catalyst interface convective fluxes are 0. On the other hand, from equation 6.13 the molar diffusive flux through the boundary layer at the liquid-solid interface must be equal to the reaction rate at the liquid-solid interface ... 

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