Convective diffusion liquid

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

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. 

The general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of c, as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function Cj(x,y,z) is essentially independent of time t, i.e. in the steady-state. Then equation (15) reduces to ... 

Let us consider the transport of one component i in a liquid solution. Any disequilibration in the solution is assumed to be due to macroscopic motion of the liquid (i.e. flow) and to gradients in the concentration c,. Temperature gradients are assumed to be negligible. The transport of the solute i is then governed by two different modes of transport, namely, molecular diffusion through the solvent medium, and drag by the moving liquid. The combination of these two types of transport processes is usually denoted as the convective diffusion of the solute in the liquid [25] or diffusion-advection mass transport [48,49], The relative contribution of advection to total transport is characterised by the nondimensional Peclet number [32,48,49], while the relative increase in transport over pure diffusion due to advection is Sh - 1, where Sh is the nondimensional Sherwood number [28,32,33,49,50]. 

CONVECTIVE DIFFUSION FROM AN UNBOUNDED LAMINARLY FLOWING LIQUID TO A PLANAR SURFACE... 

Figure 4. Schematic representation of the convective-diffusion problem for an active plane parallel to the direction of flow dealt with in Section 4.1. The liquid flow extends up to a — oo, where its free velocity is v in the direction of increasing y. The leading edge of the plane is the segment x — 0, y — 0, 0 < z

CONVECTIVE DIFFUSION FROM A CHANNELLED LAMINARLY FLOWING LIQUID... 

CONVECTIVE DIFFUSION FROM A LIQUID TO A MOVING SPHERICAL BODY... 

It should be highlighted that equation (47) holds for solid particles. In the case of liquid particles, e.g. with emulsions, the convective diffusion process is very different due to interfacial momentum transfer which gives rise to a different velocity profile. Consequently, convective diffusion to/from a liquid particle is more effective than that for a solid particle. Starting again from equation (43),... 

The convective diffusion theory was developed by V.G. Levich to solve specific problems in electrochemistry encountered with the rotating disc electrode. Later, he applied the classical concept of the boundary layer to a variety of practical tasks and challenges, such as particle-liquid hydrodynamics and liquid-gas interfacial problems. The conceptual transfer of the hydrodynamic boundary layer is applicable to the hydrodynamics of dissolving particles if the Peclet number (Pe) is greater than unity (Pe > 1) (9). The dimensionless Peclet number describes the relationship between convection and diffusion-driven mass transfer ... 

Two approaches can be used for the analysis of turbulent mass transfer near a liquid-fluid interface. One has the time-averaged convective diffusion equation as the starting point. For obtaining in that procedure an equation for... 

Figure 5.3-11. Scheme of the two-compartment convective-diffusion model and equations for the end zones in BSCR (tested with a liquid tracer) [47],... 

For the flow direction and coordinate system given in Fig. 5 b, and assuming diffusion perpendicular to the solid-liquid interface, the general convective diffusion equation in Cartesian coordinates is 38 401 ... 

In gases and liquids, the rates of these diffusion processes can often be accelerated by convective flow. For example, the copper sulfate in the tall bottle can be completely mixed in a few minutes if the solution is stirred. This accelerated mixing is not due to diffusion alone, but to a combination of diffusion and convection. Diffusion still depends on the random molecular motions that take place over small molecular distances. The convective stirring is not a molecular process, but a macroscopic process which moves portions of the fluid over longer distances. After this macroscopic motion, diffusion mixes the newly adjacent portions of the fluids. 

Membrane. In discussing transport across a membrane, we consider only the case where the pressures on both sides of the membrane are substantially the same so that no liquid is being pressed across it and there is no transport by convection. Diffusion is then the only mechanism... 

Although the diffusion layer model is the most commonly used, various alterations have been proposed. The current views of the diffusion layer model are based on the so-called effective diffusion boundary layer, the structure of which is heavily dependent on the hydrodynamic conditions, fn this context, Levich [102] developed the convection-diffusion theory and showed that the transfer of the solid to the solution is controlled by a combination of liquid flow and diffusion. In other words, both diffusion and convection contribute to the transfer of drug from the solid surface into the bulk solution, ft should be emphasized that this observation applies even under moderate conditions of stirring. 

At any interface there will be boundary layers of liquid not subject to convection. Diffusion of materials to and from the interface will be determined by the thickness of such layers. In previous studies of solvent extraction and partitioning, a wide variety of techniques to set up the liquid-liquid interface have been employed. These are illustrated in Fig. 

Convection, diffusion, and dispersion can only describe part of the processes occurring during transport. Only the transport of species that do not react at all with the solid, liquid or gaseous phase (ideal tracers) can be described adequately by the simplified transport equation (Eq. 94). Tritium as well as chloride and bromide can be called ideal tracers in that sense. Their transport can be modeled by the general transport equation as long as no double-porosity aquifers are modeled. Almost all other species in water somehow react with other species or a solid phase. These reactions can be subdivided into the following groups, some of which have already been considered in the previous part of the book. 

Yeager and co-workers [379, 419] studied the kinetics of peroxide decomposition on dispersed oxide powders by measuring changes in the convective-diffusion limiting current for peroxide oxidation at a rotating gold electrode immersed in the liquid dispersion. The current decay was observed to be proportional to the peroxide concentration decay due to the catalytic decomposition process. On perovskite [379] and spinel oxides [419], it follows first-order kinetics described by the equation... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

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