Diffusive-convective characteristics

Figures l(a, b) represent phase relationships for cases in which solid solutions exist, a common occurrence in metallic and semiconductor systems. In such cases solidification does not cause complete separation and the degree of separation depends not only on the equilibrium relationships represented by Figures l(a, b) but also on the convective-diffusive characteristics of the system. If the addition of solute lowers the melting point as in Figure la, then k0 < 1 on the other hand if the melting point is raised by adding solute, ko > 1 as in Figure lb.

A detailed review of zone melting and its applications has been given recently by Shaw (2). In the present paper we shall confine our attention primarily to the convective-diffusive characteristics of such systems, and we shall strive primarily to obtain a sound qualitative understanding of their behavior. 

S. Murakami, S. Kato, S. Nagano, Y. Tanaka. Diffusion characteristics of airborne particles with gravitational settling in a convective-dominant indoor flow field. ASHRAE Transactions. 98(1), 1992, 82-97. 

A more rigorous treatment takes into account the hydrodynamic characteristics of the flowing solution. Expressions for the limiting currents (under steady-state conditions) have been derived for various electrodes geometries by solving the three-dimensional convective diffusion equation ... 

In convective diffusion to a rotating disk, the characteristic velocity V0 is given by the product of the disk radius r, as a characteristic dimension of the system, and the radial velocity co, so that the Reynolds number is given by the equation... 

Shah and Nelson [33] introduced a convective mass transport device in which fluid is introduced through one portal and creates shear over the dissolving surface as it travels in laminar flow to the exit portal. They demonstrated that this device produces expected fluid flow characteristics and yields mass transfer data for pharmaceutical solids which conform to convective diffusion equations. 

The effective diffusivities determined from limiting-current measurements appear at first applicable only to the particular flow cell in which they were measured. However, it can be argued plausibly that, for example, rotating-disk effective diffusivities are also applicable to laminar forced-convection mass transfer in general, provided the same bulk electrolyte composition is used (H8). Furthermore, the effective diffusivities characteristic for laminar free convection at vertical or inclined electrodes are presumably not significantly different from the forced-convection diffusivities. 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

Originally, the concept of the Prandtl boundary layer was developed for hydraulically even bodies. It is assumed that any characteristic length L on the particle surface is much greater than the thickness (<5hl) of the boundary layer itself (L > Ojil) Provided this assumption is fulfilled, the concept can be adapted to curved bodies and spheres, including real drug particles. Furthermore, the classical ( macroscopic ) concept of the hydrodynamic boundary layer is valid solely for high Reynolds numbers of Re>104 (14,15). This constraint was overcome for the microscopic hydrodynamics of dissolving particles by the convective diffusion theory (9). 

A reciprocal proportionality exists between the square root of the characteristic flow rate, t/A, and the thickness of the effective hydrodynamic boundary layer, <5Hl- Moreover, f)HL depends on the diffusion coefficient D, characteristic length L, and kinematic viscosity v of the fluid. Based on Levich s convective diffusion theory the combination model ( Kombi-nations-Modell ) was derived to describe the dissolution of particles and solid formulations exposed to agitated systems [(10), Chapter 5.2]. In contrast to the rotating disc method, the combination model is intended to serve as an approximation describing the dissolution in hydrodynamic systems where the solid solvendum is not necessarily fixed but is likely to move within the dissolution medium. Introducing the term... 

The effectiveness of deep-bed filters in removing suspended particles is measured by die value of die filter coefficient which in turn is related to the capture efficiency of a single characteristic grain of the bed. Capture efficiencies are evaluated in the present paper for nil cases of practical importance in which London forces and convective-diffusion serve to transport particles to the surface of a spherical collector immersed in a creeping How field. Gravitational forces are considered in some cases, but the general results apply mainly to submicron or neutrally buoyant particles suspended in a viscous fluid such as water. Results obtained by linearly superimposing the in-... 

The objectives of the present paper are to (1) compute the rate of deposition of particles onto a spherical collector in a creeping flow field for all situations in which London forces and convective-diffusion act as transport mechanisms, (2) identify limiting behaviors according to the relative values of the characteristic parameters for each mechanism, (3) establish the physical conditions in which each of the limiting cases is valid, and (4) test the accuracy of the additivity rule. 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

If u0 is the average velocity in a system where both molecular diffusion and convective diffusion are taking place, L is a characteristic length, and c0 is a representative concentration, then Eq. 10.16 can be put in dimensionless form by making the following substitutions Ux = uju0, X = x/L, C = c/c0, etc. Equation 10.16 becomes... 

Axial Convective Diffusion. The variation in width, length and direction of individual channels formed by the interstices of the packing give rise to a dispersion which can be characterized by the dimensionless Bodenstein number. Bo, which is a similar number as the Peclet number but with the particle diameter as characteristic dimension... 

By examining these characteristic dimensionless numbers, it is possible to appreciate possible interactions of different processes (convection, diffusion, reaction and so on) and to simplify the governing equations accordingly. A typical dimensionless form of the governing equation can be written (for a general variable, (p) ... 

We point out that the existence of a small or large parameter is a fundamental characteristic feature of many problems of physicochemical hydrodynamics. Indeed, as was already noted, convective diffusion in fluids is characterized by large Schmidt numbers, which is related to the characteristic values of physical constants. In the corresponding singularly perturbed problems, there exist narrow... 

The effect of retardation of the transport of adsorbing ions passing the diffuse part of the double layer becomes clearer when considering a steady transport in one dimension. It is difficult to realise a strictly one-dimensional steady transport in an experiment. But the conditions of a one dimensional steady transport are approximately realised under the conditions of convective diffusion when the thickness of the diffusion layer 5q is much less than the characteristic geometric dimension of the system L... 

If adsoiption processes proceed under conditions of convective diffusion, an important characteristic parameter, the diffusion layer thickness, 5d enters the discussion (cf Section... 

Each of these is a ratio of a convective transfer rate to the corresponding diffusion rate of transfer. Dimensionless analysis indicates that, for fixed geometry and constant properties, the Nusselt number and the Sherwood number depend on the Reynolds number (forced convection), Rayleigh number (natural convection), flow characteristics, Prandtl number (heat transfer), and Schmidt number (mass transfer). 

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