Boundary layer thickness fluid dynamics

Because D is independently determined, and p is obtainable from initial conditions and thermod5mamic equilibrium, the problem of determining the convective dissolution rate now becomes the problem of estimating the boundary layer thickness. In fluid dynamics, the boundary layer thickness appears in a dimensionless number, the Sherwood number Sh ... 

When the fluid approaches the sphere from above, the fluid initially contacts the sphere at 0 = 0 (i.e., the stagnation point) because polar angle 6 is defined relative to the positive z axis. This is convenient because the mass transfer boundary layer thickness Sc is a function of 6, and 5c = 0 at 0 = 0. In the laminar and creeping flow regimes, the two-dimensional fluid dynamics problem is axisymmetric (i.e., about the z axis) with... 

Flow of fluid over the surface is characterized by the formation of a hydro-dynamic or velocity boundary layer, which is defined as the thin layer of fluid over which the velocity of the fluid varies from no-slip zero velocity at the surface to the outer stream velocity over the thickness of the boundary layer y = 5. Because of the effect of viscosity, fluid flow slows down near the stationary solid surface and maintains the no-slip fluid-solid interface boundary conditions. The flow is assumed to be viscous within the boundary layer and inviscid outside the boimdary layer. The boundary layer thickness increases in the downstream x-direction and results in a varying x-component velocity profile u(y) as shown in the figure. 

At 700°C and 1 atm this leads to a diffusion constant of 0.81 cm /s The flow field around a superheater tube is very complex involving both laminar and turbulent boundary layers and the estimation of the local boundary layer thicknesses (velocity, diffusion and thermal boundary layers) around the tube requires computer simulations with computational fluid dynamic (CFD) software packages. However, for this rough analysis an average value of the thermal boundary layer thickness around the tube is enough and can be estimated if the average Nusselt number around the tube is known... 

Again, in practice reality is more complex than these simple descriptions, although the concepts remain the same - fluid mechanics is used to control the boundary layer thickness such that the growth rate is constant under conditions of gas-transport limited growth. For real CVD reactors a complete fluid dynamics model is coupled to a film growth model to obtain an optimal susceptor design to fit in the reactor tube (which may be rectangular or round or some other shape as desired). 

A boundary layer is formed between the two phases (fluid and solid). This is a stagnant film that represents a layer of less movement of the fluid and hence builds up a zone with resistance to mass transfer. The mass transfer coefficient and generally the mass transfer rate depend on the fluid dynamics of the system. Higher fluid velocities significantly reduce the thickness of the film. 

Initially it was assumed that no solution movement occurs within the diffusion layer. Actually, a velocity gradient exists in a layer, termed the hydro-dynamic boundary layer (or the Prandtl layer), where the fluid velocity increases from zero at the interface to the constant bulk value (U). The thickness of the hydrodynamic layer SH is related to that of the diffusion layer... 

Using formulas (1.2.9), one can estimate the perturbations caused by the rotating disk in the fluid remote from the disk surface. It follows from the boundary conditions (1.2.3) that the pressure, as well as the radial and the angular velocity, is not perturbed as 2 - 00. However, the remote dimensionless axial velocity is not zero, u(oo) = -0.886. This is the rate at which the disk draws the ambient fluid. Figure 1.1 shows that the pressure and the radial and angular velocities are perturbed only near the disk surface, in the so-called dynamic boundary layer. The thickness of this layer is independent of the radial coordinate and is approximately equal to S =... 

Again in the case of flowing liquids, the supply of the cathodic reagents or of the inhibitors to the surface depends on the fluid dynamic conditions, and it may have contrasting effects. The general increase in the thickness of the boundary layer can reduce the supply of the cathodic reactant and that of the inhibitor, and, within the dead spaces, a cathodic process can be substituted for another, such as in the case of aerated and sulfatic waters, where anaerobic conditions that are established in the dead spaces permit the development of sulfate-reducing bacteria. 

A comprehensive fluid-dynamical solution for single phase flow was given in the Chin 1986 work. There, a flow model with coupled cake growth and filtrate invasion was constructed, comprised of a three-layer Darcy flow the mudcake, the rock with filtrate, and the rock with reservoir fluid. Mass and pressure continuity were enforced at interfaces. At the cake surface, where it is exposed to borehole fluids, the well pressure was specified and a constitutive model for empirical cake buildup was invoked (e.g., see Collins, 1961). Cake thickness was allowed to increase with time in the moving boundary value problem formulation, thus slowing the rate of formation invasion. In the third-layer farfield, the reservoir pressure was assumed to be known. Once pressures... 

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