Boundary-layer

However, the transition Reynolds number depends on free-stream turbulence and may range from 3 x 105 to 3 x 106. The laminar boundary layer thickness 8 is a function of distance from the leading edge  

The total drag on the plate of length L and width b for a laminar boundary layer, including the drag on both surfaces, is  

For non-Newtonian power law fluids (Acrivos, Shah, and Peterson, AIChEJ., 6, 312-317 [I960] Hsu, AIChEJ., 15, 367-370 [1969]), 

Here the second term accounts for the laminar leading edge of the boundary layer and assumes that the critical Reynolds number is 500,000. 

The behavior of the gas as it flows down the tube is controlled by fluid mechanics and a complete investigation wouldbe lengthy and outside the scope of this book. It is enough to say that the Reynolds number, which is a dimensionless parameter that characterizes the 

In the case of laminar flow, the velocity of the gas at the deposition surface (the inner wall of the tube) is zero. The boundary is that region in which the flow velocity changes from zero at the wall to essentially that of the bulk gas away from the wall. This boundary layer starts at the inlet of the tube and increases in thickness until the flow becomes stabilized as shown in Fig. 2.4b. The reactant gases flowing above the boundary layer have to diffuse through this layer to reach the deposition surface as is shown in Fig. 2.3. 

The thickness of the boundary layer, A, is inversely proportional to the square root of the Reynolds number as follows  

X = distance from inlet in flow direction l = viscosity 

This means that the thickness of the boundary layer increases with lower gas-flow velocity and with increased distance from the tube inlet. 

Hence the dimensionless concentration distribution can be expressed in the following way  

Two convective diffusion regimes are similar if the Reynolds, Peclet, and interception nunibers are the same. 

The local rate of particle transfer by diffusion to the surface of the body is 

Setting the local ma.ss transfer coefficient k = J/n and rearranging, the result is 

The particle transfer coefficient k has dimensions of velocity and is often called the deposition velocity. At a given location on the collector surface the dimensionless group kL/D, known as the Sherwood number, is a function of the Reynolds. Peclet, and interception numbers. Rates of particle deposition measured in one fluid over a range of values of Pe, Re, and R can be u.sed to predict deposition rate.s from another fluid at the same values of the dimensionless groups. In some cases, it is convenient to work with the Schmidt number Sc = u/D = Pe/Re in place of Pe as one of the three groups, because Sc depends only on the nature of the fluid and the suspended particles. 

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Besides the material based characteristics, the difference of density of the used particle/substrate combination is a very important criterion. The difference of density influences the contrast of the radiographic tests. Tungsten carbides were used as mechanically resistant particles and titanium based alloys as substrate. The substrate material is marked by an advantageous relation of strength to density. This material is often used in aeronautics, astronautics, and for modification of boundary layers. The density of tungsten carbide (15.7 g/cm ) is about 3.5 times higher than the density of titanium (4.45-4.6 g/cm ). 

The rate of dissolving of a solid is determined by the rate of diffusion through a boundary layer of solution. Derive the equation for the net rate of dissolving. Take Co to be the saturation concentration and rf to be the effective thickness of the diffusion layer denote diffusion coefficient by . 

The rate of physical adsorption may be determined by the gas kinetic surface collision frequency as modified by the variation of sticking probability with surface coverage—as in the kinetic derivation of the Langmuir equation (Section XVII-3A)—and should then be very large unless the gas pressure is small. Alternatively, the rate may be governed by boundary layer diffusion, a slower process in general. Such aspects are mentioned in Ref. 146. 

A quantitative treatment for the depletive adsorption of iogenic species on semiconductors is that known as the boundary layer theory [84,184], in which it is assumed that, as a result of adsorption, a charged layer is formed. Doublelayer theory is applied, and it turns out that the change in surface potential due to adsorption of such a species is proportional to the square of the amount adsorbed. The important point is that very little adsorption, e.g., a 0 of about 0.003, can produce a volt or more potential change. See Ref. 185 for a review. 

It is essential for the rotating-disc that the flow remain laminar and, hence, the upper rotational speed of the disc will depend on the Reynolds number and experimental design, which typically is 1000 s or 10,000 rpm. On the lower lunit, 10 s or 100 rpm must be applied in order for the thickness of tlie boundary layer to be comparable to that of the radius of the disc. 

Besides the expressions for a surface derived from the van der Waals surface (see also the CPK model in Section 2.11.2.4), another model has been established to generate molecular surfaces. It is based on the molecular distribution of electronic density. The definition of a Limiting value of the electronic density, the so-called isovalue, results in a boundary layer (isoplane) [187]. Each point on this surface has an identical electronic density value. A typical standard value is about 0.002 au (atomic unit) to represent electronic density surfaces. 

Schlichting, H., 1968. Boundary-Layer Theory, McGraw-Hill, New York. 

CATALYSTS - REGENERATION - FLUID CATALYTIC CRAC KING UNITS] (Vol 5) Turbulent boundary layer model... 

The relationship between heat transfer and the boundary layer species distribution should be emphasized. As vaporization occurs, chemical species are transported to the boundary layer and act to cool by transpiration. These gaseous products may undergo additional thermochemical reactions with the boundary-layer gas, further impacting heat transfer. Thus species concentrations are needed for accurate calculation of transport properties, as well as for calculations of convective heating and radiative transport. 

Another concept sometimes used as a basis for comparison and correlation of mass transfer data in columns is the Clulton-Colbum analogy (35). This semi-empirical relationship was developed for correlating mass- and heat-transfer data in pipes and is based on the turbulent boundary layer model... 

External Fluid Film Resistance. A particle immersed ia a fluid is always surrounded by a laminar fluid film or boundary layer through which an adsorbiag or desorbiag molecule must diffuse. The thickness of this layer, and therefore the mass transfer resistance, depends on the hydrodynamic conditions. Mass transfer ia packed beds and other common contacting devices has been widely studied. The rate data are normally expressed ia terms of a simple linear rate expression of the form... 

Fig. 3. An overview of atomistic mechanisms involved in electroceramic components and the corresponding uses (a) ferroelectric domains capacitors and piezoelectrics, PTC thermistors (b) electronic conduction NTC thermistor (c) insulators and substrates (d) surface conduction humidity sensors (e) ferrimagnetic domains ferrite hard and soft magnets, magnetic tape (f) metal—semiconductor transition critical temperature NTC thermistor (g) ionic conduction gas sensors and batteries and (h) grain boundary phenomena varistors, boundary layer capacitors, PTC thermistors.

Some empirical equations to predict cyclone pressure drop have been proposed (165,166). One (166) rehably predicts pressure drop under clean air flow for a cyclone having the API model dimensions. Somewhat surprisingly, pressure drop decreases with increasing dust loading. One reasonable explanation for this phenomenon is that dust particles approaching the cyclone wall break up the boundary layer film (much like spoiler knobs on an airplane wing) and reduce drag forces. 

Fig. 4. Boundary layer development around a circular cylinder where A represents the point of separation.

Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

Fig. 5. Entrance flows in a tube or duct (a) separation at sharp edge (b) growth of a boundary layer (illustrated for laminar flow).

Seldom is the temperature difference across the wall thickness of an item of equipment known. Siace large temperature gradients may occur ia the boundary layers adjacent to the metal surfaces, the temperature difference across the wall should not be estimated from the temperatures of the fluids on each side of the wall, but from the heat flux usiag equation 27... 

The most widely used approach to channel flow calculations assumes a steady qua si-one-dimensional flow in the channel core, modified to account for boundary layers on the channel walls. Electrode wall and sidewall boundary layers may be treated differently, and the core flow may contain nonuniformities. 

In the flow models these effects are confined to the boundary layers, maintaining the vaHdity of the qua si-one-dimensional flow model. The flow is... 

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