Viscosity characteristic length from

D is a characteristic length jG, jL are the volume fluxes of the gas and liquid, respectively and m, C are constants. For turbulent flow, m is equal to unity. The value of C is found to depend on the design of the ends of the tubes and the way in which the liquid and gas are added and extracted. It may have values ranging from 0.725 to 1. For viscous flow in a liquid, m and C are functions of the dimensionless inverse viscosity, NF, where... 

The specific surface area of an industrial-sized continuous stirred tank reactor (CSTR) can be calculated from the reactor dimensions. However, it is difficult to estimate the effect of the formation of bubbles and of the stirrer-induced vortex at low melt viscosity. The calculation of the characteristic length of diffusion in a high-viscosity finishing reactor with devices for the generation of thin films with respective high specific surface areas is more complex. 

As follows from the hydrodynamic properties of systems involving phase boundaries (see e.g. [86a], chapter 2), the hydrodynamic, Prandtl or stagnant layer is formed during liquid movement along a boundary with a solid phase, i.e. also at the surface of an ISE with a solid or plastic membrane. The liquid velocity rapidly decreases in this layer as a result of viscosity forces. Very close to the interface, the liquid velocity decreases to such an extent that the material is virtually transported by diffusion alone in the Nernst layer (see fig. 4.13). It follows from the theory of diffusion transport toward a plane with characteristic length /, along which a liquid flows at velocity Vo, that the Nernst layer thickness, 5, is given approximately by the expression,... 

The quantity / is usually called the characteristic length of the interaction potential, and is also employed in the more realistic wave-mechanical treatment. Many authors employ the symbol, a, which is equal to /-1. The form of the molecular interaction potential can be determined in terms of a suitable model, from experimental measurements of the temperature dependence of the viscosity of a gas, whence the characteristic length can be estimated. 

Transformation from laminar to turbulent flow is characterized by a significant value of a dimensionless quantity known as Reynolds number (Re). Reynolds number relates inertial and viscous forces as represented in Equation 4.2 where p is the fluid density, vs is the fluid velocity, I is the characteristic length (cross section flow in a channel) and p is the fluid viscosity. 

X. - lernperalute of tire fluid sufficiently far from the surface, "C Lc = characteristic length of the geomelry, ni V = kinematic viscosity of the fluid, mVs... 

As always, we need to separate the real from the imaginary. A material s viscosity N is real, its coefficient for self-diffusion K is real, its characteristic length R in nanometers or micrometers is a real attribute of the material, with R = 2(NKy as in Chapter 12. By contrast, a quadrant arc of radius R is imaginary and even more so is a swarm of such arcs, as in Figure 11.5. Then what microstructure is real, when R equals say 200 nm ... 

This formula for the electroosmotic velocity past a plane charged surface is known as the Helmholtz-Smoluchowski equation. Note that within this picture, where the double layer thickness is very small compared with the characteristic length, say alX t> 100, the fluid moves as in plug flow. Thus the velocity slips at the wall that is, it goes from U to zero discontinuously. For a finite-thickness diffuse layer the actual velocity profile has a behavior similar to that shown in Fig. 6.5.1, where the velocity drops continuously across the layer to zero at the wall. The constant electroosmotic velocity therefore represents the velocity at the edge of the diffuse layer. A typical zeta potential is about 0.1 V. Thus for = 10 V m" with viscosity that of water, the electroosmotic velocity U 10 " ms, a very small value. 

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