Turbulent kinematic viscosity

We imagine a distribution of a which is characterized by an amplitude o0 and a length scale L which exceeds the maximum scale of the turbulent pulsation, l. We denote the pulsating velocity by u the turbulent coefficient of diffusion, the coefficient of thermal conductivity and the effective turbulent kinematic viscosity are all expressed by the formula k = ul. For an initial uniform distribution of a, obviously,... 

If we assume the equality of turbulent kinematic viscosities of the two phases, the z component of equation of motion for the liquid phase can... 

Viscosity of the continuous phase, kg/m s Molecular kinematic viscosity of the liquid, m /s Turbulent kinematic viscosity, m /s... 

Cartesian coordinate vector (x, y, z) Molecular thermal diffusivity Turbulent thermal diffusivity Molecular kinematic viscosity Turbulent kinematic viscosity Karman constant Mass density See Eq. (26)... 

The shear stress r is related to the mean axial velocity of liquid through the sum of the molecular and turbulent kinematic viscosities ... 

Fig. 31. Turbulent kinematic viscosity v, as a function of column diameter Dy for bubble columns vi increases rapidly with Dy, but is little influenced by t/c Note that Ua = 1.9-93 cm/sec.

The correlation shown in Figs. 32 and 33 as curve A-A is the turbulent kinematic viscosity of the emulsion phase of fluidized cracking catalyst beds estimated by an indirect method explained in Section IV,C in relation to axial dispersion of the emulsion. The correlation is given in cm-sec units by... 

Fig. 32. Turbulent kinematic viscosity I l as a function of and Ug- Line A-A is for FCC-catalyst beds and line B-B for bubble columns of low-viscosity liquid. 

The turbulent kinematic viscosity v, has been introduced in the basic relation, Eq. (3-2), and is correlated with the operational variables through the experimental data (cf. Figs. 31-33) with a simplifying assumption that Vt is constant radially as an experimentally adjustable parameter. 

In the bubble column the velocity profile of recirculating liquid is shown in Fig. 27, where the momentum of the mixed gas and liquid phases diffuses radially, controlled by the turbulent kinematic viscosity Pf When I/l = 0 (essentially no liquid feed), there is still an intense recirculation flow inside the column. If a tracer solution is introduced at a given cross section of the column, the solution diffuses radially with the radial diffusion coefficient Er and axially with the axial diffusion coefficient E. At the same time the tracer solution is transported axially Iby the recirculating liquid flow. Thus, the tracer material disperses axially by virtue of both the axial diffusivity and the combined effect of radial diffusion and the radial velocity profile. 

When the velocity profile of the emulsion phase is similar to that of the liquid phase in a bubble column, Eq. (4-11) will apply to the fluidized catalyst bed. This similarity seems to be well justified as mentioned in Sections III,A,4-5, although there is no direct calculation of the turbulent kinematic viscosity from the measurement of velocity profile in the fluidized catalyst bed. 

The turbulent kinematic viscosity vt of the fluidized catalyst bed has been determined, as Eq. (3-3 la), from the use of axial dispersion coefficient This is a natural consequence of the analogy between the bubble column and the fluidized catalyst bed of good fluidity (such as in fluidized catalytic cracking). The mean gas holdup (Fig. 36) and the mean bubble velocity along the bed axis (Fig. 37) are reasonably well predicted by applying Eq. (3-3 la) for the fluidized cracking catalyst bed. 

By defining SCj j-b have replaced the problem of estimating with the problem of estimating the turbulent kinematic viscosity. To estimate z t rb e need to know the velocity profiles between the interface and the bulk fluid phase. For simple flow situations. 

The Navier-Stokes equation is written here for a Cartesian two-dimensional coordinate system where i and j represent the two axes. Accordingly, vi and vj are the velocity components in the directions i and j. P is the hydrostatic pressure, and v and vt are the moleciflar and the turbulent kinematic viscosity, respectively. For systems involving forced convection, the fluid flow equations are typically decoupled from the electrochemical process, and can be solved separately. 

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