Application to turbulent flows

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

Zijlema M (1996) On the construction of a third-order accurate monotone convection scheme with applications to turbulent flows in general domains. Int J Numer Methods Fluids 22 619-641... 

The application to pipe flow is not strictly valid because u (= fRjp) is constant only in regions close to the wall. However, equation 12.34 appears to give a reasonable approximation to velocity profiles for turbulent flow, except near the pipe axis. The errors in this region can be seen from the fact that on differentiation of equation 12.34 and putting y = r, the velocity gradient on the centre line is 2.5u /r instead of zero. 

Numerous turbulent mass-transfer relationships are given in Eqs. (39)-(50), Table VII. Although the most important ones in practical applications are those for channels and tubes, several other configurations also have been investigated because of their hydrodynamic interest. Generally, it is not possible to predict mass-transfer rates quantitatively by recourse to turbulent flow theory. An exception to this is for the region of developing mass transfer, where a Leveque-type correlation between the mass-transfer coefficient and friction coefficient/can be established ... 

On the theoretical side, Dmitriev and Bonchkovskaya (D8) have shown that in principle turbulence should spread from waves. Kapitsa (K9) has calculated a general tensor quantity, termed the coefficient of wavy transfer, which is applicable to any flow with periodic disturbances, such as pulsations or surface waves. This treatment predicts an appreciable increase in the rates of heat and mass transfer in wavy films, though this increase does not appear to be as large as that observed experimentally under certain conditions. 

Equation (6.6) is notably named the Darcy equation [8]. It is applicable to both laminar flow, Re < 2,000, and to turbulent flow, Re > 10,000. Two restrictions govern this equation, however ... 

AP equation arising from simultaneous turbulent kinetic and viscous energy losses that is applicable to all flow types. Ergun s equation relates the pressure drop per unit depth of packed bed to characteristics such as velocity, fluid density, viscosity, size, shape, surface of the granular solids, and void fraction. The original Ergun equation is ... 

LIFS is now ready to begin being seriously applied to turbulent flows. For some species, sufficient information already exists to obtain quantitative results of direct applicability, although a major effort to collect and collate collision data must continue. Reliable equipment is available... 

The laminar stationary flow of an incompressible viscous liquid through cylindrical tubes can be described by Poiseuille s law this description was later extended to turbulent flow. Flowing patterns of two immiscible phases are more complex in microcapillaries various patterns of liquid-liquid flow are described in more detail in Chapter 4.3, while liquid-gas flow and related applications are discussed in Chapter 4.4. 

It turns out that Eq. (5-56) can also be applied to turbulent flow over a flat plate and in a modified way to turbulent flow in a tube. It does not apply to laminar tube flow. In general, a more rigorous treatment of the governing equations is necessary when embarking on new applications of the heat-trans-fer-fluid-friction analogy, and the results do not always take the simple form of Eq. (5-56). The interested reader may consult the references at the end of the chapter for more information on this important subject. At this point, the simple analogy developed above has served to amplify ouf understanding of the physical processes in convection and to reinforce the notion that heat-transfer and viscous-transport processes are related at both the microscopic and macroscopic levels. 

Application of the Governing Equations to Turbulent Flow 123 Prandtl s Mixing-Length Model... 

Application of the Governing Eqnations to Turbulent Flow 127 Finally, integrating with respect to y+, we obtain a logarithmic velocity profile ... 

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