Rate of kinetic energy dissipation

Te can be interpreted as the time necessary to decrease significantly the size of the structure having initially size A. Another interpretation is that it is the time constant for the rate of kinetic energy dissipation when the cascade is in dynamic equilibrium (steady state), and a third interpretation could be that it is an estimate of a time necessary to transfer the kinetic energy from the scale A to the scale of dissipation. 

Where e is the rate of kinetic energy dissipation per unit mass and Cx is of order 1. Equation (36) is valid for drops whose diameter falls within the inertial subrange of turbulence, < d < L, where L is the integral scale of turbulence and q = is the Kolmogorov microscale of turbulence. 

Turbulence reinforces viscous dissipation phenomena. We have shown in section 2.8 of Chapter 2 (equation [2.45]) that the rate of kinetic energy dissipation per unit mass is, at any point in the flow ... 

We have seen in Chapters 2 and 4 that turbulence dramatically strengtherrs the dissipation of kinetic energy. This property results from the definition of the instantaneous rate of kinetic energy dissipation per unit volume ... 

Expression [8.14] of the rate of kinetic energy dissipation encompasses within it the property whereby a turbulent flow dissipates more than a lamirrar flow. For a laminar flow with characteristic scales of velocity Ums and length f, the rate of dissipation derived from [8.13] is eiam Hums I (tV Denoting by etu the rate of dissipation [8.14] in a turbulent flow leads to ... 

In the case of laminar flow in a pipe, work is done by the shear stress component rTX and the rate of doing work is the viscous dissipation rate, that is the conversion of kinetic energy into internal energy. The rate of viscous dissipation per unit volume at a point, is given by... 

In turbulent flow, there is direct viscous dissipation due to the mean flow this is given by the equivalent of equation 1.98 in terms of the mean values of the shear stress and the velocity gradient. Similarly, the Reynolds stresses do work but this represents the extraction of kinetic energy from the mean flow and its conversion into turbulent kinetic energy. Consequently this is known as the rate of turbulent energy production ... 

The propagation of pressure waves such as acoustic wave, shock wave, and Prandtl-Meyer expansion through a gas-solid suspension is a phenomenon associated primarily with the transfer of momentum although certain processes of energy transfer such as kinetic energy dissipation and heat transfer between gas and solids almost always occur. Typical applications of the pressure wave propagation include the measurements of the solids concentration and flow rate by use of acoustic devices as well as detonation combustion such as in a rocket propellant combustor or in the barrel of a gun. 

It is of interest to note that if the convection and diffusion terms are negligible in the turbulence kinetic energy equation, i.e., if the rate of production of kinetic energy is just equal to the rate of dissipation of turbulence kinetic energy, Eq. (5.62) reduces to ... 

Pulsations of less scale possess significantly less energy and are not able to deform particles of disperse phase. Pulsations of big scale carry the elements of disperse phase and do not deform their surface. The fundamental problem under estimation of disperse inclusions of multiphase systems in tubular turbulent apparatus according to (1.23) is calculation of rate of turbulence kinetic energy dissipation e. It requires the development of model describing disperse processes in turbulent flows. 

In this equation, I is the unit tensor, is the pseudo-Fourier fluctuating kinetic energy flux, and y is the dissipation rate of granular energy due to inelastic particle-particle collisions. In the KTGF, coUisions are assumed binary and quasi-instantaneous and do not take long-term and multiple particle contact into account (which is the case in the dense part of the fluidized bed). To correct for this shortcoming, the solids phase viscosity Ps and the solids phase pressure are split up into a kinetic part and a frictional part. 

This quantity is always negative. Therefore, it is justified in referring to it as the rate of dissipation of kinetic energy, as it can only contribute to decrease kinetic energy. 

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