Turbulent kinetic energy dissipation rate

Turbulent kinetic energy dissipation rate density (m s ) = Turbulent viscosity (kg m sec )... 

The source terms in the equation for the turbulent kinetic energy dissipation rate are implemented through the source terms Sp and Sc in the following way ... 

Various models for bubble breakage and coalescence rates are presented in the literature. These rates usually depend on physical properties, such as densities, viscosities and surface tension, and on turbulence properties, most commonly the turbulent kinetic energy dissipation rate. To calculate local bubble size distributions, also local physical properties and turbulence level should be used. This can be done via CFD (Alopaeus et al. 1999,2002 Keskinen and Majander 2000). 

So the turbulent kinetic energy dissipation rate at the end of the jet can be calculated from quantities that are known the jet velocity at the nozzle, the nozzle diameter, and the path length. 

The gas phase turbulent kinetic energy dissipation rate equation yields. 

The second length scale characterizing turbulence is that over which molecular effects are significant it can be introduced in terms of a representative rate of dissipation of velocity fluctuations, essentially the rate of dissipation of the turbulent kinetic energy. This rate of dissipation, which is given by the symbol e0, is... 

Step 4 Set the boundary conditions as follows. The centerline, inlet velocity, and exit velocity/pres sure are set as in the laminar case slip/symmetry, v = 2, Normal flow/ Pressure, p = 0. The wall boundary condition, though, is set to the Logarithmic wall function. This is an analytic formula for the velocity, turbulent kinetic energy, and rate of dissipation, as determined by experiment (Deen, 1998, pp. 527-528). 

Here r is the position vector, v is the velocity vector, and k and e are the local turbulence kinetic energy and rate of dissipation, respectively. 

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. 

Variable e is the rate of turbulent kinetic energy dissipation per unit mass, which, based on [8.14], can be written as ... 

In Chapter 8, we characterized the large scales of turbulence by introducing the tnrbnlent velocity Mnns and the integral scale It. These two quantities are used to quantify two key processes of turbulence, the turbulent diffusion coefficient Kt Urtcdt (equation [8.29] of Chapter 8) and the kinetic energy dissipation rate in a turbulent flow ... 

Let us first consider the case of steady-state turbulence (the case of unsustained turbulence is discussed later). An energy source has to inject some energy at a rate equal to the rate s of turbulent kinetic energy dissipation. That is what ideally occurs in a perfectly stirred reactor of volume V (Figure 11.3) in which a moving... 

The starting point of Kolmogorov s theory is eqnation [11.1], which stipnlates that the rate of turbulent kinetic energy dissipation is independent of viscosity and hence depends only on Ums and It ... 

It was shown by Taylor [ 160] that an analysis of the dissipation term occurring within the turbulent kinetic energy balance equation (derived later in this chap) shows that in isotropic turbulence the energy dissipation rate is equal to ... 

The turbulent kinetic energy is calculated from equation 41. Equation 43 defines the rate of energy dissipation, S, which is related to the length scale via... 

Turbulent velocity fluctuations ultimately dissipate their kinetic energy through viscous effects. MacroscopicaUy, this energy dissipation requires pressure drop, or velocity decrease. The ener dissipation rate per unit mass is usually denoted . For steady ffow in a pipe, the average energy dissipation rate per unit mass is given by... 

The rate of dissipation of turbulent kinetic energy, s, is more difficult to measure. 

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

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