Rate of turbulent kinetic energy dissipation

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 ... 

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 ... 

As an example, for steady, incompressible, and isothermal turbulent flows using the k- model, the independent equations are (1) the continuity equation, Eq. (5.61) (2) the momentum equation, Eq. (5.65) (3) the definition of the effective viscosity, /xeff (combination of Eq. (5.64) and Eq. (5.72)) (4) the equation of turbulent kinetic energy, Eq. (5.75) and (5) the equation for the dissipation rate of turbulent kinetic energy, Eq. (5.80). Thus, for a three-dimensional model, the total number of independent equations is seven. The corresponding independent variables are (1) velocity (three components) (2) pressure (3) effective viscosity (4) turbulent kinetic energy and (5) dissipation rate of turbulent kinetic energy. Thus, the total number of independent variables is also seven, and the model becomes solvable. 

Polzin et al. (1995) were able to find a scaling of the dissipation rate of turbulent kinetic energy in terms of the frequency distribution of energy within the deep-ocean internal wave field for wave fields that differed from the Garret-Munk model. 

The critical parent particle diameter defines the minimum particle size for a given dissipation rate of turbulent kinetic energy for which breakage can occur. The minimum daughter diameter defines the distance over which the turbulent normal stresses just balance the confinement forces of a parent particle of size d. The minimum diameter, therefore, gives the minimum length over which the underlying turbulence can pinch off a piece of the parent... 

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... 

Kronecker symbol thickness of shearing layer, m Dissipation rate of turbulent kinetic energy, m s Karman constant Viscosity, kg m s ... 

Daughter bubbles can be generated by introducing a mother bubble into a highly turbulent liquid flow. Martinez-Bazan et al. [9] found that the bubble size probability density function of the daughter bubbles depends not only on the size of the mother bubble, but also on the value of the dissipation rate of turbulent kinetic energy. 

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

In whichever approach, the common denominator of most operations in stirred vessels is the common notion that the rate e of dissipation of turbulent kinetic energy is a reliable measure for the effect of the turbulent-flow characteristics on the operations of interest such as carrying out chemical reactions, suspending solids, or dispersing bubbles. As this e may be conceived as a concentration of a passive tracer, i.e., in terms of W/kg rather than of m2/s3, the spatial variations in e may be calculated by means of a usual transport equation. 

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 ... 

This form is appealing because the first term in F.-,/2 can be interpreted as a gradient diffusion of turbulent kinetic energy, and the second is negative-definite (suggestive of dissipation of turbulence energy). However, the rate of entropy production is proportional to... 

For multiphase flow processes, turbulent effects will be much larger. Even operability will be controlled by the generated turbulence in some cases. For dispersed fluid-fluid flows (as in gas-liquid or liquid-liquid reactors), the local sizes of dispersed phase particles and local transport rates will be controlled by the turbulence energy dissipation rates and turbulence kinetic energy. The modeling of turbulent multiphase flows is discussed in the next chapter. 

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