Kinetic energy, turbulent

One-equation models relax the assumption that production and dissipation of turbulence are equal at all points of the flow field. Some effects of the upstream turbulence are incorporated by introducing a transport equation for the turbulence kinetic energy k (20) given by... 

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... 

Eigure 20 compares the predictions of the k-Q, RSM, and ASM models and experimental data for the growth of the layer width 5 and the variation of the maximum turbulent kinetic energy k and turbulent shear stress normalized with respect to the friction velocity jp for a curved mixing layer... 

The fundamental principle of Hquid disiategration Hes ia the balance between dismptive and cohesive forces. The common dismptive forces ia atomizer systems iaclude kinetic energy, turbulent fluctuation, pressure fluctuation, iaterface shearing, friction, and gravity. The cohesive forces within the Hquid are molecular bonding, viscosity, and surface tension. 

The balanced equation for turbulent kinetic energy in a reacting turbulent flow contains the terms that represent production as a result of mean flow shear, which can be influenced by combustion, and the terms that represent mean flow dilations, which can remove turbulent energy as a result of combustion. Some of the discrepancies between turbulent flame propagation speeds might be explained in terms of the balance between these competing effects. 

A j, = Model constant for reaction n C = Concentration of species i (mole m ) k = Turbulent kinetic energy density (m s )... 

HOTM AC/RAPTAD contains individual codes HOTMAC (Higher Order Turbulence Model for Atmospheric Circulation), RAPTAD (Random Particle Transport and Diffusion), and computer modules HOTPLT, RAPLOT, and CONPLT for displaying the results of the ctdculalinns. HOTMAC uses 3-dimensional, time-dependent conservation equations to describe wind, lempcrature, moisture, turbulence length, and turbulent kinetic energy. 

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

Figure 15.5 Measured and simulated turbulent kinetic energies (normalized with the impeller tip speed) at the impeller plane in a stirred tank reactor (From [17]).

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. 

Therefore, it is yet to be clarified whether the description of the particle destruction process with Eqs. (2-4) or the simpfification in the estimate of energy dissipation from the measured turbulent kinetic energy produces these differ-... 

As a result, the turbulent-flow field in a stirred vessel may be far from isotropic and homogeneous. Some of the cornerstones of turbulence theory, however, start from the assumption that production and dissipation of turbulent kinetic energy balance locally. In many chemical engineering flows, this... 

Fig. 3. Velocity vector fields and levels of turbulent kinetic energy in a plane midway between two baffles in a stirred vessel, according to LDA data (a) a RANS-based simulation (b) and two LES (c) and (d). Reproduced with permission from Hartmann et al. (2004a).

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. 

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