Particle turbulent kinetic energy

The particle turbulent kinetic energy is governed by its own transport equation. Similarly to the derivation of the -equation, the p-equation is given by... 

Furthermore, it is assumed that there is no particle fluctuation on the wall so that the boundary condition of particle turbulent kinetic energy at the wall is given by... 

One of the earliest models for turbulence modulation in homogeneous dilute particleladen flows is Hinze s model (1972), in which the assumption of vortex trapping of particles is employed. On the basis of this model, the particle turbulent kinetic energy, kp, is determined by the local gas turbulent kinetic energy k as... 

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. 

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-... 

Fluctuating components of gas velocity are selected from a Gaussian distribution with variance derived from the local value of turbulent kinetic energy. Integration of the particle equation of motion yields the particle position at any given instant in time. 

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. 

Subscript 1 indicates continuous phase and 2 indicates dispersed phase. Cd is a parameter of the standard k-s model (0.09), k is turbulent kinetic energy and si is turbulent energy dissipation rate. The eddy lifetime seen by dispersed phase particles will in general be different from that for continuous phase fluid particles due to the so-called crossing-trajectory effect (Csnady, 1963). This can be expressed in the form ... 

To determine the energy contained in eddies of different scales, a distribution function of the kinetic energy for eddies in turbulent flows is required. A Maxwellian distribution function may be a natural and consistent choice as the eddy velocity is assumed to follow this distribution [66], but Luo and Svendsen [74] preferred an empirical energy-distribution density function for fluid particles in liquid developed by Angelidou et al [1[. The turbulent kinetic energy distribution of eddies with size A is approximated as follows ... 

The daughter bubble size is thus limited by two constraints. The capillary pressure constraint states that if the dynamic pressure of the turbulent eddy Pc v x exceeds the capillary pressure aijd", the fluid particle is deformed and finally breaks up resulting in a minimum breakage fraction /vm, (or bubble size dj min) [69]. d denotes the diameter of the smaller daughter size (or two times the minimum radius of curvature). When breakage occurs, the d3mamic pressure induced by the eddy turbulence kinetic energy satisfy the criterion ... 

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... 

In this section the application of multiphase flow theory to model the performance of fluidized bed reactors is outlined. A number of models for fluidized bed reactor flows have been established based on solving the average fundamental continuity, momentum and turbulent kinetic energy equations. The conventional granular flow theory for dense beds has been reviewed in chap 4. However, the majority of the papers published on this topic still focus on pure gas-particle flows, intending to develop closures that are able to predict the important flow phenomena observed analyzing experimental data. Very few attempts have been made to predict the performance of chemical reactive processes using this type of model. 

Proper boundary conditions are generally required for the primary variables like the gas and particle velocities, pressures and volume fractions at all the vessel boundaries as these model equations are elliptic. Moreover, boundary conditions for the granular temperature of the particulate phase is required for the PT, PGT and PGTDV models. For the models including gas phase turbulence, i.e., PGT and PGTDV, additional boundary conditions for the turbulent kinetic energy of the gas phase, as well as the dissipation rate of the gas phase and the gas-particle fluctuation covariance are required. The... 

Slurry bubble column reactor for methanol and other hydrocarbons productions from synthesis gas is an issue of interest to the energy industries throughout the world. Computational fluid dynamics (CFD) is a recently developed tool which can help in the scale up. We have developed an algorithm for computing the optimum process of fluidized bed reactors. The mathematical technique can be applied to gas solid, liquid-solid, and gas-liquid-solid fluidized bed reactors, as well as the LaPorte slurry bubble column reactor. Our computations for the optimum particle size show that there is a factor of about two differences between 20 and 60 pm size with maximum granular-like temperature (turbulent kinetic energy) near the 60 pm size particles. 

The research presented opens a new area for flow control in combustion systems — mixing by active feedback control. Considered herein is a two-dimensional (2D) jet, and simple control strategies are proposed that, with small control effort, generate a large increase in turbulent kinetic energy of the jet flow. The control is applied by microjets or microflaps at the lip of the jet and requires only the meeisurement of pressure at the jet lip. It is demonstrated that the controller enhances mixing of massless particles, particles with mass, and a passive scalar. 

Since m 2 is proportional to the total turbulent kinetic energy, the total energy of the turbulence is important in the early dispersion. After long times the largest eddies will contribute to Ru and Ru will not go to zero until the particle can escape the influence of the largest eddies. From its definition, K,j has the dimensions of a diffusivity, since as t —> oo... 

In this context the number of particles depositing for given hydraulic conditions (C — Ceq) must be determined. This means that the characteristics of the particulates must be correlated to local parameters describing the turbulent flow field. This leads to a critical sedimentation velocity i s.cr for a particle. It is derived on the basis of an energy balance the potential energy loss attributable to settling in a non-turbulent system must equal the turbulent kinetic energy that must be imparted on the particle in a turbulent flow system to prevent sedimentation (29). 

Ceq = equilibrium concentration of the suspended matter Co = initial concentration of the suspended matter Cd = coefficient of drag D = depth of the flow (m) ds = diameter of particle resp. floe (m) g= gravitational acceleration (m/sec ) k = turbulent kinetic energy per unit mass (m /sec )... 

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. 

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