Local instantaneous velocities

For multiphase flows perturbed by the presence of particles to obtain a turbulence like behavior the local instantaneous velocity of the continuous phase can for example be decomposed adopting the Reynolds averaging procedure (i.e., other methods including time-, volume-, ensemble-, and Favre averaging have been used as well) and expressed as Vg = Vg- - < v >g, where v(. is the fluctuating component of the continuous phase velocity. Introducing the peculiar velocity for the dispersed phase this relation can be re-arranged as ... 

For undisturbed turbulent flows the local instantaneous velocity of the continuous phase have been decomposed in various ways, not necessarily in accordance with the familiar Reynolds - and Favre averaging procedures. 

Measurements of local instantaneous velocity, density, and mass flow of phases of a gas-solid suspension are needed in determining transport properties, validating theoretical predictions, and formulating design procedures. 

Once position versus time data are obtained and stored on a PC hard disk, time differentiation of displacements yields local instantaneous velocities. Ensemble averaged velocity, rms velocity and other turbulence quantities are computed after acquiring the data for a sufficient length of time (e.g. several hours) at fixed operating conditions. 

The population balance in equation 2.86 employs the local instantaneous values of the velocity and concentration. In turbulent flow, there are fluctuations of the particle velocity as well as fluctuations of species and concentrations (Pope, 1979, 1985, 2000). Baldyga and Orciuch (1997, 2001) provide the appropriate generalization of the moment transformation equation 2.93 for the case of homogeneous and non-homogeneous turbulent particle flow by Reynolds averaging... 

Equation 2.101 enables calculation of local average quantities such as moments of the particle size distribution. Baldyaga and Orciuch (2001) review expressions for local instantaneous values of particle velocity and diffusivity of particles, Z)pT, required for its solution and recover the distribution using the method of Pope (1979). 

The instantaneous velocity is then used to estimate the instantaneous local static pressure using Bernoulli s equation of the following form ... 

Eulerian equations for the dispersed phase may be derived by several means. A popular and simple way consists in volume filtering of the separate, local, instantaneous phase equations accounting for the inter-facial jump conditions [274]. Such an averaging approach may be restrictive, because particle sizes and particle distances have to be smaller than the smallest length scale of the turbulence. Besides, it does not account for the Random Uncorrelated Motion (RUM), which measures the deviation of particle velocities compared to the local mean velocity of the dispersed phase [280] (see section 10.1). In the present study, a statistical approach analogous to kinetic theory [265] is used to construct a probability density function (pdf) fp cp,Cp, which gives the local instantaneous probable num-... 

Table XXII shows some of the parameters used to characterize the microstructure of the combustion wave. In general, the parameters can be divided into two groups. The first group characterizes the shape of the combustion front, and includes the local [F(y,f)] and average [F(r)] front profiles, as well as the front dispersion, CTp, which is a measure of roughness of the combustion front. The second group describes the combustion front propagation at the microscopic level. For this, the instantaneous, U(y,t) and average, U, velocities of the combustion wave, as well as the dispersion of the instantaneous velocities, ar calculated.

Figure 19 shows, as an example, the evolution and propagation of bubbles in a 2D gas-fluidized bed with a heated wall. The bubbles originate from an orifice near the heated right wall (air injection velocity through the orifice s 5.25 m/s, which corresponds to 2 Uj. The instantaneous axial profile of the wall-to-bed heat transfer coefficient is included in Fig. 19. From this figure the role of the developing bubble wake and the associated bed material refreshment along the heated wall, and its consequences for the local instantaneous heat transfer coefficient, can be clearly seen. In this study it became clear that CFD based models can be used as a tool (i.e., a learning model) to gain insight into complex system behavior.

To understand atmospheric dispersion, therefore, it is necessary to understand the nature of turbulent flows and the structure of the atmosphere near the Earth s surface. Turbulent flows exhibit apparently random fluctuations in local velocity and pressure (Mathieu and Scott 2000). These fluctuations appear over a wide range of length and time scales. Reynolds (1895) drew an analogy between the behavior of the velocity in a turbulent flow and the velocity of the individual molecules in a gas, leading to the definition of the instantaneous velocity at a point as having a mean and a fluctuating component ... 

V local instantaneous mixture velocity, mass average velocity (m/s) modified mass diffusion velocity for species s (m/s) species velocity of molecules of species s generated by chemical reaction (m/s)... 

Equation (57) employs the local instantaneous values of velocity and concentration. To describe the effects on the process of turbulent fluctuations of the particle velocity as well as species and particle concentrations, one can use Reynolds-averaged form of the population balance... 

In most cases one is interested in fluid flows at scales that are much larger than the distance between the molecules. The value of the molecular mean free path in air at room temperature and 1 atm of pressure is A = 6.7 x 10-8 m and in water A = 2.5 x 10-10 m. When the Knudsen number - defined as the ratio of the molecular mean-free-path to a characteristic length scale of the flow (e.g. the size of the smallest eddies) - is small, the fluid can be described as a continuous medium in motion. In this continuum approximation the flow can be characterized by the velocity field v(x, t) representing the instantaneous velocity of infinitesimal fluid elements at time t and at position x. Fluid elements represent small volumes of fluid that are much smaller than the smallest characteristic scale of the flow, but sufficiently large to contain a large number of molecules so that a well defined local velocity exists and molecular fluctuations can be neglected. 

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