Velocity density functions

Panicles entrained in the airstream deposit along the airway as a function of size, density, airstream velocity, and breathing frequency. Sizes of rougjily spherical or irregularly shaped particles arc commonly characterized by relating the settling velociiy of the particle to that of an idealized spherical particle. For example, an irregular particle which settles at the same rate as a 5 pm spherical particle has a mean mass aerodynamic diameter (MMAD) of. 5 pm. Since spherical particle mass, is a function of particle diameter, J... 

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

Anderson (A2) has derived a formula relating the bubble-radius probability density function (B3) to the contact-time density function on the assumption that the bubble-rise velocity is independent of position. Bankoff (B3) has developed bubble-radius distribution functions that relate the contacttime density function to the radial and axial positions of bubbles as obtained from resistivity-probe measurements. Soo (S10) has recently considered a particle-size distribution function for solid particles in a free stream ... 

Fig. 6 shows the FFT spectrum for calculated bed pressure drop fluctuations at various centrifugal accelerations. The excess gas velocity, defined by (Uo-U ,, was set at 0.5 m/s. Here, 1 G means numerical result of particle fluidization behavior in a conventional fluidized bed. In Fig. 6, the power spectrum density function has typical peak in each centrifugal acceleration. However, as centrifugal acceleration increased, typical peak shifted to high frequency region. Therefore, it is considered that periods of bubble generation and eruption are shorter, and bubble velocity is faster at hi er centrifugal acceleration. 

Since the histogram gives a probability density function of the particle position, the correlation in the velocities Vy and V2j in the j-direction causes the change in the shape of the histogram plotted against Vy and V2j, due to the different coefficient y — Pj in... 

The spin-displacement density function, c (z, Z), and the normalized displacement distribution function, P(z, Z), can be converted readily into the joint spin-velocity density function, q(z, vn), and the normalized velocity distribution function, P(z, vn), respectively, with the net velocity vn defined as vn = Z/A. Once the velocity density function is determined for each of the volume elements, the superficial average velocity, v, is calculated by [23] ... 

The porosity cj)(z) for the voxel at z, is determined as described in Section 4.1.4.1. Note that the observed spin-displacement density function, q(z, vn), is not actually associated with the intrinsic value of spin density, q(z), due to the transverse and longitudinal relaxation. However, this does not affect the calculated average velocity given in Eq. (4.1.23) because the spin density terms in the denominator and nominator cancel each other. 

An example of the spin-velocity density function is demonstrated in Figure 4.1.6. A velocity imaging experiment was performed on water flowing through a 6-mm diameter tube. The velocity density function was spatially resolved along the axial direction of the tube, denoted by z in the figure. It is observed that the velocity density function has a steep peak at zero velocity when the fluid is not flowing, but is shifted to a positive velocity when the flow rate was increased to 2.5 mL min-1. 

In solving open channel flow equations, the THINC I code (Zernick et al., 1962) was the first calculational technique capable of satisfactorily assigning inlet flows to the assemblies within a semiopen core. In the THINC I approach, it was recognized that the total pressure distribution at the top of the core region is a function of inlet pressure, density, and velocity distributions. This functional dependence can be expressed as,... 

Molecules in a gas are in constant motion at speeds on the order of the speed of a rifle bullet at equilibrium there is no net flow of gas and the motion is random. This motion produces collisions of the molecules with the walls of the vessel containing the gas, with a change in momentum of the gas molecule resulting from each collision. This change in momentum produces a force per unit area, or pressure on the wall. Consider those molecules with the component of velocity in the x direction between the value of vx and vx + dvx. The x direction is defined as the direction normal to the wall. The fraction of molecules with the x component of velocity in this range, denoted dN(vx)/N, is given by the density function, f(vx), where... 

A partial differential equation is then developed for the number density of particles in the phase space (analogous to the classical Liouville equation that expresses the conservation of probability in the phase space of a mechanical system) (32>. In other words, if the particle states (i.e. points in the particle phase space) are regarded at any moment as a continuum filling a suitable portion of the phase space, flowing with a velocity field specified by the function u , then one may ask for the density of this fluid streaming through the phase space, i.e. the number density function n(z,t) of particles in the phase space defined as the number of particles in the system at time t with phase coordinates in the range z (dz/2). 

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where... 

In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function /s(r, v), which can be defined as the number density of the solid particles in the 6D coordinate and velocity space. That is, /s(r, v, t) dv dr is the average number of particles to be found in a 6D volume dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by... 

For a fixed point in space x and a given instant t, the random velocity field Ui(x, t) can be characterized by a one-point probability density function (PDF) fufiVi x, t) defined by4... 

An alternative method for describing the SGS velocity fluctuations is to define the filter density function (FDF) first proposed by Pope (1990) ... 

The resolved velocity U would then be found from an LES simulation, and the LES velocity PDF (defined in Section 4.2) would be written in terms of the unresolved velocity uL Alternatively, the filtered density function (FDF) approach can be used with a variant... 

Note that the RANS formulation used in (B.44) and (B.45) can easily be extended to the LES, as outlined in Section 5.10. Moreover, by following the same steps as outlined above, DQMOM can be used with the joint velocity, composition PDF transport equation. Finally, the reader can observe that the same methodology is applicable to more general distribution functions than probability density functions. Indeed, DQMOM can be applied to general population balance equations such as those used to describe multi-phase flows. 

Anand, M. S., A. T. Hsu, and S. B. Pope (1997). Calculations of swirl combustors using joint velocity-scalar probability density function methods.. 47.4.4 Journal 35, 1143-1150. 

Gicquel, L. Y. M., P. Givi, F. A. Jaberi, and S. B. Pope (2002). Velocity filtered density function for large eddy simulation of turbulent flows. Physics of Fluids 14, 1196-1213. 

Erratum Application of the velocity-dissipation probability density function model to inhomogeneous turbulent flows [Phys. Fluids A 3, 1947 (1991)]. Physics of Fluids A Fluid Dynamics 4, 1088. 

Once mass flux is expressed as a function of density and velocity, it becomes... 

Hybrid compounds containing heterocyclic nitramine and em-dinitro functionality represent a class of high performance energetic materials. Such compounds frequently exhibit higher heats of formation, crystal density, detonation velocity and pressure, and better oxygen balance compared to analogous aromatic compounds. 

The present study is to elaborate on the computational approaches to explore flame stabilization techniques in subsonic ramjets, and to control combustion both passively and actively. The primary focus is on statistical models of turbulent combustion, in particular, the Presumed Probability Density Function (PPDF) method and the Pressure-Coupled Joint Velocity-Scalar Probability Density Function (PC JVS PDF) method [23, 24]. 

The observed flame features indicated that changing the atomization gas (normal or preheated air) to steam has a dramatic effect on the entire spray characteristics, including the near-nozzle exit region. Results were obtained for the droplet Sauter mean diameter (D32), number density, and velocity as a function of the radial position (from the burner centerline) with steam as the atomization fluid, under burning conditions, and are shown in Figs. 16.3 and 16.4, respectively, at axial positions of z = 10 mm, 20, 30, 40, 50, and 60 mm downstream of the nozzle exit. Results are also included for preheated and normal air at z = 10 and 50 mm to determine the effect of enthalpy associated with the preheated air on fuel atomization in near and far regions of the nozzle exit. Smaller droplet sizes were obtained with steam than with both air cases, near to the nozzle exit at all radial positions see Fig. 16.3. Droplet mean size with steam at z = 10 mm on the central axis of the spray was found to be about 58 /xm as compared to 81 pm with preheated air and 96 pm with normal unheated air. Near the spray boundary the mean droplet sizes were 42, 53, and 73 pm for steam, preheated air, and normal air, respectively. The enthalpy associated with preheated air, therefore, provides smaller droplet sizes as compared to the normal (unheated) air case near the nozzle exit. Smallest droplet mean size (with steam) is attributed to decreased viscosity of the fuel and increased viscosity of the gas. 

This equation shows that for a given collection efficiency, the precipitator size is inversely proportional to particle drift velocity and directly proportional to gas flow rate. Increasing the gas density (migration velocity is a function of gas viscosity) by reducing its temperature or increasing the pressure will reduce the precipitator size. However, theory does not account for gas velocity. This is a variable that influences particle re-entrainment and the drift velocity. This typically requires an ESP design at lower velocities than predicted in theory. 

Abbreviations MD, molecular dynamics TST, transition state theory EM, energy minimization MSD, mean square displacement PFG-NMR, pulsed field gradient nuclear magnetic resonance VAF, velocity autocorrelation function RDF, radial distribution function MEP, minimum energy path MC, Monte Carlo GC-MC, grand canonical Monte Carlo CB-MC, configurational-bias Monte Carlo MM, molecular mechanics QM, quantum mechanics FLF, Hartree-Fock DFT, density functional theory BSSE, basis set superposition error DME, dimethyl ether MTG, methanol to gasoline. 

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