Density and velocity

A unifonn monoenergetic beam of test or projectile particles A with nnmber density and velocity is incident on a single field or target particle B of velocity Vg. The direction of the relative velocity m = v -Vg is along the Z-axis of a Cartesian TTZ frame of reference. The incident current (or intensity) is then = A v, which is tire number of test particles crossing unit area nonnal to the beam in unit time. The differential cross section for scattering of the test particles into unit solid angle dO = d(cos vji) d( ) abont the direction ( )) of the final relative motion is... 

Momentum Flow Meters. Momentum flow meters operate by superimposing on a normal fluid motion a perpendicular velocity vector of known magnitude thus changing the fluid momentum. The force required to balance this change in momentum can be shown to be proportional to the fluid density and velocity, the mass-flow rate. 

All parameters of interest with respect to fragmentation will be discussed. The extent of damage or injury caused by these fragments is, however, not covered in this volume. (Parameters of the terminal phase include first, fragment density and velocity at impact, and second, resistance of people and structures to fragments.)... 

The emphasis in this chapter is on the generalization of piston flow to situations other than constant velocity down the tube. Real reactors can closely approximate piston flow reactors, yet they show many complications compared with the constant-density and constant-cross-section case considered in Chapter 1. Gas-phase tubular reactors may have appreciable density differences between the inlet and outlet. The mass density and thus the velocity down the tube can vary at constant pressure if there is a change in the number of moles upon reaction, but the pressure drop due to skin friction usually causes a larger change in the density and velocity of the gas. Reactors are sometimes designed to have variable cross sections, and this too will change the density and velocity. Despite these complications, piston flow reactors remain closely akin to batch reactors. There is a one-to-one correspondence between time in a batch and position in a tube, but the relationship is no longer as simple as z = ut. 

A priori, neither the value of O nor the values of density and velocity are known at the faces of the control volume. They have to be determined via interpolation from their values at neighboring nodes. A simple approximation would be... 

For the computation of compressible flow, the pressure-velocity coupling schemes previously described can be extended to pressure-velocity-density coupling schemes. Again, a solution of the linearized, compressible momentum equation obtained with the pressure and density values taken from a previous solver iteration in general does not satisfy the mass balance equation. In order to balance the mass fluxes into each volume element, a pressure, density and velocity correction on top of the old values is computed. Typically, the detailed algorithms for performing this task rely on the same approximations such as the SIMPLE or SIMPLEC schemes outlined in the previous paragraph. 

So far, the majority of granular matter studies by NMR/MRI have used liquid state proton measurements in solid materials. Because proton signals are relatively insensitive to chemical environment through the chemical shift effect and because the physical environments are relatively similar in all liquids, the resonance frequency and the NMR linewidth are not good indicators of granular parameters such as particle density and velocity. 

H, p, V = mean values in AZ of enthalpy, density, and velocity, respectively w = flow exchange rate per unit length by mixing Kmz = pressure loss coefficient for channel m in interval Z P = pressure... 

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,... 

Now the pressure drops along the pipe because of energy dissipation (e.g., friction), just as for an incompressible fluid. However, because the density decreases with decreasing pressure and the product of the density and velocity must be constant, the velocity must increase as the gas moves through the pipe. This increase in velocity corresponds to an increase in kinetic energy per unit mass of gas, which also results in a drop in temperature. There is a limit as to how high the velocity can get in a straight pipe, however, which we will discuss shortly. 

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... 

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

Density and velocity can change as the fluid flows along the axial or z direction. There are now two independent variables time t and position z. Density and... 

Wang, Y. Wang, C. Y. Comparing variable density with constant density models for polymer electrolyte fuel cells. ECEC Technical Report 2003-02-, Penn State University University Park, PA, 2003. See also Modeling polymer electrolyte fuel cells with large density and velocity changes. J. Electrochem. Soc., in press. 

Assume that to the left of the combustor outlet boundary, rr = 0, there exists a stationary solution of the Euler equations p = po, P = Po, u = uq, where po, po, and Mo are the constant pressure, density, and velocity. Flow velocity has a single nonzero component, mq, along the x axis. The flow is assumed subsonic, i.e., M = uq/cq < 1, where cq is the speed of sound. We consider the solution of the nonstationary Euler equations and linearize the problem in the vicinity of the stationary solution by assuming that... 

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. 

W.L. Murray J. Plant, "A Method for the Study of Solid Explosives and Other Solid (Including Porous) Materials When Subjected to Shock Waves , 4tbONRSympDeton (1965), 355-65. Description is given of a method for determining simultaneously the pressure, density and velocity in shock waves in solid (including porous) materials. The shock was poduced in a cylinder of material using the solid "barrier technique which is usually... 

Table 3.3 Some aluminized explosive formulations and their density and velocity of detonation .

For this purpose, a new type of "unified model atmospheres" has been developed at the Munich Observatory by R. Gabler (1986) and A. Wagner (1986) in cooperation with J. Puls, A. Pauldrach and R.P. Kudritzki. These NLTE model atmospheres are spherically extended, in radiative equilibrium, and include the density and velocity distribution of radiation driven winds. The spectra of H and He lines are then calcu-... 

The pressure is 60Torr, at which the saturation temperature of steam is 106°F. The superheat of 24°F is neglected in figuring the vapor density and velocity. 

Table I indicated that detonations were characterized by pressure increases, density increases, and velocity decreases while deflagrations were characterized by pressure decreases, density decreases, and velocity increases. The magnitude of these changes are required to be detennined experimentally at the present time, even though their limits may be obtained analytically. The experimental results of all investigators cannot be discussed here hence, only selected data showing the trends of pressure, density, and velocity changes (and their interrelationship) are presented.

In the broadest sense, I found the analogy with fluid mechanics to be very helpful. Just as kinematics provides the geometrical framework of fluid mechanics by exploring the motions that are possible, so also stoicheiometry defines the possible reactions and the restrictions on them without saying whether or at what rate they may take place. When dynamic laws are imposed on kinematic principles, we arrive at equations of motion so, also, when chemical kinetics is added to stoicheiometry, we can speak about reaction rates. In fluid mechanics different materials are distinguished by their constitutive relations and allow equations for the density and velocity to be formulated thence, various flow situations are examined by adding appropriate boundary conditions. Similarly, the chemical kinetics of the reaction system allow the rates of reaction to be expressed in terms of concentrations, and the reactor is brought into the picture as these rates are incorporated into appropriate equations and their boundary conditions. 

It should be pointed out, that two dynamites of the same strength do not necessarily produce the same blasting action in mines, quarries, etc. This is due to the fact that props other than strength, particularly density and velocity of detonation, have a distinct influence on their performance... 

The general character of the motion is shown in Fig. 1. A shock wave propagates in an unperturbed cold gas the maximum compression, which is dependent on the adiabatic index of the gas, is reached at the front of this wave. The velocity of propagation of the shock wave and the mass velocity at the front are related in an elementary way to the pressure of the shock wave. Behind the front the pressure, density, and velocity decrease.1... 

According to our conceptions in a detonation wave the pressure, density and velocity of propagation fall as the chemical reaction progresses the more slowly the chemical reaction runs, the slower this drop in pressure and the wider the zone of increased pressure and increased velocity. 

Impaction is caused by the inertial mass of the traveling aerosol particles that forces them to move in a straight-line direction even when the flow of the inhaled air transporting them is bent around a curvature. Hence the particles tend to deposit on obstacles placed in the path of their travel. The inertial mass depends on particle size, density, and velocity. The stopping distance S of a particle having mass mP and initial velocity v0iP is defined according to... 

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