Number density of particles

Number density operator, total, 452 Number density of particles, 3 Numbers, representation in digital computation, 50 Numerical analysis, 50 field of, 50... 

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). 

For particles of equal mass, we thus have esps = mn with n the local number density of particles. From the KTGF, the time evolution of the granular temperature is given by... 

Integrating to find the flux through the disc with p as the number density of particles ... 

In terms of the common features of colloids just listed, nanoparticles with a clean surface have tendency to amalgamate when placed on a TEM sample mesh that is evident in Figures 9.4.16 and 9.4.17. Furthermore, the particles often grow by collision (not by Ostwald ripening because metal ions are hard to dissolve in organic liquids) in the suspension state when the number density of particles in... 

If f(a,z>t) denotes the number density of particles with diameter a at position z and time t, the continuity equation is... 

The sedimentation process causes an increasing relative number density of particles with increasing size at the sampling altitudes. Applicable correction can be calculated. 

Consider an axially symmetric cloud of spherical particles in the atmosphere. If wind and diffusion effects are neglected, the only change will be caused by gravitational settling of individual particles. Denote by f(a,x,t) the number density of particles with diameter a at position z and time t. The continuity equation is... 

Barr et al. (2003) performed an analysis of the impact of phytogenic aerosol (PhA) which is defined as forming mainly due to monoterpene oxidation (primarily, a- and /3-pinenes), on the radiative regime of the ABL over the forest in the eastern part of Canada. In the forest ecosystem the level of emissions to the atmosphere of biogenic hydrocarbons is moderate, with the concentration of a- and /3-pinenes constituting about 1.6 ppb. NMHC oxidation resulted in the formation of PhA at a number density of particles of about 5 108 cm 3. For a given concentration and size distribution of aerosol, its impact on the short-wave radiation transfer in the ABL was assessed. 

In the context of dilute solutions, suspensions, or vapors, we can safely regard the historically earliest idealizations of inverse-sixth-power interactions. These are for the conditions that hold in a van der Waals gas. The medium is vacuum the a s and /i s of individual particles already introduced are now those of atoms or small molecules. The vapor is so dilute that its dielectric response is that of the vacuum plus very small contributions proportional to the number density of particles. 

In several examples for gases and dilute suspensions, we expand the dielectric response e around its vacuum value of 1 or around its pure-solvent value em, respectively, for the suspending medium. In those cases, the dimensionless x for the gas or for the suspension as a whole will be proportional to the number density of particles (units 1/length3), and the contribution to the polarizability from individual particles will have volume units (length3). 

Summing the rate of formation of k-size aggregates and the rate of reaction of k- size aggregates to form larger aggregates, the rate of change of the number density of particles with volume vt is... 

In astrophysics, particle sizes in protoplanetary disks are usually described in the context of a distribution, dn = f(a)da, where a is the particle radius and (in is the number density of particles with radii between a and a + da. Note that in cosmochemistry, grain size usually refers to the largest diameter of a monomer. The description is often associated with two assumptions. First, it is often assumed that the distribution function, /(a), is close to being a power law in the ISM, except at very large or very small sizes (Weingartner Draine 2001), and power laws are therefore often used in protoplanetary disk studies. Second, the definition of a particle radius assumes that the particles are spherical - a convenience for converting the size distribution to an opacity law. 

The population balance is used as a method of accounting for particles as they go throng a process, such as grinding, dassiiication, crystallization, a [regation, or grain growth. This chapter is devoted to the development of population balances, because it is of fundamental importance to several other of the diapters in this book. The chapter draws heavily on the excellent text Theory of Particulate Processes by Randolph and Larson [1]. The number density of particles N(L) (with units of number of particles per unit volume) is equal to the integral of the population 7)q(L) from size L to L + AL and is defined as... 

FIGURE 7 Schematic diagram of gas phase reactor with (a) number density of particles, (b) average diameter of particles, and (c) temperature of flame. 

Problem 10.2. Determine the Half-Life for Doublet Formation for Various Initial Number Densities of Particles in Water... 

Likewise, if the number of particles in the container is doubled, the impacts with the wall will double. For a general case, we need to consider not the absolute number of particles but the number of particles per unit volume (the number density of particles), which can be represented by N/V, the number of particles N divided by the volume V (in m3). Thus ZA is expected to depend directly on N/V. That is, ZA N/V. 

---