Velocity, constant distribution functions

The quantities n, V, and (3 /m) T are thus the first five (velocity) moments of the distribution function. In the above equation, k is the Boltzmann constant the definition of temperature relates the kinetic energy associated with the random motion of the particles to kT for each degree of freedom. If an equation of state is derived using this equilibrium distribution function, by determining the pressure in the gas (see Section 1.11), then this kinetic theory definition of the temperature is seen to be the absolute temperature that appears in the ideal gas law. 

Perhaps the most simple flow problem is that of laminar flow along z through a cylindrical pipe of radius r0. For this so-called Poiseuille flow, the axial velocity vz depends on the radial coordinate r as vz (r) — Vmax [l (ro) ] which is a parabolic distribution with the maximum flow velocity in the center of the pipe and zero velocities at the wall. The distribution function of velocities is obtained from equating f P(r)dr = f P(vz)dvz and the result is that P(vz) is a constant between... 

Such expressions can be extended to permit the evaluation of the distribution of concentration throughout laminar flows. Variations in concentration at constant temperature often result in significant variation in viscosity as a function of position in the stream. Thus it is necessary to solve the basic expressions for viscous flow (LI) and to determine the velocity as a function of the spatial coordinates of the system. In the case of small variation in concentration throughout the system it is often convenient and satisfactory to neglect the effect of material transport upon the molecular properties of the phase. Under these circumstances the analysis of boundary layer as reviewed by Schlichting (S4) can be used to evaluate the velocity as a function of position in nonuniform boundary flows. Such analyses permit the determination of material transport from spheres, cylinders, and other objects where the local flow is nonuniform. In such situations it is not practical at the present state of knowledge to take into account the influence of variation in the level of turbulence in the main stream. 

Here n designates the density or distribution function j the diffusion current vd the apparent velocity, namely, the drift velocity, of a Brownian particle and D the diffusion constant. Equation (1) is a continuity equation while Eq. (2) is simply Fick s law augmented by a definition of vd. 

The Boltzmann distribution of the populations of a collection of molecules at some temperature T was discussed in Section 8.3.2. This distribution, given by Eq. 8.46 or 8.88, was expressed in terms of the quantum mechanical energy levels and the partition function for a particular type of motion, for instance, translational, vibrational, or rotational motion. It is useful to express such population distributions in other forms, particularly to obtain an expression for the distribution of velocities. The velocity distribution function basically determines the (translational) energy available for overcoming a reaction barrier. It also determines the frequency of collisions, which directly contributes to the rate constant k. 

The degree of conversion inside this volume is constant, but the MWD function qw(n, r), where n is the degree of polymerization, depends on r. This is a reflection of different reaction time in the various layers of the polymer. The residence time distribution function f(r) for the reactive mass in a reactor is determined from rheokinetic considerations, while the MWD for each microvolume qw(n,t) is found for various times t from purely kinetic arguments. The values t and r in the expressions for qw are related to each other via the radial distribution of axial velocity. 

Temperature along the Riser Height Temperature (T) affects the cracking (k.) and deactivation (VO constants (or functions). It also influences the gas density and, as a consequence, the flow rates, the superficial velocities, and the residence time of the gas in the riser. Its effect on ki makes the product distribution (gas, coke, etc.) dependent on T. 

Here, u[0) — I g(r) is the two-particle radial distribution function, E (r) and u (r) are the electric and velocity fields at position r generated by a single neigboring sphere located at the origin subject to the applied electric field Eoq. These fields are expressed by Eqs. (23) and (24) for r>a. Inside the neighbouring sphere (r < a both E and u are constant ... 

In this equation, h is Planck s constant divided by 2tt, V is the crystal volume, T is temperature, fej, is Boltzmann s constant, phonon frequency, is the wave packet, or phonon group velocity, t is the effective relaxation time, n is the Bose-Einstein distribution function, and q and s are the phonon wave vector and polarization index, respectively. 

In this subsection we put a particle at an origin in a velocity field u(r) and study effects of flow on a stationary density profile n,i(r). When there is no flow u(r) = o,n (r) is obviously given by niq(r), with g(r) a radial distribution function. Due to the flow u(r), this equilibrium distribution is distorted and from the distortion we can calculate some transport coefficients, like viscosity g and friction constant C, as we show below. 

Exercise 9.9.4. Show that the distribution function of residence times for laminar flow in a tubular reactor has the form 2z /Zp, where tp is the time of passage of any fluid annulus and the minimum time of passage. Diffusion and entrance effects may be neglected. Hence show that the fractional conversion to be expected in a second order reaction with velocity constant k is 2B[1 + j lnu5/(5 + 1)] where B = akt n and a is the initial concentration of both reactants. (C.U.)... 

Carrica et al [24] further assumed that the distribution function, the particle velocities, the particle mass exchange rate and the breakage and coalescence probability are assumed constant in each group. Under these restrictions, the balance equation yields ... 

Kuwabara function for particle deposition in porous media particle radial distribution function pair correlation function for particles of types a and / constant appearing in the pair correlation function velocity parameters used in conjunction with EQMOM mean velocity difference used to approximate vi - Vp I... 

The average velocity of a gas molecule is determined by the molecular weight and the absolute temperature of the gas. Air molecules, like many other molecules at room temperature, travel with velocities of about 500 m s"1 but there is a distribution of molecular velocities. This distribution of velocities is explained by assuming that the particles do not travel unimpeded but experience many collisions. The constant occurrence of such collisions produces the wide distribution of velocities. The quantitative treatment was carried out by Maxwell in 1859, and somewhat later by Boltzmann. The phenomenon of collisions leads to the concept of a free path, that is the distance traversed by a molecule between two successive collisions with other molecules of that gas. For a large number of molecules, this concept must be modified to a mean free path which is the average distance travelled by all molecules between collisions. For molecules of air at 25°C, the mean free path X at 1 mbar is 0.00625 cm. It is convenient therefore to use the following relation as a scaling function ... 

CONCEPTS More about the effect of collisions on distribution functions microscopic theory of dielectric loss The Debye theory can define a distribution function which obeys a rotational diffusion equation. Debye [22, 23] has based his theory of dispersion on Einstein s theory of Brownian motion. He supposed that rotation of a molecule because of an applied field is constantly interrupted by collisions with neighbors, and the effect of these collisions can be described by a resistive couple proportional to the angular velocity of the molecule. This description is well adapted to liquids, but not to gases. 

The constants Ci, C2, and C3 in Eq. (101) are not yet determined. We fix their values and consequently arrive at a unique solution for d>i by requiring that the functions n, u, and T appearing in the local equilibrium distribution function be the actual values of the local density, mean velocity, and temperature of the gas. That is, we require that... 

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