Collision density

The importance of the nuclear reaction rates (the number of neutron-nucleus collisions per unit volume per unit time) to the problems of reactor physics was discussed in the preceding chapters. The purpose of the present chapter is to establish the relationship between the various reaction rates and the neutron population and the nuclear characteristics of the reactor materials. The ideas developed in this connection will be used to examine, in greater detail than was done in Chap. 1, the requirements for a self-sustaining chain reaction. 

The expected number of collisions per unit volume per unit time can be expressed in terms of the neutron density and the appropriate cross section for the type of reaction in question. In order to introduce the necessary concepts for this calculation, let us examine the following simplified situation. Consider a medium which is supporting a neutron gas of a specific kinetic energy. Thus, all the neutrons in this system have the same speed, but their directions of motion are randomly distributed, We should like to compute the average number of neutron-nucleus collisions which occur in time di in the element of volume dr around some space point r in this medium. This quantity may be expressed in terms of the following factors  

The quantity F(r,y, ) is called the collision density for the nuclear reaction described by the cross section 2(y). No subscript has been assigned to 2 so as to keep the derivation general. Equation (3.4) shows clearly the dependence of the reaction rate (or collision density) on the neutron density n(r,y,/). This relationship was stated previously in the introduction to Chap. 1. 

In dealing with specific types of reactions, we add the appropriate subscripts to the cross section according to the notation of (2.40). Thus, for example, 

The macroscopic cross sections 2(y) may be written in terms of the nuclear densities in the usual way [cf. Eq. (2.39)]. In general, these densities will be spatially variant functions and so should be properly written in the form N(j). The resulting equation for the collision density is then 

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In conclusion, for condensed phases molecular rotations have quite a short lifetime, because of collisions. The eventual oscillations induced by the electric field are then dissipated in the liquid state leading to vibration. At collision densities corresponding to liquids the frequency of the collisions become comparable with the frequency of a single rotation, and because the probability of a change in rotational state on collision is high, the time a molecule exists in a given state is small. It is, therefore, obvious that the electric field cannot induce organization in condensed phases such as in the liquid state. 

Counting molecules, collision frequency, collision density MB speed distribution and averages... 

Considering such binary collisions between billiard ball molecules of masses mi and m2 in a gas mixture at rest in a uniform steady state, the number of collisions per unit volume and time is given by (2.168) after dividing by dr and dt. Integrating the result over all values of Vl, c and C2, we obtain the collision density representing the total number of collisions per unit volume and time between molecules of type mi and molecules of type m2 ... 

Note that the number of collisions between pairs of similar molecules mi cannot be calculated by simply changing the suffix 2 to 1 in the given collision density relation for unlike molecules (2.171), because in this case one counts... 

Moreover, if there is a total of ns molecules of s per unit volume in addition to the r molecules, the collision density Zsr denoting the total number of s — r collisions per unit volume per unit time is ... 

If only the s type of molecules are present, the collision density Zss represents the collisions of s molecules with one another. The expression for this density resembles that for Zgr but a pre-factor 1/2 has to be introduced to avoid counting each collision twice ... 

Models for the collision densities were derived assuming that the mechanisms of the bubble-bubble and eddy-bubble collisions are analogous to collisions between molecules as in the kinetic theory of gases [16]. 

The parameterization of the particle collision densities was obviously performed emplo3ung elementary concepts from the kinetic theory of gases, thus the derivation of the source term closures at the microscopic level have been followed by some kind of averaging and numerical discretization by a discrete numerical scheme [f6, 92, 118]. 

In kinetic theory the collision density of a single particle is defined as the number of collisions per unit time and length (diameter) ... 

This relation represents a rough collision density model for a dispersion containing only one type of particles. 

Kolev [46] discussed the validity of these relations for fluid particle collisions considering the obvious discrepancies resulting from the different nature of the fluid particle collisions compared with the random molecular collisions. The basic assumptions in kinetic theory that the molecules are hard spheres and that the collisions are perfectly elastic and obey the classical conservation laws do not hold for real fluid particles because these particles are deformable, elastic and may agglomerate or even coalescence after random collisions. The collision density is thus not really an independent function of the coalescence probability. For bubbly flow Colella et al [15] also found the basic kinetic theory assumption that the particles are interacting only during collision violated, as the bubbles influence each other by means of their wakes. 

Prince and Blanch [92] modeled bubble coalescence in bubble columns considering bubble collisions due to turbulence, buoyancy, and laminar shear, and by analysis of the coalescence probability (efficiency) of collisions. It was assumed that the collisions from the various mechanisms are cumulative. The collision density resulting from turbulent motion was expressed as a function of bubble size, concentration and velocity in accordance with the work of Smoluchowski [111] ... 

To calculate the collision density an estimate of the length of the relative velocity between a pair of unlike particles, taking into account the distribution of particle velocities, is required. From kinetic theory of gases, we know that Urei,i,2 = ( 1 + be justified by a detailed calculation with... 

The collision density relation proposed by Prince and Blanch [92] for use in the framework of a discrete solution method was deduced from (9.12) and... 

The collision density formula was written in a discrete form consistent with the numerical scheme adopted. The number densities of the bubbles were thus defined as the number of bubbles per unit mixture volume and not as a probability density in accordance with the kinetic theory concept. 

Hagesffither et al [29, 30] did not make any firm conclusion on the relative importance of the various collision density contributions, as the turbulent bubble velocity closures used by them are at best inaccurate. 

Substituting the expression for the number density of eddies (9.41) into (9.36), the collision density for the eddies of size between A and d with particles of size di can be expressed as ... 

As for the collision density in the macroscopic model formulation, the average collision frequency of fluid particles is usually described assuming that the mechanisms of collision is analogous to collisions between molecules as in the kinetic theory of gases. The volume average coalescence frequency, ac d d, Y), can thus be defined as the product of an effective swept volume rate hc d d, Y) and the coalescence probability, pc d d, Y) (e.g., [16, 92, 114, 39, 46, 118]) ... 

B di, j) eddy-bubble collision rate uic modified particle collision density (- ... 

In the collision density, the most commonly used integral transport formulation, the flux and adjoint equations can be written as follows ... 

The birth-rate density formulation is sometimes 26) referred to as the collision density formulation. We shall distinguish between the two. 

The proportionality constant C2 is the ratio of the total neutron, density to the total collision density. Using Eq. (72) in Eq. (58) we get another relation between the importance functions ... 

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