9.13. As already noted, a large proportion of the energy from fission is available in the form of heat within a very short distance of the fission event. The total rate of heat generation is proportional to the fission rate, i. e., to 2^ф or ІУоуф, where oyis the microscopic fission cross section, ф is the neutron flux, and N is the number of fissile nuclei per unit volume of fuel. If N remains uniform throughout a particular reactor or reactor region, as an initial approximation, the thermal source function, expressed
as the power density, may be taken to be the same as the spatial distribution of the neutron flux.

9.14. The overall flux distribution in the reactor core can be calculated by the methods described in Chapter 3 and 4. For a reflected cylindrical reactor in which the fuel is distributed uniformly, the flux distribution in the core and reflector can be represented approximately by

Ф, /„ .л_ r TTZ ^ = ^2.405-) cos-,

where фтах is the flux at the center of the core where it is assumed to have its maximum value; J0 is the zero-order Bessel function of the first kind, r and z are the radial and vertical coordinates, as in Fig. 3.11, and R’ and Hf are the effective radius and height, respectively, of the reactor, including an allowance for the reflector.

9.15. The average neutron flux in the core is given by

1 f R Г1/2Н

Фау = p2„ ф2Ttr dr dz, (9.2)

ttR2H Jo J — i/2h

where R and H are the actual radius and height of the fuel-containing region. Upon substituting the expression for ф from equation (9.1) and performing the integration, the result is

ф_ 2rr’4-405£)

фтах " 2.405R2 ’ ttH

where Jx is the first-order Bessel function.

9.16. If the concentration of fissile material is uniform throughout the reactor core, the power density may be taken to be proportional to the neutron flux.[2] It then follows that the reciprocal of equation (9.3) is equal to Fmax/Pav? i e., the ratio of the maximum to the average power density. For a reflected cylindrical reactor, R/R’ and НІН’ may be taken as roughly 5/6, and then

^ax (reflected) — 2.4.

* av

For a bare reactor, i. e., one without a reflector, R/R’ and ШН’ are both unity; then equation (9.3), after inversion, yields

—™-a* fbare) = —^405— . Z = з 54 Pav } 2/1(2.405) 2

If in a bare cylindrical reactor the arrangement of fuel elements or control rods (or both) is such that the flux is uniform in the radial direction, the factor containing the Bessel function is unity, and Pmax/Pav is u/2, i. e.,

1.57. Similar calculations have been made for reactors of other shapes, and the results of some core geometries are given in Table 9.1. It may be noted that, because of the arguments presented in §3.54, based on the similarity of the curves in Fig. 3.12, a cosine distribution of the flux may be assumed in all cases, e. g., even in the radial direction of a cylindrical reactor.