9.163. We have mentioned that a nuclear heat flux factor is developed from appropriate nuclear calculations. It can be defined as the ratio of the maximum local fuel rod linear power density to the average core-wide linear power density. This assumes that there are no deviations from man­ufacturing specifications in fuel pellets and rods, densification effects, or uncertainties in calculations and the core monitoring system. Therefore, it is necessary to use subfactors to account for these considerations.

9.164. Most subfactors used in reactor core design tend to fall into two classes. First, there are factors arising from random-type uncertainties, such as those resulting from manufacturing operations, and second, factors introduced to account for inadequate knowledge. Random uncertainties lend themselves to statistical treatment, but those based on inadequate knowledge, e. g., flow distribution and intermixing of the coolant, are gen­erally combined by multiplying them together. The multiplication of subfac­tors implies that the corresponding events will occur at the same time and at the same place in the core; this is a very conservative assumption. For example, subfactors accounting for flow and mixing uncertainties are design dependent and are not treated statistically. On the other hand, the statis­tical combination of subfactors is justified when there is a body of exper­imental or manufacturing data to provide meaningful information. How­ever, it is sometimes possible to establish arbitrary confidence limits for a given uncertainty and to treat the resulting subfactor statistically, as will be seen.

9.165.

Statistics theory provides procedures for the analysis and pre­diction of data which are widely used in the quality control of manufacturing processes. For the present purpose, a simple discussion is adequate. In considering a large number of measurements of a particular quantity, the dispersion of the measurements about a mean value is often expressed as the standard deviation a; this is the square root of the arithmetic mean of the squares of the deviations of each value Xt from the arithmetic mean

where n is the number of measurements. If the deviations are random in character, the measurements will exhibit a so-called normal (Gaussian) distribution which is symmetrical about the mean value. In such a distri­bution, it can be shown that 99.73 percent of the values will lie within the range of ±3a. In other words, there are only 2.7 chances in 1000 that the actual value will be outside the range. In view of this high probability, the ± 3cr variation from the average is taken as the basis for calculating en­gineering hot-channel subfactors.

9.166. In order to combine standard deviations arising from variations in several different quantities, e. g., ax, a2, °з> etc., to obtain the overall standard deviation a, the relationship

O’ = [(сгі)2 + (o"2)2 + (0‘s)2 + • • -]° 5

is used. A similar expression is, of course, applicable to the 3a values employed in the determination of engineering hot-channel factors. An application of this procedure will be given in the next section.

9.167. In obtaining the nuclear heat flux factor for a PWR, it is usually helpful to assume that the radial and axial power distribution are separable. The radial factor F%H, also called the enthalpy rise hot-channel factor, is effectively the maximum-to-average ratio of the total power generated (or coolant enthalpy increase) in a (vertical) flow channel to that for all chan­nels. The axial factor F* is the ratio of the maximum to average power density in the axial (vertical) direction; it is derived from calculations utilizing the principles in §9.146 et seq. with a more realistic distribu­tion—rather than the sinusoidal form—of the heat source. The product of the radial and axial factors gives the heat flux nuclear hot-spot (or peaking) factor, F%; thus,

FNq = F& x F*

It is the ratio of the maximum heat flux, i. e., at the hot spot in the reactor, to the core average based on ideal design parameters. This must be mul­tiplied by the engineering hot-channel factor to obtain the overall peaking factor for the heat flux.

9.168. The engineering hot-channel factor is a combination of a number of subfactors arising from variations in the density, diameter, and enrich­ment of the uranium dioxide fuel pellets contained in the fuel rod, and the diameter of the rod with its cladding. The engineering subfactors may be treated statistically since they are based on manufacturing quality control data. The combination of these subfactors, in terms of variations of ±3a from the average, then gives the heat flux engineering hot-channel factor

Fq. Some values of the engineering subfactors for a pressurized-water reactor and their statistical combination are quoted in Table 9.2; each subfactor represents the ratio of maximum-to-average heat flux associated with variations of ±3a in the indicated quantities.[16]

9.169. For the fuel rod considered in Table 9.2, a reasonable computed value for F%H is 1.55 and F? is also about 1.55; hence, the heat flux hot — channel nuclear factor is

Fq = F%H x F"

= 1.55 x 1.55 = 2.40.

The total heat flux hot-channel factor Fq is the product of the nuclear and engineering factors, so that

F = FN x FE

L q * q * q

= 2.40 x 1.04 = 2.50.

In a typical PWR, the average flux for the whole core is in the vicinity of 6 x 105 W/m2; hence, the peak heat flux at the “hottest” spot in the hot channel is (2.5)(6 x 105) = 1.5 x 106 W/m2.