Approximate Methods for Inverting Generating Functions from the Pál-Bell Equations for Low Source Problems

Abstract

A number of approximate probability distribution functions (pdf’s) for the neutron density are examined with reference to low source startup. The most accurate method for determining the safe source strength, to reduce the likelihood of a rogue transient during startup, is that arising from the Pál-Bell equations. When these equations are extended to include space and energy dependence the numerical work becomes extensive. A pdf is developed which gives results that compare favorably with those from the exact solution but requires very much less numerical work. The method is applicable to space- and energy-dependent problems. Extensive numerical examples are given of the new method and of others which have been proposed over the years. In addition, we explore other approximations, unrelated to the generating function, that can lead to substantial computational savings. We have additionally described the principles behind, and provided a simple review of, the low source algorithm from which anyone unfamiliar with low source concepts can benefit.

Keywords:

• Pál-Bell
• low source
• generating functions

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Approximate Methods for Inverting Generating Functions from the Pál-Bell Equations for Low Source Problems

Nomenclature

$⟨C_{iS}\left(t,R|s\right)⟩$ =	=	mean value of precursor density when a source of strength S is present
$f[G]$ =	=	generating function of prompt neutrons emitted from fission
$F_0(E)$ and $F_i(E)$ =	=	fission neutron energy spectra from prompt and delayed neutrons, respectively
$\tilde{G}=$	=	1 - G
$G_{di}\left(z,t,R|{\mathbf{r}}_0,s\right)$ =	=	generating function of the delayed neutron pdf emitted by i’th delayed species
$G_S(z,t)$ =	=	generating function of $P_S(n,t)$
$G\left(z,t,R|{\mathbf{r}}_0,E_0,s\right)$ =	=	generating function of $P\left(n,t,R|{\mathbf{r}}_0,\right.$ $\left.E_0,s\right)$
$⟨N_S\left(t,R|s\right)⟩$ =	=	mean value of neutron density when a source of strength S is present
$\bar{n}(t)\mathrm{o}\mathrm{r}⟨N(t)⟩$ =	=	mean value of neutron density
$P\left(n,t,R|{\mathbf{r}}_0,E_0,s\right)$ =	=	pdf of neutrons at time t in region R arising from a single neutron of energy $E_0$ at position ${\mathbf{r}}_0$ at time s
$P_S(n,t)$ =	=	probability distribution for neutron population
$Q(n^{\ast },t)$ =	=	cumulative generating function
$z_0$ =	=	root of Eq. (13)

Greek

$\mu =⟨n(n-1)⟩$	=	where $⟨...⟩$ implies stochastic average
$\sigma (t{)}^2\mathrm{o}\mathrm{r}{\sigma }_N^2$ =	=	variance of neutron density

Further definitions may be found in Appendices A and B.

Acknowledgments

The author wishes to thank C. M. Cooling, G. Winter, and M. D. Eaton for helpful comments and constructive criticism in the course of this work.