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Nuclear Reactor Digitization Project

Raymond L. Murray Reactor Project Notebook

Prepared for the North Carolina State University Science and Technology Electronic Text Center

The lineation of the manuscript has been maintained and all end-of-line hyphens have been preserved.

Keywords in the header are a local Science and Technology Electronic Text Center scheme to aid in establishing analytical groupings.

The question was raised in a recent meeting on the amount of depression of
flux produced by a control rod in a reactor. In this note, calculations of flux

Assumptions.
a water solution of NH/N

235 = 400.
of 1" is located along the central axis of the reactor.

Theory.
Reactor without control rod.
The solution of the reactor equation in cylindrical coordinates r,

l/r d/dr (r dφ/dr) + K

2 φ = 0 ----(1)

is φ = φ

o Jo(Kr)

where φ

o is the central flux and Jo(Kr) is the 1st order Bessel function
of the argument Kr. In order for φ to go to zero at the critical radius

R, K = 2.405/R.

The constant K depends on the composition of the reactor solution

cording

2 = lnk/L2+τ

where k is the infinite multiplication constant, L is the thermal

fusion

235 = 400, K is
approximately 0.128, so that R = 2.405/0.128 = 18.8 cm.

Reactor with control rod.
Since the thermal flux goes to zero at the surface of the control rod,

of radius taken as a, the solution of equation (l) must be written as a

linear combination of the bessel functions J

o, and No (using the
tion

Ernde). Thus assume

1 = A Jo (Kr) + B No (Kr)

where the prime distinguishes the case with a rod, and apply the boundary

conditions φ

1 = o at r = a and r = R1.
This yields the condition

Since the method of calculation involves two separate reactors, with only a
common active solution, there is no preferable mode of normalization for

son

distribution with a rod are plotted in the attached graph so that the ordinates

agree near the walls. The flux is affected strongly for about a third of the way