Memorandum from Raymond L. Murray to Dr. Clifford K. Beck

%MurNBneutronflux083051; ]> Memorandum from Raymond L. Murray to Dr. Clifford K. Beck Machine readable transcription Murray, Raymond L. Creation of machine-readable version: Russell S. Koonts Creation of digital images: Russell S. Koonts Conversion to TEI.2-conformant markup: Russell S. Koonts ca. 9 kilobytes NCSU Libraries Raleigh, NC Modern English, MurNBneutronflux083051

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23 October, 2000

Nuclear Reactor Digitization Project

Raymond L. Murray Reactor Project Notebook

Illustrations have been included from the print version. Scanned by Russell Koonts with Photoshop 5.0 software. Memorandum from Raymond L. Murray to Dr. Clifford K. Beck Raymond L. Murray

3 pp. Manuscript copy consulted UA 105.16

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Library of Congress Subject Headings August 30, 1951 English LCSH manuscript 24-bit color: 400 dpi Memorandum from Raymond L. Murray to Dr. Clifford K. Beck Typescript 3 pp. August 30, 1951 MurNBneutronflux083051

NCSC-33 August 30, 1951 TO: Clifford K. Beck FROM: Raymond L. Murray SUBJECT: Disturbance of neutron flux in reactor by a control rod CC: A. C. Menius, Jr., Arthur W. Waltner, and Newton Underwood

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 distributions for a simple reactor rod geometry are reviewed.

Assumptions. 1. The reactor is a bare, infinitely long circular cylinder, containing a water solution of NH/N235 = 400. 2. A control rod, black to thermal neutrons, with arbitrarily chosen diameter of 1" is located along the central axis of the reactor. 3. One group diffusion theory is applicable to the estimate of critical size and flux distribution.

Theory. (a) Reactor without control rod. The solution of the reactor equation in cylindrical coordinates r, l/r d/dr (r dφ/dr) + K2 φ = 0 ----(1) is φ = φo Jo(Kr) where φo 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 ac- cording to K2 = lnk/L2+τ where k is the infinite multiplication constant, L is the thermal dif- fusion length, and τ is the "age." For the case NH/N235 = 400, K is approximately 0.128, so that R = 2.405/0.128 = 18.8 cm. (b) 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 Jo, and No (using the nota- tion of Johnke and Ernde). Thus assume

φ1 = A Jo (Kr) + B No (Kr) where the prime distinguishes the case with a rod, and apply the boundary conditions φ1 = o at r = a and r = R1. This yields the condition

from which the critical radius R1 is computed. Since one of the constants A and B is arbitrary, we take B =+Jo (Ka). Thus A = -No (Ka) and φ1 = - No (Ka) Jo (Kr) + Jo (Ka) No (kr) For the case of K 0.l28, Ka 0.163, No (Ka) = -1.215, Jo (Ka) = 0.994. Thus, if we let KR1 = x No(x)/Jo(x) = -1.215/0.994 = -1.222 It x = 3.08, this equation is satisfied. The central radius is thus R1 = 3.08/0.128 = 24.1 cm, larger, as expected, than that for, the reactor without a rod. The flux is given by φ1 = 1.215 Jo (0.128r) + 0.994 No (0.128r) ----(3)

Since the method of calculation involves two separate reactors, with only a common active solution, there is no preferable mode of normalization for compari- son. In order to illustrate the rod effect, a simple Bessel function and the 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 out.