%MurNBrodpower041651; %MurNBrodpower041651forms; ]> Memorandum from R. L. Murray and A. C. Menius, Jr. to C. K. Beck and Reactor Committee, with Appendix [a machine-readable transcription] Murray, Raymond L. Menius, A. C., Jr. Creation of machine-readable version: Russell S. Koonts Creation of digital images: Russell S. Koonts Conversion to TEI.2-conformant markup: NCSU Science and Technology Electronic Text Center. ca. 9 kilobytes NCSU Libraries. Raleigh, NC. Modern English, MurNBrodpower041651

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 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 R. L. Murray and A. C. Menius, Jr. to C. K. Beck and Reactor Committee, with Appendix <author>Raymond L. Murray</author> <author>A. C. Menius, Jr.</author> <respStmt> <resp/> <name/> </respStmt> </titleStmt> <editionStmt> <p></p> </editionStmt> <extent>3 pp.</extent> <publicationStmt> <publisher></publisher> <pubPlace></pubPlace> <date></date> <idno>Manuscript copy consulted: NCSU Libraries call number UA105.16</idno> </publicationStmt> <seriesStmt> <p/> </seriesStmt> <notesStmt> <note></note> </notesStmt> </biblFull> </sourceDesc> </fileDesc> <encodingDesc> <projectDesc> <p>Prepared for the NCSU Libraries Science and Technology Electronic Text Project.</p> </projectDesc> <editorialDecl> <p>Spell-check and verification made against printed text using Notetab spell checker.</p> <p>The lineation of the manuscript has been maintained and all end-of-line hyphens have been preserved.</p> <p>The images exist as archived TIFF images, one or more JPEG versions for general use, and thumbnails.</p> <p id="ETC">Keywords in the header are a local Electronic Text Center scheme to aid in establishing analytical groupings.</p> </editorialDecl> <refsDecl> <p>ID elements are given for each page element and are composed of the text's unique cryptogram and the given page number, as in NEprop033050a for the title page of Clifford Beck's Proposal.</p> </refsDecl> <classDecl> <taxonomy id="LCSH"> <bibl> <title>Library of Congress Subject Headings April 16, 1951 English non-fiction; prose LCSH 24-bit color; 400 dpi Memorandum from R. L. Murray and A. C. Menius, Jr. to C. K. Beck and Reactor Committee, with Appendix Typescript 3 pp. April 16, 1951 MurNBrodpower041651

April 16, 1951NCSC-12 TO: C. K. Beck and Reactor Committee FROM: R. L. Murray and A. C. Menius, Jr. SUBJECT: Control Rod-Power Characteristic

For purposes of instrument and control-rod design it is necessary to know the effect of changes in control- rod position in the reactor power level. Such data are normally obtained empirically; but lacking a precise graph from Los Alamos, a rough theoretical calculation was made.

The power is shown to be approximately proportional to the excess reactivity; the latter in turn is given by an S-shaped curve which is similar to a displaced sine curve. The variation with position of rod of the power is shown in the attached figure, adjusted to fit a total value of 20,000 microres.

The arguments loading to this graph are given in the appendix of this note.

APPENDIX Derivation of Excess Reactivity Curve

The effect of a cadmium or boron control rod is to depress the neutron flux, usually to zero at the boundary. If such a rod were inserted axially in a circular cylinder of length much greater than the diameter, the neutron flux would be represented, in a bare reactor, by the function

where Jo is the Bessel function and the dimensions are given in the sketch, Fig. 1.

(This is to be contrasted with the undisturbed flux

The critical condition for such a system would be written

where M2 is an effective migration length for neutrons.

(Undisturbed, the k is given by the same relation without the a)

In a finite cylindrical bare pile, there is an axial sinusoidal variation in flux density,

, "modulating" the flux formulas given above; see Fig. 2.

The approximation is now made that the effective k of a finite cylinder with a rod partially inserted is the average of the undisturbed value kv and the disturbed value kd, with weighting factors pro- portional to the axial flux that is affected, namely the respective areas under the flux curve

It is easy to show that the effective k by this criterion is

If r1 is the excess reactivity value of the rod and the "rod-in" position corresponds to r = 0, then the excess reactivity r at any

distance the rod is pulled up, Z, is given by

In order to translate this into a power graph it is necessary only to note that if the pile is just critical at zero power with the rod in and the solution at the temperature of the cooling water, for any other steady state at power P the excess reactivity due to the rod must just balance the drop due to the solution temperature rise, to again reach k = 1.

Calculations on heat removal and experimental data from Los Alamos indicate an essentially linear rise in allowed power with temperature. Thus P/P1 = T/T1 where (P1,T1) is the maximum operating point, and (P,T) is any other.

If the temperature effect on reactivity is written

where α is the temperature coefficient, then the rod-power correlation is
This is plotted for the following assumed values of the constants Pl = 10 kw α = 240 μre/°C rl = 1.7 x 104 μre T1 = 70°C

Thus