Memorandum from Raymond L. Murray to Clifford K. Beck

%MurNBequilibrium000000; %MurNBequilibrium000000forms; ]> Memorandum from Raymond L. Murray to 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, MurNBequilibrium000000

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25 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 Clifford K. Beck Raymond L. Murray

6 pp. Manuscript copy consulted UA 105.16

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Library of Congress Subject Headings English LCSH manuscript 24-bit color: 400 dpi Memorandum from Raymond L. Murray to Clifford K. Beck Typescript 6 pp. February, 1952 MurNBequilibrium000000

TO: Clifford K. Beck FROM: Raymond L. Murray CC: Reactor Committee SUBJECT: Equilibrium of Radioactive Product of Intermittent Reactor Operation.

Problem: An element or radioactive half-life λ1, is generated at a rate g during the operating time τ of a reactor; no production takes place during an idle time (k-1)τ the cycle is repeated with period kτ. A daughter product of half-life λ2 is liberated continuously from the accumulation of parent. The equilibrium level of the latter and the amount of daughter activity at various times in the cycle is to be determined.

Analysis: The solution of the differential equation for the parent during reactor operation

(1) At the end of a time τ, the accumulation is:

During the period thereafter, simple decay takes place, according to

The variable t inserted in this equation is taken as zero at time τ.

If the reactor has been operating for a long time, the number of parent atoms is the sum of contributions from all previous cycles, ie, the sum of terms like (2) with times t, t + kτ, t + 2kτ,....., t + ikτ. inserted, as may be seen from Figure 1.

The sum of the form
, so that
Several features of this equation are of interest.

1. The solution is applicable to time after the termination of reactor operation from o to (k-l)τ, i.e. until operation begins again. At that time a term of the form of equation (1) must be added. The peak level is, of course g B/λ, where B is the expression in parentheses. Figure 2 indicates the trend schematically.

2. When one cycle has elapsed, t = kτ, the peak is again reached, as may be demonstrated by substitution. (Appendix 1)

3. The minimum level occurs at t = (k-l)τ, and is thus a factor e-λ1(k-1)τ lower than the peak.

4. The average number (or activity) of the parent atoms over a cycle is exactly that expected if the generation rate were assumed to be continuous at a level 1/k times the actual generation rate. (Appendix 2)

Accumulated activity of daughter over a period k .

Assume that the daughter product grows during the interval between successive times of shutting down the reactor, ie. between times t = o and t = kτ.

The amount collected is the solution of the equation

As shown in Appendix 3, the result is the sum of two terms, the first applicable to times in the range t = otot = kτ, the second starting at t = (K-1)τ and going to t = kτ

---(3) At the end of the next reactor operation t kτ, the activity, A2 = N2λ2, is

Application of theory to the case of the mass 133 cycle.

Half-lives: parent I133 22 hr = 0.917d daughter Xe133 5.3d Note; as shown inAppendix 4, thepredecessor ofI133 can be assumedto decay instantly. Decay constants: λ1= 0.693/0.917 = 0.756 d-1λ2 = 0.693/5.3 = 0.131 d-1 Operating time τ = 6 hours = 0.25d.thusk = 4, kτ = 1d

Xe133 activity from equation (3) obtained by calculating different constants and terms.
λ1τ = 0.189, e-λ1τ = 0.828Kλ1τ - 0.756, e-kλ1τ = 0.470 ∴ B = 0.324 λ2-λ1 = -0.525 -.625λ2τ = 0.0327, e-λ2τ = 0.968Kλ2τ = 0.131, e-kλ2τ = 0.877

A2/g 1/-0.625[.131x0.324(0.8280.470-0.877)+0.131(1-0.828)-0.756(1-0.968)] (0.0424)(-0.407) (0.131)(0.172) -(0.756)(.032) -0.0172 0.0225 -0.0241 = -0.0188 = 0.0333 A2/g 0.0301

Evaluate g, using the fact that mass 133 appears with 4.5% yield in fission, and assuming a 10kw reactor power level.

thus the accumulated Xe activity is (0.03330.0301)(365) = 12.211.0 curies

Appendix 1 Proof that solution is periodic. N1)total
origin of time at τsubstitute t=kτ origin of time at 0substitute τ

the starting peak value, at t=0.

Appendix 2 Proof that average number of parent atoms is 1/k times continuous equilibrium case

= g/λ1 1/k

Appendix 3 Solution of Daughter Equation

where N1 = g/λ1Be-λ1t + g/λ1(1-e-λ1t) t=0 to t=kτ t=(k-1)τ to kτusing range t=0 to t=τ

Separate solution into two parts, superpose contribution of 1st and 2nd terms in N1. 1st N2

when t=0, N2=0,
N2

2nd N2

when t=0, N2=0,
N2

or simplifying N2

For t=0 to t=(K-1)τ, use only N1. (off-time) for t=(K-1)τ to t=Kτ, use N1 + N2 (on-time)

Appendix 4. Justification of neglect of predecessors of I133. Actual chain:

consider the 60 m decay from continuously generated Te. No = g/λo(1-e-λot) By the previous theory the peak equilibrium level of this element occurs at the end of the operating period τ

λo = 0.693/1/24 = 16.6, λoτ = 4.15, kλoτ = 16.6 ∴Bo

thus the activity is within 1.6% of being at its equilibrium value Ao = λoNo = g.

Murray TO: Clifford K. Beck FROM: Raymond L. Murray CC: Reactor Committee Errata: Equilibrium of Radioactive Product of Intermittent Reactor Operation.

The following typographical errors in the original note of the above title have been found: Page 2.Equation for N2. Delete the λ2 in the denominator of the first term. Equation for A2)Kτ. Multiply B by λ2. Third line from bottom should read λ2 - λ1 = -0.625. Page 3.A2/g = 1/-0.625[(0.131)(0.324)(0.470 - 0.877) + (0.131) (1 - 0.828) - 0.756(1 - 0.968)] = 0.030 ----------------- The accumulated activity is (0.030)(365) = 11 curies.