%MurNBdecay123151; ]> 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, MurNBdecay123151

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 24 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 December 31, 1951 English LCSH manuscript 24-bit color: 400 dpi Memorandum from Raymond L. Murray to Dr. Clifford K. Beck Typescript 3 pp. December 31, 1951 MurNBdecay123151

Murray December 31, 1951 TO: Dr. Clifford K. Beck FROM: Raymond L. Murray CC: Reactor Committee

Since you pointed out that it was impractical to exhaust the contaminated gases from the reactor continuously (over a 24 hr. period.), it has been necessary to reconsider the analysis reported earlier. In addition, an alternative "transit time" method of estimating necessary holding volumes was proposed by Dr. Underwood that gave, a quite different form to the theory and calculated result. In this note it is shown that such qualitative approaches to the problem do not give the correct answer. An extension of the previous theory to the case of interrupted gas exhaust is provided.

The final conclusion that is reached is that a factor of Xe concentration attenuation of greater than 10 is achieved by 8 tanks of 100 gallon capacity, assuming 6 hours per day 10 Kw reactor operation with 100 ml/hr air flow. A smaller total volume could be used if the number of tanks was increased and the individual tank size decreased.

Transit Time Treatment Of Holding System.

Assumptions: Fluid is discharged at a rate v into and out of a container of volume V. Two possible modes of transfer through the container are (a) steady flow, (b) flow with complete mixing.

Case (a) Regardless of the dimensions of the system, the time ¯t for a given sample of fluid to traverse the system is given by V/v. (Assume a length of total path L, area A, then ¯t = L/u where u is the flow speed. However u A = v, so ¯t = LA/v = V/v).

Case (b) The rate at which particles of fluid leave the system is dN/dt=-ρv where ρ is the no. per unit volume of the selected group, and is N/V. Thus dN/dt = -Nf where f = v/V. Let ψ(t)dt = |dN|/No be the distribution function for transit times. Since N = Noe-ft, ψ(t) = fe-ft. The mean time is

By simple analogy with the computation of mean free paths by integration, ¯t = l/f = V/v as in case (a)

Extension to Radioactive Deacy: The concentration of exhaust fluid would be expected to be reduced by a factor e-λ¯t if the mean holding time is ¯t. Thus Cf/Co the ratio of concentrations would be

This result would be obtained whether V was composed of n smaller tanks of volume Vl,or one large tank of volume n Vl.

The transit time analysis is applicable to a case in which the radioactive gas decays into a stable gas which also continues through the system of containers. In the actual situation, the Xe decays into Cs, which will be largely trapped by the water chamber walls, etc. The fallacy that is embodied in the transit time approach is thus readily seen. The average transit time would have to include all those Cs atoms which are stopped; ie., have infinite lifetime in the system. Thus it appears that the differential equations approach is more realistic, and so far as is determined, correct. The revision in calculation by the latter method follows.

REVISED METHOD OF CALCULATION OF SERIES HOLDING EFFECTS

In an earlier report, the reduction of Xe concentration in a sequence of holding tanks was calculated on the assumption that the discharge was continuous. The result found was that there was an attenuation of a factor

where λ is the decay constant in inverse days, and f was the ratio of air volume accumu- lated and discharged per day v to the volume of individual tanks, and n was the number of such tanks. Since it was decided to release active air only during the period the reactor is operating, a new approach to the problem was made.

1. Air containing Xe is delivered to the first tank, volume V of the holding system, at a concentration Co, at a rate v liters per day, for the reactor ope- ration time. If the total activity to be eliminated is A, then Co = A/vτ 2. The concentration Cl of discharge from the first tank during the period τ is governed by the equation dCl/dt = Cof - Cl (λ+ f) where f is the ratio v/V. 3. The solution of the above equation, applicable for o < t < τ is

4. The concentration in the tank subsequent to time τ is given by
5. The contributions of all previous operation cycles may be added by forming the sum
where the integer i signifies a shift in time of i days necessary to pick up previous days' effects. The geometric series is summed as follows:
Thus the concentration for continued but intermittent operation is a factor 1/1-e-λ than that due to one cycle. 6. The maximum value of concentration ratio Cl/Co is thus
Where B is the function indicated. 7. It is assumed by analogy with the previous derivation that the effect of n tanks is to raise this factor to the power n. Example:v = 100 ml/min = 144 liters/dayV = 100 gallons = 378.5 litersλ = 0.331 days-1f = v/V = 0.380

λl = 0.511 τ = 0.25 days (6 hours) λlτ = 0.128
Thus B =

= 0.978 f/λl = 0.744 Cl/Co = 0.728

In order to achieve en attenuation of 10, the number of tanks must be that for which (0.728)n = 0.1 or

or conservatively n = 8.

It is interesting to note that the predicted attenuation by transit time theory is attained only by an infinite array of infinitesimal tanks, as shown below: Let

Fix the total volume VT = n V, so that the above ratio becomes

the limit of which as n goes to

One other case besides that of continuous discharge may be analyzed by the same method: Assume that contaminated air is let in for a period τ, but the system is flushed with pure air for a period 1 - τ. The multiplying factor B now involves the "total" decay constant λl = λ + f in the denominator rather then just λ. The attenuation per tank is reduced by

which makes the system much more favorable. With 8 tanks, an additional reduction of (0.3)8 = 6.6 x 10-5 is obtained. Qualitatively, the reason for this improvement is that "physical" decay characterized by f is much stronger than radioactive decay.

Memo to C.K. Beck, <hi rend="underline">Effectiveness of Series Holding System for Radioactive<lb/>Gases</hi>