Memorandum from Raymond L. Murray to Dr. Clifford K. Beck
3 pp.
December 31, 1951


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=-[rho]v
where [rho] is the no. per unit volume of the selected group, and is N/V. Thus dN/dt = -Nf
where f = v/V. Let [psi](t)dt = |dN|/No be the distribution function for transit times.
Since N = Noe-ft, [psi](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-[lambda]¯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


In an earlier reporta1, 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 [lambda]
is the decay constant in inverse days, and f was the ratio of air volume accumu-
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.

Thus B = = 0.978
f/[lambda]l = 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:

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 [tau], but the system
is flushed with pure air for a period 1 - [tau].


a1Memo to C.K. Beck, Effectiveness of Series Holding System for Radioactive