FROM:

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

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

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

Extension to Radioactive

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

follows.

REVISED METHOD OF CALCULATION OF SERIES HOLDING EFFECTS

In an earlier report^{a1}, 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

is the decay constant in inverse days, and f was the ratio of air volume

lated

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 reactorope- time. If the total activity to be eliminated is A, then Co = A/v[tau]

ration - 2. The concentration Cl of discharge from the first tank during the period [tau]

is governed by the equation dCl/dt = Cof - Cl ([lambda]+ f) where f is the ratio v/V. - 3. The solution of the above equation, applicable for o < t < [tau] is

- 4. The concentration in the tank subsequent to time [tau] 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-[lambda] 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/day V = 100 gallons = 378.5 liters [lambda] = 0.331 days-1 f = v/V = 0.380

[lambda]l = 0.511 [tau] = 0.25 days (6 hours) [lambda]l[tau] = 0.128

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

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 [tau], but the system

is flushed with pure air for a period 1 - [tau].

- The multiplying factor B now involves the "total" decay constant

[lambda]l = [lambda] + f in the denominator rather then just [lambda]. 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.

Notes:

^{a1}Memo to

Gases