Memorandum from Raymond L. Murray to Clifford K. Beck
Typescript
4 pp.
1953
MurNBholdingtanks000053


[page 1]

Murray

TO: Clifford K. Beck
FROM: Raymond L. Murray
SUBJECT: Effectiveness of Series Holding System for Radioactive Gases.

The principle of operation of the system sketched in Figure I is as follows:

1. An active gas such as Xe is introduced into the first of a sequence of
chambers. Initially, the tank is full of air only. A water seal isolates
the tanks.

2. The discharge to the next tank is composed of air and Xe in the proportions
characteristic of the mixture at that particular time.

3. The gas exhausted to the stack by the pressure due to air flow is at a low
concentration. Since the air diluent volume is conserved, the absolute amount
of Xe discharged is correspondingly small.

Theory of Approach to Equilibrium.

Let v be the volume of air containinr the discharged active material that is
introduced per second to the system. This must be the same as the volume sent
to the stack, for steady operation.

Let V be the volume of a given tank, of which a number n may be used.

Let C1 be the concontration of Xe (number of atoms/unit volume) in air at any
point in the system. That sent in initially is taken as Co.

Set up the differential equation for Xe added and removed on a concentration
basis. (See Appendix 1)

Where C is the ratio v/V

Considor also decay, which adds a term -C1[lambda]

and let [lambda]¹ be an "effective" decay constant = [lambda] + f.

It is assumed that the Xe itself is of negligible volume in comparison with the
air at any time and does not perturb the volume from which concentration is
computed. Thus

For which the solution is, by the usual methods,

The highest concentration is smaller than that injected, even at .


[page 2]

The balance equation for the second tank is

or

As shown in the appendix 2, the solution is

In general, for the nth tank, as shown in appendix 3,

which is soon to involve the first n terms of the series expansion of e[lambda]1t.
That the concentration goes to zero for all times in an infinite sequence of
tanks is verified. The drop is made more rapid by virtue of pa rtial cancel-
lation
of terms on the right side of the equation.

Numerical Example

Assume that 12 curies of Xe are available for discharge by the time one day
has elapsed from the termination of a six hour operting period of the reactor.

These will decay negligibly during the discharge period of one full day, and
in fact it is conservative to take the maxime value.

Assume that a set of n tanks of volume V are arranged to accept this; let the
ratio of daily volume v of air that comes from the recombines to dilute the
Xe to the tank volume be f = v/V.

The concentration of material exhausted to the stack is at time t large

The maximum value of the ratio at time t Cn/Co can be no larger than the coef-
ficient
, so it may be used for calculation purposes.

Evaluate [lambda] + f for a typical example:
10 cc/min for 6 hours gives 10 x 360 = 3600 cc = 3.6 liters.
This is to be released over a day's period.
Assume the tanks have a volume of 10 gallons, ~ 40 liters, then f ~ 0.1.

The discharged activity is of concentration , which is tabulated
along with the activity itself for various numbers of tanks,


[ Appendix]

Appendix 1

Derivation of Differential Equations

Let N be the total number of Xe atoms in a container of volume V, and define
the concentration C1, as N/V. The number added par unit time is CoV where
Co is the number per unit volume in the intake; the number removed per unit
time is Cv. Thus

or
the effect of decay is to bring in a term -N[lambda] in the first equation of -C[lambda] in
the second

Appendix 2

Solution of equation for C2





[therefore]
At


[ Appendix]

Appendix 3

Solution of equation for C3






[therefore]
By extension it is clear that the solution for n tanks is