TO: Dr. Clifford K. Beck
FROM: Raymond L. Murray
SUBJECT: Heat from Fission Product Radioactivity in Reactor
CC: Reactor Committee

Mr. John Dee prepared the following report for our use, which appears to
answer the question very nicely.

The 10 Kw water boiler operates continuously up to a shutdown. It is neces-
sary to determine the power from fission product decay after this time.

Reference:	1. Goodman, Vol. I, p.243
	2. K. Way, Phys. Rev. 70, 115 (1946)

K. Way gives the following expression for energy dissipation from fission products
by [beta] and [gamma] radiation.

E[beta] and [gamma] = 2.66 t-1.2 Mev/sec-fission		10<t<107 seconds
	(goodman)	(within factor of 2)
Solution:

The contribution to power at time t by products produced in the interval dT is:

The total is the integral:

In terms of watts:

If the reactor has been operating for say 6 months, then To in seconds is 6 x 30 x
86,400 = 15,550,000 seconds.

Evaluating E at t = 10 seconds:

	at t=100 seconds:

Conclusions:

At 100 seconds after shutdown, the maximum [beta] and [gamma] power from decay of fission
products is 500 watts assuming:

• 1. K Way's formula is incorrect adversely by a factor of 2.
• 2. All [beta] and [gamma]'s are absorbed in reactor.
  Since 1/2 Kw is removed from reactor by the surroundings, the solution does not
  boil.

That this estimate is reasonable may be seen by the following considerations.
(Goodman,I, p. 240) Of the 200 Mev produced by a fission, ~ 22 Mev is due to
fission product decay. "About half of this last figure is emitted as neutrino
energy." Therefore, only ~11 Mev is available as [gamma] and [beta] energy from fission
products. That is, if equilibrium is nearly established between the fission
products and a 10 Kw power level, then the steady state decay contribution will be
approximately 11/200 x 10 Kw = 550 watts. This decays rapidly upon reactor shutdown
and represents a maximum "decay power" in a reactor not exceeding 10 Kw.

Also, the 1/2 Kw conduction by surrounds is conservative in that an increase
in fuel temperature above normal operation increases the heat removed by surround-
ing almost proportionately with the change in the total temperature drop. That
is, if normally the conditions are 85°-25°C removing .5 Kw of heat, then at
100°-25°C the heat removal will be ~ 75/60 x .5 = .625 Kw.

This is the maximum heat removal by surroundings if the boiling point is
100°C. However, the boiling point will be higher due to the involatile solute and
may be computed by an elementary method given in "Outlines of Physical Chemistry"
by Farrington Daniels, page 231, John Wiley and Sons, 1948.

If there is some question about the closeness of the power production to
heat conduction shortly after shutdown it may be shown that if the contents of the
water boiler is assumed to be 1 ft³ of water, then a net gain of 500 watts for
a period of almost one hour is required to raise the temperature to boiling. At
one hour