Chapter 3 



Effects of Time and Mixing Characteristics 



37 



\t-XN 



shall take R, the world rate of fission, as at 

 the present time (^ = 0) and increasing linearly 

 from the present time until it reaches the 1000 

 ton rate in 50 years. We shall further assume 

 continuous stripping of fission products into 

 the sea, and examine the transient character- 

 istics of long-lived and a short-lived fission 

 product. 



The rate of increase of a fission product in 

 the sea is given by: 



dN ,{R\ 



where {R/t) is a constant by virtue of the 

 assumed linear increase from R = 0. N is now 

 the amount of a fission product in the sea at any 

 time t. We thus have: 



dN+{xN-{jR/t)tyt = 



Multiplication by e'^^ makes the equation exact, 

 and the solution is: 



Evaluating the constant from N = at / = 0, we 

 have the general solution: 



N, = i|[A/- (1-^-0} (9) 



where N^ is the amount in the sea at the time /, 

 Multiplication by A. to give the activity is seen 

 to give an equation of the same form as (5) 

 for the steady state amount in reactors, except 

 that in (9) both R and / are variables, with 

 R/t being constant. 



We take i?=:0 at the present time, increas- 

 ing linearly to 1000 tons U-^Yyear in 50 years. 

 As noted previously, this rate is equivalent to 

 2.2x10^ megacuries of fission, and thus R/t — 

 4.4 X 10* megacuries/year. Thus the activity of 

 a fission product in the sea at any time / is 

 given by: 



At^AAxlO'Uxt-il-e-^t)'] (10) 

 A 



where A^ is in megacuries, A = yrs-^, / is in 

 years, and / is the fission yield. We tabulate 

 below the increasing activity in the sea for a 

 long-lived and a short-lived isotope with con- 

 tinuous stripping into the sea. 



Activity (megacuries) in the sea 



SfOO 1131 



/i/2 = 28}/ t-,/^_ — Sd 



t (years) 



/ = 0.05 



/ = 0.028 



At 50 years, when the fission rate of 1000 

 tons/year is reached, the Sr^o activity is half 

 the amount which would be in steady state with 

 this fission rate with an irradiation time of 1 

 year (see below and Table 1 ) . If R continues 

 to increase at the same rate, the steady state Sr^** 

 activity for constant R is reached in about 100 

 years, and thereafter the activity increases lin- 

 early at a rate given by: At = 2200{t — A0), the 

 mean life of Sr^o being 40 years. The factor 

 (l_^-\f) grows in to 95 per cent at 3 mean 

 lives or 4 half-lives. 



With a constant fission rate of 1000 tons 

 U-^5y'year, irradiation time one year, and no 

 cooling time, the I^^^ steady state activity in 

 the sea would be 2000 megacuries (calculated 

 as in Table 1, but with no cooling time) . With 

 the linear increase of fission rate and continu- 

 ous stripping as shown above, this level is sur- 

 passed in two years. These data illustrate rather 

 strikingly how rapidly the short half-life iso- 

 topes build up to secular equilibrium with an 

 increasing fission rate. Sr^** does not equal the 

 P^^ activity until after 100 years of dumping 

 into the sea, under the above conditions. For 

 all species which have grown into secular equi- 

 librium with the increasing fission rate, the ac- 

 tivity ratios in the sea are simply given by the 

 fission yield ratios. 



IV. Steady state fission product spectrum in a 

 homogeneous, rapidly mixed sea 



The first three columns of Table 1 list all 

 the fission products of any significance, together 

 with their half-lives and fission yields. Col- 

 umns 4 and 5 show the total amounts of each 

 isotope in the sea, in metric tons and mega- 

 curies of activity respectively, in secular equi- 

 librium with a fission rate of 1000 tons U-^5 



