Chapter 3 



Effects of Time and Mixing Characteristics 



39 



cules removed in the interval / and t-\-dt, i.e., 

 the number of molecules with a reservoir life- 

 time equal to /. 



The number of molecules of the original 

 particular set of N^ which are removed in any 

 interval dt is simply given by the concentration 

 of such molecules in the reservoir, multiplied 

 by the total flux from the reservoir, i.e.: 



N' 

 dN'=--^dt 



which yields on integration N =Nq exp 



Substituting for dN' and then for N' in the 

 integral expression for t, and integrating be- 

 tween t = and infinity, we obtain: 



N 



and from the expression for N' one sees that 

 T, the average life, is also the time required 

 for the original number of N^ particular mole- 

 cules to be reduced to l/e times the initial 

 number, r is thus formally equivalent to a 

 radioactive mean life. 



In our particular model we are assuming the 

 rate of removal to be dependent only on the 

 total amount of substance, N, in the reservoir, 

 so that the outgoing flux is given by <^=zkN. 

 In such cases we see that j—X/k, just as the 

 radioactive mean life is equal to 1/A. The total 

 removal rate of a radioactive isotope from a 

 reservoir is of course the sum of the physical 

 removal rate and the radioactive decay; t as 

 defined above refers only to the residence time 

 relative to physical removal. 



The symbols used in the following discussion 

 are listed below, where / refers to the subscripts 

 m and d for the mixed layer and deep sea. 



N^, Fj.' mass and volume, respectively, of 

 water in reservoir /. 



Nj*=r amount of any fission product in reser- 

 voir /'. 

 /ifj = activity of any fission product in reser- 

 voir / ( = xNi), in megacuries. 



^1= activity of any fission product per unit 

 volume of sea water in reservoir ;', 



y^i = exchange rate constant, = fraction of 

 material in reservoir / removed per 

 year. 



Ti= residence time in reservoir / relative 

 to physical removal, —l/ki. 



w = average depth of mixed layer of the 

 sea (taken as 100 meters). 



D = average depth of the ocean (taken as 

 3800 meters) . 



We assume that the fission products are intro- 

 duced into the deep sea after the 100 day cool- 

 ing period, the disposal rate or flux of a given 

 fission product isotope, termed ^, being equal 

 to the steady-state total activity in the sea Ag 

 as given by equation (11) . </> is thus in "mega- 

 curies of flux," = atoms/sec divided by 3.7 X 

 10^°. We wish to ask what steady-state activity 

 per unit volume of water will be in the mixed 

 layer, as a function of the rate of cross-thermo- 

 cline exchange of sea water and fission products. 



The water balance between the reservoirs is 

 given by: 



k,,N„, = kaNa 



or, neglecting density differences which are not 

 important for these calculations, 



^ = — ^ (12) 



The fission products are introduced into the 

 deep sea with a rate of introduction for any 

 give isotope 0. The radioactive balance in the 

 two reservoirs is then given by: 



Deep sea: 4> + K,Nn, = kJS!i+xN^ 



Mixed layer : k^N^ = k„,N* + AN,' 



Total : </> = A (N,; -}.N*a)=A, = A,^ -f- A, 



From (12) and (14) 



NJ 



(13) 

 (14) 



(15) 



— "^"^ _L \ 



or: 



^d _ 



D-m 



A, 



+ \ra 



Thus for a stable element (A = 0) the partition- 

 ing is simply statistical. From (15): 



-^m = 



D/m + \Ta 



(16) 



which gives the total activity of any fission 

 product in the mixed layer as a function of 

 decay constant, relative sizes of the mixed layer 

 and deep sea, and exchange rate between the 

 reservoirs as given by ra- 



Various estimates of the value to be assigned 

 to Td may be obtained from the separate papers 

 by Wooster and Ketchum, and by Craig, in this 

 report, and are discussed in relation to this 



