(2) Even if such a value of k was possible, the neutrons multiply

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1 CHANGE OF REACTOR Nuclear Thery - Curse 227 POWER WTH REACTVTY CHANGE n this lessn, we will cnsider hw neutrn density, neutrn flux and reactr pwer change when the multiplicatin factr, k, r the reactivity, Sk, change. Fr the mment, we will ignre the effects f the accumulatin f fissin prducts in the fuel and the effects f using up the U-235 in the fuel. We are nly cncerned with hw the reactr pwer changes immediately fllwing a change in reactivity. This will enable us t decide, later, hw changes in pwer can be achieved safely, ie, it will help us t determine hw reactr pwer can be regulated. Effect f Reactivity n Neutrn Multiplicatin n the previus lessn, we saw hw neutrns multiplied at a fantastic rate when the multiplicatin factr was equal t 2 r the reactivity was 1000 milli-k. This is, f curse, an extreme case fr tw reasns: - (1) A multiplicatin factr f 2 wuld be impssible t achieve in practice because neutrn lsses culd nt be cut dwn t this extent. (2) Even if such a value f k was pssible, the neutrns multiply s fast that the pwer wuld increase 1000 times in nehundredth f a secnd. This type f pwer increase wuld be impssible t cntrl and is, in fact, an explsive rate f increase. Let us nw lk at mre practical values f k and cmpare the neutrn multiplicatins fr varius values f k. Fig. 1 shws hw the neutrns multiply fr values f k frm t r fr reactivity values frm 0.5 milli-k t 3 milli-k. When.!i k = 0.5 mk, the neutrn ppulatin r neutrn density is dubled in 1400 ne~trn generatins. When Ek = 1 mk, the neutrn ppulatin r density is dubled in 700 neutrn generatins, trebled in 1100 generatins and is increased by a factr f 4 in 1400 generatins. When 6 k = 2 mk, the neutrn density is dubled in 350 generatins and has t increase 15 times its riginal value in 1350 generatins. July 1967 (R-2) - 1 -

2 L ' -~ _- f- uenerat10ns 590 1~ , t ,--...l..---'--~-- k = 1.000, bk = Omk k = , ~k = 0.5 mk. -~--,-,_.._----_. ' -_.._ ,---- k = ~k 1 mk :..., k = 1.00a bk=2mk k = Fig. 1 Finally, when S k = J mk, the neutrn density is dubled in nly 250 generatins and, in 950 generatins, the neutrn density wuld be 17 times its riginal value. S, as k, and the reactivity increase in value, the number f neutrns grws prgressively faster and faster. Als fr any ne value f the reactivity, the number f neutrns prduced in successive equal numbers f generatins, gradually increases. eg, when bk = 3 mk

3 New additinal neutrns prduced in first 250 generatins is 1.12 New additinal neutrns prduced in secnd 250 generatins is 2.36 New additinal neutrns prduced in third 250 generatins is 5.02 New additinal neutrns prduced in furth 250 generatins is 10.5 n cmparisn, when Sk = 2 mk, New additinal neutrns prduced in first 250 generatins is 0.65 New additinal neutrns prduced in secnd 250 generatins is 1.07 New additinal neutrns prduced in third 250 generatins is 1.76 New additinal neutrns prduced in furth 250 generatins is 2.9 Figure 2 shws, graphically, hw the neutrn density increases fr varius values f reactivities. Fig. 2 The shape f each curve is the same as the shape f the envelpe f the crrespnding neutrn diagram in Fig. 1 Such curves are knwn; EXPONENTAL curves. Thus we can say that the neutrns ppulatin in a reactr increases expnentially

4 f we had started with a neutrn instead f just ne neutrn, then the neutrn we had initially. Therefre, increases expnentially. density no' in a reactr, same wuld apply t every the neutrn density als Thus if the neutrn density is n, then after N neutrn generatins: - where "e" is the "expnential e" (e = 2.718) and a is sme factr depending n the reactivity r the value f k. n fact, a = k. That is, n = n en. k f S k = 0.5 mk = and N = 1400 f 0 k = 3 mk = and N = 240 n = n = n eo. 7 = 2 01 n 0 n e O 72 = 2 05 n 0 As can be seen, these are mre accurate figures than thse used in Fig. 1. Effect f Reactivity n Neutrn Flux and Reactr Pwer We have seen that bth the neutrn flux and the pwer level, in a reactr, are prprtinal t the neutrn density. Therefre, reactr pwer and the neutrn flux fllw the same expnential law as neutrn density. S we can write: - P = P e N.8k fr the pwer and tj = tj e N.k fr the flux On pltting either pwer r flux against the number f neutrn generatins, N, we get the same shape curves as in Fig. 2. n wrds, what happens is that, if 0' k is psitive, mre fissins take place than is just required t maintain the chain reactin. The neutrn density, therefre, increases and still mre neutrns becme available t cause further fissins. Thus, the neutrn density increases, the rate f fissining therefre increases and cnsequently, the pwer increases. All these increases are expnential

5 Effect f Negative Reactivity Changes Suppse that a reactr was perating at steady pwer s that it was just critical (ie, k = 1, 6k = 0). Nw suppse that k was suddenly reduced belw 1, say t k = k, is nw , r -1 mk. What happens? The reactivity, Obviusly, the chain reactin can n lnger be sustained s that the neutrn density starts t decrease. We n lnger have ne neutrn frm each fissin causing a further fissin. S the reactr pwer decreases and the flux increases. The neutrn ppulatin decreases and we g frm right left, in Fig. 1, instead f frm left t right. The neutrn density, neutrn flux and reactr pwer again change expnentially except that Skis nw negative. ie, n = n en. 5 k eg, if 5 k = -1 mk = and N = 1000 and P = P en. 6 k S the pwer decreases t 37% f P in 1000 neutrn generatins if Sk = -5 mk = and N = 1000 P = P e- 5 = P r 0.7% P The mre negative reactivity is intrduced (ie, the lwer k is made) the faster the pwer decreases. ASSGNMENT 1. Hw is reactr pwer affected by: - (a) increasing the psitive reactivity in a reactr? (b) increasing the negative reactivity in a reactr? 2. (a) The reactivity is +3 mk. The number f neutrns prduced in the first 250 neutrn generatins is 2.12, the number prduced in the secnd 250 generatins is 4.48, in the third 250 generatins, 9.5 and in the furth ~50 generatins, 20. What d these figures illustrate? - 5 -

6 2. (b) f the reactivity is reduced t zer at the end f these 1000 neutrn generatins, what wuld be the number f neutrns prduced during the next 250 generatins? (c) f the multiplicatin factr, k, is made equal t 0.997, at the end f these 1250 generatins, what wuld be the number f neutrns at the end f the next 500 generatins? 3. Write dwn the expnential equatins cnnecting the neutrn density, neutrn flux and neutrn pwer with reactivity. A. Williams - 6 -