Neutron Transport Equation - Nuclear Engineering at McMaster ...
Neutron Transport Equation - Nuclear Engineering at McMaster ...
Neutron Transport Equation - Nuclear Engineering at McMaster ...
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Week 3 – <strong>Neutron</strong> <strong>Transport</strong> <strong>Equ<strong>at</strong>ion</strong> 7 - 9<br />
S S E<br />
( r, , , t)<br />
EXTERNAL = Ω<br />
EQ. 12<br />
SUMMING UP<br />
Summing these up and plugging into EQ. 6 gives the Boltzmann <strong>Transport</strong> <strong>Equ<strong>at</strong>ion</strong>:<br />
∂ n<br />
=<br />
⎧<br />
−<br />
⎫<br />
′<br />
t<br />
p<br />
+ −∇⋅<br />
∂ ∫∫<br />
f<br />
<br />
s<br />
⎩ S ⎭<br />
E′<br />
O<br />
+<br />
vn ′ ′ ⎨v′ ( 1 ß′<br />
) ∑ Σ f ⎬ dΩ′ dE′<br />
nv<br />
N<br />
∑<br />
i=<br />
1<br />
λ<br />
i<br />
C − vn ( + s)<br />
+<br />
i i Σa<br />
Σ S<br />
EXTERNAL<br />
Similarly, the delayed neutron precursor concentr<strong>at</strong>ion equ<strong>at</strong>ion is:<br />
EQ. 13<br />
⎛∂<br />
C<br />
⎜<br />
⎝ ∂ t<br />
i<br />
⎞<br />
⎟<br />
⎠<br />
+<br />
∫∫O E′<br />
( β i<br />
v Σ<br />
f ′)<br />
= −λ<br />
Ci<br />
′ ′′′ vn dΩ′<br />
dE′<br />
i<br />
EQ. 14<br />
Although these equ<strong>at</strong>ions are very general and difficult to solve directly, they embody several<br />
assumptions, as already noted. We’ll make a few more approxim<strong>at</strong>ions to get to the stage where<br />
we can solve these equ<strong>at</strong>ions on a routine basis.<br />
Note th<strong>at</strong> the capability of modern computers has made a dram<strong>at</strong>ic difference to reactor physics.<br />
Over the past four decades we have progressed to the point th<strong>at</strong> many approxim<strong>at</strong>ions th<strong>at</strong> were<br />
essential in the 1940-1960 time frame can now simply be discarded. However, some of these<br />
approxim<strong>at</strong>ions are still used, both to improve the economics of analysis and to permit easier<br />
understanding of results. Detailed models produce a vast array of numerical d<strong>at</strong>a. Summaries<br />
and approxim<strong>at</strong>ions enhance our understanding of these d<strong>at</strong>a.<br />
Assumptions so far:<br />
1) t' = t<br />
2) r' = r<br />
3) no st<strong>at</strong>istical fluctu<strong>at</strong>ions<br />
4) no sinks or sources other than those listed<br />
5) no neutron - neutron interactions<br />
THINGS TO CONTEMPLATE<br />
1) Can you find any other assumptions<br />
2) This continuity equ<strong>at</strong>ion is similar to the conserv<strong>at</strong>ion of mass in<br />
thermalhydraulics. Why do we not also generally consider the<br />
conserv<strong>at</strong>ion of momentum and energy *<br />
3) Expand on the implic<strong>at</strong>ions of t' = t, r' = r.<br />
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