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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 />

H;|Violeta 1 – Word \web\<strong>Neutron</strong> <strong>Transport</strong> <strong>Equ<strong>at</strong>ion</strong>.doc

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