Finding power change due to prompt jump

[doublemint]

I am trying to figure out what happens to the power when a prompt jump occurs. From Nuclear Reactor Analysis by Duderstadt and Hamilton, a prompt jump approximation can be done to yield the following equation:
[itex]\frac{P_{2}}{P_{1}}[/itex] = [itex]\frac{\beta-\rho_{1}}{\beta-\rho_{2}}[/itex]

Now the question is for a CANDU reactor that has 0.7% enriched uranium. If the delayed neutron fraction for U235 is 0.00682 then for the CANDU reactor the fraction of delayed neutron is 0.00682*0.007=4.77E-5. (Not sure if I did this part correctly).

Ignoring fast fission of U238, there was a step increase of +3mk in an initially critical reactor, then the power change is just:
[itex]\frac{P_{2}}{P_{1}}[/itex] = [itex]\frac{4.77E-5-0}{4.77E-5-0.003}[/itex]
but this yields a negative ratio which does not make sense..I am thinking I calculated the delayed neutron fraction incorrectly.

Any help is appreciated!

 

[Astronuc]

I am trying to figure out what happens to the power when a prompt jump occurs. From Nuclear Reactor Analysis by Duderstadt and Hamilton, a prompt jump approximation can be done to yield the following equation:
[itex]\frac{P_{2}}{P_{1}}[/itex] = [itex]\frac{\beta-\rho_{1}}{\beta-\rho_{2}}[/itex]

Now the question is for a CANDU reactor that has 0.7% enriched uranium. If the delayed neutron fraction for U235 is 0.00682 then for the CANDU reactor the fraction of delayed neutron is 0.00682*0.007=4.77E-5. (Not sure if I did this part correctly).

Ignoring fast fission of U238, there was a step increase of +3mk in an initially critical reactor, then the power change is just:
[itex]\frac{P_{2}}{P_{1}}[/itex] = [itex]\frac{4.77E-5-0}{4.77E-5-0.003}[/itex]
but this yields a negative ratio which does not make sense..I am thinking I calculated the delayed neutron fraction incorrectly.

Any help is appreciated!

This part: "If the delayed neutron fraction for U235 is 0.00682 then for the CANDU reactor the fraction of delayed neutron is 0.00682*0.007=4.77E-5." is not correct. The delay neutron fraction should be weighted according to the fraction in which fissions occur, not the enrichment fraction. The fraction of fissions would be a function of enrichment, or atomic fraction, AND the fission cross-section.

β_eff = w(U235) β(U235) + w(U238) β(U238). If 90% of fissions occur in U235 and 10% of fissions occur in U238, then assuming β(U235)= 0.00650 and β(U238)= 0.0157, then β_eff = 0.9*0.0065 + 0.1*0.0157 = 0.0074.

Normally, positive reactivity insertion is nowhere near β (1$), but rather in cents.

 

[doublemint]

That makes more sense. How would the fraction of fission function look like?
Would it be something like this:

w(U235) = [itex]\frac{0.007 * \sum_{f,U235}}{0.007 * \sum_{f,U235}+0.993 * \sum_{f,U238}}[/itex]

where [itex]\sum_{f,U235}=582.6b[/itex] and [itex]\sum_{f,U238}=1b[/itex]

 

[Astronuc]

That makes more sense. How would the fraction of fission function look like?
Would it be something like this:

w(U235) = [itex]\frac{0.007 * \sum_{f,U235}}{0.007 * \sum_{f,U235}+0.993 * \sum_{f,U238}}[/itex]

where [itex]\sum_{f,U235}=582.6b[/itex] and [itex]\sum_{f,U238}=1b[/itex]

Not quite. One has to weight it according to [itex]\int \Sigma_i(E) \phi(E) dE[/itex] for each fissile species, i.

In fact, β = β(E), because the fission product distribution changes with incident neutron energy and favors a slightly more symmetric fission.

For LWRs, about 8 to 10% of fissions occur in U-238, because the fast flux is about an order of magnitude greater than the thermal flux, which partially compensates for the low cross-section in the fast energy range.

I believe the HWR flux spectrum is slightly more epithermal than for LWRs. Somewhere I have some flux spectra that I should be able to post.

 

[doublemint]

It is a little more complicated than I hoped it will be. But it is good to know!

Although, I am working on an assignment which deals with something similar. I am given the concentration of 3 fissile isotopes in a CANDU core burning at a power P and asked to find the 6 delayed neutron parameters for each isotope and the total delayed fraction of the core. I am given a table similar to the one in Dunderstadt and Hamilton to work with. The delayed constants would stay the same so there is no work needed, but as for calculating the fractions, no other information is given such as the flux spectra you have mentioned.

So I'm curious if your method is the only way, or is there a simpler, approximated, approach to solving it.