How to calculate the absorption rate density?

[Dam08]

I was doing my homework and really got stuck on one question. Not looking for just an answer as that is useless, I really want to understand how to do this problem as I have no idea how to involve some of the givens.
UO_2 is a common fuel for reactors. If the fuel is enriched to 4 atomic percent U-235 and the density is 10.98g/cc, calculate the absorption rate density and the fission rate density in the Uranium mixture, assuming that the 2200m/s flux is 2x10^12 n/cc*s and the fuel temperature is 600 degrees Celsius.

 

[Simon Bridge]

Welcome to PF;
the key to understanding these things is to start from the definition of what the terms mean... in English (or whatever language you are best at).
Then express the same concept mathematically... and try to put that expression in terms you are provided in the problem.

 

[Dam08]

I know the definition of the terms but the textbook does a poor job of providing any type of applicable examples.

 

[QuantumPion]

It helps to keep in mind the units of the quantities to understand how to approach the problem. You are trying to calculate fissions per unit volume. You are given enough information to figure out how many U-235's there are per unit volume and the neutron flux (neutrons/cm2/s). All you need to calculate is the the macroscopic cross section which has units of 1/cm.

I'm a little confused about what the flux is in the problem. Flux is in units of neutrons/cm^2/s, not cc. Are they giving you the flux or the neutron number density? Remember your units and that number density is neutrons/cc, flux is neutrons/cm2/s. They also give you velocity in cm/s, so you can calculate flux from the number density if that is the case.

Start by figuring out the number density of U-235. You can calculate this based on the fuel density, molecular weight, and enrichment percent.

Then, calculate the macroscopic fission cross section. You just need the number density and the microscopic cross section (listed in your textbook). There is a temperature correction formula to apply to the xsection to account for the fuel temperature.

Now you have the macroscopic fission cross section and the flux. The reaction rate is simply the product of the two.

 

[Astronuc]

The textbook should be clear on how to do atomic density of a compound, e.g., UO₂, and how the reaction rate is the product of the macroscopic cross section (for the reaction) and the flux, i.e., ΣΦ. The 2200 m/s represents the most probable speed of thermal neutrons at 293.6 K and is equivalent to a neutron kinetic energy of 0.0253 eV. That's more or less at cold, zero power (CZP) conditions with a fresh core, before the pumps heat the core to hot, zero power (HZP) conditions.

Given the density of a material, one calculates the molecular density of UO₂, and from that, that atomic density of each constituent (nuclide). Note that ²³⁵U can absorb a thermal neutron and still not fission, some of the time.

There are absorption cross sections, for each nuclide, but only fission cross section for fissile isotopes - in the thermal spectrum.

Not to confuse the issue, there U-238 fissions at fast energies, but that's not part of this problem.