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  • MICHAEL SHORT: I think I might actually

  • use all 16 colors today.

  • Oh no, this is the most satisfying day.

  • Whereas Tuesday was probably the most mathematically intense,

  • because we developed this equation right here,

  • today is going to be the most satisfying,

  • because we are going to cancel out just about every term,

  • leaving a homogeneous, infinite reactor criticality condition.

  • So we will go over today, how do you

  • go from this, to what is criticality in a reactor?

  • So I want to get a couple of variables up over here

  • to remind you guys.

  • We had this variable flux of r, e,

  • omega, t in the number of neutrons

  • per centimeter squared per second traveling

  • through something.

  • And we also had its corresponding non-angular

  • dependent term, on just r, e, t, if we don't care

  • what angle things go through.

  • We've got a corresponding variable called current.

  • So I'll put this as flux.

  • We have current j, r, e, omega, t, and its corresponding,

  • we don't care about angle form.

  • And today, what we're going to do

  • is first go over this equation again so that we

  • understand all of its parts.

  • And there are more parts here than are in the reading.

  • If you remember, that's because I

  • wanted to show you how all of these terms are created.

  • Just about every one of these terms,

  • except the external source and the flow through some surface,

  • has the form of some multiplier, times the integral

  • over all possible variables that we care about,

  • times a reaction rate d stuff, where this reaction

  • rate is always going to be some cross-section, times some flux.

  • So when you look at this equation using that template,

  • it's actually not so bad.

  • So let's go through each of these pieces right here.

  • And then we're going to start simplifying things.

  • And this board's going to look like some sort of rainbow

  • explosion.

  • But all that's going to be left is a much simpler form

  • of the neutron diffusion equation.

  • So we've got our time-dependent term right here,

  • where I've stuck in this variable flux,

  • instead of the number of neutrons n,

  • because we know that flux is the number of neutrons,

  • times the speed at which they're moving.

  • And just to check our units, flux

  • should be in neutrons per centimeters squared per second.

  • And n is in neutrons per cubic centimeter.

  • And velocity is in centimeters per second.

  • So the units check out.

  • That's why I made that substitution right there.

  • And this way, everything is in terms of little fi, the flux.

  • We have our first term here.

  • I think I'll have a labeling color.

  • That'll make things a little easier to understand.

  • Which is due to regular old fission.

  • In this case, we have new, the number of neutrons created

  • per fission, times chi, the sort of fission birth spectrum,

  • or at what energy the neutrons are born.

  • Over 4pi would account for all different angles in which they

  • could go out, times the integral over our whole control volume,

  • and all other energies in angles.

  • If you remember now, we're trying

  • to track the number of neutrons in some small energy group, e,

  • traveling in some small direction, omega.

  • And those have little vector things on it

  • at some specific position as a function of time.

  • So in order to figure out how many neutrons are entering

  • our group from fission, we need to know,

  • what are all the fissions happening

  • in all the other groups?

  • I've also escalated this problem a little bit

  • to not assume that the reactor's homogeneous.

  • So I've added an r, or a spatial dependence

  • for every cross-section here, which

  • means that as you move through the reactor,

  • you might encounter different materials.

  • You almost certainly will, unless your reactor

  • has been in a blender.

  • So except for that case, you would actually

  • have different cross sections in different parts of the reactor.

  • So all of a sudden, this is starting

  • to get awfully interesting, or messy depending on what

  • you want to think about it.

  • There is the external source, which is actually

  • a real phenomenon, because reactors

  • do stick in those californium kickstarter sources.

  • So for some amount of time, there

  • is an external source of neutrons,

  • giving them out with some positional energy angle

  • and time dependence.

  • So let's call this the kickstarter source.

  • There's this term right here, the nin reactions.

  • So these are other reactions where it's absorb a neutron,

  • and give off anywhere between 2 and 4 neutrons.

  • Beyond that, it's just not energetically

  • possible in a fission reactor.

  • But don't undergo fission.

  • They have their own cross sections, their own birth

  • spectrum.

  • And I've stuck in something right here,

  • if we have summing over all possible i, where you have

  • this reaction be n in reaction, where 1 neutron goes in,

  • and i neutrons come out.

  • You've got to multiply by the number of neutrons

  • per reaction.

  • For fission, that was new.

  • For an nin reaction, that's just i.

  • But otherwise, the term looks the same.

  • You have your multiplier, your birth spectrum, your 4pi,

  • your integral over stuff, your unique cross-section,

  • and the flux.

  • And these two together give you a reaction rate.

  • I've just written all of the differentials as d stuff,

  • because it takes a lot of time to write those

  • over and over again.

  • And then we have our photo fission term,

  • where gamma rays of sufficiently high energy

  • can also induce fission external to the neutrons.

  • The term looks exactly the same.

  • There's going to be some new for photofission,

  • some birth spectrum for photofission,

  • some cross section for photofission, and the same flux

  • that we're using everywhere else.

  • Then we had what's called the in scattering term, where neutrons

  • can undergo scattering, lose some energy,

  • and enter our group from somewhere else.

  • That's why we have those en omega primes,

  • because it's some other energy group.

  • And we have to account for all of those energy groups.

  • That's why we have this integral there.

  • And it looks very much the same.

  • There's a scattering cross-section.

  • And that should actually be an e prime right there.

  • Make sure I'm not missing any more

  • of those inside the integral.

  • That's all good.

  • That's a prime, good.

  • There's also a flux.

  • And then there was this probability function

  • that a given neutron starting off an energy

  • e prime omega prime, ends up scattering into r

  • energy, e and omega.

  • So this would be the other one.

  • And this would be r group.

  • But otherwise, the term looks very much the same.

  • And that takes care of all the possible gains of neutrons

  • into r group.

  • The losses are a fair bit simpler.

  • There is reaction of absolutely any kind.

  • Let's say this would be the total cross-section, which

  • says that if a neutron undergoes any reaction at all,

  • it's going to lose energy and go out of our energy group d, e.

  • Notice here that these are all--

  • these energies and omega is our all r group,

  • because we only care about how many neutrons in r group

  • undergo a reaction and leave.

  • And the form is very simple--

  • integrate over volume, energy, and direction,

  • times a cross-section, times a flux,

  • just like all the other ones.