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MICHAEL SHORT: So today is going to be the last day of neutron

physics.

As promised, we're going to talk about what

happens as a function of time when you perturb the reactor,

like you all did about a month ago.

Did any of you guys notice the old-fashioned analog panel

meter that said, reactor period, when you were doing your power

manipulations?

We're going to do that today.

And you're going to explore that on the homework.

So I'm arranging for all of your actual power manipulation

traces to be sent to you.

So each one, you'll have your own reactor data.

You'll be able to describe the reactor period

and see how well it fits our infinite medium single group

equations, which it turns out is not very well.

But that's OK, because you'll get to explain the differences.

First, before we get into transients

I wanted to talk a bit about criticality and perturbing it.

So let's say we had our old single group kit criticality

relation.

And I'd like to analyze, just intuitively or mentally

with you guys, a few different situations.

Let's say we're talking about a light water

reactor or a thermal reactor, like the MIT reactor, or pretty

much all the reactors we have in this country.

What sort of things could you do to perturb it?

And how would that affect criticality?

For example, let's say you shoved in a control rod.

Let's take the simplest scenario.

Control rods in.

What would happen to each of the terms

in the criticality condition?

And then, what would happen to k effective?

So let's just go one by one.

Does nu ever change, ever?

Actually, yeah, it does.

Over time, you'll start--

that nu right there, remember, that's

a nu bar, number of neutrons produced per fission.

As you start to consume U238 add neutrons.

And as you guys saw through a complicated chain of events

on the exam, eventually make plutonium 239,

which is a fissile fuel.

The nu for 238 is actually different than the nu for 239.

So I don't want to say that nu never changes.

It's just that shoving the control rods into the reactor

is not going to change nu.

But it does change slowly over time as you build up plutonium.

What about sigma fission?

If this were a blended homogeneous reactor

or a reactor in a blender, what would happen to sigma fission

as you then shove in an absorbing material?

Does it change?

AUDIENCE: No.

MICHAEL SHORT: You say, no.

And I'm going to add here homogeneous.

So in this case, remember if we define the average sigma

fission as a sum--

I'll add bits to it--

of each material's volume fraction, or let's say

atomic fraction, times each material's sigma fission,

if we throw nu materials into the reactor,

then this homogeneous sigma fission

does change when we put materials in

or take materials out.

So you guys want to revise your idea?

AUDIENCE: Yes.

MICHAEL SHORT: Yes, thank you.

There's only one other choice.

Now the question is, by how much?

If you put in a control rod where let's say the control

rod's sigma fission would be equal to zero,

but volume would be equal to small.

Can't be any more specific than that.

How much of an effect do you think

you'll have on sigma fission?

AUDIENCE: Small.

MICHAEL SHORT: Very small.

So let's say a little down arrow like that.

What about sigma absorption?

The volume is still small, but a control rod by definition sigma

absorption equals huge.

So what do you think?

AUDIENCE: It's going to increase.

MICHAEL SHORT: It's going to increase a little or a lot?

AUDIENCE: A lot.

MICHAEL SHORT: Quite a bit.

Now let's look at the diffusion constant.

And remember that the diffusion constant is 1 over 3 sigma

total, minus the average cosine scattering angle

sigma scattering.

What do you think is going to happen to the neutron diffusion

coefficient as you throw in an absorbing material?

Something that's got an enormous absorption cross-section

is also going to have an enormous total cross-section,

because sigma total is sigma absorption plus sigma

scattering.

And sigma scattering doesn't change that much.

But if sigma absorption goes up, sigma total goes up.

If sigma total goes up, then what happens

o the diffusion coefficient?

AUDIENCE: Decrease.

MICHAEL SHORT: Yep, it's got a decrease.

And how does inserting a control rod change the geometry?

AUDIENCE: It doesn't.

MICHAEL SHORT: Very, very close.

Yeah, you're right.

The control rod better not change the geometry,

but what I do want to remind you of is

that this buckling term includes-- let's say,

this was a one dimensional infinite slab

Cartesian reactor.

That little hot over there means we have some extrapolation

distance.

Remember, if we were to draw our infinite reactor

with the thickness A and we wanted

to draw a flux profile on top of that,

it would have to be symmetric about the middle.

And let's say we had our axis of this is x and this is flux.

Flux can't go to zero right at the edge of the reactor,

because that would mean that no neutrons were literally

leaking out.

So there's going to be some small extrapolation

distance equal to about two times

the diffusion coefficient.

So the geometric buckling is actually

pi over the reactor geometry, plus 2 times

the diffusion coefficient.

And if the diffusion coefficient goes down,

but it's also very, very small compared

to the geometric buckling, how much does the buckling change

and by how much?

And in what direction?

AUDIENCE: It increases very slightly.

MICHAEL SHORT: Increases very slightly.

So the buckling might increase very slightly.

What's the overall net effect on k effective?

AUDIENCE: It goes down.

MICHAEL SHORT: Should go down, you would hope.

If you put a control rod in, it should

make k effective go down, because there's

a little decrease here.

Things kind of cancel out there.

But the big one is putting an absorption material,

like a control rod in, should make k effective go down.

And that's the most intuitive one,

but you can work out one term at a time what's

generally going to happen.

So let's now look at some other scenarios

for the same criticality condition.

I'll just rewrite it so that we can mess it all up again.

Now we want to go for the case of boil or void your coolant.

And now we're getting into the concept of different feedback

mechanisms.

We've already talked once about how

raising the temperature of something

tends to increase cross-sections in certain ways.

But now let's say, what would happen

if you boil your coolant?

If things got really hot and the water started to boil.

What do you want to happen to k effective?

You want it to increase?

AUDIENCE: Decrease.

MICHAEL SHORT: Decrease, thank you.

You want it to decrease, or else you'd get a Chernobyl.

And we'll talk about how that happened in a week or two.

So now let's reason through each one of these.

Let's assume that nu doesn't change

when you boil the coolant.

What about sigma fission of the whole reactor?

You're taking a little bit of material out of the reactor

by taking liquid water, which is fairly dense,

and making it gaseous water, which is less dense.

So overall, there are more fissile atoms in the reactor

proportionately when the coolant is boiled away

than when it's not.

So what happens to sigma fission?

The average sigma fission for the reactor

will go up ever so slightly.

Probably not enough to matter.

What about sigma absorption?

If the coolant disappears.

AUDIENCE: It goes down.

MICHAEL SHORT: Yeah, water is an absorber.

Hydrogen and oxygen-- really just hydrogen--

have some pretty non-negligible absorption coefficients.

And if those go away, then you're

losing a bit of absorber, aren't you?

Actually it's interesting.

Oxygen is the lowest thermal cross-section of any element.

So we can treat it as pretty much transparent.

Now how about the diffusion coefficient?

We've got the formula for it up there.

If all of a sudden your neutrons don't

have much to moderate from--

there's not much to moderate your neutrons.

Yeah.

AUDIENCE: Your scattering just disappears.

MICHAEL SHORT: Your scattering just disappears, right?

But so does some of your total cross-section.

So chances are, those neutrons are

going to go farther before they undergo any given

collision because there's no water in the way.

So you'd expect neutron diffusion to go up.

And what about geometric buckling?

Diffusion goes up, then the geometric buckling--

I'm just going to make it really small.

But the net effect here, once again, k effective goes down.

We didn't talk about anything to do with the actual temperature

effects on the cross-sections.

This is just a density thing on the coolant itself.

So let's now look at that.

What about if you have some power spike,

raised fuel temperature?

I'll write it again, so we can mess it up again.

So let's say you raise the fuel temperature.

And that's going to cause every cross-section effectively

to increase if you're doing this average scenario.

Let's talk a little bit about why.

It's not as simple as just saying,

the cross-sections go up.

So let's say we had two different temperatures,

cold and hot.

So this would be your sigma fission cold.

And this would be your sigma fission hot.

For cold, sigma fission looks something like that.

And as the temperature goes up, these

resonances, which I'll just label right here--

resonances being specific energy is

where the absorption suddenly goes up, suddenly goes

down will actually decrease in height.

But they'll start to spread out more.

That's about as well as I can draw it very crudely.

And same thing goes not just for sigma fission,

but for sigma anything, including absorption, including

total whatever you want.

And so if your goal is to get your neutrons

from the fast region where they're

born into the thermal region where you get fission,

broadening these cross-sections makes it more likely

that if the neutron loses any amount of energy,

it's going to hit one of these big resonance regions

and get absorbed or taken away before it gets a chance