字幕列表 影片播放 列印英文字幕 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. 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.