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 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 gonna 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 our I Omega T in the number of neutrons per centimeter squared per second travelling through something and we also had its corresponding non angular dependent term on just R et 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 Omega T and it's corresponding we don't care about angle form and today what we're gonna 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 boards going to look like something rainbow explosion but all that's gonna 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 V at which they're moving and just to check our units flux should be in neutrons per centimeter squared per second and n is in neutrons per cubic centimeter and velocity is in centimeters per second so the unit's check out that's why I made that substitution right there and this way everything is in terms of little Phi 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 a regular old fission in this case we have knew the number of neutrons created per fission times Chi the sort of the fission births spectrum or at what energy the neutrons are born over 4pi to account for all different angles in which they could go out times the integral over our whole control volume and all other energies and 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 divisions happening in all the other groups I've also escalated this problem a little bit to not assume that the reactors 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 reactors been in a blender so it 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 want to think about it there's 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 n i n reactions so these are other reactions where it's absorb a neutron and give off anywhere between 2 & 4 neutrons beyond that there 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're summing over all possible I where you have this reaction D and I an reaction where one 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 ni n reaction that's just AI 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 photo fission some birth spectrum for photo fission some cross-section for photo fission 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 Ian Omega Prime's because it's some other energy 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 those should actually be it should 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 at energy E prime Omega prime ends up scattering into our energy in Omega so this would be the other one and this would be our group but otherwise the term looks very much the same and that takes care of all the possible gains of neutrons into our group the losses are a fair bit simpler there's 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 de notice here that these are all these energies and omegas are all our group because we only care about how many neutrons in our group undergo a reaction and leaf and the form is very simple integrate over volume energy in and Direction times a cross section times the flux just like all the other ones then the only difference and one right here is what we'll call leakage these are neutrons moving out of whatever control surface that we're looking at and this can be some arbitrarily complex control surface in 3d I don't really know how to draw a blob in 3d but at every point on that glob there's going to be a normal vector and you can then take the current of neutrons traveling out that normal vector and figure out how much of that is actually leaving our surface DS the one problem we had is that everything here is in terms of volume volume volume surface so we don't have all the same terms because once we have everything in the same variables we can start to make some pretty crazy simplifications the last thing we did is we invoked the divergence theorem that says that the surface integral of some variable f DS is the same as the volume integral of the divergence of that variable DV so I remember there was some snickering last time because you probably haven't seen this since was it 1801 or 1802 1802 okay that makes sense because divergence usually has more than one variable associated with it I'll include the dot because that's what makes it divergence so we can then rewrite this term let's start our simplification colors that's our divergence theorem so let's get rid of it in this form and call it minus integral over all that stuff then we'll have del dot little J to be careful R Omega T so for every step I'm going to use a different color so you can see which simplification led to how much crossing stuff out and so like I said this board is gonna look like a rainbow explosion but then we'll rewrite it at the end and it's gonna look a whole lot simpler so now let's start making some simplifications let's say you're an actual reactor designer and all you care about is how many neutrons are here of the variables here which one do you think we care the least about angle I mean do we really care which direction the neutrons are going no we pretty much care where are they and are they causing fission or getting absorbed so let's start our simplification board and in blue neglect angle this is where it starts to get fun so in this case we will just perform the Omega integral over all angles we just neglect angle here we forget the Omega integral forget Omega there away goes the 4 pi because we've integrated over all 4 pi star ad ins or all solid angle and let's just keep going forget the 4 pi forget the Omega forget the Omega forget the 4 PI and the Omega and the Omega same thing here forget the Omega in the scattering kernel forget it in the flux forget it there forget it there and there as well okay we've now completely eliminated one variable and all we had to do is ditch the for PI's and one of the integrals what next we're tracking right now every possible position every possible energy at every possible time if you want to know what is your flux going to be in the reactor at steady state what variable do you attack next yep time so let's just say this reactor is at steady state that's going to invoke a few things for one it's going to ditch the entire steady state term we're gonna get rid of all the T's in all the fluxes this shouldn't take too long to do I think that's all of them and the third thing is if this reactor is at steady state chances are we've taken our Kickstarter source out because we just needed it to get it going but the reactor should be self-sustaining once it's at steady state so let's just get rid of our source term so you just want to make sure I didn't miss any here okay next up go with green what else do you think we can simplify about this problem well if you look far enough away from the reactor we can make an assumption that the reactor is roughly homogeneous in some cases it's not so good of an assumption like very close to anything that has a huge absorption cross-section now I want to explain the physics behind this if the neutrons travel a very long distance through any group of materials then those materials will appear to be roughly homogeneous to the neutrons if however the neutrons travel through something that's very different from the materials around it then that homogeneous assumption breaks down so in what locations in a nuclear reactor do you think you cannot treat the system is homogeneous where do the properties of materials suddenly change by a huge amount yeah Luke control rods right so let's say it's bad for control rods where else how about the fuel all of a sudden you're moving from a bunch of structural materials where Sigma fission equals zero to the fuel where Sigma fission like you saw in the test can be like five hundred barns which even though it's got a very small exponent in front of it ten to the minus 22 centimeters squared it's still pretty significant so this assumption breaks down around the control rods and around the fuel but we can get around this let's analyze the simplest crazy as possible reactor which would be a molten salt fueled reactor it's just a blob of 700 Celsius goo that's got its fuel coolant and control rods all built in so if we assume that the reactor is homogeneous which is a pretty good assumption for molten salt fueled reactors because the fuels dissolved in the coolant and it builds up its own fission product poisons so it's got some of its own control rods kind of built in usually will have other extra ones too but whatever then we can start to really simplify things if we get rid of any homogeneity assumptions we cannot necessarily get rid of the R in the flux because even if the reactors homogeneous it still might have boundaries so you might be able to approximate it as just a cylinder or a slab of uniform materials but if we were to get rid of the R's in the flux term that would mean that as we graph flux as a function of distance it would look like that including infinitely far away from the reactor and now is that true absolutely not so I don't want to leave that up for anyone we'll fill in what these graphs look like a little later just leave them there for now we can get rid of some of the other ARS though like these cross-sections if the reactor is actually homogeneous then the cross-section is the same everywhere because the materials are the same everywhere so we can get rid of the ARS here the R's here and there and there and that's it I think don't think I missed any good next up if this reactors homogeneous then does it really matter at which location were taking this balance does it really matter which little volume element we're looking at we say these equations are will call them volume identical which means if this same equation is satisfied at any point in the reactor we don't need to do the volume integral over the whole reactor it's not like it's gonna change anywhere we go so forget the volume integrals hopefully you guys see where I'm going with this and I've never I've never tried teaching it like this rainbow explosion before but I'm kind of excited to see how it turns out so already like 2/3 of the stuff that we had written are gone what's the only variable left that we can go after and what's the only color left that I haven't really used energy yeah so we can make a couple of us different assumptions this equation as it is is not yet really analytically solvable because a lot of these energy dependent terms don't have analytical solutions or even forms like the cross-sections but we can start attacking energy hopefully this is different enough from white yeah is that big enough difference for you guys to see good ok we can start doing this in a few different ways I want to mention what they are and then we're gonna do the easiest one so the way it's done for real like in the computational reactor physics group is you can discretize the energy discretize the energy into a bunch of little energy groups so you can write this equation for every little energy group and assume that along this energy scale ranging from your maximum energy to probably thermal energy yeah oh to five okay let's do this clearly with thick chalk there we go you can then discretize into some little energy group let's say that's eg I that's eg I plus one and so on and so on and depending on the type of reactor that you're looking at and the energy resolution that you need you choose the number of energy groups accordingly does anyone happen to know for a light water reactor how many energy groups do you think we need to model a light water reactor the answer might surprise you it's just two actually all we care about so let's say this would be for the general case all we care about for a light water reactor is are your neutrons thermal or are they not because the neutrons that are not thermal are not contributing to fission that much they are just a little bit and you can account for those but pretty much they're not once the neutrons slow down to get thermal in the range from like let's say about an e V to that temperature took a surprising amount of time to write with sidewalk chalk then you've got things that are about 500 or a thousand times more likely to undergo fission and so all you care about is the neutrons are all born they're all born right about here and they scatter and bounce around and you don't care because they're just in this not thermal region and when they enter the thermal region you start tracking them because those are the ones that really count for fission and if you actually look up the specifications for the ap1000 this is a modern reactor under construction in many different places in the world when you see how do they do the neutronic analysis two group approximation so this isn't just an academic exercise to make it easier for sophomores to understand this is actually something that's done for real reactors so if you ever felt like I'm making it too simple no no this is I'm simplifying it down to what's really done and I will get you that specification so you can see what westinghouse says like this is how we design the reactor we made a two group simplification in many cases so you can discretize you can forget it which we're going to call the one group approximation or you can well let's say two group is the other one that we're actually gonna tackle we're gonna do this one forget energy but we're not really going to forget energy because you can't just pick an energy and pick a cross-section and say okay that's the cross-section we're going to use if most cross-sections have the following form if this is log of energy and this is log of Sigma and it goes something like that what energy do you pick go ahead tell me which energy do you pick anyone want to wager a guess the ones before after the big squigglies I don't think that's correct because if you do it this then you're gonna weigh underestimate vision if you do it here you're gonna weigh overestimate vision or whatever rank we have the reaction you have we didn't say which reaction this is the rest of you who are silent and afraid to speak up you're actually correct I wouldn't actually pick any single value here what you need to do is find some sort of average cross-section for whatever reaction that accurately represents the number of reactions happening in the system and in order to do that you have to come up with some average cross-section for whatever reaction you have by integrating over your whole energy range of the energy dependent cross-section as a function of energy times your flux de over this look familiar from 1801 or two as well what's the average value of some function a little bit well we'll bring it back here now so retrieve it from cold storage in your memories because this is how actual cross-sections are averaged for whatever energy range you're picking you know what I'm gonna make this a little more general I won't say zero I'll just say your minimum energy for your group so now this equation is general for the multi group and the one group and two group method for whatever cross-section you want to pick and whatever energy range you're looking at you take the actual data and perform an average for the fast and thermal delineation where let's say this is fast and this is thermal you would have two different averages maybe this average would be right there you know what let's use white so it actually has some contrast so this would be one value of the cross-section and maybe the next average would be right there so you simplify this absolutely non analytical form of your complicated cross-section to just a couple of values maybe we'll call that average Sigma fast and we'll call that average Sigma thermal so using this analogy and this color we can then say we're gonna take an average new an average CAI get rid of the energies because we can perform the same energy average integration on every quantity with energy dependence so all we do is we put a bar there ditch the energies it's the energies and let's just say that flux is going to be what it is same thing here yeah same thing there and there and there they're there they're here and here and here and there's a cross-section there's an energy there's an energy there is a cross-section we don't care about those anymore and there's a couple of other implications of this energy simplification what is the birth spectrum now what's the probability that a neutron is born in our energy group which contains all energies one yeah okay so forget the Chi and that one and that one and what about this scattering kernel what's the probability that a neutron scatters from any other energy which is already in our group into our group which contains all energies yeah scattering no longer matters when you do the one group approximation because if the neutron loses some of its energy it's still in our energy group because our energy group contains all energies so forget the scattering kernel and forget the energy integrals what are we actually left with not much there's no green in here yet good because I need to do one more thing there is no more green oh we did green we did time okay green red orange is if the orange I used damn it okay we've used those purple no we've used that oh my god we've used both blues bright yellow yeah yes chi is the fission birth spectrum the probability that a neutron is born at any given energy but because all neutrons are born in our energy group which contains all energies then that just becomes one and goes away there's there's no birth spectrum because they're just born in our group yes that makes sense yeah okay I think I found the reaction we'll only color besides black on a black chalkboard and white which we already have that I have left we also have a slightly darker shade of gray but I'm literally out this worked out awesome because there's one more thing that we want to deal with what do we even have left all right what is the one term that is not in all of the same variables as the others that current that J what do we do about that so we're going sorry yeah the F e to e so I'll actually I'll recreate some of our variables here because there's a lot of them so our F of e prime to e is what's called the scattering kernel and that's the probability that a neutron scatters from some other energy Group E Prime in two hours in a about de and Chi of e is the fission birth spectrum and just for completeness knew of e is our Neutron multiplier or neutrons per fission I think that gives a pretty complete explanation of what's up here so now let's figure out how to deal with the current term this is when we make one of the biggest approximations here and go from what's called the neutron transport equation which is a fully accurate physical model of what's really going on to the neutron diffusion equation and this is where it gets really fun you don't assume that neutrons are subatomic particles that are whizzing about and knocking off of everything else you then treat the neutrons kind of like a gas or like a chemical and you just say that it follows the laws of diffusion again this works out very well except for places where cross sections suddenly change like near control rods or near fuel but for most of the reactor especially if we have a molten salt fuel reactor we can invoke what's called ficks law this sound familiar to anyone ficks law of diffusion three or nine one or five eleven one it's it is that it's the change of a chemical on down a density or concentration gradient so yeah you've got the idea what ficks law says is that the current or let's say the diffusion current or the Neutron current is going to be equal to some diffusion coefficient times the gradient of whatever a chemical concentration you've got let me put the C in there so right here this would be the current I'll label in a different color this would be your variable of interest maybe C is for concentration or Phi is for flux oh that reminds me where are those bars on our flux which term did we do energy whereas my slightly blue over here all of these Phi's become Capitol because we've gotten rid of all the angular and energy in everything dependence Oh angular dependence neglect Omega that should be dark blue mega goes away and the flux has become capital so many terms to keep track of luckily you will never have to and then the J becomes a capital J miss any Phi's here no because that one was already gone cool alright so we can use ficks law and transform the current into something related to flux and what we're saying here is that we're getting rid of the true physics which is that there's some fixed Neutron current and we're saying that neutrons behave kind of like a gas or a chemical in solution and so in yellow we can ditch our current related term and rewrite it we don't have any integrals left as negative del squared Phi I think the only variable left is our not too bad now we have a second order linear differential equation describing the flow of neutrons in this system it's we actually have something that we can solve for flux I think it's time to rewrite it wouldn't you say this has been fun so let's rewrite what's left actually see everything there will write it in boring old white so we have no transient dependence and we have left Sigma fission times flux as a function of our no source and we have our Neutron and I n reactions of oh we forgot our new Sigma fission then we have our I Sigma fission from n I n times flux next term we have photo fission so we have a new from gamma rays times Sigma fission from gamma rays times flux next up we have well last last simplification to make we have scattering and we have total cross-section when we said forget about energy and our scattering kernel becomes one and that's light blue gonna make one more modification to this board do we care about scattering at all anymore whatsoever because scattering doesn't change the number of neutrons left so we can then take these two terms and just call it Sigma absorption times flux because if we take scattering minus the total cross-section it's like saying all that's left if you don't scatter is you absorb and if you remember all add to the energy pile we said that our total cross-section is scattering plus absorption and absorption could be vision and capture and capture could be lets say capture with nothing happens Plus these end I and reactions plus any other capture reaction that does something so we're going to use this cross section identity right here with a couple of minus signs on it and say well scattering minus total leaves you with negative absorption to simplify terms I'll leave that up there for everyone to see so then we have scattering and total just becomes minus Sigma absorption times Phi of R and we're left with what was that current that becomes plus there's Adi missing in there isn't there a yellow D minus D okay and that's it yes D is the diffusion coefficient right here so we're assuming that neutrons diffused like a gas or a chemical with some diffusion coefficient and so we'll define what that is Oh probably next class because we have seven minutes yeah Luke uh-huh the see right here that's whatever variable we're tracking so let's call that flux or let's call it an the number of neutrons because flux is just number of neutrons times velocity so let's say that the concentration was the concentration of neutrons and we just multiplied by their velocity to get flux so it's almost like we can say that the concentration of neutrons is directly related to the flux and that way we have everything in flux and that's the entire Neutron diffusion equation yep this is for one group with all the assumptions we made right here homogeneous what other assumptions did we make steady state and we already neglected that and I think that's enough qualifiers for this but it's directly from this equation right here that we can develop what's called our criticality condition under what conditions is the reactor critical so then in this case by critical we're gonna have some variable called K effective which defines whether that's defines the number of neutrons produced over the number of neutrons consumed and if K effective equals 1 then we say that the reactor is critical that means that exactly the number of neutrons produced by regular fission ni n reactions and photo fission equals exactly the number of neutrons absorbed in the anything and that leaked out so let's relabeled our terms in the same font that we did here so this would be the fission term this would be n i n reactions this would be photo fission this would be absorption this would be leakage how many neutrons get out of our finite boundary and if you remember when we started out we said we're gonna make the neutron balance equation equal to gains minus losses and through our rainbow explosion simplification we've done exactly that these are your gains these are your losses when gains minus losses equals zero the reactors in perfect balance yep leakage comes out to be negative despite the plus sign here and that's actually intentional that's because neutrons travel down the concentration gradient so let's say we're gonna draw an imaginary flux spectrum that's gonna be quite correct and I'm doing all of those features for a reason but let's look at the concentration gradient right here leakage is positive when your flux gradient is negative that's why the sign is flipped right there so a positive diffusion term means you have neutrons leaking out down a negative concentration gradient because if you look at the slope here the change in X is positive and the change in flux is negative so the slope is negative concentration gradients negative that's why the sign is the opposite of what you may expect and the same thing goes for chemical or gaseous or any other kind of diffusion that's come glad you asked because that's always a point of confusion is why is there that plus sign that's intentional and that's correct cool so it's in yeah Sean in that case if you really explicitly write our losses would it be - absorption let's put some parentheses on here equals zero and a minus and when we say plus leakage we have that plus sign in there so I'm not gonna put any parentheses up here because that wouldn't be correct but what I can say is that gains minus losses have to be in perfect balance to have a K effective equal to one anyone else have any questions before I continue the explanation cool let's say you're producing more neutrons than you're destroying that's what we call supercritical so I just did an interview for this K through 12 outreach program and they said should be people be afraid when something quote-unquote goes critical sounds scary emotionally right and the answer is absolutely not if your reactor goes critical it's turned on and it's in perfect balance that's exactly what you want so going critical is not a scary thing it means we have control if something goes supercritical it doesn't necessarily mean it's out of control reactors can be very slightly supercritical and still in control because of what's called delayed neutrons which I will not introduce today because we have two minutes if a reactor has a K effective of less than 1 we call that subcritical so it's important to note that the the nuclear terminology that's kind of leaked out into our vernacular is not physically correct in the way that it's used words like critical or me are used to incite emotions and bring about fear when to a nuclear engineer critical means in perfect control in balance like you would expect are in equilibrium that all sounds kind of nice makes you calm down a little bit yeah so we can put one last term in front of our criticality condition we can take either the gains or the losses move the equal sign and zero over a little bit and put a 1 over K effective here this then perfectly describes the difference between the gains and the losses in a reactor so if the gains equal the losses then k effective must equal 1 and the reactor has got to be in balance if there are more gains than losses which means if you are producing more neutrons than you're consuming then k effective must be greater than 1 for this equation to still equal zero because this equation must be satisfied so if you're making more neutrons your k effectives got to be greater than 1 so you have a less than 1 multiplier in front and on the opposite side if you're losing more neutrons than you're gaining your K effective has to be less than 1 to make this equation balanced going along with all these definitions right here so it's exactly 505 of I've given you delivered promised blackboard of Lucky Charms and we've hit perfect spot which is the one group homogeneous steady-state neutron diffusion equation from which we can develop our criticality conditions and solve this much simpler equation to get the flux profiles that I've started to draw here so I want to stop here and take any questions on any of the terms you see here yeah the disparate energy groups and then when we're doing the one group you're actually just treating sperm one fast together we are that's right yep so a lot of reactors at least thermal reactors where you only care if neutrons are thermal or not two groups enough when you have a one group or a two group equation these are fairly analytically solvable things and get to any more groups than that and yes they're analytically solvable but it gets horrible and that's why we have computers to do the sorts of repetitive calculations over and over again once we've solved the one group equation I'll then show you intuitive ways to write but not solve the equations for multi group equations sure any other questions so like I promise we didn't stay complex for long because there's basically nothing left yeah oh that's a partial derivatives yep that yeah there we go yeah so this is saying a change in Neutron population or the partial derivative of n with respect to T cuz n varies with space energy angle time and anything else you could possibly think about equals the gains minus the losses I think this is worthy of a t-shirt if any of you guys would like to update the department shirts to properly take into account photo fission external sources and ni n reactions I think it would make for a much more impressive thing because we kind of printed an oversimplification before it's too bad and we definitely had room on the shirt there was like room on the sides and on the sleeves yeah keep going on that might have to be long-sleeved I think they'll be pretty soon yeah okay if no one else has any immediate questions you'll have plenty of time tomorrow because the whole goal tomorrow is gonna be to solve this equation that's only gonna take like 20 minutes so we can do a quick review of the simplification of the neutron transport equation solve the neutron diffusion equation if you have questions we'll spend time to answer them there and if you don't we'll move on to writing multi group equations and also Friday for recitation it's electron microscope time so now that you guys have learned different electron interactions with matter you're going to see them so we're going to be analyzing a couple of different pieces of materials that a couple of you are gonna get to select and we're going to image them with electrons to show how you can beat the wavelength of light imaging limit like I told you before we're gonna produce our own x-ray spectra to analyze them elementally where you'll see the bremsstrahlung you'll see the characteristic Peaks and you'll see a couple of other features that I'll explain too so get ready for some SEM tomorrow