1 00:00:01,090 --> 00:00:03,460 The following content is provided under a Creative 2 00:00:03,460 --> 00:00:04,850 Commons license. 3 00:00:04,850 --> 00:00:07,060 Your support will help MIT OpenCourseWare 4 00:00:07,060 --> 00:00:11,150 continue to offer high-quality educational resources for free. 5 00:00:11,150 --> 00:00:13,690 To make a donation or to view additional materials 6 00:00:13,690 --> 00:00:17,650 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,650 --> 00:00:18,550 at ocw.mit.edu. 8 00:00:22,740 --> 00:00:25,350 PROFESSOR: So Tuesday, we developed the largest equation 9 00:00:25,350 --> 00:00:27,720 that you'll probably ever use at MIT. 10 00:00:27,720 --> 00:00:31,170 Thursday, we destroyed it down to a rather manageable size. 11 00:00:31,170 --> 00:00:33,690 And today, we are going to solve it and actually 12 00:00:33,690 --> 00:00:36,420 show you guys how to work out different reactor problems all 13 00:00:36,420 --> 00:00:40,020 the way from simple one-group homogeneous reactors 14 00:00:40,020 --> 00:00:43,450 to expanding intuitively, not directly, from the mathematics 15 00:00:43,450 --> 00:00:47,100 to solve, or to pose and solve, two-group reactor 16 00:00:47,100 --> 00:00:50,460 problems like the one they did on the AP 1000 reactor 17 00:00:50,460 --> 00:00:53,070 where they separate the neutrons into a fast 18 00:00:53,070 --> 00:00:54,570 and a thermal group. 19 00:00:54,570 --> 00:00:56,970 And I wanted to put up where we left off. 20 00:00:56,970 --> 00:01:03,150 We have some equation describing fission, n, n reactions, photo 21 00:01:03,150 --> 00:01:06,810 fission, absorption, and leakage or diffusion 22 00:01:06,810 --> 00:01:10,080 as the big balance equation for how many neutrons are there 23 00:01:10,080 --> 00:01:11,070 in some reactor. 24 00:01:11,070 --> 00:01:15,480 And I think last time, I had some example reactors 25 00:01:15,480 --> 00:01:21,250 up on the board ranging from an infinite slab 26 00:01:21,250 --> 00:01:24,520 with a thickness a, to a cylinder with-- we'll call it 27 00:01:24,520 --> 00:01:27,220 a thickness a and a height z. 28 00:01:31,200 --> 00:01:33,990 And the question that we want to be able to answer 29 00:01:33,990 --> 00:01:41,170 is, if we draw a graph of flux versus x through this reactor, 30 00:01:41,170 --> 00:01:49,483 and in this case, it would be r and z, 31 00:01:49,483 --> 00:01:50,900 what do these functions look like? 32 00:01:50,900 --> 00:01:53,155 What is the form of this flux? 33 00:01:53,155 --> 00:01:54,530 And so today, from this equation, 34 00:01:54,530 --> 00:01:56,000 we're actually going to solve it. 35 00:01:56,000 --> 00:01:56,810 So bear with me. 36 00:01:56,810 --> 00:01:59,583 There'll be maybe 20 minutes of remaining derivation, 37 00:01:59,583 --> 00:02:01,500 and then I'm going to teach you how to use it. 38 00:02:01,500 --> 00:02:03,140 So today's class is going to be more 39 00:02:03,140 --> 00:02:04,807 like a recitation of how do you actually 40 00:02:04,807 --> 00:02:07,880 use this equation instead of getting to it. 41 00:02:07,880 --> 00:02:09,560 First of all, I want to simplify things, 42 00:02:09,560 --> 00:02:10,685 which is going to make it-- 43 00:02:13,270 --> 00:02:14,065 de-escalate. 44 00:02:17,470 --> 00:02:22,390 If we simplify things and don't worry about these NIN reactions 45 00:02:22,390 --> 00:02:25,690 and photo fission, we have the equation that you'll actually 46 00:02:25,690 --> 00:02:26,740 see in your reading. 47 00:02:26,740 --> 00:02:28,570 I'm going to drop them for now-- 48 00:02:28,570 --> 00:02:32,020 and I realize I'm missing one little gamma-- 49 00:02:32,020 --> 00:02:33,460 because they're just extra terms. 50 00:02:33,460 --> 00:02:36,610 They were instructive in writing the neutron transport equation 51 00:02:36,610 --> 00:02:38,200 because all the terms looked similar, 52 00:02:38,200 --> 00:02:41,520 but now they're just kind of extra things for us to write. 53 00:02:41,520 --> 00:02:44,880 And the last thing that isn't in units of flux 54 00:02:44,880 --> 00:02:48,630 is this Laplacian operator, for those who don't remember. 55 00:02:53,630 --> 00:02:56,360 And this Laplacian operator takes different forms 56 00:02:56,360 --> 00:03:00,200 in different dimensions and different coordinate systems. 57 00:03:00,200 --> 00:03:05,930 For 1d Cartesian, it's pretty easy. 58 00:03:05,930 --> 00:03:11,050 It's just double derivative, let's say in x. 59 00:03:11,050 --> 00:03:22,700 For cylindrical, it's significantly uglier, 60 00:03:22,700 --> 00:03:26,740 plus d squared over d r squared. 61 00:03:26,740 --> 00:03:28,450 So this is what the Laplacian operator 62 00:03:28,450 --> 00:03:33,460 looks like in the case of a finite cylindrical reactor. 63 00:03:33,460 --> 00:03:36,820 So first we're going to focus on the infinite slab 64 00:03:36,820 --> 00:03:41,190 case right here because it's a lot easier to solve, 65 00:03:41,190 --> 00:03:42,060 analytically. 66 00:03:42,060 --> 00:03:43,560 And then we'll show you how it would 67 00:03:43,560 --> 00:03:45,367 go for the cylindrical reactor which 68 00:03:45,367 --> 00:03:47,700 looks a lot more like the reactors you'll see everywhere 69 00:03:47,700 --> 00:03:50,440 else in the world. 70 00:03:50,440 --> 00:03:54,680 So let's start doing a little bit of rearrangement 71 00:03:54,680 --> 00:04:03,510 and isolate that Laplacian, so we can then subtract d del 72 00:04:03,510 --> 00:04:05,230 squared phi from each side. 73 00:04:12,490 --> 00:04:13,800 OK. 74 00:04:13,800 --> 00:04:18,060 And that cancels those out. 75 00:04:18,060 --> 00:04:20,339 And then we've got something on the left hand, is 76 00:04:20,339 --> 00:04:23,250 in del squared phi, and on the right side 77 00:04:23,250 --> 00:04:24,990 has all in units of flux. 78 00:04:24,990 --> 00:04:29,150 So then we can divide everything by the flux. 79 00:04:34,640 --> 00:04:36,500 And if we then cancel out all the flux 80 00:04:36,500 --> 00:04:40,210 terms, all two of them, we're left 81 00:04:40,210 --> 00:04:41,620 with something awfully simple. 82 00:04:41,620 --> 00:04:45,670 And the last thing we can do is divide by d on both sides. 83 00:04:48,867 --> 00:04:49,450 And I'm sorry. 84 00:04:49,450 --> 00:04:51,117 I had to cancel that flux because that's 85 00:04:51,117 --> 00:04:52,450 the way that goes. 86 00:04:52,450 --> 00:04:57,080 The d's cancel, and I'll redraw what we have right here. 87 00:04:57,080 --> 00:05:03,970 So we have minus del squared flux over flux 88 00:05:03,970 --> 00:05:08,170 equals constants. 89 00:05:08,170 --> 00:05:10,180 And what are they actually? 90 00:05:10,180 --> 00:05:15,390 It'd be 1 over k times new sigma fission 91 00:05:15,390 --> 00:05:20,100 minus sigma absorption over d. 92 00:05:20,100 --> 00:05:23,070 Everyone remember why we're putting these bars 93 00:05:23,070 --> 00:05:25,380 on the cross sections in d and everything? 94 00:05:25,380 --> 00:05:26,603 Can someone just tell me? 95 00:05:26,603 --> 00:05:29,070 AUDIENCE: Averaged? 96 00:05:29,070 --> 00:05:30,070 PROFESSOR: That's right. 97 00:05:30,070 --> 00:05:34,180 We've averaged overall energy, so some average cross-section 98 00:05:34,180 --> 00:05:39,190 would be the average from our minimum to our maximum energy 99 00:05:39,190 --> 00:05:46,770 of that cross-section as a function of energy times flux, 100 00:05:46,770 --> 00:05:47,780 over the range itself. 101 00:05:56,912 --> 00:05:59,370 So by this, we're saying, we're averaging the cross-section 102 00:05:59,370 --> 00:06:02,970 somehow over the whole energy range. 103 00:06:02,970 --> 00:06:09,650 Now, I ask you guys, what sort of functions in Cartesian space 104 00:06:09,650 --> 00:06:11,765 happen to have its double derivative over itself 105 00:06:11,765 --> 00:06:12,515 equal to constant? 106 00:06:12,515 --> 00:06:15,976 AUDIENCE: Exponentials. 107 00:06:15,976 --> 00:06:16,967 PROFESSOR: Is what? 108 00:06:16,967 --> 00:06:17,925 AUDIENCE: Exponentials. 109 00:06:17,925 --> 00:06:23,380 PROFESSOR: Exponentials, or sine and cosine. 110 00:06:23,380 --> 00:06:25,810 You've basically said the same thing two different ways, 111 00:06:25,810 --> 00:06:29,890 yeah, because the plus and minus exponentials can be 112 00:06:29,890 --> 00:06:32,230 rewritten in sines and cosines. 113 00:06:32,230 --> 00:06:35,650 So if we now assume that our flux solution has 114 00:06:35,650 --> 00:06:46,060 to take the form of, let's call it a cosine bx plus c cosine, 115 00:06:46,060 --> 00:06:47,680 what's the next constant? 116 00:06:47,680 --> 00:06:48,940 Fx. 117 00:06:48,940 --> 00:06:51,610 I'm sorry, that's a sine. 118 00:06:51,610 --> 00:06:53,610 I don't want to use d again for obvious reasons, 119 00:06:53,610 --> 00:06:54,790 and e stands for energy. 120 00:06:54,790 --> 00:06:58,380 So that's the next letter that we haven't really used. 121 00:06:58,380 --> 00:07:03,290 The flux has got to take this shape or else this condition 122 00:07:03,290 --> 00:07:04,320 is not satisfied. 123 00:07:04,320 --> 00:07:06,930 So this is the easy way of solving differential equations, 124 00:07:06,930 --> 00:07:10,010 which is guess the solution from previous knowledge 125 00:07:10,010 --> 00:07:11,870 or experience or something. 126 00:07:11,870 --> 00:07:15,800 So based on this, if we were to draw a flux profile, 127 00:07:15,800 --> 00:07:20,750 let's say, right here was x equals 0, one of those terms 128 00:07:20,750 --> 00:07:23,475 has got to go away by reasons of symmetry. 129 00:07:23,475 --> 00:07:24,600 What do you think it'll be? 130 00:07:28,725 --> 00:07:30,100 Which of those halves right there 131 00:07:30,100 --> 00:07:34,800 is symmetric about the x-axis right here? 132 00:07:34,800 --> 00:07:37,048 Sine or the cosine? 133 00:07:37,048 --> 00:07:38,390 AUDIENCE: Cosine. 134 00:07:38,390 --> 00:07:39,200 PROFESSOR: Cosine. 135 00:07:39,200 --> 00:07:40,040 Yep. 136 00:07:40,040 --> 00:07:45,930 The sines, it can be inverted around your coordinate system, 137 00:07:45,930 --> 00:07:46,990 but it's not symmetric. 138 00:07:46,990 --> 00:07:49,060 It's not a mirror image. 139 00:07:49,060 --> 00:07:54,770 So actually, the sine term goes away 140 00:07:54,770 --> 00:07:56,600 and we've got to have some solution which 141 00:07:56,600 --> 00:08:06,780 looks like a times cosine b of x because right here our d flux 142 00:08:06,780 --> 00:08:11,060 dx should equal 0 at x equals 0. 143 00:08:11,060 --> 00:08:13,730 Now, I've intentionally drawn the flux 144 00:08:13,730 --> 00:08:17,000 not to go to zero at the very edge of the reactor. 145 00:08:17,000 --> 00:08:18,680 If the flux automatically went to zero 146 00:08:18,680 --> 00:08:20,138 at the edge of the reactor, there'd 147 00:08:20,138 --> 00:08:21,590 be no need to shield it, right? 148 00:08:21,590 --> 00:08:27,420 So there are some neutrons streaming out of this reactor. 149 00:08:27,420 --> 00:08:29,720 And this distance right here is actually 150 00:08:29,720 --> 00:08:35,990 equal to two times the diffusion constant. 151 00:08:35,990 --> 00:08:40,929 Let me get rid of the stuff we don't need anymore. 152 00:08:40,929 --> 00:08:42,990 This is what we call the extrapolation length. 153 00:08:51,330 --> 00:08:52,943 And now I should also mention, I'm not 154 00:08:52,943 --> 00:08:54,360 going to derive it because I think 155 00:08:54,360 --> 00:08:56,027 we've done enough deriving for one week, 156 00:08:56,027 --> 00:08:57,480 but I'll just give you the form. 157 00:08:57,480 --> 00:08:59,220 This diffusion constant can actually 158 00:08:59,220 --> 00:09:09,180 be expressed in terms of other cross sections 159 00:09:09,180 --> 00:09:18,560 where this mu nought is what's known as the average scattering 160 00:09:18,560 --> 00:09:19,190 angle cosine. 161 00:09:24,950 --> 00:09:28,820 And it approximately equals 2/3 times 162 00:09:28,820 --> 00:09:31,520 the average atomic mass of whatever 163 00:09:31,520 --> 00:09:32,990 it's scattering off of. 164 00:09:32,990 --> 00:09:34,730 So this d that I've just introduced 165 00:09:34,730 --> 00:09:37,610 from some physical analogy actually has 166 00:09:37,610 --> 00:09:40,820 an expression from cross sections and the material 167 00:09:40,820 --> 00:09:42,290 properties that you can look up. 168 00:09:42,290 --> 00:09:45,710 So we've now turned it into some sort of condition 169 00:09:45,710 --> 00:09:48,560 where everything can be looked up in the Janis library, 170 00:09:48,560 --> 00:09:51,260 or whatever cross section library you have, 171 00:09:51,260 --> 00:09:53,540 plus whatever number densities you actually 172 00:09:53,540 --> 00:09:54,420 have in your reactor. 173 00:09:54,420 --> 00:09:59,300 So the picture's starting to get very real and very physical. 174 00:09:59,300 --> 00:10:06,860 So now, if we assume that phi takes the form of a cosine bx, 175 00:10:06,860 --> 00:10:08,430 let's plug it in. 176 00:10:08,430 --> 00:10:10,420 So if we rewrite this expression-- 177 00:10:10,420 --> 00:10:15,340 I think I'll need another color for a substitute-- 178 00:10:20,530 --> 00:10:25,580 so we'll take a cosine bx and stick it in there 179 00:10:25,580 --> 00:10:26,690 and stick it in there. 180 00:10:29,270 --> 00:10:33,200 And we'll end up with, minus del squared cosine 181 00:10:33,200 --> 00:10:37,850 is going to look like a times b squared 182 00:10:37,850 --> 00:10:46,490 cosine bx over a cosine bx equals those constants. 183 00:10:52,058 --> 00:10:54,180 It's starting to get very simple. 184 00:10:54,180 --> 00:10:57,290 Keep the bars on there because they're quite important. 185 00:10:57,290 --> 00:11:03,260 And if we cancel things out, the cosine bx's go away, 186 00:11:03,260 --> 00:11:06,610 the a's go away, and we're left with whatever 187 00:11:06,610 --> 00:11:07,970 is inside that cosine. 188 00:11:07,970 --> 00:11:12,470 B squared equals a bunch of material properties. 189 00:11:19,090 --> 00:11:22,060 There's no information in here about the geometry 190 00:11:22,060 --> 00:11:23,590 of our reactor. 191 00:11:23,590 --> 00:11:26,590 There's only the material properties, 192 00:11:26,590 --> 00:11:29,920 whereas over here, this constant b, 193 00:11:29,920 --> 00:11:34,210 I'm going to add a little g to it which stands for geometry. 194 00:11:34,210 --> 00:11:36,310 And we've now set up a condition where 195 00:11:36,310 --> 00:11:39,130 if you know the geometry and the materials in the reactor, 196 00:11:39,130 --> 00:11:42,610 you can solve for the final unknown which is its k 197 00:11:42,610 --> 00:11:44,020 effective. 198 00:11:44,020 --> 00:11:47,340 How critical or not is this reactor? 199 00:11:47,340 --> 00:11:51,843 So what would b have to be to make this cosine valid? 200 00:11:57,140 --> 00:12:00,530 So what would bg equal if we have the form of this flux 201 00:12:00,530 --> 00:12:01,100 like so? 202 00:12:09,020 --> 00:12:10,310 I'll give you a hint. 203 00:12:10,310 --> 00:12:15,710 If the cosine goes to zero right here at some reactor 204 00:12:15,710 --> 00:12:20,060 length, a over 2, plus some extrapolation length, 2d, 205 00:12:20,060 --> 00:12:24,740 then how do we make the cosine equal to zero at this point? 206 00:12:24,740 --> 00:12:25,970 What does bg have to equal? 207 00:12:29,449 --> 00:12:32,940 AUDIENCE: Pi halves over 2d? 208 00:12:32,940 --> 00:12:34,580 PROFESSOR: Quite close. 209 00:12:34,580 --> 00:12:41,970 Bg would have to equal pi over a over 2 plus 2d, such that 210 00:12:41,970 --> 00:12:43,880 when you substitute-- 211 00:12:43,880 --> 00:12:45,930 or over 2, right, yeah. 212 00:12:49,250 --> 00:12:52,950 Oh, yeah, OK-- so that when you substitute x equals a over 2 213 00:12:52,950 --> 00:12:58,335 plus 2d into there, this cosine evaluates out to zero. 214 00:12:58,335 --> 00:13:01,360 Does that makes sense? 215 00:13:01,360 --> 00:13:02,040 Cool. 216 00:13:02,040 --> 00:13:06,230 So now, we have bg. 217 00:13:06,230 --> 00:13:07,940 There's nothing here but geometry 218 00:13:07,940 --> 00:13:09,950 plus a little bit of extrapolation length 219 00:13:09,950 --> 00:13:10,640 right here. 220 00:13:10,640 --> 00:13:10,800 Yeah? 221 00:13:10,800 --> 00:13:12,520 AUDIENCE: Shouldn't there be an over 2? 222 00:13:12,520 --> 00:13:13,730 PROFESSOR: Oh, yeah. 223 00:13:13,730 --> 00:13:17,370 There should be an over 2, so that when 224 00:13:17,370 --> 00:13:22,260 you plug in x equals a over 2 plus 2d, you get pi over 2, 225 00:13:22,260 --> 00:13:24,570 and cosine of pi over 2 is 0. 226 00:13:24,570 --> 00:13:25,500 Yep. 227 00:13:25,500 --> 00:13:26,550 Cool. 228 00:13:26,550 --> 00:13:28,560 So now we have our bg. 229 00:13:28,560 --> 00:13:31,830 So we substitute that back in there. 230 00:13:31,830 --> 00:13:35,220 Let's just continue to call it bg since we have it over there. 231 00:13:35,220 --> 00:13:38,520 And we have this much simpler expression, bg squared equals 1 232 00:13:38,520 --> 00:13:45,500 over k mu sigma fission plus sigma absorption over d. 233 00:13:45,500 --> 00:13:47,660 Then, where's our rearranging color? 234 00:13:47,660 --> 00:13:49,580 We can multiply everything by d. 235 00:13:53,620 --> 00:13:56,720 And let's see. 236 00:13:56,720 --> 00:13:58,280 That should have been a minus. 237 00:13:58,280 --> 00:14:00,277 Copied myself over wrong. 238 00:14:00,277 --> 00:14:03,940 The d's go away. 239 00:14:03,940 --> 00:14:08,570 You can then add sigma absorption to each side. 240 00:14:11,276 --> 00:14:12,940 And then those go away. 241 00:14:16,250 --> 00:14:25,350 And then, you can multiply everything by k, 242 00:14:25,350 --> 00:14:28,440 and those go away. 243 00:14:28,440 --> 00:14:35,910 And finally, divide everything by sigma a plus d bg squared. 244 00:14:41,610 --> 00:14:45,660 And what we're left with is our criticality condition. 245 00:14:45,660 --> 00:14:50,900 Our k, our criticality, is pretty simple, 246 00:14:50,900 --> 00:14:57,690 nu sigma fission over absorption plus leakage. 247 00:14:57,690 --> 00:14:59,670 So finally, after all that derivation, we've 248 00:14:59,670 --> 00:15:01,890 arrived at some intuitive result. 249 00:15:01,890 --> 00:15:04,470 Remember we said that k, the criticality, or the k 250 00:15:04,470 --> 00:15:11,150 effective, is the ratio of gains to losses of neutrons. 251 00:15:11,150 --> 00:15:12,910 So that's exactly what we have. 252 00:15:12,910 --> 00:15:16,233 The only gain mechanism right here is by fission, 253 00:15:16,233 --> 00:15:17,650 and the loss mechanisms are either 254 00:15:17,650 --> 00:15:20,500 absorption by the stuff in the reactor or leakage 255 00:15:20,500 --> 00:15:23,920 outside of the reactor. 256 00:15:23,920 --> 00:15:25,570 And so this is actually how you tell 257 00:15:25,570 --> 00:15:28,060 when the reactor's in perfect balance, 258 00:15:28,060 --> 00:15:31,660 is if this condition is satisfied, and if it equals 1, 259 00:15:31,660 --> 00:15:34,705 then the reactor is critical. 260 00:15:34,705 --> 00:15:36,330 So now, we can start to play with this. 261 00:15:36,330 --> 00:15:39,360 And let's say we started off with a critical reactor 262 00:15:39,360 --> 00:15:43,920 and all of a sudden, we were to boost the absorption. 263 00:15:43,920 --> 00:15:47,590 What should happen to k effective? 264 00:15:47,590 --> 00:15:49,630 If you start observing more, right, it 265 00:15:49,630 --> 00:15:51,250 should go sub-critical. 266 00:15:51,250 --> 00:15:54,100 Now, mathematically speaking, that means this denominator 267 00:15:54,100 --> 00:15:54,920 gets larger. 268 00:15:54,920 --> 00:15:57,220 So this ratio gets smaller. 269 00:15:57,220 --> 00:16:01,340 And therefore, k effective has to go down, as well. 270 00:16:01,340 --> 00:16:02,350 Now that's an easy case. 271 00:16:04,960 --> 00:16:07,810 Let's start exploring some of the more interesting ones. 272 00:16:07,810 --> 00:16:12,450 Let's say you raise the temperature in the reactor. 273 00:16:12,450 --> 00:16:17,430 Do we necessarily know what's going to happen next? 274 00:16:17,430 --> 00:16:18,890 Let's work it out. 275 00:16:18,890 --> 00:16:22,040 So most cross sections, when you raise the temperature, 276 00:16:22,040 --> 00:16:25,190 will actually go down in value due to a process called 277 00:16:25,190 --> 00:16:28,190 Doppler broadening that you'll learn about in 22.05. 278 00:16:28,190 --> 00:16:30,560 But suffice to say for now that cross sections tend 279 00:16:30,560 --> 00:16:32,420 to go down with temperature. 280 00:16:32,420 --> 00:16:36,050 The most important reason why is because a cross section is 281 00:16:36,050 --> 00:16:39,920 a number density times a microscopic cross section, 282 00:16:39,920 --> 00:16:45,930 and if the temperature goes up, then the density goes down, 283 00:16:45,930 --> 00:16:48,670 and the number density goes down. 284 00:16:48,670 --> 00:16:50,670 And if the number density goes down, 285 00:16:50,670 --> 00:16:52,500 the macroscopic cross-section goes down. 286 00:16:52,500 --> 00:16:54,660 The atoms just spread out from each other. 287 00:16:54,660 --> 00:16:59,490 So regardless of what happens at the microscopic cross-section, 288 00:16:59,490 --> 00:17:02,010 which I'll leave to Ben and Cord to teach you next year, 289 00:17:02,010 --> 00:17:04,710 we know that the macroscopic cross-section goes down 290 00:17:04,710 --> 00:17:06,512 because it gets less dense. 291 00:17:06,512 --> 00:17:07,970 So let's try and work out now, what 292 00:17:07,970 --> 00:17:09,137 would happen to k effective? 293 00:17:09,137 --> 00:17:13,819 So what will happen to nu if we raise the temperature? 294 00:17:13,819 --> 00:17:15,510 Nothing, let's hope. 295 00:17:15,510 --> 00:17:16,760 What happens to sigma fission? 296 00:17:20,424 --> 00:17:22,500 This goes down a bit. 297 00:17:22,500 --> 00:17:24,125 But what about sigma absorption? 298 00:17:27,930 --> 00:17:30,380 Sigma absorption is going to go down. 299 00:17:30,380 --> 00:17:32,690 Does bg change? 300 00:17:32,690 --> 00:17:35,240 Has the geometry of the reactor changed? 301 00:17:35,240 --> 00:17:35,927 Probably not. 302 00:17:35,927 --> 00:17:38,010 Might have thermally expanded by a few nanometers, 303 00:17:38,010 --> 00:17:40,700 but let's just say, it doesn't change at all. 304 00:17:40,700 --> 00:17:42,950 What about the diffusion constant? 305 00:17:42,950 --> 00:17:45,000 Let's work that out. 306 00:17:45,000 --> 00:17:47,520 Sigma total is going to go down. 307 00:17:47,520 --> 00:17:49,750 Sigma scattering is going to go down. 308 00:17:49,750 --> 00:17:51,750 So probably what's going to happen 309 00:17:51,750 --> 00:17:54,220 is, this diffusion constant is going 310 00:17:54,220 --> 00:17:57,690 to go up, which means that if the atoms spread out more, 311 00:17:57,690 --> 00:18:00,610 neutrons will move farther, on average. 312 00:18:00,610 --> 00:18:02,387 Hopefully, that makes intuitive sense, 313 00:18:02,387 --> 00:18:03,970 because if the cross-sections go down, 314 00:18:03,970 --> 00:18:07,660 then a neutron can move farther before an average interaction. 315 00:18:07,660 --> 00:18:12,280 So the diffusion constant is probably going to go up. 316 00:18:12,280 --> 00:18:13,530 Which way does k effective go? 317 00:18:16,860 --> 00:18:17,680 You're correct. 318 00:18:17,680 --> 00:18:20,830 No one said anything because you can't really say anything. 319 00:18:20,830 --> 00:18:23,200 So it depends on the relative amounts 320 00:18:23,200 --> 00:18:25,000 that these increase or decrease. 321 00:18:25,000 --> 00:18:27,820 So depending on what you choose for your materials, 322 00:18:27,820 --> 00:18:30,580 you can have what's called positive or negative 323 00:18:30,580 --> 00:18:33,940 temperature feedback, which means in some conditions 324 00:18:33,940 --> 00:18:36,040 or scenarios, what you want to happen 325 00:18:36,040 --> 00:18:40,970 is that if the temperature goes up, k effective should go down, 326 00:18:40,970 --> 00:18:43,040 but not necessarily so. 327 00:18:43,040 --> 00:18:46,550 Depending on what you use, you can actually have situations 328 00:18:46,550 --> 00:18:49,970 where raising the temperature raises k effective, 329 00:18:49,970 --> 00:18:53,443 and that is some seriously bad news and is actually outlawed. 330 00:18:53,443 --> 00:18:55,610 You can't design a reactor with positive temperature 331 00:18:55,610 --> 00:18:57,862 coefficients. 332 00:18:57,862 --> 00:19:00,070 So this is the first little taste of reactor feedback 333 00:19:00,070 --> 00:19:03,102 is, now that we've written this criticality condition, 334 00:19:03,102 --> 00:19:04,810 we can start to explore what happens when 335 00:19:04,810 --> 00:19:07,120 you start probing the reactor. 336 00:19:07,120 --> 00:19:11,470 So let's say, what happens if you just add more reactor? 337 00:19:16,270 --> 00:19:18,400 In this case-- where's my green? 338 00:19:18,400 --> 00:19:20,890 All the way over there-- 339 00:19:20,890 --> 00:19:25,690 without changing the materials, what happens when 340 00:19:25,690 --> 00:19:29,060 you make the reactor bigger? 341 00:19:29,060 --> 00:19:31,190 What increases, decreases or stays the same? 342 00:19:34,750 --> 00:19:35,970 Let's just work it through. 343 00:19:35,970 --> 00:19:37,920 Does nu change? 344 00:19:37,920 --> 00:19:39,000 Sigma fission? 345 00:19:39,000 --> 00:19:39,500 No. 346 00:19:39,500 --> 00:19:41,250 Sigma absorption? 347 00:19:41,250 --> 00:19:42,770 D? 348 00:19:42,770 --> 00:19:43,850 How does bg change? 349 00:19:49,150 --> 00:19:52,480 Bg decreases, and as you'd expect, 350 00:19:52,480 --> 00:19:55,760 if you add more reactor to your reactor, 351 00:19:55,760 --> 00:19:57,812 the k effective should increase. 352 00:19:57,812 --> 00:19:59,270 And so this, hopefully, is starting 353 00:19:59,270 --> 00:20:01,250 to follow some intuitive pattern. 354 00:20:01,250 --> 00:20:04,590 With a given criticality condition, in some situations, 355 00:20:04,590 --> 00:20:07,760 you can work out, will the reactor gain or lose power? 356 00:20:07,760 --> 00:20:09,590 Speaking of, where's the power? 357 00:20:13,013 --> 00:20:16,930 Where'd it go? 358 00:20:16,930 --> 00:20:17,513 Yeah? 359 00:20:17,513 --> 00:20:19,365 AUDIENCE: The kinetic energy of neutrons? 360 00:20:19,365 --> 00:20:21,060 PROFESSOR: So yes, the power comes 361 00:20:21,060 --> 00:20:23,430 from the kinetic energy of neutrons, 362 00:20:23,430 --> 00:20:25,970 but where did the power go in our expression? 363 00:20:25,970 --> 00:20:26,470 Yeah? 364 00:20:26,470 --> 00:20:28,158 AUDIENCE: Power's not dependent on criticality. 365 00:20:28,158 --> 00:20:29,420 PROFESSOR: That's right. 366 00:20:29,420 --> 00:20:30,770 That's exactly right. 367 00:20:30,770 --> 00:20:32,360 And it follows directly from the math. 368 00:20:32,360 --> 00:20:34,580 This a got canceled away. 369 00:20:34,580 --> 00:20:36,030 It doesn't matter. 370 00:20:36,030 --> 00:20:39,030 You can actually have what's called a zero power reactor. 371 00:20:39,030 --> 00:20:41,510 So the power of the reactor and its criticality 372 00:20:41,510 --> 00:20:43,340 are not necessarily linked. 373 00:20:43,340 --> 00:20:45,860 You can have a reactor that is critical while producing 374 00:20:45,860 --> 00:20:47,340 tons of power. 375 00:20:47,340 --> 00:20:50,210 You can also have a reactor that is critical while producing-- 376 00:20:50,210 --> 00:20:53,030 I won't say zero, but an infinitesimally small amount 377 00:20:53,030 --> 00:20:53,792 of power. 378 00:20:53,792 --> 00:20:55,250 And they actually have built these. 379 00:20:55,250 --> 00:20:57,590 They're great test systems for testing 380 00:20:57,590 --> 00:20:59,900 our knowledge of neutron physics because you've 381 00:20:59,900 --> 00:21:03,190 got a reactor that's producing maybe 10 watts of power. 382 00:21:03,190 --> 00:21:06,080 It's easy to cool by blowing a fan on it, let's say. 383 00:21:06,080 --> 00:21:08,060 But you can still measure the neutron flux 384 00:21:08,060 --> 00:21:10,730 in different places and test how well your codes are working 385 00:21:10,730 --> 00:21:13,820 with a much safer configuration than sticking probes 386 00:21:13,820 --> 00:21:15,730 into a gigawatt commercial reactor. 387 00:21:15,730 --> 00:21:16,420 Yep? 388 00:21:16,420 --> 00:21:19,300 AUDIENCE: So [INAUDIBLE] steady state reactor, 389 00:21:19,300 --> 00:21:24,100 how are you [INAUDIBLE] if it's not really at the steady state? 390 00:21:24,100 --> 00:21:27,203 PROFESSOR: That would push the reactor out of steady state. 391 00:21:27,203 --> 00:21:28,620 Indeed, so on Tuesday, we're going 392 00:21:28,620 --> 00:21:32,170 to start covering transience, and if k effective 393 00:21:32,170 --> 00:21:34,030 become something other than one, the reactor 394 00:21:34,030 --> 00:21:35,950 is no longer in steady state. 395 00:21:35,950 --> 00:21:38,800 It's not in equilibrium because the gains and the losses 396 00:21:38,800 --> 00:21:40,120 are not equal to each other. 397 00:21:40,120 --> 00:21:42,640 And at that point, the power will start to change, 398 00:21:42,640 --> 00:21:46,870 what you guys all saw when you manipulated the reactor power. 399 00:21:46,870 --> 00:21:49,930 So since you brought it up, does anybody remember, 400 00:21:49,930 --> 00:21:53,880 if we draw as a function of time, 401 00:21:53,880 --> 00:21:57,460 let's say the reactor power was cruising along, 402 00:21:57,460 --> 00:22:01,770 and right at the time is now, you withdrew a control rod. 403 00:22:01,770 --> 00:22:03,480 What happened when you guys did that? 404 00:22:03,480 --> 00:22:06,625 Anyone, because you all did it. 405 00:22:06,625 --> 00:22:07,690 It went up. 406 00:22:07,690 --> 00:22:08,250 OK. 407 00:22:08,250 --> 00:22:09,028 And then what? 408 00:22:11,720 --> 00:22:16,840 When you stopped withdrawing the control rod, did it level out? 409 00:22:16,840 --> 00:22:18,525 So everyone, tell me what happened. 410 00:22:18,525 --> 00:22:19,770 AUDIENCE: It slowed down. 411 00:22:19,770 --> 00:22:23,100 PROFESSOR: It slowed down the increase, 412 00:22:23,100 --> 00:22:24,900 but it didn't stop going up. 413 00:22:24,900 --> 00:22:26,220 Kind of freaky. 414 00:22:26,220 --> 00:22:31,650 So this is why I had you guys do that power ramp because just 415 00:22:31,650 --> 00:22:34,410 controlling a reactor is not as simple as, remove the control 416 00:22:34,410 --> 00:22:37,200 rod, you remove a certain amount of reactivity 417 00:22:37,200 --> 00:22:40,770 because there are time-dependent effects 418 00:22:40,770 --> 00:22:47,210 due to delayed neutrons, neutrons that 419 00:22:47,210 --> 00:22:50,150 aren't immediately released after fission 420 00:22:50,150 --> 00:22:53,237 that can have a large effect on how you control your reactor. 421 00:22:53,237 --> 00:22:55,070 And then if you wanted to decrease it again, 422 00:22:55,070 --> 00:22:56,653 let's say you put the control rod back 423 00:22:56,653 --> 00:22:59,570 into its original position, the power 424 00:22:59,570 --> 00:23:01,730 would not come back to its original position. 425 00:23:01,730 --> 00:23:06,110 But then, eventually, it would start to coast down 426 00:23:06,110 --> 00:23:08,923 and probably go beneath its original position 427 00:23:08,923 --> 00:23:11,090 at which point you have to constantly be controlling 428 00:23:11,090 --> 00:23:13,250 those control rods to keep it in what 429 00:23:13,250 --> 00:23:15,320 I'll call dynamic equilibrium. 430 00:23:15,320 --> 00:23:19,940 You never really hit static equilibrium unless it's off. 431 00:23:19,940 --> 00:23:22,520 As I went to a seminar a couple weeks ago and said, 432 00:23:22,520 --> 00:23:25,190 I don't study biological organisms in static equilibrium 433 00:23:25,190 --> 00:23:27,470 because that's better known as a dead organism. 434 00:23:27,470 --> 00:23:28,790 They're not very interesting. 435 00:23:28,790 --> 00:23:31,520 But dynamic equilibrium sure is, for them and for us. 436 00:23:34,220 --> 00:23:38,590 So with this process of getting the single-group balance 437 00:23:38,590 --> 00:23:42,230 equations, I'd like to generalize this 438 00:23:42,230 --> 00:23:44,300 to the two-group balance equations. 439 00:23:44,300 --> 00:23:46,430 And this is something you can actually use. 440 00:23:46,430 --> 00:23:49,190 In every case, we're going to say, 441 00:23:49,190 --> 00:23:52,460 let's put our gains on the left and put our losses 442 00:23:52,460 --> 00:23:58,340 on the right, if we want to have this reactor in equilibrium. 443 00:23:58,340 --> 00:24:00,410 And now we'll separate our equations 444 00:24:00,410 --> 00:24:06,700 into the fast and thermal regions of neutron energy. 445 00:24:06,700 --> 00:24:14,350 So we'll call those f, and we'll call the thermal ones th. 446 00:24:14,350 --> 00:24:18,370 So using this model of our neutron diffusion equation, 447 00:24:18,370 --> 00:24:22,180 what are the gains of neutrons into the fast spectrum? 448 00:24:27,664 --> 00:24:29,086 AUDIENCE: Straight from fission. 449 00:24:29,086 --> 00:24:30,711 PROFESSOR: Yeah, straight from fission. 450 00:24:30,711 --> 00:24:32,020 So how do we write that? 451 00:24:32,020 --> 00:24:33,770 This process that we're going through now, 452 00:24:33,770 --> 00:24:36,320 this is where recitation really begins because this 453 00:24:36,320 --> 00:24:39,200 is how I want to show you guys how to approach a problem, 454 00:24:39,200 --> 00:24:41,720 let's say, a one-sentence statement like, give me 455 00:24:41,720 --> 00:24:44,490 the flux anywhere in a two-group reactor. 456 00:24:44,490 --> 00:24:46,770 This is how we go about it. 457 00:24:46,770 --> 00:24:50,150 So how do you equationally put the neutron gains from fission? 458 00:24:53,260 --> 00:24:55,063 What terms do we have up there right now? 459 00:24:55,063 --> 00:24:56,896 AUDIENCE: Your neutron multiplication factor 460 00:24:56,896 --> 00:24:57,854 and your cross section. 461 00:24:57,854 --> 00:24:59,178 PROFESSOR: Yep. 462 00:24:59,178 --> 00:25:00,106 Yep. 463 00:25:00,106 --> 00:25:04,240 You'll have your nu, your neutron multiplication factor. 464 00:25:04,240 --> 00:25:09,790 And now, we're actually going to split every cross section 465 00:25:09,790 --> 00:25:17,895 into its fast and thermal energy ranges 466 00:25:17,895 --> 00:25:20,270 because now we're actually splitting that energy, like we 467 00:25:20,270 --> 00:25:22,640 did when I drew that crazy cross section. 468 00:25:22,640 --> 00:25:27,200 Let's see, we had log of e versus log of sigma, 469 00:25:27,200 --> 00:25:31,640 and they all follow roughly that formula. 470 00:25:31,640 --> 00:25:33,350 And we split it and said, if we want 471 00:25:33,350 --> 00:25:36,870 to draw an average cross section, 472 00:25:36,870 --> 00:25:38,950 it would look something like this. 473 00:25:38,950 --> 00:25:42,440 And that would be our sigma thermal 474 00:25:42,440 --> 00:25:45,800 and this would be our sigma fast. 475 00:25:45,800 --> 00:25:47,820 So that's what we're doing here. 476 00:25:47,820 --> 00:25:49,790 So now it gets a little more complicated 477 00:25:49,790 --> 00:25:53,480 because both fast and thermal neutrons 478 00:25:53,480 --> 00:25:55,730 can contribute to fission. 479 00:25:55,730 --> 00:25:57,890 So how do we write this in terms of equations? 480 00:26:02,111 --> 00:26:04,720 AUDIENCE: [INAUDIBLE] 481 00:26:04,720 --> 00:26:06,220 PROFESSOR: We only want the neutrons 482 00:26:06,220 --> 00:26:09,970 that are born into the fast region, the fast gains. 483 00:26:09,970 --> 00:26:12,370 That doesn't mean you don't have to consider where 484 00:26:12,370 --> 00:26:14,740 are the thermal neutrons, because it's mostly 485 00:26:14,740 --> 00:26:17,590 those thermal neutrons that, when they get absorbed and make 486 00:26:17,590 --> 00:26:20,910 fission, create fast neutrons. 487 00:26:20,910 --> 00:26:26,090 So what we'd really need is sigma fission fast 488 00:26:26,090 --> 00:26:29,390 times our fast flux because we're 489 00:26:29,390 --> 00:26:32,090 going to split every variable into its fast and thermal 490 00:26:32,090 --> 00:26:35,410 parts, plus-- 491 00:26:35,410 --> 00:26:38,310 let's put a parentheses there-- 492 00:26:38,310 --> 00:26:44,080 sigma fission thermal phi thermal. 493 00:26:44,080 --> 00:26:46,045 So do you guys see what I've done here? 494 00:26:46,045 --> 00:26:47,420 We're assuming that every neutron 495 00:26:47,420 --> 00:26:49,820 is born in the fast group, where we're cutting 496 00:26:49,820 --> 00:26:52,910 this off at around 1 ev. 497 00:26:52,910 --> 00:26:54,830 And we are assuming that no neutrons 498 00:26:54,830 --> 00:26:58,710 are born below 1 ev, which is a very good assumption. 499 00:26:58,710 --> 00:27:01,580 So in this case, both the fast and the thermal fluxes 500 00:27:01,580 --> 00:27:04,933 contribute to creating fast neutrons. 501 00:27:04,933 --> 00:27:06,725 Is there any other source of fast neutrons? 502 00:27:11,450 --> 00:27:13,230 Good, because I don't know of one either. 503 00:27:13,230 --> 00:27:14,690 OK what about losses? 504 00:27:17,270 --> 00:27:21,860 By what mechanisms can neutrons leave the fast group? 505 00:27:21,860 --> 00:27:22,360 Yeah? 506 00:27:22,360 --> 00:27:23,332 AUDIENCE: Aren't they absorbed? 507 00:27:23,332 --> 00:27:24,040 PROFESSOR: Yeah. 508 00:27:24,040 --> 00:27:24,980 They can be absorbed. 509 00:27:24,980 --> 00:27:28,085 So how do I write that? 510 00:27:28,085 --> 00:27:30,390 AUDIENCE: Sigma af-- 511 00:27:30,390 --> 00:27:31,030 PROFESSOR: af-- 512 00:27:31,030 --> 00:27:33,158 AUDIENCE: --times the flux fast. 513 00:27:33,158 --> 00:27:34,450 PROFESSOR: Times the fast flux. 514 00:27:34,450 --> 00:27:37,990 So only neutrons in the fast flux group 515 00:27:37,990 --> 00:27:40,670 will leave the fast flux group by absorption. 516 00:27:40,670 --> 00:27:43,270 And what's the other mechanism that we had in our neutron 517 00:27:43,270 --> 00:27:44,260 diffusion equation? 518 00:27:44,260 --> 00:27:46,280 AUDIENCE: Scattering. 519 00:27:46,280 --> 00:27:47,780 PROFESSOR: Yeah, actually, so that's 520 00:27:47,780 --> 00:27:50,030 not in the diffusion equation, but you are right. 521 00:27:50,030 --> 00:27:52,670 That's the missing piece that is going to be the hard part, so. 522 00:27:52,670 --> 00:27:55,047 Let's add that in now. 523 00:27:55,047 --> 00:27:56,630 So there's going to be some scattering 524 00:27:56,630 --> 00:28:03,500 from the fast to the thermal group, times our fast flux. 525 00:28:03,500 --> 00:28:05,270 So not every scattering event will 526 00:28:05,270 --> 00:28:08,330 cause the neutron to leave the fast group, but some of them 527 00:28:08,330 --> 00:28:09,145 will. 528 00:28:09,145 --> 00:28:10,520 So we have to figure out, what is 529 00:28:10,520 --> 00:28:13,010 the proportion of those neutrons that 530 00:28:13,010 --> 00:28:17,720 will scatter from the fast group to the thermal group? 531 00:28:17,720 --> 00:28:20,420 For the case of hydrogen, it's pretty easy 532 00:28:20,420 --> 00:28:22,910 because the probability of a neutron landing anywhere 533 00:28:22,910 --> 00:28:26,960 from zero to e, starting off at energy ei, 534 00:28:26,960 --> 00:28:33,160 if we had our scattering kernel, is a constant. 535 00:28:33,160 --> 00:28:36,940 So that's not too hard. 536 00:28:36,940 --> 00:28:41,060 And then last, what other way can we lose neutrons 537 00:28:41,060 --> 00:28:42,719 from the fast group? 538 00:28:42,719 --> 00:28:44,825 AUDIENCE: Leakage. 539 00:28:44,825 --> 00:28:45,450 PROFESSOR: Yep. 540 00:28:45,450 --> 00:28:45,960 Leakage. 541 00:28:45,960 --> 00:28:48,210 They can leave the reactor, and we 542 00:28:48,210 --> 00:28:55,770 can write that as a d fast bg squared flux. 543 00:28:55,770 --> 00:28:59,230 Make sure everything has bars that needs them. 544 00:28:59,230 --> 00:28:59,730 OK. 545 00:29:02,850 --> 00:29:04,840 Now, using the same sort of logic, let's-- 546 00:29:04,840 --> 00:29:05,340 Yeah, Luke? 547 00:29:05,340 --> 00:29:06,340 AUDIENCE: What's the bg? 548 00:29:06,340 --> 00:29:10,157 How is that different from [INAUDIBLE]?? 549 00:29:10,157 --> 00:29:10,990 PROFESSOR: It's not. 550 00:29:10,990 --> 00:29:12,220 It's the same. 551 00:29:12,220 --> 00:29:15,800 It is the same bg that describes the geometry of the reactor. 552 00:29:15,800 --> 00:29:18,250 AUDIENCE: I guess what's the subscript b of g? 553 00:29:18,250 --> 00:29:20,784 PROFESSOR: G means geometry. 554 00:29:20,784 --> 00:29:21,284 Yep. 555 00:29:25,432 --> 00:29:26,390 And you had a question? 556 00:29:26,390 --> 00:29:28,750 AUDIENCE: Yeah, just in the last flux, [INAUDIBLE] fast flux. 557 00:29:28,750 --> 00:29:29,833 PROFESSOR: Yes, thank you. 558 00:29:29,833 --> 00:29:30,710 That is a fast flux. 559 00:29:30,710 --> 00:29:31,693 Yep. 560 00:29:31,693 --> 00:29:33,860 But it's important to note that this flux right here 561 00:29:33,860 --> 00:29:35,450 is not a fast flux. 562 00:29:35,450 --> 00:29:36,860 We'll get back to that soon. 563 00:29:36,860 --> 00:29:39,290 Now, using the same sort of logic, 564 00:29:39,290 --> 00:29:42,230 let's write the gains and losses in the thermal group. 565 00:29:42,230 --> 00:29:44,600 So what is the only source of neutrons 566 00:29:44,600 --> 00:29:47,455 into the thermal energy group? 567 00:29:47,455 --> 00:29:49,830 I want to hear from someone who hasn't said anything yet. 568 00:29:57,730 --> 00:29:59,074 So Jared, what would you say? 569 00:29:59,074 --> 00:30:01,407 Either Jared because I haven't heard from either of you. 570 00:30:01,407 --> 00:30:02,690 AUDIENCE: [INAUDIBLE] 571 00:30:02,690 --> 00:30:03,482 PROFESSOR: You did. 572 00:30:03,482 --> 00:30:04,400 OK, then you. 573 00:30:04,400 --> 00:30:05,510 I'm sorry. 574 00:30:05,510 --> 00:30:06,010 All right. 575 00:30:06,010 --> 00:30:07,490 Yeah, you said the no power thing. 576 00:30:07,490 --> 00:30:09,173 Thank you. 577 00:30:09,173 --> 00:30:12,230 AUDIENCE: So could you, like if something 578 00:30:12,230 --> 00:30:14,330 is absorbed in the fast spectrum, 579 00:30:14,330 --> 00:30:15,680 jump down to the thermal? 580 00:30:15,680 --> 00:30:16,560 PROFESSOR: Close. 581 00:30:16,560 --> 00:30:18,480 I want to replace one word in what you said. 582 00:30:18,480 --> 00:30:21,138 If something is blank in the fast spectrum, 583 00:30:21,138 --> 00:30:22,680 it goes down to the thermal spectrum. 584 00:30:22,680 --> 00:30:23,130 AUDIENCE: Scattered. 585 00:30:23,130 --> 00:30:24,213 PROFESSOR: Yes, scattered. 586 00:30:24,213 --> 00:30:27,420 Every neutron that leaves the fast group by scattering 587 00:30:27,420 --> 00:30:30,534 enters the thermal group also by scattering. 588 00:30:34,330 --> 00:30:38,730 And in this case, we want to have the fast flux appear here 589 00:30:38,730 --> 00:30:41,160 because the number of neutrons entering the thermal group 590 00:30:41,160 --> 00:30:43,755 depends on how many scatter out of the fast group. 591 00:30:43,755 --> 00:30:44,380 Yeah, Luke? 592 00:30:44,380 --> 00:30:47,090 AUDIENCE: Would you ever scatter up into the fast group? 593 00:30:47,090 --> 00:30:48,050 PROFESSOR: You'll see. 594 00:30:48,050 --> 00:30:50,480 Yes. 595 00:30:50,480 --> 00:30:52,010 Yeah, great. 596 00:30:52,010 --> 00:30:53,810 I just gave something away. 597 00:30:53,810 --> 00:30:54,740 Yes. 598 00:30:54,740 --> 00:30:57,320 You can, but no, you usually don't. 599 00:30:57,320 --> 00:30:59,138 So we would consider that once neutrons 600 00:30:59,138 --> 00:31:01,430 enter the thermal group, they're at thermal equilibrium 601 00:31:01,430 --> 00:31:03,590 with the stuff around them, and up scattering 602 00:31:03,590 --> 00:31:05,990 is rarely a possibility. 603 00:31:05,990 --> 00:31:06,560 You'll see. 604 00:31:09,980 --> 00:31:11,950 Yeah, quite soon actually. 605 00:31:11,950 --> 00:31:13,160 Don't worry. 606 00:31:13,160 --> 00:31:15,980 Not like quiz you'll see, but you'll see. 607 00:31:15,980 --> 00:31:17,513 Yeah. 608 00:31:17,513 --> 00:31:19,180 I've already got some stuff planned out. 609 00:31:19,180 --> 00:31:21,520 It's going to be a part of the homework question. 610 00:31:21,520 --> 00:31:23,410 So now what loss mechanisms do we have? 611 00:31:26,254 --> 00:31:27,676 AUDIENCE: Leakage. 612 00:31:27,676 --> 00:31:28,810 PROFESSOR: Yeah, leakage. 613 00:31:28,810 --> 00:31:33,610 So we're going to have some separate thermal diffusion 614 00:31:33,610 --> 00:31:36,640 coefficient because that diffusion coefficient depends 615 00:31:36,640 --> 00:31:38,110 on the cross sections which depends 616 00:31:38,110 --> 00:31:45,430 on the groups you're in, times the same geometry, times phi 617 00:31:45,430 --> 00:31:46,130 thermal. 618 00:31:46,130 --> 00:31:48,020 And what's the only other mechanism of loss? 619 00:31:48,020 --> 00:31:48,980 AUDIENCE: Absorption. 620 00:31:48,980 --> 00:31:50,048 PROFESSOR: Absorption. 621 00:31:53,036 --> 00:31:54,680 We've got to clear a. 622 00:31:57,840 --> 00:32:01,964 Why is there no scattering from the thermal group? 623 00:32:01,964 --> 00:32:04,820 AUDIENCE: Didn't you say it was very rare to have it 624 00:32:04,820 --> 00:32:06,697 scatter up to the fast group? 625 00:32:06,697 --> 00:32:08,030 PROFESSOR: I'd say even simpler. 626 00:32:08,030 --> 00:32:09,447 Once you're at the bottom, there's 627 00:32:09,447 --> 00:32:11,350 no more lower you can go. 628 00:32:11,350 --> 00:32:13,100 So in neutronics, when you hit the bottom, 629 00:32:13,100 --> 00:32:14,475 you don't say, throw me a shovel. 630 00:32:14,475 --> 00:32:17,510 You say, you're at the final energy group. 631 00:32:17,510 --> 00:32:19,970 So now, what we'd like to be able to do 632 00:32:19,970 --> 00:32:25,740 is, last thing we want to stick in is our k effective, 633 00:32:25,740 --> 00:32:28,380 our criticality, because in reality, 634 00:32:28,380 --> 00:32:32,520 this is kind of what we want to know in terms of the geometry 635 00:32:32,520 --> 00:32:33,900 and the materials in the reactor. 636 00:32:33,900 --> 00:32:38,370 So if we know what we make it out of and how big to make it, 637 00:32:38,370 --> 00:32:41,100 we should be able to get those in balance such 638 00:32:41,100 --> 00:32:43,020 that k effective equals 1. 639 00:32:43,020 --> 00:32:47,765 So the only really unknown here besides the flux unknowns is k. 640 00:32:47,765 --> 00:32:49,890 And the reason I don't care about the flux unknowns 641 00:32:49,890 --> 00:32:51,340 is, they're going to go away soon. 642 00:32:51,340 --> 00:32:51,840 Yeah? 643 00:32:51,840 --> 00:32:56,370 AUDIENCE: Does the thermal also have the [INAUDIBLE] over k? 644 00:32:56,370 --> 00:32:58,440 PROFESSOR: Absolutely, because the k effective 645 00:32:58,440 --> 00:33:02,220 is on the bottom of the total original sources of neutrons. 646 00:33:02,220 --> 00:33:04,350 Just like, let's see. 647 00:33:04,350 --> 00:33:06,180 That was one group. 648 00:33:06,180 --> 00:33:06,720 Yeah. 649 00:33:06,720 --> 00:33:08,430 So I'd say right now, this accounts 650 00:33:08,430 --> 00:33:10,050 for the production of all neutrons, 651 00:33:10,050 --> 00:33:13,168 and everything else down the chain is losses. 652 00:33:13,168 --> 00:33:13,710 Yeah, Monica? 653 00:33:13,710 --> 00:33:19,202 AUDIENCE: Do we assume that all neutrons [INAUDIBLE]?? 654 00:33:19,202 --> 00:33:20,660 PROFESSOR: We know, experimentally, 655 00:33:20,660 --> 00:33:23,690 that they tend to be born between 1 and 10 mev, 656 00:33:23,690 --> 00:33:26,978 but since you asked, let's escalate the problem. 657 00:33:26,978 --> 00:33:29,270 And then we will de-escalate very quickly, just to say, 658 00:33:29,270 --> 00:33:31,010 let's do a thought experiment, right? 659 00:33:31,010 --> 00:33:33,950 Let's say some of the neutrons were born thermal. 660 00:33:33,950 --> 00:33:38,930 What would we have to add to this expression? 661 00:33:38,930 --> 00:33:40,930 There's one variable missing that's not anywhere 662 00:33:40,930 --> 00:33:43,550 on these boards, but was there on Thursday and Tuesday. 663 00:33:46,987 --> 00:33:47,779 AUDIENCE: Spectrum? 664 00:33:47,779 --> 00:33:48,779 PROFESSOR: That's right. 665 00:33:48,779 --> 00:33:50,340 Chi, the birth spectrum. 666 00:33:50,340 --> 00:33:53,760 So if some neutrons are born thermal, 667 00:33:53,760 --> 00:33:59,720 then we would have to add a Chi fast here, 668 00:33:59,720 --> 00:34:02,810 and we would have to add a Chi thermal to say, 669 00:34:02,810 --> 00:34:07,070 this is the proportion of neutrons born fast or thermal, 670 00:34:07,070 --> 00:34:11,909 times nu times sigma fission fast, 671 00:34:11,909 --> 00:34:16,592 phi fast plus sigma fission thermal, phi thermal. 672 00:34:16,592 --> 00:34:18,300 And I'm not writing nice because I'm just 673 00:34:18,300 --> 00:34:20,795 going to erase it in a second, but to go with your thought 674 00:34:20,795 --> 00:34:22,170 experiment, this is what it would 675 00:34:22,170 --> 00:34:24,719 look like if some of the neutrons were born thermal. 676 00:34:24,719 --> 00:34:26,645 Perfectly fine thing to model. 677 00:34:26,645 --> 00:34:28,020 Doesn't happen much in real life, 678 00:34:28,020 --> 00:34:30,850 but great exam question for next year. 679 00:34:30,850 --> 00:34:31,614 Thank you. 680 00:34:31,614 --> 00:34:33,040 AUDIENCE: Next year. 681 00:34:33,040 --> 00:34:34,239 PROFESSOR: Next year. 682 00:34:34,239 --> 00:34:36,281 I'm not going to give you your own exam question. 683 00:34:36,281 --> 00:34:37,850 That's just too easy for you. 684 00:34:37,850 --> 00:34:42,659 So for now, let's forget about that stuff 685 00:34:42,659 --> 00:34:47,040 and stick with the most realistic situation. 686 00:34:47,040 --> 00:34:48,389 Ah, running out of room already. 687 00:34:48,389 --> 00:34:48,889 OK. 688 00:34:52,010 --> 00:34:53,730 That's for next week. 689 00:34:53,730 --> 00:34:57,410 So let's forget about that. 690 00:34:57,410 --> 00:34:59,100 What do we do next? 691 00:34:59,100 --> 00:35:02,270 We have two equations and three unknowns. 692 00:35:02,270 --> 00:35:03,530 Interesting. 693 00:35:03,530 --> 00:35:04,370 Or do we really? 694 00:35:07,730 --> 00:35:10,850 Well, for one thing, if we can get that top equation 695 00:35:10,850 --> 00:35:14,240 all in terms of one of the fluxes, either fast or thermal, 696 00:35:14,240 --> 00:35:16,310 then every term is in terms of a flux 697 00:35:16,310 --> 00:35:18,630 and they can all be divided out. 698 00:35:18,630 --> 00:35:20,690 So let's take one of these equations 699 00:35:20,690 --> 00:35:23,210 and substitute in so that we get everything 700 00:35:23,210 --> 00:35:25,100 in terms of only one flux. 701 00:35:25,100 --> 00:35:28,580 So let's say, the top one, which has got the k in it, 702 00:35:28,580 --> 00:35:30,950 has one instance of phi thermal. 703 00:35:30,950 --> 00:35:33,920 So let's isolate phi thermal in terms of everything else. 704 00:35:33,920 --> 00:35:37,190 So we have that thermal equation right there. 705 00:35:37,190 --> 00:35:43,620 So we have sigma scattering from fast to thermal times 706 00:35:43,620 --> 00:35:49,670 fast flux equals two things times phi thermal, which 707 00:35:49,670 --> 00:35:59,060 is d thermal bg squared plus sigma absorption thermal. 708 00:35:59,060 --> 00:36:00,760 We're actually not that far away. 709 00:36:00,760 --> 00:36:03,420 So all we do is, we divide each side-- 710 00:36:03,420 --> 00:36:05,600 where's my simplifying color-- 711 00:36:05,600 --> 00:36:06,400 substitute. 712 00:36:06,400 --> 00:36:08,560 That's not it. 713 00:36:08,560 --> 00:36:11,260 Rearrange. 714 00:36:11,260 --> 00:36:32,720 Divide everything by this stuff, and those cancel out. 715 00:36:32,720 --> 00:36:37,600 And we're left with an expression for phi thermal 716 00:36:37,600 --> 00:36:40,300 which we can now plug into that top equation. 717 00:36:40,300 --> 00:36:44,130 So we're like one step away from the final answer. 718 00:36:44,130 --> 00:36:46,360 There, everything's still visible. 719 00:36:46,360 --> 00:36:51,970 And so now we end up with 1 over k times 720 00:36:51,970 --> 00:36:59,400 nu sigma fission fast, fast, plus 721 00:36:59,400 --> 00:37:05,720 sigma fission thermal times this expression, 722 00:37:05,720 --> 00:37:13,340 sigma scattering fast to thermal, phi flux 723 00:37:13,340 --> 00:37:16,325 over d thermal. 724 00:37:16,325 --> 00:37:18,700 I don't usually spend this much of the class with my back 725 00:37:18,700 --> 00:37:21,810 to you, but this is pretty mathematically intense, 726 00:37:21,810 --> 00:37:25,170 so I apologize for that. 727 00:37:25,170 --> 00:37:34,550 And equals sigma absorption fast, fast flux, plus d fast 728 00:37:34,550 --> 00:37:39,342 bg squared fast flux. 729 00:37:39,342 --> 00:37:40,050 That's not a bar. 730 00:37:40,050 --> 00:37:40,710 That has one. 731 00:37:40,710 --> 00:37:41,610 That has one. 732 00:37:41,610 --> 00:37:42,260 That does. 733 00:37:42,260 --> 00:37:42,950 That's good. 734 00:37:42,950 --> 00:37:43,980 OK. 735 00:37:43,980 --> 00:37:47,940 Now every single term here is in terms of fast flux. 736 00:37:47,940 --> 00:37:51,960 So we can just cancel them from every single term here. 737 00:37:51,960 --> 00:37:54,150 And now we're left with an expression 738 00:37:54,150 --> 00:37:58,470 for k effective that's just in terms of material properties 739 00:37:58,470 --> 00:38:01,020 and geometry for the two-group problem. 740 00:38:01,020 --> 00:38:03,160 We're only one step away. 741 00:38:03,160 --> 00:38:06,240 So if we multiply everything by k and divide everything 742 00:38:06,240 --> 00:38:13,400 by this stuff, we'll just have a sigma absorption 743 00:38:13,400 --> 00:38:19,740 plus d fast bg squared. 744 00:38:19,740 --> 00:38:22,820 That would equal k. 745 00:38:22,820 --> 00:38:25,970 And just like that is the criticality condition 746 00:38:25,970 --> 00:38:28,760 for a two-energy group homogeneous 747 00:38:28,760 --> 00:38:32,780 reactor of any geometry. 748 00:38:32,780 --> 00:38:35,960 All that matters to define the geometry 749 00:38:35,960 --> 00:38:38,390 is, what's this bg squared? 750 00:38:38,390 --> 00:38:41,480 So this case works for an infinite slab reactor. 751 00:38:41,480 --> 00:38:44,150 It works for an actual right cylindrical reactor. 752 00:38:44,150 --> 00:38:46,460 You just have to sell for or look up 753 00:38:46,460 --> 00:38:49,410 the correct buckling term, this bg squared, 754 00:38:49,410 --> 00:38:51,290 which I'll tell you now, we refer 755 00:38:51,290 --> 00:38:58,500 to as buckling or geometric buckling, 756 00:38:58,500 --> 00:39:00,575 and you've got the solution to this. 757 00:39:00,575 --> 00:39:02,700 Let's just check to see what we actually have here. 758 00:39:02,700 --> 00:39:06,730 We have nu material property, material property. 759 00:39:06,730 --> 00:39:10,650 All of those are material properties except for the bg's. 760 00:39:10,650 --> 00:39:13,710 So this tells you how to design a reactor, physically, 761 00:39:13,710 --> 00:39:15,870 and in terms of which materials to make sure 762 00:39:15,870 --> 00:39:17,410 that it's critical. 763 00:39:17,410 --> 00:39:19,920 And if we look at what this looks like here, 764 00:39:19,920 --> 00:39:24,660 again, it's a ratio of gains to losses because eventually, 765 00:39:24,660 --> 00:39:28,150 the losses right here, these are the losses from the fast group. 766 00:39:28,150 --> 00:39:30,250 These are the losses from the thermal group. 767 00:39:30,250 --> 00:39:32,160 These are the gains in the fast group, 768 00:39:32,160 --> 00:39:36,815 noting that some of the neutrons born in the fast group 769 00:39:36,815 --> 00:39:38,190 scatter out of the thermal group, 770 00:39:38,190 --> 00:39:39,990 but don't leave the reactor. 771 00:39:39,990 --> 00:39:42,930 So again, it turns out to a gains over losses ratio. 772 00:39:42,930 --> 00:39:45,300 And there you have it. 773 00:39:48,620 --> 00:39:49,860 So I want to stop at 10 of-- 774 00:39:49,860 --> 00:39:50,060 Yeah. 775 00:39:50,060 --> 00:39:51,952 AUDIENCE: Did we drop the scattering term 776 00:39:51,952 --> 00:39:54,320 from the fast equation? 777 00:39:54,320 --> 00:39:57,870 PROFESSOR: It should be-- did we? 778 00:39:57,870 --> 00:40:00,340 Yeah. 779 00:40:00,340 --> 00:40:01,580 Let's stick it in right here. 780 00:40:06,020 --> 00:40:09,290 So we'll just also stick it here. 781 00:40:16,690 --> 00:40:17,647 There. 782 00:40:17,647 --> 00:40:20,230 And that flux goes away because it was in terms of everything. 783 00:40:20,230 --> 00:40:20,920 Yeah. 784 00:40:20,920 --> 00:40:23,670 There we go. 785 00:40:23,670 --> 00:40:24,477 Thank you. 786 00:40:24,477 --> 00:40:25,730 OK. 787 00:40:25,730 --> 00:40:27,980 But again, this represents losses 788 00:40:27,980 --> 00:40:30,920 on the bottom, gains on the top, just like any other k 789 00:40:30,920 --> 00:40:33,200 effective. 790 00:40:33,200 --> 00:40:36,590 So I wanted to stop here at 10 of, 5 of, and answer 791 00:40:36,590 --> 00:40:38,150 any and all questions you guys have 792 00:40:38,150 --> 00:40:41,300 about going from the neutron transport equation 793 00:40:41,300 --> 00:40:43,850 all the way to something that you could solve, and then 794 00:40:43,850 --> 00:40:45,380 start to play around with to say, 795 00:40:45,380 --> 00:40:47,990 what happens if I switch isotopes? 796 00:40:47,990 --> 00:40:50,060 What happens if I raise the temperature? 797 00:40:50,060 --> 00:40:52,070 What happens if a chunk falls off of the reactor 798 00:40:52,070 --> 00:40:53,820 and it gets smaller? 799 00:40:53,820 --> 00:40:54,320 Yeah. 800 00:40:54,320 --> 00:40:59,743 AUDIENCE: We got the equation for [INAUDIBLE].. 801 00:40:59,743 --> 00:41:00,729 PROFESSOR: Yes. 802 00:41:00,729 --> 00:41:02,208 AUDIENCE: [INAUDIBLE]? 803 00:41:06,170 --> 00:41:07,300 PROFESSOR: Somewhat. 804 00:41:07,300 --> 00:41:11,930 We can assume that for considerably long enough times, 805 00:41:11,930 --> 00:41:14,330 and to a neutron, a long time could be like seconds, 806 00:41:14,330 --> 00:41:18,740 that the time and the spatial form of the flux 807 00:41:18,740 --> 00:41:21,050 are separable which is something that we'll talk about 808 00:41:21,050 --> 00:41:22,310 on Tuesday. 809 00:41:22,310 --> 00:41:24,650 But, if you remember, one of the major assumptions 810 00:41:24,650 --> 00:41:27,480 we made in the neutron transport equation was steady state. 811 00:41:27,480 --> 00:41:29,120 We got rid of any transient effects. 812 00:41:29,120 --> 00:41:32,210 We'll bring them back, now that we have a way simpler case, 813 00:41:32,210 --> 00:41:32,750 on Tuesday. 814 00:41:38,895 --> 00:41:39,395 Yeah, Luke? 815 00:41:39,395 --> 00:41:43,597 AUDIENCE: [INAUDIBLE] step, the plus scattering-- 816 00:41:43,597 --> 00:41:44,930 PROFESSOR: From fast to thermal. 817 00:41:44,930 --> 00:41:51,430 AUDIENCE: Is that also supplied by the sigma [INAUDIBLE]?? 818 00:41:57,058 --> 00:41:58,350 PROFESSOR: Where is that going? 819 00:41:58,350 --> 00:42:00,532 AUDIENCE: It must be in the denominator, 820 00:42:00,532 --> 00:42:02,720 right, because it was over on the right side? 821 00:42:02,720 --> 00:42:03,797 PROFESSOR: Let's see. 822 00:42:03,797 --> 00:42:06,130 Oh, yeah, we divided by all the stuff on the right side, 823 00:42:06,130 --> 00:42:07,240 didn't we? 824 00:42:07,240 --> 00:42:07,840 OK. 825 00:42:07,840 --> 00:42:10,370 So that shouldn't be there. 826 00:42:10,370 --> 00:42:13,240 But it should be there because we divided by everything 827 00:42:13,240 --> 00:42:16,020 on the right side. 828 00:42:16,020 --> 00:42:17,850 Let's just check that really carefully. 829 00:42:17,850 --> 00:42:22,560 So it should have been-- 830 00:42:22,560 --> 00:42:23,890 no, that's the thermal one. 831 00:42:23,890 --> 00:42:25,730 So we're not worrying about that. 832 00:42:25,730 --> 00:42:26,230 Yep. 833 00:42:26,230 --> 00:42:28,370 So it would just end up here. 834 00:42:28,370 --> 00:42:28,870 Yeah. 835 00:42:28,870 --> 00:42:30,733 Good point. 836 00:42:30,733 --> 00:42:32,813 Cool. 837 00:42:32,813 --> 00:42:34,230 Let's talk a little bit about what 838 00:42:34,230 --> 00:42:36,105 I'd want you guys to be able to do with this. 839 00:42:36,105 --> 00:42:38,890 So what would I want you to be able to do on the homework 840 00:42:38,890 --> 00:42:40,290 and on an exam? 841 00:42:40,290 --> 00:42:42,120 With the neutron transport equation, 842 00:42:42,120 --> 00:42:43,190 recite it from memory. 843 00:42:43,190 --> 00:42:44,280 Well, not really. 844 00:42:44,280 --> 00:42:47,370 But if I were to give you the neutron transport equation, 845 00:42:47,370 --> 00:42:50,857 I'd maybe want you to explain what some of the terms mean, 846 00:42:50,857 --> 00:42:52,440 or tell me how you would get the data, 847 00:42:52,440 --> 00:42:55,710 or explain one of the simplification steps 848 00:42:55,710 --> 00:42:58,080 and justify why you think it's OK because we actually 849 00:42:58,080 --> 00:43:01,560 wrote out the justification for every step on the board. 850 00:43:01,560 --> 00:43:05,220 Or explain, for example, what's the physical reason that we 851 00:43:05,220 --> 00:43:08,820 can solve the neutron transport equation with this diffusion 852 00:43:08,820 --> 00:43:11,610 approximation? 853 00:43:11,610 --> 00:43:14,140 And in which regions does that approximation break down? 854 00:43:14,140 --> 00:43:16,120 So can anyone tell me, from yesterday, 855 00:43:16,120 --> 00:43:19,720 where is the diffusion equation a bad approximation 856 00:43:19,720 --> 00:43:20,750 of the flux? 857 00:43:20,750 --> 00:43:21,250 Yep. 858 00:43:21,250 --> 00:43:24,705 AUDIENCE: Near the control rods or the fuel. 859 00:43:24,705 --> 00:43:26,580 PROFESSOR: Near the control rods or the fuel, 860 00:43:26,580 --> 00:43:30,850 or anywhere else where cross sections change all of a sudden 861 00:43:30,850 --> 00:43:34,540 because diffusion describes long distance steady state 862 00:43:34,540 --> 00:43:36,100 solutions across places, and where 863 00:43:36,100 --> 00:43:41,890 things change drastically, diffusion breaks down. 864 00:43:41,890 --> 00:43:44,080 Because we assumed here that the neutrons behaved 865 00:43:44,080 --> 00:43:47,230 like an ideal gas or some chemical species 866 00:43:47,230 --> 00:43:49,727 with no neutron to neutron interactions, 867 00:43:49,727 --> 00:43:52,060 because the mean free path length for those interactions 868 00:43:52,060 --> 00:43:55,420 is like, what did we say, 10 to the 8th centimeters, 869 00:43:55,420 --> 00:43:56,440 so a megameter? 870 00:43:56,440 --> 00:43:57,280 Yeah. 871 00:43:57,280 --> 00:44:01,420 I love using those sorts of terminologies. 872 00:44:01,420 --> 00:44:02,940 1,000 kilometers before a neutron 873 00:44:02,940 --> 00:44:05,560 would hit another neutron. 874 00:44:05,560 --> 00:44:07,510 Or I might ask you to, let's say, 875 00:44:07,510 --> 00:44:09,430 reduce the neutron diffusion equation 876 00:44:09,430 --> 00:44:13,060 and come up with a simple criticality condition. 877 00:44:13,060 --> 00:44:15,400 Or let's say, if you were to make a physical change 878 00:44:15,400 --> 00:44:17,290 to the reactor, tell me if you think 879 00:44:17,290 --> 00:44:18,790 it would go more or less critical, 880 00:44:18,790 --> 00:44:20,590 and what would happen next? 881 00:44:20,590 --> 00:44:22,870 Or I could give you a different physical situation, 882 00:44:22,870 --> 00:44:25,990 like the up scattering scenario, which I will, 883 00:44:25,990 --> 00:44:30,350 and ask you to pose and maybe solve these equations, 884 00:44:30,350 --> 00:44:34,820 or at least get forms of the criticality condition. 885 00:44:34,820 --> 00:44:37,070 I'm not going to ask you to get tons of flux equations 886 00:44:37,070 --> 00:44:39,170 because that's all 22.05 is about, 887 00:44:39,170 --> 00:44:42,110 is doing this sort of neutron physics. 888 00:44:42,110 --> 00:44:46,230 But I want to make sure that you walk into that class prepared. 889 00:44:46,230 --> 00:44:48,540 Plus we've been kind of heavy on the-- 890 00:44:48,540 --> 00:44:50,040 you know, this class, the name of it 891 00:44:50,040 --> 00:44:53,075 is Intro to Nuclear Engineering and Ionizing Radiation. 892 00:44:53,075 --> 00:44:55,200 And so far, we've been pretty heavy on the ionizing 893 00:44:55,200 --> 00:44:56,900 radiation and physics. 894 00:44:56,900 --> 00:44:58,650 So this is where the engineering comes in. 895 00:44:58,650 --> 00:45:01,170 Assuming you have some material properties, 896 00:45:01,170 --> 00:45:03,870 you can now pick them to create a reactor 897 00:45:03,870 --> 00:45:07,310 in perfect equilibrium. 898 00:45:07,310 --> 00:45:08,460 Yeah, Kristin? 899 00:45:08,460 --> 00:45:10,327 No? 900 00:45:10,327 --> 00:45:12,660 So did anyone else have any questions about the material 901 00:45:12,660 --> 00:45:14,670 or about what I might ask you to do with it? 902 00:45:17,020 --> 00:45:17,520 Yeah. 903 00:45:17,520 --> 00:45:19,103 AUDIENCE: You said this equation would 904 00:45:19,103 --> 00:45:22,370 hold for any geometry just based on [INAUDIBLE] neutrons. 905 00:45:22,370 --> 00:45:23,260 PROFESSOR: Yep. 906 00:45:23,260 --> 00:45:26,260 So all that you would do differently 907 00:45:26,260 --> 00:45:29,970 is, right here, when we had that Laplacian operator, 908 00:45:29,970 --> 00:45:34,450 we took the one-dimensional case of an infinite reactor 909 00:45:34,450 --> 00:45:36,430 in finite and one dimension, which 910 00:45:36,430 --> 00:45:40,330 meant the Laplacian operator is just double derivative of x. 911 00:45:40,330 --> 00:45:42,820 But you could pose the equation in cylindrical coordinates 912 00:45:42,820 --> 00:45:45,640 and say, well, let's say now you had an infinite cylinder 913 00:45:45,640 --> 00:45:47,650 reactor, you wouldn't necessarily 914 00:45:47,650 --> 00:45:51,750 have sines and cosines that would satisfy this relation. 915 00:45:51,750 --> 00:45:54,570 Anyone happen to know what you'd have? 916 00:45:54,570 --> 00:45:57,390 The sines and cosines of the cylindrical world, 917 00:45:57,390 --> 00:46:00,530 called Bessel functions. 918 00:46:00,530 --> 00:46:03,910 So these are the sorts of, in cylindrical geometry equations, 919 00:46:03,910 --> 00:46:05,753 that behave similar to sines and cosines 920 00:46:05,753 --> 00:46:07,420 with kind of regular routes and that you 921 00:46:07,420 --> 00:46:09,760 can describe in a similar way. 922 00:46:09,760 --> 00:46:11,980 But I'm not going to get you guys into that. 923 00:46:11,980 --> 00:46:14,230 I'll just say, OK, there exists solutions 924 00:46:14,230 --> 00:46:16,230 that you can look up in the cylindrical case. 925 00:46:16,230 --> 00:46:20,270 And I would not make you derive them by hand because, 926 00:46:20,270 --> 00:46:21,710 what's the point? 927 00:46:21,710 --> 00:46:23,840 Again I'm not here to drill your-- 928 00:46:23,840 --> 00:46:25,760 can you do the same math over and over again? 929 00:46:25,760 --> 00:46:28,730 I want to make sure that you can intuitively understand, 930 00:46:28,730 --> 00:46:30,440 what's a k effective? 931 00:46:30,440 --> 00:46:32,545 In a sentence, it's gains over losses. 932 00:46:32,545 --> 00:46:34,670 What happens when you push that out of equilibrium? 933 00:46:34,670 --> 00:46:37,610 Or what physical situations could push that out 934 00:46:37,610 --> 00:46:41,160 of equilibrium? 935 00:46:41,160 --> 00:46:43,110 So any other questions for you guys? 936 00:46:43,110 --> 00:46:43,625 Yeah. 937 00:46:43,625 --> 00:46:46,400 AUDIENCE: Just curious on the cylindrical graph 938 00:46:46,400 --> 00:46:49,073 we have there, what would the graph in flux look like? 939 00:46:49,073 --> 00:46:50,740 PROFESSOR: It would look pretty similar. 940 00:46:50,740 --> 00:46:54,670 In r, it would kind of come down like that. 941 00:46:54,670 --> 00:46:58,300 It would always be symmetric about the center for symmetry 942 00:46:58,300 --> 00:47:03,630 arguments, and in z, it would look kind of like that. 943 00:47:03,630 --> 00:47:09,450 And so actually in the end, the form of flux in r and z 944 00:47:09,450 --> 00:47:12,420 comes out as the first Bessel function. 945 00:47:16,100 --> 00:47:19,820 Let's say, that's a times the-- 946 00:47:19,820 --> 00:47:21,320 what would you call it -- 947 00:47:21,320 --> 00:47:29,364 times cosine of this distance is z, so pi is z over-- 948 00:47:29,364 --> 00:47:31,450 I'm going to have to add a subscript. 949 00:47:34,820 --> 00:47:39,370 And there'd be some constant a in there. 950 00:47:39,370 --> 00:47:42,280 So what you can assume for multi-dimensional reactors 951 00:47:42,280 --> 00:47:44,320 is that the dimensions are separable. 952 00:47:44,320 --> 00:47:47,790 So the r part is solved separately from the z part. 953 00:47:47,790 --> 00:47:51,380 And that comes right from the Laplacian operator right here. 954 00:47:51,380 --> 00:47:59,620 If you assume that some flux in r and z 955 00:47:59,620 --> 00:48:04,180 can be written like the r part as a function of r 956 00:48:04,180 --> 00:48:07,930 times the z part as a function of z, 957 00:48:07,930 --> 00:48:10,300 then the solution gets a lot easier to deal with. 958 00:48:10,300 --> 00:48:11,680 But this is not something I'd ask 959 00:48:11,680 --> 00:48:15,240 you to do in any coordinates but Cartesian because those are 960 00:48:15,240 --> 00:48:18,240 more intuitive, and you'll get plenty of the other stuff 961 00:48:18,240 --> 00:48:19,290 later. 962 00:48:19,290 --> 00:48:20,750 AUDIENCE: What's the name of those functions again? 963 00:48:20,750 --> 00:48:22,542 PROFESSOR: They're called Bessel functions. 964 00:48:31,303 --> 00:48:33,720 So if you want to look up, there's a little bit about them 965 00:48:33,720 --> 00:48:36,780 in the reading, but it's one of the more advanced topics 966 00:48:36,780 --> 00:48:39,300 that I'm not going to have you guys responsible for. 967 00:48:39,300 --> 00:48:41,050 Much rather it be you'd be able to tell me 968 00:48:41,050 --> 00:48:43,870 what happens if you compress the reactor 969 00:48:43,870 --> 00:48:46,720 or raise its temperature, or pull out a control rod, 970 00:48:46,720 --> 00:48:51,080 or raise the pumping speed and cool down the water, 971 00:48:51,080 --> 00:48:52,848 or something like that. 972 00:48:52,848 --> 00:48:54,890 So there'll be plenty of those kinds of questions 973 00:48:54,890 --> 00:48:57,963 on the homework to help reinforce your intuition, 974 00:48:57,963 --> 00:49:00,380 as well as some of the noodle scratches will be developing 975 00:49:00,380 --> 00:49:03,890 a criticality reaction or equation for a more 976 00:49:03,890 --> 00:49:06,170 complex system than the one I've just shown you here, 977 00:49:06,170 --> 00:49:09,880 but using the same methodology and the same ideas.