1 00:00:00,890 --> 00:00:03,260 The following content is provided under a Creative 2 00:00:03,260 --> 00:00:04,650 Commons license. 3 00:00:04,650 --> 00:00:06,860 Your support will help MIT OpenCourseWare 4 00:00:06,860 --> 00:00:10,950 continue to offer high-quality educational resources for free. 5 00:00:10,950 --> 00:00:13,490 To make a donation or to view additional materials 6 00:00:13,490 --> 00:00:17,450 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,450 --> 00:00:18,350 at ocw.mit.edu. 8 00:00:21,410 --> 00:00:22,785 MICHAEL SHORT: Well, as promised, 9 00:00:22,785 --> 00:00:24,830 we're gonna cover no new material today. 10 00:00:24,830 --> 00:00:27,512 We've just hit the end of part two of the course, 11 00:00:27,512 --> 00:00:28,970 which I think you'll agree with me, 12 00:00:28,970 --> 00:00:31,890 was probably the most technically challenging part. 13 00:00:31,890 --> 00:00:34,460 Who would disagree, I wonder? 14 00:00:34,460 --> 00:00:35,880 I didn't think so. 15 00:00:35,880 --> 00:00:39,060 So I wanted a bit of review and help you guys out with problem 16 00:00:39,060 --> 00:00:40,100 set seven. 17 00:00:40,100 --> 00:00:42,257 Since I've got a couple of fun problems, 18 00:00:42,257 --> 00:00:44,090 I think they're fun because they're actually 19 00:00:44,090 --> 00:00:47,330 fairly realistic, like the last one, will it blend, 20 00:00:47,330 --> 00:00:49,310 the AP 1000 edition. 21 00:00:49,310 --> 00:00:53,120 Well, you'll actually analyze if given the actual specification 22 00:00:53,120 --> 00:00:55,280 sheet of an AP1000 reactor, which 23 00:00:55,280 --> 00:00:57,830 is a modern reactor that's being built now, can 24 00:00:57,830 --> 00:01:02,300 you determine it's k-effective using the the two energy group 25 00:01:02,300 --> 00:01:03,690 approximation. 26 00:01:03,690 --> 00:01:07,520 And just to show you guys that this isn't crazy, 27 00:01:07,520 --> 00:01:11,480 I've got the AP1000 spec sheet up here. 28 00:01:11,480 --> 00:01:15,580 So who got a chance to take a look at p-set seven? 29 00:01:15,580 --> 00:01:18,060 Good, that's a lot of hands. 30 00:01:18,060 --> 00:01:20,607 All right, I highly recommend everyone look 31 00:01:20,607 --> 00:01:21,440 at it ahead of time. 32 00:01:21,440 --> 00:01:23,065 Because it is the doozy, and it will be 33 00:01:23,065 --> 00:01:24,630 the last doozy of this course. 34 00:01:24,630 --> 00:01:27,373 So part three of the course is lighter. 35 00:01:27,373 --> 00:01:28,790 Because most of your other courses 36 00:01:28,790 --> 00:01:31,370 are going to go nuts after Thanksgiving, as far as I know, 37 00:01:31,370 --> 00:01:32,450 right? 38 00:01:32,450 --> 00:01:35,000 Yeah, this one's not. 39 00:01:35,000 --> 00:01:38,870 So I'm doing my best to equalize your total course 40 00:01:38,870 --> 00:01:41,300 load this semester by making this course crazy now 41 00:01:41,300 --> 00:01:44,000 and lighten up when final season comes. 42 00:01:44,000 --> 00:01:45,467 AUDIENCE: The real MVP. 43 00:01:48,890 --> 00:01:51,680 MICHAEL SHORT: So I've got the spec sheet for the AP1000 right 44 00:01:51,680 --> 00:01:53,738 here. 45 00:01:53,738 --> 00:01:55,280 That actually goes over a description 46 00:01:55,280 --> 00:01:57,113 of what the core is like, the materials that 47 00:01:57,113 --> 00:01:59,210 are used, how many fuel rods there are, 48 00:01:59,210 --> 00:02:01,070 how many assemblies there are, basically 49 00:02:01,070 --> 00:02:03,170 what the core is made of. 50 00:02:03,170 --> 00:02:05,810 And what I want to do is jump to the end where they actually 51 00:02:05,810 --> 00:02:08,570 talk about the analytical techniques used 52 00:02:08,570 --> 00:02:09,939 in the core design. 53 00:02:09,939 --> 00:02:12,320 And what I want to show you is for the nuclear design 54 00:02:12,320 --> 00:02:15,620 of the core to get axial power distributions, 55 00:02:15,620 --> 00:02:18,350 look at that, two-group diffusion theory. 56 00:02:18,350 --> 00:02:20,720 The same stuff that we just learned right here. 57 00:02:20,720 --> 00:02:23,060 For axial power distribution control rod 58 00:02:23,060 --> 00:02:26,060 worth's one-dimensional two-group diffusion theory-- 59 00:02:26,060 --> 00:02:28,490 these are problems that you can solve with the stuff 60 00:02:28,490 --> 00:02:31,303 that we've done over the past week and a half. 61 00:02:31,303 --> 00:02:32,720 They have a little more complexity 62 00:02:32,720 --> 00:02:35,360 in that they keep all the spatial variance in there. 63 00:02:35,360 --> 00:02:38,600 So probably done with computers, however they 64 00:02:38,600 --> 00:02:40,280 make the same equations that you do. 65 00:02:40,280 --> 00:02:42,890 So Westinghouse is making the same assumptions 66 00:02:42,890 --> 00:02:44,700 that we made in this course. 67 00:02:44,700 --> 00:02:47,576 They have a lot more complexity in that they're-- 68 00:02:47,576 --> 00:02:49,670 you know, we just had cross sections 69 00:02:49,670 --> 00:02:53,480 or a macroscopic cross sections as a function of energy, 70 00:02:53,480 --> 00:02:56,960 but you may also think about it as a function of position 71 00:02:56,960 --> 00:02:58,700 and a function of temperature, since, 72 00:02:58,700 --> 00:03:00,950 as we started alluding to, on-- 73 00:03:00,950 --> 00:03:04,700 it wasn't today-- on Wednesday-- no, Tuesday. 74 00:03:04,700 --> 00:03:06,307 What day is it now? 75 00:03:06,307 --> 00:03:06,890 It's Thursday. 76 00:03:06,890 --> 00:03:09,080 Thank you, so it was definitely on Tuesday. 77 00:03:09,080 --> 00:03:11,240 We started talking about how cross sections change 78 00:03:11,240 --> 00:03:12,510 with temperature. 79 00:03:12,510 --> 00:03:15,200 And so really if we want to go crazy on that neutron transport 80 00:03:15,200 --> 00:03:17,000 equation, these cross sections will 81 00:03:17,000 --> 00:03:20,587 be functions of energy, temperature, and position. 82 00:03:20,587 --> 00:03:22,670 And so that's how this reactor would have actually 83 00:03:22,670 --> 00:03:23,420 been designed. 84 00:03:23,420 --> 00:03:25,640 But you're going to do a simpler approximation 85 00:03:25,640 --> 00:03:30,970 and take all of the information about the core in an AP1000, 86 00:03:30,970 --> 00:03:33,800 blend it, so homogenize it, figure out 87 00:03:33,800 --> 00:03:36,620 what the average atomic fractions of all 88 00:03:36,620 --> 00:03:39,110 the different things in it are, and calculate 89 00:03:39,110 --> 00:03:42,490 its k-effective, which I think is a pretty cool problem to do. 90 00:03:42,490 --> 00:03:44,750 I haven't seen it done in the courses here. 91 00:03:44,750 --> 00:03:46,490 But I want to see how it turns out. 92 00:03:46,490 --> 00:03:47,990 And you might find it's surprisingly 93 00:03:47,990 --> 00:03:50,180 different from one. 94 00:03:50,180 --> 00:03:52,550 Because we make a lot a lot of simplifications 95 00:03:52,550 --> 00:03:55,100 that actually matter quite a bit. 96 00:03:55,100 --> 00:03:57,680 But to help you parse this spec sheet, 97 00:03:57,680 --> 00:04:01,786 well, actually why don't I show you the problem first. 98 00:04:01,786 --> 00:04:04,110 And we'll go through a few of the different things 99 00:04:04,110 --> 00:04:07,320 that I've simplified the problem to make it not so tedious. 100 00:04:07,320 --> 00:04:10,470 But I also want to make sure you understand what to do. 101 00:04:10,470 --> 00:04:15,060 So the simple statement is calculate k-effective 102 00:04:15,060 --> 00:04:19,769 of the AP1000 using two-group diffusion theory, which 103 00:04:19,769 --> 00:04:22,620 means you've got a criticality condition 104 00:04:22,620 --> 00:04:25,170 from the two-group approximation. 105 00:04:25,170 --> 00:04:28,260 And I'd like to go over what that is right now. 106 00:04:28,260 --> 00:04:31,880 So let's write out, if we had a two group-- 107 00:04:31,880 --> 00:04:33,450 so we have two energy group equations 108 00:04:33,450 --> 00:04:36,075 for gains and losses of neutrons in the fast and thermal group. 109 00:04:36,075 --> 00:04:37,220 What would it look like? 110 00:04:37,220 --> 00:04:38,220 What are the gain terms? 111 00:04:41,761 --> 00:04:43,620 AUDIENCE: Sigma [INAUDIBLE] fast. 112 00:04:43,620 --> 00:04:45,410 MICHAEL SHORT: Yep. 113 00:04:45,410 --> 00:04:49,220 There's going to be some average nu times sigma 114 00:04:49,220 --> 00:04:55,190 fission fast times flux fast plus sigma 115 00:04:55,190 --> 00:05:00,650 fission thermal times flux thermal. 116 00:05:00,650 --> 00:05:02,990 Any other sources of neutrons into the fast group? 117 00:05:07,322 --> 00:05:08,030 I don't think so. 118 00:05:10,550 --> 00:05:12,710 What about sinks? 119 00:05:12,710 --> 00:05:15,535 How do neutrons leave the fast group? 120 00:05:15,535 --> 00:05:16,410 AUDIENCE: Absorption. 121 00:05:16,410 --> 00:05:17,950 MICHAEL SHORT: Yep, by absorption. 122 00:05:22,520 --> 00:05:23,425 How else? 123 00:05:23,425 --> 00:05:24,300 AUDIENCE: Scattering. 124 00:05:24,300 --> 00:05:28,250 MICHAEL SHORT: Yeah, scattering from the fast 125 00:05:28,250 --> 00:05:29,300 to the thermal group. 126 00:05:31,940 --> 00:05:32,570 How else? 127 00:05:32,570 --> 00:05:33,370 AUDIENCE: Leakage. 128 00:05:33,370 --> 00:05:34,470 MICHAEL SHORT: Leakage. 129 00:05:34,470 --> 00:05:38,270 So there'll be some diffusion constant fast times 130 00:05:38,270 --> 00:05:41,400 some fast geometric buckling squared. 131 00:05:41,400 --> 00:05:46,680 And, yeah, I think that's it. 132 00:05:46,680 --> 00:05:50,980 No, that needs a phi as well fast. 133 00:05:50,980 --> 00:05:53,900 OK, cool. 134 00:05:53,900 --> 00:05:55,610 What about the thermal group? 135 00:05:55,610 --> 00:05:58,065 What are the sources of thermal neutrons? 136 00:05:58,065 --> 00:05:58,940 AUDIENCE: Scattering. 137 00:05:58,940 --> 00:06:00,700 MICHAEL SHORT: Yep, scattering from the fast group. 138 00:06:00,700 --> 00:06:02,230 So this same term right here. 139 00:06:06,220 --> 00:06:11,470 Fast to thermal times phi fast. 140 00:06:11,470 --> 00:06:13,906 And what are the losses? 141 00:06:13,906 --> 00:06:15,398 AUDIENCE: Leakage and absorption. 142 00:06:15,398 --> 00:06:17,190 MICHAEL SHORT: Leakage and absorption, they 143 00:06:17,190 --> 00:06:19,640 look pretty familiar. 144 00:06:19,640 --> 00:06:29,460 Thermal phi, thermal plus D thermal Bg squared phi thermal. 145 00:06:29,460 --> 00:06:31,420 And that's a "t." 146 00:06:31,420 --> 00:06:32,750 OK. 147 00:06:32,750 --> 00:06:35,060 The hard part in this problem is going 148 00:06:35,060 --> 00:06:41,380 to be doing these averages. 149 00:06:41,380 --> 00:06:44,260 This is the part that we haven't explicitly done on the board 150 00:06:44,260 --> 00:06:45,280 and I want to show you. 151 00:06:45,280 --> 00:06:45,780 Yeah? 152 00:06:45,780 --> 00:06:49,540 AUDIENCE: Are those Ts or Fs, the first term 153 00:06:49,540 --> 00:06:50,830 of the second half. 154 00:06:50,830 --> 00:06:51,220 MICHAEL SHORT: This one? 155 00:06:51,220 --> 00:06:51,920 AUDIENCE: The top equation. 156 00:06:51,920 --> 00:06:52,960 MICHAEL SHORT: The top equation. 157 00:06:52,960 --> 00:06:54,166 AUDIENCE: Yeah, right there. 158 00:06:54,166 --> 00:06:56,620 MICHAEL SHORT: Uh, that should be a-- 159 00:06:56,620 --> 00:06:59,440 I'll make the Fs really curly. 160 00:06:59,440 --> 00:06:59,940 Curly. 161 00:07:03,790 --> 00:07:05,320 And those are straight Ts. 162 00:07:05,320 --> 00:07:07,670 These are curly Fs. 163 00:07:07,670 --> 00:07:11,410 OK, great, thank you. 164 00:07:11,410 --> 00:07:14,480 So the hard part is going to be doing cross section averages. 165 00:07:14,480 --> 00:07:16,480 We've just kind of written them as, hey, they're 166 00:07:16,480 --> 00:07:18,410 average cross sections. 167 00:07:18,410 --> 00:07:20,080 And an average cross section would 168 00:07:20,080 --> 00:07:24,160 look something like the integral from a minimum 169 00:07:24,160 --> 00:07:29,440 to a maximum of the cross section as a function of energy 170 00:07:29,440 --> 00:07:34,940 times the flux over the same integral 171 00:07:34,940 --> 00:07:36,080 without the cross section. 172 00:07:43,618 --> 00:07:45,410 This is where I want to point out something 173 00:07:45,410 --> 00:07:47,702 that I want you to remember for the rest of this course 174 00:07:47,702 --> 00:07:48,900 and the rest of your life. 175 00:07:48,900 --> 00:07:50,567 You don't have to do things analytically 176 00:07:50,567 --> 00:07:52,970 if you don't want to, unless it's explicitly 177 00:07:52,970 --> 00:07:54,445 stated that you have to. 178 00:07:54,445 --> 00:07:56,570 So part of this piece, that is to drill in the idea 179 00:07:56,570 --> 00:07:59,390 that it's the future, we have computers. 180 00:07:59,390 --> 00:08:02,968 And you can do numerical integration with data. 181 00:08:02,968 --> 00:08:04,760 Remember on one of the second problem sets, 182 00:08:04,760 --> 00:08:07,160 I showed you guys the web plot digitizer? 183 00:08:07,160 --> 00:08:10,950 How you can extract information from a printed graph? 184 00:08:10,950 --> 00:08:12,950 Well, a lot of times, what you'll already have 185 00:08:12,950 --> 00:08:14,840 is that data. 186 00:08:14,840 --> 00:08:16,220 And you'll have to then integrate 187 00:08:16,220 --> 00:08:19,190 that numerically in something as simple as Excel 188 00:08:19,190 --> 00:08:21,830 or as complicated as Matlab or something worse-- 189 00:08:21,830 --> 00:08:24,025 whatever tool you choose to use. 190 00:08:24,025 --> 00:08:26,150 I'll be using Excel because it's kind of the lowest 191 00:08:26,150 --> 00:08:27,175 common denominator. 192 00:08:27,175 --> 00:08:28,550 And so to show you guys you don't 193 00:08:28,550 --> 00:08:30,680 need to know any fancy software to actually solve 194 00:08:30,680 --> 00:08:32,159 these problems. 195 00:08:32,159 --> 00:08:35,700 So let's go through getting some of this data right now. 196 00:08:35,700 --> 00:08:38,909 Let's say that the cutoff between fast and thermal-- 197 00:08:38,909 --> 00:08:41,900 so if we were doing a fast cross section-- 198 00:08:41,900 --> 00:08:45,700 the cutoff would be at 1 eV. 199 00:08:45,700 --> 00:08:49,150 And our max would be, let's say, 10 MeV, 200 00:08:49,150 --> 00:08:51,160 which would be the top of the fission birth 201 00:08:51,160 --> 00:08:54,740 spectrum or the chi spectrum. 202 00:08:54,740 --> 00:09:00,430 1 eV, and that would be 10 MeV. 203 00:09:00,430 --> 00:09:03,730 So all that's left is you need tabulated values 204 00:09:03,730 --> 00:09:07,170 for the cross sections and the fluxes. 205 00:09:07,170 --> 00:09:10,440 And then you can perform this numerical integral. 206 00:09:10,440 --> 00:09:16,130 I have given you tabulated values for the fluxes, wherever 207 00:09:16,130 --> 00:09:16,630 that is. 208 00:09:16,630 --> 00:09:17,290 Yep. 209 00:09:17,290 --> 00:09:22,000 Use the attached AP1000 tabulated neutron flux profile, 210 00:09:22,000 --> 00:09:24,900 which opens, that's awesome. 211 00:09:24,900 --> 00:09:28,860 So I've given you the approximate neutron flux 212 00:09:28,860 --> 00:09:30,570 in neutrons per centimeter squared 213 00:09:30,570 --> 00:09:34,388 per second as a function of neutron energy. 214 00:09:34,388 --> 00:09:36,180 And so you can tell for low neutron energy, 215 00:09:36,180 --> 00:09:39,360 there aren't any ultra-cold neutrons in this problem, 216 00:09:39,360 --> 00:09:41,790 though there are in the problem right before. 217 00:09:41,790 --> 00:09:43,380 Because, remember, the "we'll see?" 218 00:09:43,380 --> 00:09:44,005 AUDIENCE: Yeah. 219 00:09:44,005 --> 00:09:45,790 MICHAEL SHORT: Now is when we'll see. 220 00:09:45,790 --> 00:09:49,490 But if we scale down, we get down to the thermal regions 221 00:09:49,490 --> 00:09:51,560 where it's in the eV levels. 222 00:09:51,560 --> 00:09:54,680 You start to see pretty significant neutron fluxes 223 00:09:54,680 --> 00:09:58,370 in the realm of 10 to the 14 neutrons per centimeter 224 00:09:58,370 --> 00:09:59,570 squared per second. 225 00:09:59,570 --> 00:10:01,807 So that value of 10 to the 14 for flux that we've 226 00:10:01,807 --> 00:10:03,390 been using in all our previous problem 227 00:10:03,390 --> 00:10:06,170 sets-- it's because that's what we actually get. 228 00:10:06,170 --> 00:10:10,970 This flux spectrum and the picture of it that we have here 229 00:10:10,970 --> 00:10:14,780 was taken from the MIT reactor because it's representative 230 00:10:14,780 --> 00:10:16,730 of a pressurized water reactor. 231 00:10:16,730 --> 00:10:18,950 The only difference here is this is the spectrum 232 00:10:18,950 --> 00:10:20,690 from the fast flux trap. 233 00:10:20,690 --> 00:10:24,980 You don't usually see that fast to thermal ratio 234 00:10:24,980 --> 00:10:26,930 in a thermal reactor. 235 00:10:26,930 --> 00:10:30,290 But it's the closest spectrum that I was easily 236 00:10:30,290 --> 00:10:31,760 able to get my hands on. 237 00:10:31,760 --> 00:10:34,370 And it's not that unrepresentative. 238 00:10:34,370 --> 00:10:36,020 And the reaction rates for things 239 00:10:36,020 --> 00:10:37,760 aren't going to be that different. 240 00:10:37,760 --> 00:10:40,250 Because most of the reaction rates, 241 00:10:40,250 --> 00:10:42,560 the cross sections down here are in the, like, 242 00:10:42,560 --> 00:10:43,880 thousands of barns level. 243 00:10:43,880 --> 00:10:46,963 And the cross sections here are in the one-barn level. 244 00:10:46,963 --> 00:10:49,130 It's pretty much any fast cross section for anything 245 00:10:49,130 --> 00:10:50,780 it's about a barn. 246 00:10:50,780 --> 00:10:52,170 That's a good rule of thumb. 247 00:10:52,170 --> 00:10:54,990 So it's not going to change total reaction rates that much. 248 00:10:54,990 --> 00:10:58,640 And that's why I'm not worried about taking the fast flux 249 00:10:58,640 --> 00:11:00,740 spectrum from the MIT reactor and pretending like 250 00:11:00,740 --> 00:11:02,660 it's the AP1000's. 251 00:11:02,660 --> 00:11:05,250 It's not horribly that far off. 252 00:11:05,250 --> 00:11:09,390 And it's at the right order of magnitude, which is important. 253 00:11:09,390 --> 00:11:13,020 So that data, I give you. 254 00:11:13,020 --> 00:11:15,263 Let's talk about how to get this data-- 255 00:11:15,263 --> 00:11:16,555 the macroscopic cross sections. 256 00:11:24,060 --> 00:11:27,020 So if you remember, a macroscopic cross section 257 00:11:27,020 --> 00:11:31,550 is a microscopic cross section times a number density. 258 00:11:31,550 --> 00:11:34,880 And that's for one single isotope. 259 00:11:34,880 --> 00:11:38,920 If you have a mix of isotopes, then your total 260 00:11:38,920 --> 00:11:42,640 averaged cross section for all different types of atoms 261 00:11:42,640 --> 00:11:44,390 is going to be a sum-- 262 00:11:44,390 --> 00:11:48,280 I'll make it very different-- 263 00:11:48,280 --> 00:11:55,330 over all your possible isotopes of the atom fraction 264 00:11:55,330 --> 00:12:05,350 of that isotope times the number density of that isotope times 265 00:12:05,350 --> 00:12:11,545 the microscopic cross section of that reaction for that isotope. 266 00:12:11,545 --> 00:12:13,480 Or, I'm sorry, the atom fraction is 267 00:12:13,480 --> 00:12:15,930 included in the number density. 268 00:12:15,930 --> 00:12:18,550 Let's just simplify this a little bit. 269 00:12:18,550 --> 00:12:20,540 The total number density of that isotope-- 270 00:12:20,540 --> 00:12:27,320 that'll be in atoms per cubic centimeter-- 271 00:12:27,320 --> 00:12:30,440 times the cross section for that particular isotope, 272 00:12:30,440 --> 00:12:37,380 which is in centimeters squared, which leads you to a 1 273 00:12:37,380 --> 00:12:39,795 over centimeter macroscopic cross section. 274 00:12:39,795 --> 00:12:41,670 And so let's say we were summing up something 275 00:12:41,670 --> 00:12:44,550 like stainless steel, which happened to be iron 276 00:12:44,550 --> 00:12:48,522 18 chrome 10 nickel. 277 00:12:48,522 --> 00:12:49,980 Not only would you have to then get 278 00:12:49,980 --> 00:12:52,740 the number densities of iron, chrome, and nickel, 279 00:12:52,740 --> 00:12:55,670 but you have to look at which isotopes there are. 280 00:12:55,670 --> 00:12:58,910 So if we wanted to get the macroscopic cross 281 00:12:58,910 --> 00:13:00,860 section for stainless steel, we'd 282 00:13:00,860 --> 00:13:04,180 have to split this into the stable isotopes of iron, 283 00:13:04,180 --> 00:13:06,560 the stable isotopes of chrome, and the stable isotopes 284 00:13:06,560 --> 00:13:09,920 of nickel and then say it's this number 285 00:13:09,920 --> 00:13:12,530 density times a cross section plus this number 286 00:13:12,530 --> 00:13:16,820 density times a cross section, and so on and so on and so on. 287 00:13:16,820 --> 00:13:18,770 The easy way to get those number densities, 288 00:13:18,770 --> 00:13:23,450 if you take the number density of your stainless steel 289 00:13:23,450 --> 00:13:27,410 times the atom fraction of that isotope, that should give you 290 00:13:27,410 --> 00:13:31,975 the number density of that isotope. 291 00:13:31,975 --> 00:13:33,995 Does this make sense to everybody? 292 00:13:36,850 --> 00:13:39,560 Does anyone not know how to get a number density of a material 293 00:13:39,560 --> 00:13:42,230 from its basic chemical properties? 294 00:13:42,230 --> 00:13:43,190 OK. 295 00:13:43,190 --> 00:13:46,520 So a number density is in atoms per cubic centimeter. 296 00:13:46,520 --> 00:13:48,170 And usually we would have something 297 00:13:48,170 --> 00:13:53,840 like its density, which would be grams per cubic centimeter. 298 00:13:53,840 --> 00:14:01,950 So if we take density in grams per centimeter cubed 299 00:14:01,950 --> 00:14:08,790 and multiply by Avogadro's number and divide by-- 300 00:14:08,790 --> 00:14:11,370 I'll just put a divide-by symbol here-- 301 00:14:11,370 --> 00:14:13,110 the molar mass-- 302 00:14:13,110 --> 00:14:18,990 Avogadro's number is given in atoms per mole-- 303 00:14:18,990 --> 00:14:26,310 and divide by-- molar mass units are given in-- 304 00:14:26,310 --> 00:14:29,530 what is it-- grams per mole-- 305 00:14:29,530 --> 00:14:35,290 so that's going to be like moles per gram. 306 00:14:35,290 --> 00:14:38,640 The grams cancel, the moles cancel, 307 00:14:38,640 --> 00:14:41,230 and you get atoms per cubic centimeter. 308 00:14:41,230 --> 00:14:44,310 So to get a number density, you can take the density 309 00:14:44,310 --> 00:14:46,920 of the material-- you know, for stainless steel it's like 8 310 00:14:46,920 --> 00:14:48,930 grams per cubic centimeter-- 311 00:14:48,930 --> 00:14:54,120 times Avogadro's number-- let's say 6 times 10 to the 23rd, 312 00:14:54,120 --> 00:14:59,460 and this would be like 8 grams per centimeter cubed or so-- 313 00:14:59,460 --> 00:15:01,620 divided by the molar mass or the average molar 314 00:15:01,620 --> 00:15:04,030 mass of stainless steel. 315 00:15:04,030 --> 00:15:12,320 I'm guessing that's around 56 grams per mole. 316 00:15:12,320 --> 00:15:15,190 And then you'll get a number density. 317 00:15:15,190 --> 00:15:17,230 And typical solid number densities 318 00:15:17,230 --> 00:15:23,260 tend to range from like 10 to the 26 to 10 to the 28. 319 00:15:23,260 --> 00:15:25,550 It'll be atoms per meter cubed. 320 00:15:25,550 --> 00:15:29,620 So it's going to be around, like, 21 to 23 atoms 321 00:15:29,620 --> 00:15:31,760 per centimeter cubed. 322 00:15:31,760 --> 00:15:34,270 So if you end up with something way outside those bounds, 323 00:15:34,270 --> 00:15:36,970 you've probably got some sort of unit or power error. 324 00:15:40,772 --> 00:15:42,230 So that'll help you check your math 325 00:15:42,230 --> 00:15:44,450 to make sure you get the number densities right. 326 00:15:44,450 --> 00:15:46,702 If you get the number densities right 327 00:15:46,702 --> 00:15:48,410 and you know the atom fractions, then you 328 00:15:48,410 --> 00:15:50,720 have the number density of each isotope 329 00:15:50,720 --> 00:15:54,770 in the number of atoms of that isotope per cubic centimeter. 330 00:15:54,770 --> 00:15:59,690 And then you multiply by your microscopic cross section 331 00:15:59,690 --> 00:16:02,160 and you get your macroscopic cross section. 332 00:16:02,160 --> 00:16:02,740 Yeah? 333 00:16:02,740 --> 00:16:04,115 AUDIENCE: Could you explain again 334 00:16:04,115 --> 00:16:05,840 what atom fraction refers to? 335 00:16:05,840 --> 00:16:07,380 MICHAEL SHORT: Yeah, atom fraction 336 00:16:07,380 --> 00:16:13,300 is a fraction between 0 and 1 of what 337 00:16:13,300 --> 00:16:18,443 proportion of the atoms in your material are that isotope. 338 00:16:18,443 --> 00:16:19,860 So again, if you're atom fractions 339 00:16:19,860 --> 00:16:21,360 are outside the bounds of 0 to 1, 340 00:16:21,360 --> 00:16:23,370 that's not physically significant. 341 00:16:23,370 --> 00:16:27,000 If your number densities are really far from those bounds, 342 00:16:27,000 --> 00:16:28,860 then they're probably not right-- 343 00:16:28,860 --> 00:16:32,370 unless you're talking about a gas or a neutron star. 344 00:16:32,370 --> 00:16:36,300 But solid matter tends to have approximately those number 345 00:16:36,300 --> 00:16:38,388 densities. 346 00:16:38,388 --> 00:16:40,830 Mm-hm? 347 00:16:40,830 --> 00:16:43,272 AUDIENCE: Since we're putting this reactor 348 00:16:43,272 --> 00:16:44,314 in a blender [INAUDIBLE]. 349 00:16:44,314 --> 00:16:45,670 MICHAEL SHORT: Mm-hm. 350 00:16:45,670 --> 00:16:48,240 AUDIENCE: So when we [INAUDIBLE] the atom 351 00:16:48,240 --> 00:16:50,220 fractions to do these calculations, 352 00:16:50,220 --> 00:16:52,630 how would we determine the fractions? 353 00:16:52,630 --> 00:16:53,960 MICHAEL SHORT: Good question. 354 00:16:53,960 --> 00:16:59,260 You'll determine those fractions from the AP1000 spec sheet. 355 00:16:59,260 --> 00:17:02,560 So in the AP1000 spec sheet, it tells you 356 00:17:02,560 --> 00:17:08,119 things like total weight of the fuel as uranium dioxide. 357 00:17:08,119 --> 00:17:11,690 And so you can go from total weight of the fuel to, let's 358 00:17:11,690 --> 00:17:14,411 say, molar or atom fraction of the fuel. 359 00:17:14,411 --> 00:17:16,369 If you know the weight of the fuel-- it's nice, 360 00:17:16,369 --> 00:17:17,960 they give you the weight of the fuel. 361 00:17:17,960 --> 00:17:19,890 They give you the weight of the clad. 362 00:17:19,890 --> 00:17:22,160 And let's see, which materials did 363 00:17:22,160 --> 00:17:24,840 we say you have to think about? 364 00:17:24,840 --> 00:17:27,589 It said you have to talk about four materials; 365 00:17:27,589 --> 00:17:31,400 the coolant or the moderator-- water; the fuel-- 366 00:17:31,400 --> 00:17:34,970 UO2; the clodding-- where you can assume pure zirconium, 367 00:17:34,970 --> 00:17:38,330 forget all the crazy zircaloids because that's just busy work; 368 00:17:38,330 --> 00:17:40,370 and structural materials-- 369 00:17:40,370 --> 00:17:42,940 assume pure iron. 370 00:17:42,940 --> 00:17:46,370 And so on this spec sheet, luckily, they just 371 00:17:46,370 --> 00:17:48,170 tell you the mass of the fuel. 372 00:17:48,170 --> 00:17:50,000 They tell you the mass of the clad. 373 00:17:50,000 --> 00:17:52,580 I do not believe they give you the mass of the water, 374 00:17:52,580 --> 00:17:56,870 but they do give you the volume of the core. 375 00:17:56,870 --> 00:17:59,620 And so you can figure out, if you've got a core 376 00:17:59,620 --> 00:18:03,880 and you subtract off the volume of the fuel 377 00:18:03,880 --> 00:18:06,400 and the cladding and the structural materials, 378 00:18:06,400 --> 00:18:09,760 all you're left with is volume of the water in the core. 379 00:18:09,760 --> 00:18:12,500 And that'll give you your total weight of the water. 380 00:18:12,500 --> 00:18:14,000 And once you have all the weights, 381 00:18:14,000 --> 00:18:16,635 then you can go to atom fractions. 382 00:18:16,635 --> 00:18:18,260 And then you've got all the information 383 00:18:18,260 --> 00:18:23,150 you need to get the macroscopic cross sections. 384 00:18:23,150 --> 00:18:24,888 Is that unclear to anybody? 385 00:18:27,520 --> 00:18:28,870 Cool. 386 00:18:28,870 --> 00:18:31,120 So we talked about how to get the N's. 387 00:18:31,120 --> 00:18:33,550 Let's show you how to get the sigmas. 388 00:18:33,550 --> 00:18:36,130 So there was a comment that came in that said please teach us 389 00:18:36,130 --> 00:18:37,983 how to use these databases. 390 00:18:37,983 --> 00:18:39,400 We just kind of throw them around. 391 00:18:39,400 --> 00:18:41,800 Well, I want to teach you how to use this database. 392 00:18:41,800 --> 00:18:45,940 Let's say we're going to get the cross sections for oxygen 393 00:18:45,940 --> 00:18:48,010 in uranium dioxide. 394 00:18:48,010 --> 00:18:51,310 That's nice and easy because there's only one stable isotope 395 00:18:51,310 --> 00:18:53,260 of oxygen you have to consider. 396 00:18:53,260 --> 00:18:55,670 It's oxygen 16. 397 00:18:55,670 --> 00:18:58,910 I highly recommend using the Java version of JANIS 398 00:18:58,910 --> 00:19:00,650 because it is a lot easier to use, 399 00:19:00,650 --> 00:19:03,075 less clunky, and more intuitive. 400 00:19:03,075 --> 00:19:05,450 So if you don't have Java on your machine-- first of all, 401 00:19:05,450 --> 00:19:08,370 it runs on everything, like phones, tablets, Linux, Mac, 402 00:19:08,370 --> 00:19:09,253 Windows whatever. 403 00:19:09,253 --> 00:19:11,420 And second of all, it'll just make your life easier. 404 00:19:11,420 --> 00:19:16,580 So a little time investment now will make the p-set take less. 405 00:19:16,580 --> 00:19:20,890 So once you have Java, it should just open cleanly. 406 00:19:20,890 --> 00:19:22,593 And it may show up with nothing. 407 00:19:22,593 --> 00:19:24,260 It may show up with something, depending 408 00:19:24,260 --> 00:19:26,540 on what you last looked at. 409 00:19:26,540 --> 00:19:30,240 In this case, it's looking at whatever last library 410 00:19:30,240 --> 00:19:30,780 I looked at. 411 00:19:30,780 --> 00:19:33,800 So let's pretend we're starting over. 412 00:19:33,800 --> 00:19:37,970 So sometimes you may just see this database as NEA. 413 00:19:37,970 --> 00:19:40,640 That's all the databases that come with JANIS. 414 00:19:40,640 --> 00:19:43,770 If we expand this, make sure that you 415 00:19:43,770 --> 00:19:46,110 go to incident neutron data. 416 00:19:46,110 --> 00:19:49,260 Because these are cross sections for neutron reactions 417 00:19:49,260 --> 00:19:51,100 that we want to go for. 418 00:19:51,100 --> 00:19:54,120 And there are a lot of databases in here. 419 00:19:54,120 --> 00:19:58,540 And in the problem set, specifically 420 00:19:58,540 --> 00:20:04,690 say to use the most recent ENDF or evaluated nuclear data file, 421 00:20:04,690 --> 00:20:07,930 just to make sure that we're all using the same data set. 422 00:20:07,930 --> 00:20:10,900 You'll notice that there are discrepancies between the data. 423 00:20:10,900 --> 00:20:12,760 So different groups have measured things 424 00:20:12,760 --> 00:20:15,300 with different uncertainties and different values. 425 00:20:15,300 --> 00:20:17,230 And so these cross sections aren't necessarily 426 00:20:17,230 --> 00:20:18,650 fundamental constants of nature. 427 00:20:18,650 --> 00:20:21,910 They're measurements of those constants with whatever 428 00:20:21,910 --> 00:20:25,780 uncertainty and error, which are two separate things, that 429 00:20:25,780 --> 00:20:28,230 could be in there. 430 00:20:28,230 --> 00:20:31,920 So let's open up the most recent ENDF library 431 00:20:31,920 --> 00:20:34,140 and click on cross sections. 432 00:20:34,140 --> 00:20:38,510 And now whatever you see here, the green squares 433 00:20:38,510 --> 00:20:42,470 are the elements that have tabulated cross sections. 434 00:20:42,470 --> 00:20:47,830 Just to make sure that we only have to consider oxygen 16, 435 00:20:47,830 --> 00:20:50,020 let's go to the table of nuclides. 436 00:20:50,020 --> 00:20:51,520 You'll never stop using this table. 437 00:20:51,520 --> 00:20:55,115 It's like the most useful thing in this class. 438 00:20:55,115 --> 00:20:56,490 And check to make sure that there 439 00:20:56,490 --> 00:20:58,543 is no other stable isotopes of oxygen 440 00:20:58,543 --> 00:20:59,710 that we have to worry about. 441 00:21:02,420 --> 00:21:05,648 As long as the internet is working. 442 00:21:05,648 --> 00:21:06,580 Huh. 443 00:21:06,580 --> 00:21:10,980 Did the immigrate to Korea website crash too? 444 00:21:10,980 --> 00:21:12,940 Interesting. 445 00:21:12,940 --> 00:21:14,440 Well, we'll let that load for a bit. 446 00:21:14,440 --> 00:21:16,690 And let's start looking at oxygen 16. 447 00:21:16,690 --> 00:21:22,387 So if you double click on the element of interest, 448 00:21:22,387 --> 00:21:24,470 it'll usually take a little while because its Java 449 00:21:24,470 --> 00:21:26,330 and it's loading cross sections. 450 00:21:26,330 --> 00:21:29,270 And then you can pick the nuclear reaction of choice 451 00:21:29,270 --> 00:21:30,070 from here. 452 00:21:30,070 --> 00:21:35,110 And unfortunately, I can't easily make this bigger. 453 00:21:35,110 --> 00:21:38,860 I will take a very quick detour and see 454 00:21:38,860 --> 00:21:43,570 if I can make these things a bit larger so you can see them. 455 00:21:43,570 --> 00:21:46,030 But if not, then whatever. 456 00:21:46,030 --> 00:21:46,590 No. 457 00:21:46,590 --> 00:21:47,090 No. 458 00:21:49,760 --> 00:21:51,800 Ah, oh, well. 459 00:21:51,800 --> 00:21:55,400 So let's say you wanted to get the elastic scattering cross 460 00:21:55,400 --> 00:22:00,300 section for oxygen. The first letter 461 00:22:00,300 --> 00:22:04,710 before the comma is going to be the incident particle. 462 00:22:04,710 --> 00:22:08,490 Notice that sometimes it says N and sometimes it says Z. N 463 00:22:08,490 --> 00:22:10,140 specifically means neutrons. 464 00:22:10,140 --> 00:22:13,860 Z means whatever incident particle you chose. 465 00:22:13,860 --> 00:22:18,610 So we know that here Z means neutrons coming in and elastic. 466 00:22:18,610 --> 00:22:21,010 That's elastic scattering. 467 00:22:21,010 --> 00:22:23,820 So we can then click on cross section. 468 00:22:23,820 --> 00:22:27,620 If you want to see what it looks like, you can check P for plot. 469 00:22:27,620 --> 00:22:30,090 And this will give you a logarithmic plot 470 00:22:30,090 --> 00:22:33,240 of the scattering cross section as a function of energy. 471 00:22:33,240 --> 00:22:36,510 You can see that it's pretty boring for oxygen. 472 00:22:36,510 --> 00:22:38,010 There aren't a lot of nucleons. 473 00:22:38,010 --> 00:22:40,020 There aren't a lot of different energy levels. 474 00:22:40,020 --> 00:22:44,000 There are not that many resonances or things going on. 475 00:22:44,000 --> 00:22:47,060 So I wouldn't be that upset if you just 476 00:22:47,060 --> 00:22:50,270 approximated the fast scattering cross section for argon 477 00:22:50,270 --> 00:22:53,030 as that value, whatever it is. 478 00:22:53,030 --> 00:22:56,270 But let's do this completely. 479 00:22:56,270 --> 00:22:59,210 Let's tabulate this. 480 00:22:59,210 --> 00:23:02,210 So if you click on T, you actually get a table of data. 481 00:23:02,210 --> 00:23:04,100 And you can choose how much data you export. 482 00:23:04,100 --> 00:23:07,823 By default, you get, like, thousands 483 00:23:07,823 --> 00:23:10,490 upon thousands of entries, which is just going to make your life 484 00:23:10,490 --> 00:23:12,290 horrible. 485 00:23:12,290 --> 00:23:15,500 So what you can do is either pick the original values 486 00:23:15,500 --> 00:23:18,440 starting at 1 eV. 487 00:23:18,440 --> 00:23:21,140 And we know we only have to go up to 10 MeV. 488 00:23:21,140 --> 00:23:23,270 So you can pick your bounds. 489 00:23:23,270 --> 00:23:27,050 And all of a sudden, there is only a few thousands 490 00:23:27,050 --> 00:23:28,340 worth of data. 491 00:23:28,340 --> 00:23:30,320 Or you can interpolate them. 492 00:23:30,320 --> 00:23:33,320 You can interpolate values either linearly 493 00:23:33,320 --> 00:23:34,460 or logarithmically-- 494 00:23:34,460 --> 00:23:37,130 I recommend logarithmic because this is such a large energy 495 00:23:37,130 --> 00:23:38,360 range-- 496 00:23:38,360 --> 00:23:42,300 and get maybe 5 values per decade. 497 00:23:42,300 --> 00:23:50,030 So between 1 and 10 MeV, you only need five numbers. 498 00:23:50,030 --> 00:23:53,310 That's still pretty intense. 499 00:23:53,310 --> 00:23:56,145 Oh, I didn't uncheck the original values. 500 00:23:56,145 --> 00:23:59,190 Ah, isn't that better? 501 00:23:59,190 --> 00:24:02,400 So now there's only 20 or 30 entries. 502 00:24:02,400 --> 00:24:04,170 It glosses over the resonances. 503 00:24:04,170 --> 00:24:05,490 It sure does. 504 00:24:05,490 --> 00:24:08,130 But are they that important? 505 00:24:08,130 --> 00:24:09,980 That's up to you guys to decide. 506 00:24:09,980 --> 00:24:11,960 You can try doing one of these calculations 507 00:24:11,960 --> 00:24:14,750 with the original values and with the interpolated values 508 00:24:14,750 --> 00:24:18,560 and see just how different they really are in two-group theory. 509 00:24:18,560 --> 00:24:21,430 And hint is, not very much. 510 00:24:21,430 --> 00:24:24,280 So then you can actually export that data. 511 00:24:27,770 --> 00:24:29,720 So either you can just copy-paste it. 512 00:24:29,720 --> 00:24:38,540 So I just highlighted, copied, start Excel, and in it comes. 513 00:24:38,540 --> 00:24:40,040 And right there is the data that you 514 00:24:40,040 --> 00:24:43,500 can start to use to do your numerical integration. 515 00:24:43,500 --> 00:24:44,940 And I don't want to give away how 516 00:24:44,940 --> 00:24:48,120 to do the numerical integration, though I 517 00:24:48,120 --> 00:24:51,090 kind of did, symbolically. 518 00:24:51,090 --> 00:24:52,620 I'd like you guys to figure out how 519 00:24:52,620 --> 00:24:56,050 to do this numerical integration mathematically, given 520 00:24:56,050 --> 00:24:57,300 that you can get the data now. 521 00:25:00,050 --> 00:25:03,880 So is there any step here that's unclear to anyone? 522 00:25:03,880 --> 00:25:06,130 Yeah? 523 00:25:06,130 --> 00:25:08,500 AUDIENCE: So are we going to have to go through, like, 524 00:25:08,500 --> 00:25:12,580 every single material, like, all their different cross 525 00:25:12,580 --> 00:25:18,100 sections to, basically, sum up and average them? 526 00:25:18,100 --> 00:25:19,510 MICHAEL SHORT: Yes, you are. 527 00:25:19,510 --> 00:25:20,985 Sounds horrible, isn't it? 528 00:25:20,985 --> 00:25:21,610 AUDIENCE: Yeah. 529 00:25:21,610 --> 00:25:24,160 MICHAEL SHORT: That's why I made some simplifications. 530 00:25:24,160 --> 00:25:29,350 So if you notice, we are simplifying the cladding as Zr. 531 00:25:29,350 --> 00:25:32,470 And we're simplifying stainless steel as pure iron. 532 00:25:32,470 --> 00:25:34,120 I don't want you guys to just do tons 533 00:25:34,120 --> 00:25:35,350 and tons of repetitive stuff. 534 00:25:35,350 --> 00:25:37,720 But I do want you to get roughly the right answer. 535 00:25:37,720 --> 00:25:41,730 AUDIENCE: For Uranium, do you want us to [INAUDIBLE] U-238 536 00:25:41,730 --> 00:25:42,387 versus 235? 537 00:25:42,387 --> 00:25:44,470 MICHAEL SHORT: I want you to answer that question. 538 00:25:44,470 --> 00:25:48,220 So how would you consider the isotopes of uranium 539 00:25:48,220 --> 00:25:49,158 in this question? 540 00:25:49,158 --> 00:25:50,950 AUDIENCE: Shouldn't it be enriched uranium? 541 00:25:50,950 --> 00:25:52,117 MICHAEL SHORT: That's right. 542 00:25:52,117 --> 00:25:56,410 It's not the natural values, it's the enriched values. 543 00:25:56,410 --> 00:26:00,512 So it will say in the spec sheet-- 544 00:26:00,512 --> 00:26:04,663 AUDIENCE: I don't want to look up enriched uranium on Google. 545 00:26:04,663 --> 00:26:06,580 MICHAEL SHORT: You don't want to be on a list? 546 00:26:06,580 --> 00:26:08,800 You don't have to look up enriched uranium on Google. 547 00:26:08,800 --> 00:26:12,150 You can look it up on JANIS. 548 00:26:12,150 --> 00:26:14,580 So if you go to JANIS, right, are you 549 00:26:14,580 --> 00:26:16,425 looking for what the enrichment level is? 550 00:26:16,425 --> 00:26:17,050 AUDIENCE: Yeah. 551 00:26:17,050 --> 00:26:21,300 MICHAEL SHORT: Oh, that's on the AP1000 spec sheet. 552 00:26:21,300 --> 00:26:22,560 Where'd it go? 553 00:26:22,560 --> 00:26:27,395 If we search for enrichment, fuel enrichment, first cycle 554 00:26:27,395 --> 00:26:28,020 weight percent. 555 00:26:31,430 --> 00:26:35,670 You can either average these or just pretend it's five. 556 00:26:35,670 --> 00:26:38,370 Because that's a pretty typical enrichment level 557 00:26:38,370 --> 00:26:43,790 is 5% atomic fraction U-235. 558 00:26:43,790 --> 00:26:46,170 Or if you want to get really technical, 559 00:26:46,170 --> 00:26:49,222 you can average these noting how many fuel assemblies are 560 00:26:49,222 --> 00:26:50,180 in each of the regions. 561 00:26:50,180 --> 00:26:51,805 But I don't really care if you do that. 562 00:26:51,805 --> 00:26:54,650 AUDIENCE: What would be the other [INAUDIBLE]?? 563 00:26:54,650 --> 00:26:56,713 MICHAEL SHORT: U-238. 564 00:26:56,713 --> 00:26:58,130 Yeah, so there's only two isotopes 565 00:26:58,130 --> 00:27:00,980 of uranium, one of oxygen, however 566 00:27:00,980 --> 00:27:04,670 many of ion and zirconium-- hint there's not that many. 567 00:27:04,670 --> 00:27:06,650 And then there is H2O. 568 00:27:06,650 --> 00:27:08,870 And there's only one hydrogen and one oxygen. 569 00:27:08,870 --> 00:27:10,820 So it's not really that much busy work. 570 00:27:10,820 --> 00:27:14,430 It's just enough for you to get to the right answer. 571 00:27:14,430 --> 00:27:15,930 So I wouldn't even call it busy work 572 00:27:15,930 --> 00:27:18,770 because there's a point to it. 573 00:27:18,770 --> 00:27:21,190 So in that way, you can determine 574 00:27:21,190 --> 00:27:22,930 all of these cross sections. 575 00:27:22,930 --> 00:27:25,420 We give you these fluxes. 576 00:27:25,420 --> 00:27:29,410 These new values you can also get from JANIS. 577 00:27:29,410 --> 00:27:32,110 They're not labeled as new, but they 578 00:27:32,110 --> 00:27:35,230 are labeled as neutron multiplication factor. 579 00:27:35,230 --> 00:27:36,850 And they're usually way down here. 580 00:27:36,850 --> 00:27:40,810 And we probably need to find a physial isotope for that. 581 00:27:40,810 --> 00:27:44,170 So let's ditch oxygen for now, and go up to uranium. 582 00:27:54,590 --> 00:27:55,580 OK. 583 00:27:55,580 --> 00:27:58,800 So let's do U-235. 584 00:27:58,800 --> 00:27:59,960 It'll pull up the data. 585 00:27:59,960 --> 00:28:06,827 And then near the bottom, new bar total, neutron production. 586 00:28:09,960 --> 00:28:11,310 Check it out. 587 00:28:11,310 --> 00:28:14,720 For pretty much all energies, until you 588 00:28:14,720 --> 00:28:18,080 get to the fast region, it's the value we talked about-- 589 00:28:18,080 --> 00:28:19,670 2.44. 590 00:28:19,670 --> 00:28:22,160 And then in the fast region, you suddenly 591 00:28:22,160 --> 00:28:24,880 can get more neutrons from fission. 592 00:28:24,880 --> 00:28:26,130 Why do you guys think that is? 593 00:28:33,230 --> 00:28:34,770 Well, what sort of additional things 594 00:28:34,770 --> 00:28:40,263 happen when your incident particle energy increases? 595 00:28:40,263 --> 00:28:42,430 Let's think back to binding energy and all the stuff 596 00:28:42,430 --> 00:28:43,722 at the beginning of the course. 597 00:28:47,670 --> 00:28:49,670 There's more different kinds of fission products 598 00:28:49,670 --> 00:28:50,378 that can be made. 599 00:28:50,378 --> 00:28:54,200 Because you're increasing the Q value of this reaction 600 00:28:54,200 --> 00:28:57,340 by increasing the initial kinetic energy. 601 00:28:57,340 --> 00:28:59,658 So there are more fission products that can be made. 602 00:28:59,658 --> 00:29:01,200 And some of them, let's say, give off 603 00:29:01,200 --> 00:29:03,220 more neutrons than others. 604 00:29:03,220 --> 00:29:06,510 So you'll be able to get your new bar total 605 00:29:06,510 --> 00:29:08,640 from this one near the bottom. 606 00:29:08,640 --> 00:29:10,350 And everything else will be near the top. 607 00:29:10,350 --> 00:29:11,725 They'll either be a fission cross 608 00:29:11,725 --> 00:29:15,220 section, an absorption cross section, a scattering 609 00:29:15,220 --> 00:29:17,300 cross section. 610 00:29:17,300 --> 00:29:19,379 How do you get your diffusion coefficients? 611 00:29:24,270 --> 00:29:29,212 They're not tabulated, but they're close. 612 00:29:29,212 --> 00:29:31,545 AUDIENCE: Yeah, isn't it based off the numbers based off 613 00:29:31,545 --> 00:29:32,790 the different cross sections we have [INAUDIBLE]?? 614 00:29:32,790 --> 00:29:34,140 MICHAEL SHORT: Yeah. 615 00:29:34,140 --> 00:29:36,550 Exactly. 616 00:29:36,550 --> 00:29:39,430 So let's render this nicely. 617 00:29:39,430 --> 00:29:41,920 The diffusion constant is 1 over 3 times 618 00:29:41,920 --> 00:29:47,140 the total cross section minus mu 0 scattering-- 619 00:29:47,140 --> 00:29:49,390 average, average, average. 620 00:29:49,390 --> 00:29:59,200 And this average cosine is about 2/3 times the atomic mass. 621 00:29:59,200 --> 00:30:01,060 So you can get these diffusion coefficients 622 00:30:01,060 --> 00:30:02,890 from tabulated data. 623 00:30:02,890 --> 00:30:09,370 The total cross section is found at the top, better known as-- 624 00:30:09,370 --> 00:30:10,810 well, it just says total. 625 00:30:10,810 --> 00:30:14,037 So you can get total cross sections here. 626 00:30:14,037 --> 00:30:16,120 The only one that might be a little tricky for you 627 00:30:16,120 --> 00:30:19,060 to find from notation is absorption. 628 00:30:19,060 --> 00:30:21,100 You're not going to see something labeled 629 00:30:21,100 --> 00:30:23,600 absorption in this database. 630 00:30:23,600 --> 00:30:32,470 However, you will see this gamma reaction. 631 00:30:32,470 --> 00:30:34,120 Or how else do you get it? 632 00:30:34,120 --> 00:30:38,660 If you take the total minus fission minus scattering, 633 00:30:38,660 --> 00:30:40,980 you're left with absorption-- 634 00:30:40,980 --> 00:30:44,160 if you don't count N2N reactions or really 635 00:30:44,160 --> 00:30:46,540 esoteric high-energy things. 636 00:30:46,540 --> 00:30:48,540 So if you can't find it, that's OK. 637 00:30:48,540 --> 00:30:50,100 Because you can calculate it. 638 00:30:50,100 --> 00:30:54,990 Because we know sigma absorption is sigma total 639 00:30:54,990 --> 00:31:01,340 minus sigma scattering minus sigma fission minus others 640 00:31:01,340 --> 00:31:04,320 that we don't care about. 641 00:31:04,320 --> 00:31:06,760 And since you're getting these anyway, 642 00:31:06,760 --> 00:31:09,200 tabulating sigma absorptions should be trivial. 643 00:31:09,200 --> 00:31:10,680 It's just an Excel subtraction. 644 00:31:10,680 --> 00:31:11,930 Yeah? 645 00:31:11,930 --> 00:31:13,843 AUDIENCE: I guess U-235 doesn't have it. 646 00:31:13,843 --> 00:31:16,330 But last night I was looking at Pu-239. 647 00:31:16,330 --> 00:31:19,337 And it has N in an inelastic. 648 00:31:19,337 --> 00:31:20,170 MICHAEL SHORT: Yeah. 649 00:31:20,170 --> 00:31:22,087 AUDIENCE: Would that be considered absorption? 650 00:31:22,087 --> 00:31:23,545 MICHAEL SHORT: Inelastic scattering 651 00:31:23,545 --> 00:31:24,760 is not considered absorption. 652 00:31:24,760 --> 00:31:25,510 AUDIENCE: OK. 653 00:31:25,510 --> 00:31:27,093 MICHAEL SHORT: So inelastic scattering 654 00:31:27,093 --> 00:31:29,170 means one neutron goes in, one neutron 655 00:31:29,170 --> 00:31:32,680 comes out, but at a very different energy level. 656 00:31:32,680 --> 00:31:38,290 So I guess we would also say minus sigma inelastic. 657 00:31:38,290 --> 00:31:41,220 And inelastic does happen for just about every other element. 658 00:31:41,220 --> 00:31:42,970 But the nice thing is, those don't turn on 659 00:31:42,970 --> 00:31:45,220 until about 1 MeV. 660 00:31:45,220 --> 00:31:47,350 So it's not going to matter much. 661 00:31:47,350 --> 00:31:49,798 But you can quantify how much it matters, and just check. 662 00:31:49,798 --> 00:31:52,090 So if you can make a justified assumption to say here's 663 00:31:52,090 --> 00:31:54,715 one of the calcs with and without inelastic scattering, 664 00:31:54,715 --> 00:31:56,090 and if they don't differ by much, 665 00:31:56,090 --> 00:31:58,360 then forget it, as long as you show me your math. 666 00:32:00,950 --> 00:32:02,150 So let's say we've got nus. 667 00:32:02,150 --> 00:32:03,200 We have cross sections. 668 00:32:03,200 --> 00:32:04,340 I give you fluxes. 669 00:32:04,340 --> 00:32:06,620 We have Ds. 670 00:32:06,620 --> 00:32:09,620 You can calculate buckling from the geometry 671 00:32:09,620 --> 00:32:14,030 of the reactor, which is given in the AP1000 spec sheet. 672 00:32:14,030 --> 00:32:15,680 The only other trick will be what's 673 00:32:15,680 --> 00:32:18,905 sigma scattering from fast to thermal? 674 00:32:18,905 --> 00:32:20,280 So you'll have to figure out what 675 00:32:20,280 --> 00:32:23,400 is the cross section, not just of scattering total, 676 00:32:23,400 --> 00:32:25,740 but the cross section or the probability 677 00:32:25,740 --> 00:32:28,810 that a fast neutron enters the thermal group. 678 00:32:28,810 --> 00:32:30,780 And I don't want to give that away either, 679 00:32:30,780 --> 00:32:34,960 but it's not terribly mathematical. 680 00:32:34,960 --> 00:32:35,680 Yeah? 681 00:32:35,680 --> 00:32:38,980 AUDIENCE: So if you only have one isotope for whatever you're 682 00:32:38,980 --> 00:32:42,810 analyzing, you don't have to go through the whole overall cross 683 00:32:42,810 --> 00:32:43,310 section. 684 00:32:43,310 --> 00:32:44,920 Can't you just do the number density 685 00:32:44,920 --> 00:32:46,234 times the number of cross sections? 686 00:32:46,234 --> 00:32:46,716 Yeah, you don't have to. 687 00:32:46,716 --> 00:32:48,350 MICHAEL SHORT: Hm, not quite. 688 00:32:48,350 --> 00:32:51,370 So even if you only have one isotope for each material, 689 00:32:51,370 --> 00:32:53,540 if you have more than one material, 690 00:32:53,540 --> 00:32:55,450 you've got to average those cross sections. 691 00:32:55,450 --> 00:32:57,970 Because this criticality criterion 692 00:32:57,970 --> 00:33:01,150 is for the entire reactor and all the stuff in it. 693 00:33:01,150 --> 00:33:02,920 That's why it's like in a blender. 694 00:33:02,920 --> 00:33:05,380 So even if iron has one isotope and zirconium has 695 00:33:05,380 --> 00:33:08,500 one isotope and uranium has one isotope, which wouldn't really 696 00:33:08,500 --> 00:33:10,690 be a reactor, then you'd still have 697 00:33:10,690 --> 00:33:14,470 to take atom fractions of those to get the total criticality 698 00:33:14,470 --> 00:33:15,700 condition. 699 00:33:15,700 --> 00:33:18,190 Speaking of which, I forgot to write where the K is. 700 00:33:24,880 --> 00:33:27,800 And since everything else here will be tabulated, 701 00:33:27,800 --> 00:33:29,510 you can solve for k-effective. 702 00:33:29,510 --> 00:33:32,600 So k-effective is the only variable unknown 703 00:33:32,600 --> 00:33:34,780 in this whole equation. 704 00:33:34,780 --> 00:33:35,280 Yeah? 705 00:33:35,280 --> 00:33:36,510 AUDIENCE: But I mean, like, for one of the other questions, 706 00:33:36,510 --> 00:33:38,770 it's, like, oh, if you had a perfectly spherical ball 707 00:33:38,770 --> 00:33:41,350 of plutonium-230i, I think it is, in that situation, 708 00:33:41,350 --> 00:33:43,350 could you just use the cross sections from JANIS 709 00:33:43,350 --> 00:33:44,487 where you don't have to account for any kind 710 00:33:44,487 --> 00:33:46,110 of other isotope [INAUDIBLE]? 711 00:33:46,110 --> 00:33:46,790 MICHAEL SHORT: That's right. 712 00:33:46,790 --> 00:33:48,390 So, yeah, let's go to the rest of the problem set, 713 00:33:48,390 --> 00:33:49,800 since you mentioned it. 714 00:33:49,800 --> 00:33:54,460 For one of the other questions, North Korean nuclear weapons. 715 00:33:54,460 --> 00:33:57,010 Why is it that just putting together 716 00:33:57,010 --> 00:33:59,080 a super critical mass of plutonium 717 00:33:59,080 --> 00:34:01,360 does not constitute an effective bomb? 718 00:34:01,360 --> 00:34:04,150 We're lucky for that, too, that making nuclear weapons 719 00:34:04,150 --> 00:34:07,270 is a lot harder than just getting nuclear material. 720 00:34:07,270 --> 00:34:10,330 And this is a lot of the reason why theirs have been 500 ton 721 00:34:10,330 --> 00:34:11,909 yields or kiloton yields-- 722 00:34:11,909 --> 00:34:13,600 duds. 723 00:34:13,600 --> 00:34:15,427 It's really, really, really, really hard. 724 00:34:15,427 --> 00:34:17,260 And I want you to think about what would you 725 00:34:17,260 --> 00:34:20,590 need to do to turn a super critical mass 726 00:34:20,590 --> 00:34:23,889 of nuclear material into an effective weapon. 727 00:34:23,889 --> 00:34:26,260 And why is it that it's so difficult to do? 728 00:34:26,260 --> 00:34:29,346 I'm actually really glad it's so difficult to do. 729 00:34:29,346 --> 00:34:31,900 It's one of the reasons that a lot of folks don't have them. 730 00:34:31,900 --> 00:34:33,699 But, yeah, if you only have one isotope 731 00:34:33,699 --> 00:34:35,500 like you do in this problem, you don't 732 00:34:35,500 --> 00:34:37,120 have to do atom fractions. 733 00:34:37,120 --> 00:34:39,100 You just take number density times 734 00:34:39,100 --> 00:34:40,840 microscopic cross section from JANIS 735 00:34:40,840 --> 00:34:42,977 and you get the macro cross sections. 736 00:34:42,977 --> 00:34:45,310 That's why it's one of the skill building problems where 737 00:34:45,310 --> 00:34:47,860 it shouldn't be that hard. 738 00:34:47,860 --> 00:34:49,900 So you'll have a criticality condition. 739 00:34:49,900 --> 00:34:52,870 You'll be able to get cross sections and number densities. 740 00:34:52,870 --> 00:34:56,739 And you'll be able to tell what that radius of the sphere 741 00:34:56,739 --> 00:35:00,020 is given the form of buckling for a sphere, which 742 00:35:00,020 --> 00:35:02,570 should be in your reading. 743 00:35:02,570 --> 00:35:03,307 Yeah? 744 00:35:03,307 --> 00:35:06,043 AUDIENCE: Where do we get our data for power manipulation? 745 00:35:06,043 --> 00:35:07,960 MICHAEL SHORT: That's being sent to you today. 746 00:35:07,960 --> 00:35:08,560 AUDIENCE: OK. 747 00:35:08,560 --> 00:35:10,810 MICHAEL SHORT: So you guys all did power manipulations 748 00:35:10,810 --> 00:35:11,890 from the reactor. 749 00:35:11,890 --> 00:35:13,450 I wanted to get them for Monday. 750 00:35:13,450 --> 00:35:16,150 The guys were busy doing an in-core experiment install. 751 00:35:16,150 --> 00:35:18,490 They've promised me the data today. 752 00:35:18,490 --> 00:35:20,470 So you guys will be able to take a look at it, 753 00:35:20,470 --> 00:35:23,050 and using the transient stuff that we talked about Tuesday, 754 00:35:23,050 --> 00:35:29,620 explain why it doesn't look linear feedback or intuitive. 755 00:35:29,620 --> 00:35:30,120 Yeah. 756 00:35:30,120 --> 00:35:33,580 So I'll be giving each of you that data today. 757 00:35:33,580 --> 00:35:36,680 We talked about nuclear weapons. 758 00:35:36,680 --> 00:35:38,480 This top one I want you to do on your own, 759 00:35:38,480 --> 00:35:41,900 because it's other repetitions of the intuitive criticality 760 00:35:41,900 --> 00:35:44,480 examples that we did on Tuesday. 761 00:35:44,480 --> 00:35:46,280 The last one we haven't talked about 762 00:35:46,280 --> 00:35:48,680 is the ultra-cold nuclear reactor or the "we'll see" 763 00:35:48,680 --> 00:35:50,240 problem. 764 00:35:50,240 --> 00:35:51,950 So here I want you to actually get 765 00:35:51,950 --> 00:35:54,800 a criticality condition for the case where you 766 00:35:54,800 --> 00:35:56,770 can have ultra-cold neutrons. 767 00:35:56,770 --> 00:35:59,760 Where your moderator is, let's say, liquid hydrogen, really, 768 00:35:59,760 --> 00:36:01,753 really cold, way below the thermal level, 769 00:36:01,753 --> 00:36:03,170 and you have to split your reactor 770 00:36:03,170 --> 00:36:07,475 into three energy groups-- fast, thermal, and ultra cold. 771 00:36:07,475 --> 00:36:09,350 And the trick to this here is somebody asked, 772 00:36:09,350 --> 00:36:12,330 can you ever have up-scattering? 773 00:36:12,330 --> 00:36:13,760 Why yes. 774 00:36:13,760 --> 00:36:15,380 If you have an ultra-cold moderator 775 00:36:15,380 --> 00:36:19,580 but you've got hot fuel, you can actually scatter up in energy 776 00:36:19,580 --> 00:36:20,990 where the surrounding atoms could 777 00:36:20,990 --> 00:36:23,150 be hotter than some of the neutrons hitting it 778 00:36:23,150 --> 00:36:25,010 and will impart energy to those neutrons 779 00:36:25,010 --> 00:36:29,890 and send it up the energy spectrum. 780 00:36:29,890 --> 00:36:30,500 Yeah? 781 00:36:30,500 --> 00:36:31,530 AUDIENCE: Yeah, about that, it says 782 00:36:31,530 --> 00:36:33,085 all fission neutrons are born fast 783 00:36:33,085 --> 00:36:35,360 and all delayed neutrons are born thermal. 784 00:36:35,360 --> 00:36:37,968 Can there be no up-scattering from ultra cold? 785 00:36:37,968 --> 00:36:40,260 MICHAEL SHORT: Well how do you get ultra cold neutrons? 786 00:36:40,260 --> 00:36:41,677 AUDIENCE: They would scatter down. 787 00:36:41,677 --> 00:36:42,552 MICHAEL SHORT: Mm-hm. 788 00:36:42,552 --> 00:36:43,969 But some of them might scatter up. 789 00:36:43,969 --> 00:36:44,870 AUDIENCE: OK, got it. 790 00:36:44,870 --> 00:36:47,187 MICHAEL SHORT: Yeah, they can scatter down 791 00:36:47,187 --> 00:36:49,520 by hitting something cold and scatter back up by getting 792 00:36:49,520 --> 00:36:52,158 knocked by a hot atom. 793 00:36:52,158 --> 00:36:53,062 AUDIENCE: OK. 794 00:36:53,062 --> 00:36:53,960 MICHAEL SHORT: Yeah. 795 00:36:53,960 --> 00:36:56,460 So it's all going to be in the formulation of the equations. 796 00:36:56,460 --> 00:36:59,700 For this problem, like 80% or 85% of the credit 797 00:36:59,700 --> 00:37:02,460 is, did you formulate the three-group equations 798 00:37:02,460 --> 00:37:04,020 correctly. 799 00:37:04,020 --> 00:37:06,360 So I want you to think about what are the actual sources 800 00:37:06,360 --> 00:37:08,850 and sinks in each case, what are the fractions 801 00:37:08,850 --> 00:37:11,610 of prompt and delayed neutrons, where did they go, 802 00:37:11,610 --> 00:37:13,440 and what terms matter where. 803 00:37:13,440 --> 00:37:17,220 So it's doing this, but for the case that we've given you here. 804 00:37:17,220 --> 00:37:20,080 Solving it is a lot of algebra, and therefore not a lot 805 00:37:20,080 --> 00:37:20,580 of credit. 806 00:37:24,860 --> 00:37:26,800 So is that clear to everybody? 807 00:37:26,800 --> 00:37:30,480 I figured this p-set was worth explaining and not just saying, 808 00:37:30,480 --> 00:37:31,350 have fun. 809 00:37:31,350 --> 00:37:35,156 You know, I'll see you on Piazza Sunday night. 810 00:37:35,156 --> 00:37:36,460 AUDIENCE: I'm there. 811 00:37:36,460 --> 00:37:39,250 MICHAEL SHORT: Yeah, that's why I would check it. 812 00:37:39,250 --> 00:37:40,850 Yeah. 813 00:37:40,850 --> 00:37:41,450 Cool. 814 00:37:41,450 --> 00:37:45,350 I've also got problem set four for everybody here. 815 00:37:45,350 --> 00:37:47,900 I want to mention a couple of quick things. 816 00:37:47,900 --> 00:37:51,140 Please, if you hand write your p-sets, which is fine, 817 00:37:51,140 --> 00:37:55,100 please make sure to scan them legibly and to write legibly. 818 00:37:55,100 --> 00:37:58,800 We can't give partial credit for things we can't read. 819 00:37:58,800 --> 00:38:00,930 And so this will be a lesson to some of you guys 820 00:38:00,930 --> 00:38:02,810 depending on who got what grade. 821 00:38:02,810 --> 00:38:04,520 There are times when you may have written 822 00:38:04,520 --> 00:38:06,110 stuff for partial credit, but we just honestly 823 00:38:06,110 --> 00:38:07,010 couldn't make it out. 824 00:38:07,010 --> 00:38:10,400 So please do make sure that your submissions are legible. 825 00:38:10,400 --> 00:38:14,090 And for things like these, these are handwritten problem sets, 826 00:38:14,090 --> 00:38:16,280 but they were scanned with either a scanner which 827 00:38:16,280 --> 00:38:18,800 does the correct contrast or writing them 828 00:38:18,800 --> 00:38:21,080 on one note, which apparently works pretty well, 829 00:38:21,080 --> 00:38:22,640 or apps like CamScanner. 830 00:38:22,640 --> 00:38:24,140 There's an app on your phone you can 831 00:38:24,140 --> 00:38:27,800 get that scans pieces of paper and automatically contrast 832 00:38:27,800 --> 00:38:28,430 enhances them. 833 00:38:28,430 --> 00:38:29,960 It knows the paper should be white 834 00:38:29,960 --> 00:38:31,712 and the writing should be black. 835 00:38:31,712 --> 00:38:32,920 And it also does it in color. 836 00:38:32,920 --> 00:38:34,905 So if you do color graphs, it recognizes 837 00:38:34,905 --> 00:38:36,530 that there are multiple colors and will 838 00:38:36,530 --> 00:38:38,090 take care of that for you. 839 00:38:38,090 --> 00:38:39,920 It only takes an additional minute, 840 00:38:39,920 --> 00:38:42,570 but it can give you double the points on a problem set 841 00:38:42,570 --> 00:38:44,290 because we can read stuff. 842 00:38:44,290 --> 00:38:47,110 So I'll give back p-set four now. 843 00:38:47,110 --> 00:38:48,580 We're working on five and six. 844 00:38:48,580 --> 00:38:51,760 The big delay in grading there was I went to Russia. 845 00:38:51,760 --> 00:38:55,390 And Russia doesn't have as much internet as we do, 846 00:38:55,390 --> 00:38:57,070 at least not where I was. 847 00:38:57,070 --> 00:38:57,970 And it was busy. 848 00:38:57,970 --> 00:38:59,910 So working on those solutions now. 849 00:38:59,910 --> 00:39:00,410 Yeah? 850 00:39:00,410 --> 00:39:02,740 AUDIENCE: For problem one, it says 851 00:39:02,740 --> 00:39:04,948 throwing quarters into the reactor actually happened. 852 00:39:04,948 --> 00:39:05,740 MICHAEL SHORT: Yep. 853 00:39:05,740 --> 00:39:07,420 AUDIENCE: When did that happen? 854 00:39:07,420 --> 00:39:11,980 MICHAEL SHORT: Oh, OK, yeah, good question. 855 00:39:11,980 --> 00:39:18,220 So problem 1, throwing quarters directly into the core 856 00:39:18,220 --> 00:39:19,420 like a wishing well. 857 00:39:19,420 --> 00:39:21,400 It used to be that back in the day, 858 00:39:21,400 --> 00:39:23,440 they would take folks on reactor tours 859 00:39:23,440 --> 00:39:24,970 to look down into the core. 860 00:39:24,970 --> 00:39:27,580 Because when the lid is off, you can see it. 861 00:39:27,580 --> 00:39:30,317 You can see the Cherenkov radiation, the nice blue light. 862 00:39:30,317 --> 00:39:31,900 You can actually see the fuel elements 863 00:39:31,900 --> 00:39:34,660 because the water is sufficient shielding for you. 864 00:39:34,660 --> 00:39:37,270 And the distance is sufficient shielding to keep you away 865 00:39:37,270 --> 00:39:40,210 from the gammas, not to mention the reactor is usually off 866 00:39:40,210 --> 00:39:41,530 when the lid is off. 867 00:39:41,530 --> 00:39:43,180 So it's not that hot. 868 00:39:43,180 --> 00:39:45,460 The problem is you can't watch what 869 00:39:45,460 --> 00:39:47,080 everyone's doing all the time. 870 00:39:47,080 --> 00:39:49,330 And somebody dropped a quarter into the reactor. 871 00:39:49,330 --> 00:39:51,400 And they were like, oh, it's a wishing well. 872 00:39:51,400 --> 00:39:54,610 Well, it took something like six dive robots to go into the core 873 00:39:54,610 --> 00:39:55,990 and fish it out. 874 00:39:55,990 --> 00:39:58,480 Because each of those robots lasts 10 minutes 875 00:39:58,480 --> 00:40:00,502 before the intense radiation fries it. 876 00:40:00,502 --> 00:40:01,960 And if you didn't find the quarter, 877 00:40:01,960 --> 00:40:04,450 you got to take him out, put down another one. 878 00:40:04,450 --> 00:40:06,460 And these are, like, radiation-hard, you know, 879 00:40:06,460 --> 00:40:08,110 narrow, whatever, diving robots. 880 00:40:08,110 --> 00:40:10,718 This story was relayed to me through someone that relayed it 881 00:40:10,718 --> 00:40:11,760 to them through whatever. 882 00:40:11,760 --> 00:40:13,468 So it's been through the telephone chain. 883 00:40:13,468 --> 00:40:16,093 But I do know that's one of the big reasons you can't look down 884 00:40:16,093 --> 00:40:17,300 in the core anymore. 885 00:40:17,300 --> 00:40:20,920 It's because folks abuse the privileges of tourists. 886 00:40:20,920 --> 00:40:21,832 Yeah. 887 00:40:21,832 --> 00:40:23,740 AUDIENCE: JANIS 4 just doesn't work for me. 888 00:40:23,740 --> 00:40:25,588 Like, I have my installed Java. 889 00:40:25,588 --> 00:40:26,880 It's like up to date and stuff. 890 00:40:26,880 --> 00:40:28,465 But it just doesn't work. 891 00:40:28,465 --> 00:40:29,590 MICHAEL SHORT: Interesting. 892 00:40:29,590 --> 00:40:30,970 AUDIENCE: What do you think I should do? 893 00:40:30,970 --> 00:40:32,200 MICHAEL SHORT: Then you could use the web version. 894 00:40:32,200 --> 00:40:32,600 AUDIENCE: OK. 895 00:40:32,600 --> 00:40:34,433 MICHAEL SHORT: Which is just for the browser 896 00:40:34,433 --> 00:40:36,090 and no plugins required. 897 00:40:36,090 --> 00:40:36,590 Yeah. 898 00:40:36,590 --> 00:40:37,790 AUDIENCE: [INAUDIBLE] 899 00:40:37,790 --> 00:40:38,782 MICHAEL SHORT: Cool. 900 00:40:38,782 --> 00:40:40,240 Any other questions about the p-set 901 00:40:40,240 --> 00:40:41,490 that we didn't cover together? 902 00:40:44,352 --> 00:40:45,810 Those looking a little more doable? 903 00:40:45,810 --> 00:40:47,960 AUDIENCE: [EXHALES]