the platonian reactor melt down on glia-6

THE PLATONIAN REACTOR MELT DOWN

ON GLIA-6

ME 201/MTH 281/ME400/CHE400 PROJECT 2010

ME201/MTH281/ME400/CHE400 STARDATE 9966.2 PAGE 1

INTRODUCTION General Description of the Project In this project, you are on a training exercise for Starfleet cadets. You are aboard the USS Enterprise, which is in the Glia starsystem orbiting the planet Glia-6. This is the fourth return of the Enterprise to Glia-6 since Captain Kirk’s near-fatal encounter with a glion there on Stardate 8378.2. Your training exercise will include a visit to the surface of Glia-6, where you will visit the ruins of the abandoned Platonian colony. The focus of your visit will be the ancient nuclear reactor which suffered a melt down during the active days of the colony. The damage was so extreme that no investigation of the ruins has yielded any information on the cause of the melt down. However, a recently discovered operations manual with an operations log for the reactor may provide the information necessary to understand the melt down. You and your fellow Academy cadets are the first to get a serious look at the manual, thereby giving you a real opportunity to achieve fame – or at least to be passed on to your fifth year at the Academy. The details of all this are given below. Practical Matters Those of you registered for ME 201 and MTH 281 may work alone, or in groups of two. Students registered for ME 400 or ChE 400 are required to work alone. In either case you may discuss the project freely with others. If you get significant help from someone else, that fact should appear as an acknowledgment or reference in your report. Any books or articles that you use also should be referenced. Your report should be brief but self-contained, so that a reader not having access to these directions will know what you have done. A typical report organization might be something like the following: Introduction (with a general description of the problem); Formulation (the detailed quantitative formulation of the problem); Results (results of the calculations, including any graphs or tables necessary to make the results clear); Summary (a summary of your conclusions); References (books, articles or people consulted). Mathematica notebooks showing calculations should be included as an Appendix to the report. Although typed reports are preferred, neatly handwritten reports are acceptable. Your project grade will be based on both the write-up (20%) and the technical content (80%). For groups of two, only one report need be submitted. Connection with the Real World You will work with the diffusion model for neutrons in the reactor. A more realistic model would require a study of the distribution of neutrons in both space and energy. The parameter values used in the project are not based on real parameters for a reactor. Finally, the neutorator boundary device of the Platonians is science fiction rather than science. It does, however, make for an interesting mathematical analysis.


Recent Visits to the Glia System The seventh visit to the Glia system (Stardate 9604.2, ME201/MTH281 F2000) was the first involving upper level cadets from Starfleet Academy. On that visit, two cadets well-versed in partial differential equations were able to avert a war with the Klingons by their skillful analysis of a cooking problem. The eighth expedition to the Glia system (Stardate 9863.4, ME201/MTH281 F2001), also with upper level cadets, included a visit to the abandoned colonies on Glia-6. There the cadets unraveled the secrets of an acoustic lock and gained access to the historical records of the colony. During the ninth visit to the Glia system (Stardate 9927.3, ME201/MTH281 F2002), a surface visit to Glia-6 was not possible because of a continuing dust storm. The Academy cadets on board the orbiting Enterprise engaged in the design of a stealth submarine, using the novel technique of flow into and out of the submarine to mask the more prominent components of the far field. The details of these last four visits are available on the web at http://www.me.rochester.edu/courses/ME201/webproj/proj.html .

The present tenth visit to the Glia system is focused on determining the cause of the melt

down of a nuclear reactor in the distant past. You and the other cadets involved will have to use all of your skills in diffusion theory and eigenvalue analysis to solve this nuclear mystery. Fortunately because of your work at the Academy in Partial Differential Sciences, you have all the requisite expertise in Sturm-Liouville theory. Who is on the Enterprise? Because the dust storm prevented surface exploration last year, most of the cadets in last year’s group were allowed to return this year to gain experience on the planet surface. The Enterprise is carrying you and five other cadets from Star Fleet Academy, along with 294 crew members. The other five cadets are Vinod, Catherine, Yud-ren, Roger, and Andi. This is Andi's first trip to the Glia system. She was removed from the traveling group last year because Captain Kirk heard her complaining about a homework grade to one of the Academy Instructors. Roger is new to the group also. He is replacing his twin brother Paul who came last year. Esmé, who was in last year’s group, has graduated and is now working as a librarian at the Academy. A seventh cadet, Carter, was removed from the group at the last minute by Captain Kirk. The Captain found out that Carter had written 431 emails to Professor Clark with questions on the Partial Differential Sciences course. The Captain put Carter in charge of answering all email questions from lower division students for the next six months. This leaves the six of you to do all of the work on the melt down project, with advice from Science Officer Spock and criticism from Captain Kirk.


THE PROJECT Cadets Discuss Reactor Diffusion and Wasabi Peas

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As our story opens, you and Catherine are having a discussion in the Cadet Lounge. It is early evening just after dinner. “It’s not going to be the same without Esmé,” you say. “Even though she drove me nuts with her crazy literary references, her work on the project was really good.” “Yes,” responds Catherine, “I wish she were back. Andi will be a good replacement though. She should be very helpful.” “Well, she is very bright, but she complains a lot. It's kind of a hobby with her,” you reply. “At least she doesn't complain about her homework grades anymore, after missing out on the training trip last year. If we can keep her focused on the work she will come up with some good ideas.” At that moment Andi marches into the lounge. As soon as she sees the two of you she says, “The food is really lousy here! A cadet could starve.” “Andi,” Catherine says, “if you could have any food item that's not available on the Enterprise, what would it be?” Andi thinks for a moment and then says, “Well, probably wasabi peas. Yeah, definitely wasabi peas. I really miss them.” You and Catherine look equally astonished. “What in all of the galaxy are wasabi peas?” you say. Before Andi can answer, Catherine says, “They are roasted peas coated with wasabi which is a kind of Japanese horseradish made from the herb Eutrema wasabi. It is an old Japanese recipe.” Wondering briefly why you are the only one who has never heard of wasabi peas, you say to Andi, “So the only thing between you and happiness is a handful of wasabi peas – is that right?” Fortunately this conversation, going nowhere, ends abruptly when Roger and Vinod appear at the lounge door. “Hey Andi,” shouts Roger. “Now you know what you missed last year!” Andi snorts and says, “Yeah. Bad food, cramped quarters, and annoyingly cheerful shipmates.”


Vinod, in his usual conciliatory mode says, “You will soon forget all of that when we get started on the project. It looks really interesting.” Andi looks doubtful but says nothing more.

Roger strides to the holoboard, clearly about to take charge of the meeting. Before he can say anything you ask quickly, directing your question to Roger and Vinod, “Have you guys ever heard of wasabi peas?”

They look a little surprised at the question, but then answer in unison, “Of course!

Everybody knows about wasabi peas.” You are annoyed with yourself for feeling so uncool over such a trivial thing as wasabi

peas and you decide it is time to move on. You say to Roger, “Give us an overview of the melt down project and what we are supposed to do.”

Roger looks pleased at the request, but again he is interrupted before he can say anything.

“Hey! Where's Yud-Ren?” asks Catherine. “Oh, he's down on deck 8 in the gym, ” you reply. “He's actually trying to reconfigure

and reprogram one of the universal exercise machines to a DDR.” Vinod, an avid student of 20th century earth history, looks puzzled. He asks, “DDR?

Deutsche Demokratische Republik?” Vinod is too nice to laugh at, but you are feeling relieved and distinctly less uncool. You

say, “Ah, no, in this case DDR means Dance Dance Revolution.” Vinod is clearly unenlightened by that. No matter, you think. He will soon see for himself, if Yud-Ren is successful, as he almost always is.

Roger says impatiently, “Could we get started? We have more important things to do!”

He launches into a long summary of the background on the melt down. “The Platonians founded a colony here long ago. All we know of their colony is from the repository which they left – a repository first opened two years ago by cadets who unraveled the secret of the Platonian acoustic locks. A few months ago, archivists working with the repository materials found the manual and log for the Platonian nuclear reactor. We of course knew about the reactor because of the remains of the melt down, which we saw yesterday.” Roger pauses to put a picture of the melt-down remains on the board (Figure 1). He continues. “According to the manual this reactor had an unusual feature – a device called a neutorator which allowed the reactor to become critical at a somewhat lower fuel density. We don't know whether this neutorator had anything to do with the melt down. It is our task to determine the cause of the meltdown.” Again Roger pauses and puts on the board a picture of the neutorator (Figure 2).

Catherine, astonished at the complexity of it, says, “What did it do? How did it work?” Roger replies, “We're not quite sure. Somehow it modified the boundary condition on

one face of the reactor. It seems to have been a combination of neutron sources, a neutron


monitoring device, and an elaborate feedback system. We do know the mathematical effect, because the equations describing the neutorator are in the manual.”

At this moment Yud-Ren, wearing a big smile, slips into the lounge and gives a thumbs-

up. A cheer goes up, much to Roger's annoyance. Yud-Ren, with some justifiable pride, says, “You can add DDR to the list of recreation time activities!”

Andi, who did not join in the cheer, says, “You'd better hope that Captain Kirk doesn't

find out about that, or we will all be in trouble.” Roger takes this opportunity to regain control of the meeting. “Listen up! I am giving

you each a copy of the reactor manual. Study it carefully tonight. Tomorrow we will start going over the theory and the calculations in detail. I want everyone back here by 800 hours sharp!”

Andi groans and says, “Why so early? We have all day tomorrow to finish the analysis.”

Unfortunately for Andi, she hasn't noticed Captain Kirk standing in the doorway of the lounge. He marches up to her with a distinctly unpleasant smile.

“A cadet who can't get up early,” says the Captain. “That's an easy problem to fix.

Report to the gym tomorrow at 600 hours. You can work out on the machine modified by your friend,” he says, pointing to Yud-Ren. The Captain strides briskly from the room, still smiling.

“How did the Captain know about the DDR?” exclaims Catherine. “Oh, he knows about everything,” says Andi. “That kind always does. I'd like to see him

try it. I’ll bet he would be the worst kind of bar hugger.” “Well, Andi,” says Roger, “at least you will be wide awake for our meeting.” Andi looks very annoyed, and makes some strange gesture with her right hand as she

leaves the lounge. “What was that gesture?” asks Roger. “Andi is expert on vulgar gestures from all of the galactic cultures,” says Catherine.

“That was probably from some obscure planet light years away.” “Kind of silly,” you say. “You can't tell a vulgar gesture from someone with arthritis

waving.” “Oh I think we know which it was,” says Roger. “Don't forget: 800 hours tomorrow.”

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The next morning you arrive early in the lounge and watch the others drift in. They all look tired from the late night reading of the Platonian reactor manual. Even Andi is uncharacteristically quiet, although you note with some satisfaction that she is still red-faced and sweating from her early morning gym session. Roger strides in at 759 in his full command


mode. “OK,” he says, “let's get going. Last night I asked Vinod to prepare a summary of the theory for the neutron distribution in the reactor. Vinod, you’re up!” Vinod, whose passion for 20th century history includes 20th century teaching techniques, ignores the holoboard, and rolls out an ancient blackboard from a small storage closet. He produces a pack of chalk and begins to write. You whisper to Catherine, “You wonder how those 20th century earth students survived this – no holograms, no movies, no color.” Catherine replies, “I know, and some of those courses had as many as four hour-long periods a week of droning lectures and squeaky chalk.” At that point Roger frowns at the two of you, so you settle down to listening 20th century style to Vinod, who in fact is moving rapidly in spite of the ancient chalk technology. Vinod explains that the reactor core was in the shape of a slab, with one dimension much less than the other two, so the simple slab theory with only one spatial variable is adequate for describing it. He has written the following boundary value problem on the board:

�

N = N(x, t), with ∂N∂t

= D∂ 2N∂x 2 + γN, for 0 < x < L and t > 0,

and with N(0,t) = 0 and N(L,t) = 0 (neutorator turned off).

“Here,” Vinod says, “N is the neutron density, D is the neutron diffusivity, L is the slab width, and γ is the neutron production coefficient. In terms of what we learned in the academy, γ = α − β, where α is the rate of neutron production per neutron present, and β is the rate of non-productive capture per neutron present. The Platonians always combined those two terms into a single coefficient γ and so we will do the same. The values of L and D were fixed and the reactor was made critical by adjusting the value of γ.” At this point Yud-Ren interrupts to ask, “What do you mean by neutorator turned off?” Vinod, with a slight sigh, says, “Yes, the neutorator. This was a peculiarly Platonian concept. It is a very complex mix of neutron sources, neutron detectors, and very high speed controllers. It was attached to one of the boundaries of the core – in our calculations we will take it to be the boundary at x = L. According to the manual, the effect of the neutorator was to alter the boundary condition in such a way that the critical value of γ was reduced. Here is what the boundary value problem looks like with the neutorator turned on.” Vinod proceeds to the board and writes the equations below.

N = N(x,t), with ∂N∂t

= D ∂ 2N∂x2 + γ N , for 0 < x < L and t > 0,

and with N(0,t) = 0, and ∂N∂x

(L,t) = βN(L,t) (neutorator turned on).

Vinod continues. “The parameter β, which is completely unrelated to the β we used in

the academy for non-productive capture, is adjustable to some extent. As the reactor log shows,


the Platonians tried only three different values of β before the melt down. They made many runs with β = 0 and many runs with β = 0.5 m-1. In both cases, they calculated the critical values of γ, which agreed very well with the actual values used to run the reactor at critical. The final entry in the log is a brief statement of their plan to make a run with β = 1.6 m-1. They also mention the puzzling fact that the calculated critical value of γ for this value of β is larger than the value for β = 0.5 m-1, contrary to their expectations. They thought that since the critical γ dropped when β increased from 0 to 0.5 m-1, there would be a further drop when β increased from 0.5 to 1.6 m-1.”

“Vinod,” says Catherine, “could you summarize for us the parameter values for the

reactor?” Vinod readily agrees, and creates the list shown below on his ancient blackboard. slab thickness L = 1.2 m neutron diffusivity D = 1.4 x 10-3 m2/s neutorator parameter β = 0.0, 0.5, and 1.6 m-1 achievable range of production coefficient

�

0 ≤ γ ≤ 0.05 s-1

“Thanks, Vinod,” says Roger as he moves purposefully to the front of the room. “We have the basic information we need. Our task is first to reproduce the critical values of γ found by the Platonians, and then to determine why there was a melt down. We are supposed to present our results to the Enterprise senior officers tomorrow morning, so we have the rest of today and tonight to get this done. I propose that we split into groups of two and get going on this right away.”

Andi, frowning, says, “Calculating the critical values of γ will be easy enough, and

should tell us something, but how can we determine what caused the melt down? It could have been almost anything – a defective neutron monitor, a malfunctioning control rod, a broken coolant pipe, or even operator negligence. It seems like a waste of time to try and guess at something we can never know. I think we should just calculate the critical γ’s and call it a day.”

You are angered beyond reason by this. You get slowly to your feet and address Andi

directly. “Just call it a day? People died in this melt down and you want to just call it a day? I suppose you think it was just a coincidence that a pipe broke the first time the reactor was run with a larger value of β. We have no right to conclude that the melt down was caused by some random and unknowable operational failure unless we are absolutely certain that it was not caused by a design flaw. We have an obligation to give our best effort to this analysis, and it damn well might take more than a day to do it right. Or have you forgotten why we have been through all this training and why we are here?”

Everyone is stunned at your outburst. Even you are a little stunned. Andi gets up and

slowly leaves the room without so much as a word or even a galactic gesture. After a brief silence, Roger concludes the meeting by asking Yud-Ren and Vinod to work together, and you and Catherine to work together. Roger agrees to coordinate the work by carrying back and forth between the groups any important new ideas. Andi is probably sulking somewhere, but that is the least of your worries right now. The pressure on all of you is enormous, because you know


that the Enterprise officers will expect results in less than 24 hours. You all leave soberly, seeking out quiet places for thought, discussion and calculation.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - It is after midnight and the Enterprise is quiet. Andi, deep in thought, is walking slowly down a corridor on deck 12, the observation deck. As she turns a corner she is startled to see Science Officer Spock. “Good evening cadet,” says Spock. “It would seem illogical for you to be walking the corridors at this hour when you have an important briefing in the very near future.” Andi replies, “I desperately want to understand the melt down of the Platonian reactor, sir, but I can’t seem to come up with anything. Do you have any suggestions at all?” Spock thinks briefly, and then says, “I do not know why there was a melt down. But I do know that if there is an explanation, you will find it only if you are willing to question seriously all of the assumptions on which your calculations are based. I wish you well in your work.” With that Spock rapidly moves on. Andi remains there, pondering his words and what they might tell her about the reactor.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - It is morning and the briefing room on deck 5 is rapidly filling up with the ship's officers.

As you look around, you see to your dismay that Andi is coming toward you. You start to think about what you should say to her. “Andi, about yesterday, I –”

Andi interrupts and says, “Forget about yesterday. What you said needed to be said. I

have been up all night, and I now know why the melt down happened.” You stare, uncomprehending, and Andi says again, “I know why the melt down happened!” She proceeds to explain her ideas to you, and you realize as you listen that Andi is right, and it is actually all very simple.

You grab Andi by the arm and say, “We have to find Roger and tell him.” After a brief

search you discover Roger engaged in an argument with an Enterprise officer about command style. The two of you pull him away from the conversation. He starts to protest but then sees by your expressions that this is something really big. Andi proceeds to explain her work to Roger, and he sees immediately the importance and correctness of it. With great excitement he asks her to make the last presentation in the morning briefing.

In the first part of the briefing, Vinod presents the reactor theory, as he did for the cadet

briefing yesterday. Then Yud-Ren presents the work that all of you have done to calculate the critical values of γ. At the end, he emphasizes the puzzling result that the critical value of γ for β = 0.5 m-1 (namely 0.00108 s-1 ) is smaller than the critical value for β = 1.6 m-1.


The Enterprise officers, who have been looking a bit like captive students in a 20th century classroom, start to stir and talk among themselves, thinking that the presentation is over. Roger then says, “Excuse me gentlemen, if I may have your attention. There is one more presentation. Andi is going to present her work on the cause of the melt down.” The room immediately goes quiet, and Andi steps up to the holoboard. Her idea is so simple that it doesn't take her long to explain why the melt down occurred. Then she moves on to estimate how long after startup the melt down occurred. The room remains very quiet when Andi has finished the time estimate. Everyone realizes that the time was so short that the reactor crew had little chance to escape. Andi returns to her seat.

Captain Kirk then goes to the front of the room. “Congratulations cadets,” he says.

“That was excellent work. To celebrate this occasion, I propose an immediate round of Saurian brandy for all.” Applause and cheers fill the room. Kirk goes on, “For everyone except Andi, that is.” You are astonished. Is this turkey still on Andi's case? Then Kirk, with a flourish, produces a large foil packet. Handing it to Andi, he says, “For Andi, nothing less than wasabi peas!”

During the social hour that follows, Andi finally works up enough nerve to approach

Captain Kirk. “Sir,” she says, “how did you know to get the wasabi peas? How did you know that I would solve the problem? I didn’t even know myself that I would solve it!”

The Captain replies, “The difference between a Captain and other officers is that the

Captain can pick out the winners and losers, sometimes even before people know which they are. Besides,” he says, grabbing a handful of peas from the now open packet, “I like wasabi peas too, so I would have been happy to keep them if you hadn't solved the problem. But I like this outcome better, especially as I am sure that you won't mind sharing with the Captain!”

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to Catherine. “Where's Andi?” you ask. Catherine replies, “I don't know. I haven't seen her since the social hour after the

briefing.” Just then Andi walks into the lounge. Frowning, she says, “There are no good books in

the ship's library, and it's way too warm in here!”

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Your Task As a minimum, your work should include the calculation of the critical value of γ for the following four states: (1) neutorator off, (2) neutorator on with β = 0, (3) neutorator on with


β = 0.5 m-1, and (4) neutorator on with β = 1.6 m-1. In addition, you are asked to reproduce Andi's work and explain why the melt-down occurred, and to estimate how long after startup it happened. Work Schedule In order to allow time for the evaluation of your work before term-end, your report on the melt down must be completed by stardate 9973.7 (local Rochester time: Monday December 13, 6:00 PM EST). If you are successful, you can look forward to a distinguished career in the Power Division of the Star Fleet Academy Engineers Corps. If you are unsuccessful, you risk having to join Carter in his six-month stint answering student email. REFERENCES The cover picture, showing a nuclear reactor core simulation on an iPhone is from the University of Utah. The software is described at http://www.unews.utah.edu/p/?r=042110-1 . Figure 1 is from the 1998 Grolier MultiMedia CD Encyclopedia. Figure 2 is a picture of the EDVAC computer from the web site (http://www.library.upenn.edu/exhibits/rbm/mauchly/jwm9.html) at the University of Pennsylvania. Figure 3 is the result of a Google search on wasabi peas. This project is a very slightly modified version of my project “Melt Down on Glia-6,” from ME 201/MTH281 fall 2003. The Platonians appeared in the Star Trek Episode “Plato's Stepchildren.” Information on this and other aspects of Star Trek from The Star Trek Encyclopedia, by Michael Okuda, Denise Okuda, and Debbie Mirek, Pocket Books, 1994.


FIGURES

Figure 1. Remains of Nuclear Melt Down on Glia-6

Figure 2. Photograph of Neutorator from Platonian Manual

Figure 3. Andi's Prize

the platonian reactor melt down on glia-6

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