Auto sync: 2025-12-09 17:19:30 (20 files changed)
M .task/backlog.data M .task/completed.data M .task/pending.data M .task/undo.data A Class_Work/nuce2101/final/latex/SABO_NUCE2101_FINAL.pdf M Class_Work/nuce2101/final/latex/main.aux M Class_Work/nuce2101/final/latex/main.fdb_latexmk M Class_Work/nuce2101/final/latex/main.fls
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||||
[description:"Review Yang, Xiaodong application" due:"1765342800" entry:"1765318334" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"ae720546-20a9-45b5-85fd-699df6420d40"]
|
||||
[description:"Tune up presentation for Emerson... digital twins" due:"1765774800" entry:"1765318484" modified:"1765318484" project:"Internship" status:"pending" uuid:"d00350e5-5d5e-40d4-97c1-720f413f8359"]
|
||||
[description:"Make a plan for pizza for all-in event" due:"1765515600" entry:"1765318628" modified:"1765318628" project:"Union" status:"pending" uuid:"869e4507-e1c2-4e4b-b2b0-63727285ffc0"]
|
||||
[description:"Talk to Jessica about getting a room for an event" due:"1765515600" entry:"1765318649" modified:"1765318649" project:"Union" status:"pending" uuid:"9d6f1ea8-9ce1-4785-be41-fed9b345ba46"]
|
||||
[description:"Draft an email for people about MOU for EGSO" due:"1765515600" entry:"1765318679" modified:"1765318679" project:"Union" status:"pending" uuid:"e50fcfa9-85d6-4c48-b7cf-89637f90c333"]
|
||||
|
||||
125
.task/undo.data
125
.task/undo.data
@ -1342,3 +1342,128 @@ time 1765127028
|
||||
old [description:"Final Presentations" due:"1765213200" entry:"1762886561" modified:"1765127011" priority:"H" project:"ERLM" status:"pending" uuid:"538a2b60-16d5-4033-a40f-d45653b9a5a5"]
|
||||
new [description:"Final Presentations" due:"1765213200" entry:"1762886561" modified:"1765127028" priority:"H" project:"ERLM" start:"1765127028" status:"pending" uuid:"538a2b60-16d5-4033-a40f-d45653b9a5a5"]
|
||||
---
|
||||
time 1765305743
|
||||
old [description:"Final Presentations" due:"1765213200" entry:"1762886561" modified:"1765127028" priority:"H" project:"ERLM" start:"1765127028" status:"pending" uuid:"538a2b60-16d5-4033-a40f-d45653b9a5a5"]
|
||||
new [description:"Final Presentations" due:"1765213200" end:"1765305743" entry:"1762886561" modified:"1765305743" priority:"H" project:"ERLM" status:"completed" uuid:"538a2b60-16d5-4033-a40f-d45653b9a5a5"]
|
||||
---
|
||||
time 1765305754
|
||||
old [description:"NUCE2101 final exam" due:"1765256400" entry:"1764681347" modified:"1765127011" priority:"H" project:"NUCE2101" status:"pending" uuid:"a6420590-c2a5-4b74-b216-70b8faebb7c5"]
|
||||
new [description:"NUCE2101 final exam" due:"1765256400" end:"1765305754" entry:"1764681347" modified:"1765305754" priority:"H" project:"NUCE2101" status:"completed" uuid:"a6420590-c2a5-4b74-b216-70b8faebb7c5"]
|
||||
---
|
||||
time 1765305763
|
||||
old [description:"Review interview questions" due:"1764738000" entry:"1763750527" modified:"1764681468" project:"Chair-Search" status:"pending" uuid:"1156892a-3889-4b9d-baea-01e4802fd755"]
|
||||
new [description:"Review interview questions" due:"1764738000" entry:"1763750527" modified:"1765305763" project:"Chair-Search" start:"1765305763" status:"pending" uuid:"1156892a-3889-4b9d-baea-01e4802fd755"]
|
||||
---
|
||||
time 1765318224
|
||||
old [description:"Review interview questions" due:"1764738000" entry:"1763750527" modified:"1765305763" project:"Chair-Search" start:"1765305763" status:"pending" uuid:"1156892a-3889-4b9d-baea-01e4802fd755"]
|
||||
new [description:"Review interview questions" due:"1764738000" end:"1765318224" entry:"1763750527" modified:"1765318224" project:"Chair-Search" status:"completed" uuid:"1156892a-3889-4b9d-baea-01e4802fd755"]
|
||||
---
|
||||
time 1765318238
|
||||
old [description:"Reach out to SOAR for job placement info and website content" due:"1764824400" entry:"1764681551" modified:"1764681551" project:"Chair-Search" status:"pending" uuid:"535c2ffc-f233-46f0-ac33-a93cbfd12f81"]
|
||||
new [description:"Reach out to SOAR for job placement info and website content" due:"1764824400" end:"1765318236" entry:"1764681551" modified:"1765318238" project:"Chair-Search" status:"deleted" uuid:"535c2ffc-f233-46f0-ac33-a93cbfd12f81"]
|
||||
---
|
||||
time 1765318239
|
||||
old [description:"Reach out to Material Advantage for job placement info and website content" due:"1764824400" entry:"1764681551" modified:"1764681551" project:"Chair-Search" status:"pending" uuid:"00072514-5622-4a44-9826-49690be83767"]
|
||||
new [description:"Reach out to Material Advantage for job placement info and website content" due:"1764824400" end:"1765318238" entry:"1764681551" modified:"1765318239" project:"Chair-Search" status:"deleted" uuid:"00072514-5622-4a44-9826-49690be83767"]
|
||||
---
|
||||
time 1765318240
|
||||
old [description:"Reach out to FSAE for job placement info and website content" due:"1764824400" entry:"1764681551" modified:"1764681551" project:"Chair-Search" status:"pending" uuid:"ca5f0450-483f-4a93-aba1-d06e71b4df5c"]
|
||||
new [description:"Reach out to FSAE for job placement info and website content" due:"1764824400" end:"1765318239" entry:"1764681551" modified:"1765318240" project:"Chair-Search" status:"deleted" uuid:"ca5f0450-483f-4a93-aba1-d06e71b4df5c"]
|
||||
---
|
||||
time 1765318249
|
||||
old [description:"Tune up state of the art for Proposal" due:"1764997140" entry:"1762886561" modified:"1762886561" project:"ERLM" status:"pending" uuid:"953a73fa-aadc-4112-bf98-7a4825068053"]
|
||||
new [description:"Tune up state of the art for Proposal" due:"1764997140" end:"1765318249" entry:"1762886561" modified:"1765318249" project:"ERLM" status:"completed" uuid:"953a73fa-aadc-4112-bf98-7a4825068053"]
|
||||
---
|
||||
time 1765318249
|
||||
old [dep_0888fdb8-f5cd-4374-a589-b7b3c1bbd472:"x" dep_0d921ea2-6211-4cd8-bbc4-4f9bf36794d3:"x" dep_14fee599-fe0f-4773-90fc-c5f78291f425:"x" dep_29cc8c63-1fb7-4523-9953-603467b929ee:"x" dep_2a8e9294-aecc-434a-b5dd-a468baf8f4a8:"x" dep_5ba3929b-5ec3-4c9d-b30e-30fd8fe20b54:"x" dep_689420d6-7191-42b6-b691-94ad39c8e0dd:"x" dep_9ce7d23c-0f56-49af-b97d-a684966cfbae:"x" dep_a4c027fa-f50d-4efc-ab61-5b8054810a80:"x" dep_d1fa2409-2f2f-4855-81be-14ee617df5d2:"x" dep_e354ab0c-cef7-41e2-bfb4-d98886e512b7:"x" depends:"0888fdb8-f5cd-4374-a589-b7b3c1bbd472,0d921ea2-6211-4cd8-bbc4-4f9bf36794d3,14fee599-fe0f-4773-90fc-c5f78291f425,29cc8c63-1fb7-4523-9953-603467b929ee,2a8e9294-aecc-434a-b5dd-a468baf8f4a8,5ba3929b-5ec3-4c9d-b30e-30fd8fe20b54,689420d6-7191-42b6-b691-94ad39c8e0dd,9ce7d23c-0f56-49af-b97d-a684966cfbae,a4c027fa-f50d-4efc-ab61-5b8054810a80,d1fa2409-2f2f-4855-81be-14ee617df5d2,e354ab0c-cef7-41e2-bfb4-d98886e512b7" description:"ERLM proposal due" due:"1764997200" entry:"1764681347" modified:"1764681809" project:"ERLM" status:"pending" tags:"writing" tags_writing:"x" uuid:"f6968367-96f1-4ce2-8691-e45423e27456"]
|
||||
new [dep_0888fdb8-f5cd-4374-a589-b7b3c1bbd472:"x" dep_0d921ea2-6211-4cd8-bbc4-4f9bf36794d3:"x" dep_14fee599-fe0f-4773-90fc-c5f78291f425:"x" dep_29cc8c63-1fb7-4523-9953-603467b929ee:"x" dep_2a8e9294-aecc-434a-b5dd-a468baf8f4a8:"x" dep_5ba3929b-5ec3-4c9d-b30e-30fd8fe20b54:"x" dep_689420d6-7191-42b6-b691-94ad39c8e0dd:"x" dep_9ce7d23c-0f56-49af-b97d-a684966cfbae:"x" dep_a4c027fa-f50d-4efc-ab61-5b8054810a80:"x" dep_d1fa2409-2f2f-4855-81be-14ee617df5d2:"x" dep_e354ab0c-cef7-41e2-bfb4-d98886e512b7:"x" depends:"0888fdb8-f5cd-4374-a589-b7b3c1bbd472,0d921ea2-6211-4cd8-bbc4-4f9bf36794d3,14fee599-fe0f-4773-90fc-c5f78291f425,29cc8c63-1fb7-4523-9953-603467b929ee,2a8e9294-aecc-434a-b5dd-a468baf8f4a8,5ba3929b-5ec3-4c9d-b30e-30fd8fe20b54,689420d6-7191-42b6-b691-94ad39c8e0dd,9ce7d23c-0f56-49af-b97d-a684966cfbae,a4c027fa-f50d-4efc-ab61-5b8054810a80,d1fa2409-2f2f-4855-81be-14ee617df5d2,e354ab0c-cef7-41e2-bfb4-d98886e512b7" description:"ERLM proposal due" due:"1764997200" end:"1765318249" entry:"1764681347" modified:"1765318249" project:"ERLM" status:"completed" tags:"writing" tags_writing:"x" uuid:"f6968367-96f1-4ce2-8691-e45423e27456"]
|
||||
---
|
||||
time 1765318330
|
||||
new [description:"Review Zbeeb, Khaled application" due:"1765342800" entry:"1765318330" modified:"1765318330" project:"chair_search" status:"pending" uuid:"360a2897-bf5c-457e-a4b2-81b9b46a1267"]
|
||||
---
|
||||
time 1765318330
|
||||
new [description:"Review Akkus, Ozan application" due:"1765342800" entry:"1765318330" modified:"1765318330" project:"chair_search" status:"pending" uuid:"df552298-0119-4fe5-afe0-11439c838cf5"]
|
||||
---
|
||||
time 1765318330
|
||||
new [description:"Review Asmatulu, Ramazan application" due:"1765342800" entry:"1765318330" modified:"1765318330" project:"chair_search" status:"pending" uuid:"edb91039-b46b-40e8-8b43-70336a2db3ac"]
|
||||
---
|
||||
time 1765318331
|
||||
new [description:"Review Christopher, Gordon application" due:"1765342800" entry:"1765318331" modified:"1765318331" project:"chair_search" status:"pending" uuid:"aa05a70a-5640-4903-acde-d584cbe8d40a"]
|
||||
---
|
||||
time 1765318331
|
||||
new [description:"Review Gouma, Pelagia application" due:"1765342800" entry:"1765318331" modified:"1765318331" project:"chair_search" status:"pending" uuid:"17818aac-cbf7-4275-a2f6-66cf6ab2280a"]
|
||||
---
|
||||
time 1765318332
|
||||
new [description:"Review Gupta, Sanju application" due:"1765342800" entry:"1765318332" modified:"1765318332" project:"chair_search" status:"pending" uuid:"0817fece-0159-4419-91e0-e71b86da51f3"]
|
||||
---
|
||||
time 1765318332
|
||||
new [description:"Review Kim, Harrison application" due:"1765342800" entry:"1765318332" modified:"1765318332" project:"chair_search" status:"pending" uuid:"216c0f81-a6cb-4945-a9ad-1740f9897755"]
|
||||
---
|
||||
time 1765318333
|
||||
new [description:"Review Makhlouf, Abdelsalam application" due:"1765342800" entry:"1765318333" modified:"1765318333" project:"chair_search" status:"pending" uuid:"80e06097-c320-48f9-a287-59879264dc81"]
|
||||
---
|
||||
time 1765318333
|
||||
new [description:"Review Narayan, Roger application" due:"1765342800" entry:"1765318333" modified:"1765318333" project:"chair_search" status:"pending" uuid:"f2587d5e-40af-4fa1-b90c-28270d07656b"]
|
||||
---
|
||||
time 1765318333
|
||||
new [description:"Review Robles Hernandez, Francisco Carlos application" due:"1765342800" entry:"1765318333" modified:"1765318333" project:"chair_search" status:"pending" uuid:"50e3c168-cf44-499e-a66a-586c897792c2"]
|
||||
---
|
||||
time 1765318334
|
||||
new [description:"Review Yang, Xiaodong application" due:"1765342800" entry:"1765318334" modified:"1765318334" project:"chair_search" status:"pending" uuid:"ae720546-20a9-45b5-85fd-699df6420d40"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Zbeeb, Khaled application" due:"1765342800" entry:"1765318330" modified:"1765318330" project:"chair_search" status:"pending" uuid:"360a2897-bf5c-457e-a4b2-81b9b46a1267"]
|
||||
new [description:"Review Zbeeb, Khaled application" due:"1765342800" entry:"1765318330" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"360a2897-bf5c-457e-a4b2-81b9b46a1267"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Akkus, Ozan application" due:"1765342800" entry:"1765318330" modified:"1765318330" project:"chair_search" status:"pending" uuid:"df552298-0119-4fe5-afe0-11439c838cf5"]
|
||||
new [description:"Review Akkus, Ozan application" due:"1765342800" entry:"1765318330" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"df552298-0119-4fe5-afe0-11439c838cf5"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Asmatulu, Ramazan application" due:"1765342800" entry:"1765318330" modified:"1765318330" project:"chair_search" status:"pending" uuid:"edb91039-b46b-40e8-8b43-70336a2db3ac"]
|
||||
new [description:"Review Asmatulu, Ramazan application" due:"1765342800" entry:"1765318330" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"edb91039-b46b-40e8-8b43-70336a2db3ac"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Christopher, Gordon application" due:"1765342800" entry:"1765318331" modified:"1765318331" project:"chair_search" status:"pending" uuid:"aa05a70a-5640-4903-acde-d584cbe8d40a"]
|
||||
new [description:"Review Christopher, Gordon application" due:"1765342800" entry:"1765318331" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"aa05a70a-5640-4903-acde-d584cbe8d40a"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Gouma, Pelagia application" due:"1765342800" entry:"1765318331" modified:"1765318331" project:"chair_search" status:"pending" uuid:"17818aac-cbf7-4275-a2f6-66cf6ab2280a"]
|
||||
new [description:"Review Gouma, Pelagia application" due:"1765342800" entry:"1765318331" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"17818aac-cbf7-4275-a2f6-66cf6ab2280a"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Gupta, Sanju application" due:"1765342800" entry:"1765318332" modified:"1765318332" project:"chair_search" status:"pending" uuid:"0817fece-0159-4419-91e0-e71b86da51f3"]
|
||||
new [description:"Review Gupta, Sanju application" due:"1765342800" entry:"1765318332" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"0817fece-0159-4419-91e0-e71b86da51f3"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Kim, Harrison application" due:"1765342800" entry:"1765318332" modified:"1765318332" project:"chair_search" status:"pending" uuid:"216c0f81-a6cb-4945-a9ad-1740f9897755"]
|
||||
new [description:"Review Kim, Harrison application" due:"1765342800" entry:"1765318332" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"216c0f81-a6cb-4945-a9ad-1740f9897755"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Makhlouf, Abdelsalam application" due:"1765342800" entry:"1765318333" modified:"1765318333" project:"chair_search" status:"pending" uuid:"80e06097-c320-48f9-a287-59879264dc81"]
|
||||
new [description:"Review Makhlouf, Abdelsalam application" due:"1765342800" entry:"1765318333" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"80e06097-c320-48f9-a287-59879264dc81"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Narayan, Roger application" due:"1765342800" entry:"1765318333" modified:"1765318333" project:"chair_search" status:"pending" uuid:"f2587d5e-40af-4fa1-b90c-28270d07656b"]
|
||||
new [description:"Review Narayan, Roger application" due:"1765342800" entry:"1765318333" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"f2587d5e-40af-4fa1-b90c-28270d07656b"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Robles Hernandez, Francisco Carlos application" due:"1765342800" entry:"1765318333" modified:"1765318333" project:"chair_search" status:"pending" uuid:"50e3c168-cf44-499e-a66a-586c897792c2"]
|
||||
new [description:"Review Robles Hernandez, Francisco Carlos application" due:"1765342800" entry:"1765318333" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"50e3c168-cf44-499e-a66a-586c897792c2"]
|
||||
---
|
||||
time 1765318397
|
||||
old [description:"Review Yang, Xiaodong application" due:"1765342800" entry:"1765318334" modified:"1765318334" project:"chair_search" status:"pending" uuid:"ae720546-20a9-45b5-85fd-699df6420d40"]
|
||||
new [description:"Review Yang, Xiaodong application" due:"1765342800" entry:"1765318334" modified:"1765318397" project:"Chair-Search" status:"pending" uuid:"ae720546-20a9-45b5-85fd-699df6420d40"]
|
||||
---
|
||||
time 1765318484
|
||||
new [description:"Tune up presentation for Emerson... digital twins" due:"1765774800" entry:"1765318484" modified:"1765318484" project:"Internship" status:"pending" uuid:"d00350e5-5d5e-40d4-97c1-720f413f8359"]
|
||||
---
|
||||
time 1765318628
|
||||
new [description:"Make a plan for pizza for all-in event" due:"1765515600" entry:"1765318628" modified:"1765318628" project:"Union" status:"pending" uuid:"869e4507-e1c2-4e4b-b2b0-63727285ffc0"]
|
||||
---
|
||||
time 1765318649
|
||||
new [description:"Talk to Jessica about getting a room for an event" due:"1765515600" entry:"1765318649" modified:"1765318649" project:"Union" status:"pending" uuid:"9d6f1ea8-9ce1-4785-be41-fed9b345ba46"]
|
||||
---
|
||||
time 1765318679
|
||||
new [description:"Draft an email for people about MOU for EGSO" due:"1765515600" entry:"1765318679" modified:"1765318679" project:"Union" status:"pending" uuid:"e50fcfa9-85d6-4c48-b7cf-89637f90c333"]
|
||||
---
|
||||
|
||||
BIN
Class_Work/nuce2101/final/latex/SABO_NUCE2101_FINAL.pdf
Normal file
BIN
Class_Work/nuce2101/final/latex/SABO_NUCE2101_FINAL.pdf
Normal file
Binary file not shown.
@ -1,4 +1,4 @@
|
||||
\relax
|
||||
\bibstyle{unsrt}
|
||||
\providecommand \oddpage@label [2]{}
|
||||
\gdef \@abspage@last{5}
|
||||
\gdef \@abspage@last{13}
|
||||
|
||||
@ -1,5 +1,5 @@
|
||||
# Fdb version 4
|
||||
["pdflatex"] 1765217744.32161 "main.tex" "main.pdf" "main" 1765217745.56007 0
|
||||
["pdflatex"] 1765234483.54968 "main.tex" "main.pdf" "main" 1765234484.51921 0
|
||||
"/etc/texmf/web2c/texmf.cnf" 1722610814.59577 475 c0e671620eb5563b2130f56340a5fde8 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc" 1165713224 4850 80dc9bab7f31fb78a000ccfed0e27cab ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
|
||||
@ -24,6 +24,7 @@
|
||||
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||||
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||||
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||||
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||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d ""
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@ -0,0 +1,187 @@
|
||||
\section*{Problem 10}
|
||||
\subsection*{Part A}
|
||||
|
||||
To me, this looks like there was a major steam leak on the secondary side,
|
||||
temporarily increasing steam demand by a lot. I think this was on the secondary
|
||||
side because a primary side loss of coolant would be captured in the pressure
|
||||
data.
|
||||
|
||||
\subsection*{Part B}
|
||||
|
||||
\subsubsection*{Subpart 1}
|
||||
|
||||
I'm going to assume a step change in reactivity, such that \(\dot \rho=0\).
|
||||
|
||||
Because we're considering the first few milliseconds of the transient, we can
|
||||
ignore the contribution of precursors and delayed neutrons. This is an
|
||||
appropriate assumption because the fastest precursor is on the order of 300ms
|
||||
half-life. Much slower than our analysis.
|
||||
|
||||
\[\frac{dn(t)}{dt} = \frac{n(t)(\rho - \beta)}{\Lambda}\]
|
||||
\[\frac{\dot n(t)}{n(t)} = \frac{(\rho - \beta)}{\Lambda}\]
|
||||
\[\frac{781.88}{26.06} = \frac{(\rho - 0.000650)}{0.00026}\]
|
||||
\[\rho = 0.008451\]
|
||||
|
||||
\subsubsection*{Subpart 2}
|
||||
I would assume rise rate decreases so rapidly because fuel temperature would go
|
||||
through the roof and Doppler broadening would add a lot of negative reactivity.
|
||||
There is not enough time for any other physics to happen other than those at the
|
||||
quantum level.
|
||||
|
||||
\subsection*{Part C}
|
||||
|
||||
At near steady state (around 800 seconds on the chart), the reactor power and
|
||||
temperature have stabilized after the initial transient. We can estimate the
|
||||
moderator temperature coefficient using the relationship between reactivity
|
||||
change and temperature change.
|
||||
|
||||
Given:
|
||||
\begin{itemize}
|
||||
\item Initial temperature: \(T_i = 600^\circ\)F
|
||||
\item Final temperature: \(T_f = 680^\circ\)F
|
||||
\item Temperature change: \(\Delta T = 80^\circ\)F
|
||||
\end{itemize}
|
||||
|
||||
At near steady state, the reactor is critical, so the net reactivity is zero.
|
||||
The reactivity balance is:
|
||||
|
||||
\[\rho_{net} = \rho_{inserted} + \alpha_m \Delta T = 0\]
|
||||
|
||||
From Part B.1, we found the inserted reactivity:
|
||||
|
||||
\[\rho_{inserted} = 0.008451 = 845.1 \text{ pcm}\]
|
||||
|
||||
Solving for the moderator temperature coefficient:
|
||||
|
||||
\[\alpha_m = -\frac{\rho_{inserted}}{\Delta T} = -\frac{845.1 \text{ pcm}}{80^\circ\text{F}}\]
|
||||
|
||||
\[\boxed{\alpha_m \approx -10.6 \text{ pcm}/^\circ\text{F}}\]
|
||||
|
||||
This value is reasonable for a PWR moderator temperature coefficient, typically
|
||||
ranging from -10 to -40 pcm/\(^\circ\)F depending on core conditions and boron
|
||||
concentration.
|
||||
\subsection*{Part D}
|
||||
|
||||
Starting with the one delayed group prompt jump approximation:
|
||||
|
||||
\[n(t) = \frac{\lambda_{eff} C(t) \Lambda + S\Lambda}{\beta - \rho}\]
|
||||
|
||||
\[\dot C(t) = \frac{n(t)\beta}{\Lambda} - \lambda_{eff} C(t)\]
|
||||
|
||||
To find \(\dot n(t)\), we take the implicit derivative of the first equation.
|
||||
Since both \(C(t)\) and \(\lambda_{eff}(t)\) can vary with time, we use the
|
||||
product rule:
|
||||
|
||||
\[\dot n(t) = \frac{\Lambda}{\beta - \rho} \frac{d}{dt}[\lambda_{eff} C(t)] - \frac{\Lambda (\lambda_{eff} C + S)}{(\beta - \rho)^2} \dot \rho\]
|
||||
|
||||
\[\dot n(t) = \frac{\Lambda}{\beta - \rho} \left[\dot \lambda_{eff} C(t) + \lambda_{eff} \dot C(t)\right] - \frac{\Lambda (\lambda_{eff} C + S)}{(\beta - \rho)^2} \dot \rho\]
|
||||
|
||||
Substitute \(\dot C(t) = \frac{n(t)\beta}{\Lambda} - \lambda_{eff} C(t)\):
|
||||
|
||||
\[\dot n(t) = \frac{\Lambda}{\beta - \rho} \left[\dot \lambda_{eff} C(t) + \lambda_{eff}\left(\frac{n(t)\beta}{\Lambda} - \lambda_{eff} C(t)\right)\right] - \frac{n(t)(\beta - \rho) + S\Lambda}{(\beta - \rho)^2} \dot \rho\]
|
||||
|
||||
Simplify:
|
||||
|
||||
\[\dot n(t) = \frac{\Lambda \dot \lambda_{eff} C(t)}{\beta - \rho} + \frac{\beta n(t)}{\beta - \rho} - \frac{\Lambda \lambda_{eff}^2 C(t)}{\beta - \rho} - \frac{n(t)\dot \rho}{\beta - \rho} - \frac{S\Lambda \dot \rho}{(\beta - \rho)^2}\]
|
||||
|
||||
From the first equation: \(\lambda_{eff} C(t) \Lambda = n(t)(\beta - \rho) - S\Lambda\), so:
|
||||
|
||||
\[\Lambda C(t) = \frac{n(t)(\beta - \rho) - S\Lambda}{\lambda_{eff}}\]
|
||||
|
||||
After substitution and algebraic manipulation, we get:
|
||||
|
||||
\[\frac{\dot n(t)}{n(t)} = \frac{\dot \rho + \frac{\dot \lambda_{eff}}{\lambda_{eff}}\left[(\beta - \rho) - \frac{S\Lambda}{n(t)}\right] + \lambda_{eff}\left(\rho + \frac{S\Lambda}{n(t)}\right)}{(\beta - \rho) + \frac{S\Lambda}{n(t)}}\]
|
||||
|
||||
Power turning occurs when \(\dot n(t) = 0\):
|
||||
|
||||
\[\dot \rho + \frac{\dot \lambda_{eff}}{\lambda_{eff}}\left[(\beta - \rho) - \frac{S\Lambda}{n(t)}\right] + \lambda_{eff}\left(\rho + \frac{S\Lambda}{n(t)}\right) = 0\]
|
||||
|
||||
The key term \(\frac{\dot \lambda_{eff}}{\lambda_{eff}}\left[(\beta - \rho) - \frac{S\Lambda}{n(t)}\right]\) arises from the implicit derivative of the product \(\lambda_{eff} C(t)\). This accounts for the changing effective decay constant during the transient.
|
||||
|
||||
\subsection*{Part E}
|
||||
|
||||
Values estimated from graphs at t = 17.2s (power peak):
|
||||
|
||||
\begin{itemize}
|
||||
\item \(\Delta T_{ave} \approx 611 - 600 = 11^\circ\)F
|
||||
\item \(\frac{d(\Delta T_{ave})}{dt} \approx \frac{620 - 600}{25 - 5} = 1.0^\circ\)F/s
|
||||
\item \(\lambda_{eff} \approx 0.13\) s\(^{-1}\)
|
||||
\item \(\dot \lambda_{eff} \approx 0.081\) s\(^{-2}\)
|
||||
\item From Part B.1: \(\rho_0 = 0.008451\) (845.1 pcm)
|
||||
\item From Part C: \(\alpha_m \approx -10.6\) pcm/\(^\circ\)F
|
||||
\item \(\beta = 0.0065\) (650 pcm)
|
||||
\end{itemize}
|
||||
|
||||
At power turning (\(\dot n(t) = 0\)):
|
||||
|
||||
\[\dot \rho + \frac{\dot \lambda_{eff}}{\lambda_{eff}}\left[(\beta - \rho) - \frac{S\Lambda}{n(t)}\right] + \lambda_{eff}\left(\rho + \frac{S\Lambda}{n(t)}\right) = 0\]
|
||||
|
||||
Assuming negligible source term \(S \approx 0\) at high power:
|
||||
|
||||
\[\dot \rho + \frac{\dot \lambda_{eff}}{\lambda_{eff}}(\beta - \rho) + \lambda_{eff} \rho = 0\]
|
||||
|
||||
The reactivity includes both moderator and fuel temperature feedback:
|
||||
|
||||
\[\rho(t) = \rho_0 + \alpha_m \Delta T_{ave} + \alpha_f \Delta T_{ave}\]
|
||||
|
||||
\[\dot \rho = (\alpha_m + \alpha_f) \frac{d(\Delta T_{ave})}{dt}\]
|
||||
|
||||
Substituting into the power turning equation:
|
||||
|
||||
\[(\alpha_m + \alpha_f) \frac{d(\Delta T_{ave})}{dt} + \frac{\dot \lambda_{eff}}{\lambda_{eff}}\left[\beta - \left(\rho_0 + \alpha_m \Delta T_{ave} + \alpha_f \Delta T_{ave}\right)\right] + \lambda_{eff} \left(\rho_0 + \alpha_m \Delta T_{ave} + \alpha_f \Delta T_{ave}\right) = 0\]
|
||||
|
||||
Solving numerically for \(\alpha_f\) using the given values:
|
||||
|
||||
\begin{align*}
|
||||
\alpha_m &= -10.6 \times 10^{-5} \text{ per }^\circ\text{F} \\
|
||||
\rho &= 0.008451 + (-10.6 \times 10^{-5})(11) + \alpha_f (11) \\
|
||||
\dot \rho &= (-10.6 \times 10^{-5} + \alpha_f)(1.0)
|
||||
\end{align*}
|
||||
|
||||
Using SymPy to solve:
|
||||
|
||||
\[\boxed{\alpha_f \approx 7.96 \text{ pcm}/^\circ\text{F}}\]
|
||||
|
||||
\textbf{Note:} This positive value is non-physical, as fuel temperature
|
||||
coefficients should always be negative due to Doppler broadening. This suggests
|
||||
the model assumptions may not be valid for this extreme transient.
|
||||
|
||||
\textbf{Physical interpretation:} The only physically reasonable explanation for
|
||||
the continued increase in average coolant temperature even as power is turning
|
||||
is that the fuel temperature is much hotter than the moderator temperature and
|
||||
continues dumping heat into the coolant via conduction. This large temperature
|
||||
gradient between fuel and moderator violates our assumption that both can be
|
||||
approximated by \(\Delta T_{ave}\). The fuel has likely reached extreme
|
||||
temperatures above normal operating
|
||||
conditions while the moderator lags significantly behind. In a real scenario,
|
||||
fuel damage or melting would likely have occurred.
|
||||
|
||||
\subsection*{Part F}
|
||||
|
||||
\textbf{Recommendation: Do NOT restart this plant without extensive inspection
|
||||
and fuel integrity assessment.}
|
||||
|
||||
The analysis reveals several concerning indicators:
|
||||
|
||||
\textbf{Evidence of severe fuel damage:}
|
||||
|
||||
\begin{itemize}
|
||||
\item The non-physical positive fuel temperature coefficient from Part E
|
||||
indicates the normal reactor physics models have broken down
|
||||
|
||||
\item The continued increase in average coolant temperature even as power was
|
||||
turning suggests fuel temperature was far exceeding moderator temperature
|
||||
|
||||
\item The extreme reactivity insertion (845 pcm) combined with the rapid power
|
||||
excursion likely caused fuel temperatures to reach damage thresholds
|
||||
|
||||
\item Power rose extremely rapidly in the first few seconds, indicating
|
||||
prompt-critical-like behavior that would cause severe thermal stress
|
||||
\end{itemize}
|
||||
|
||||
The fuel almost certainly experienced temperatures well
|
||||
beyond design limits. Cladding integrity is highly suspect. Restart should be
|
||||
prohibited until a comprehensive inspection confirms the core is safe to
|
||||
operate, and additional care should be paid to the chemistry of the coolant to
|
||||
look for zirconium or zirconium-irradiated products.
|
||||
|
||||
@ -13,7 +13,7 @@ negative reactivity to the system.
|
||||
+ \frac{\partial \rho_{net}}{\partial Power} dPower
|
||||
\]
|
||||
|
||||
But with ingoring fuel temperature feedback and no boron effects,
|
||||
But with ignoring fuel temperature feedback and no boron effects,
|
||||
|
||||
\[\rho_{net} = -10 [\frac{\text{pcm}}{^\circ F}]dT
|
||||
+ \frac{\partial \rho_{net}}{\partial H} dH
|
||||
|
||||
@ -1,5 +1,47 @@
|
||||
\section*{Problem 5}
|
||||
\subsection*{Part A}
|
||||
|
||||
Xenon-135 is primarily produced in a reactor as a fission product or by the
|
||||
decay of Iodine-135, which is a direct product of uranium fission. Xenon-135
|
||||
decays naturally over time or can absorb neutrons to become other isotopes with
|
||||
a much smaller neutron cross section.
|
||||
|
||||
|
||||
|
||||
\subsection*{Part B}
|
||||
|
||||
Xenon-135:
|
||||
|
||||
The second reactor will have a lower concentration of xenon-135 in the reactor
|
||||
core. Xenon-135 will be generated by the decay of iodine-135, but will also
|
||||
decay naturally to other elements over time as well as being removed through
|
||||
neutron absorption. If the reactors are shut down after a year, the xenon-135 in
|
||||
both will spike with the full power reactor having a larger spike, and it will
|
||||
take longer for the 100\% power reactor to be able to be restarted until the
|
||||
extra xenon-135 has had time to decay away.
|
||||
|
||||
Samarium-149:
|
||||
|
||||
The second reactor will also have a lower concentration of samarium-149 for a
|
||||
similar reason as the xenon-135 case. There will be less fissile products to
|
||||
decay into samarium-149 in the low power reactor. If the reactors are powered
|
||||
off, however, both reactors will develop additional samarium-149 concentrations
|
||||
that will not decay away naturally as samarium-149 is stable. It will take more reactivity
|
||||
to restart the full power reactor from shutdown as the samarium-149
|
||||
concentration will be much higher than the low power reactor.
|
||||
|
||||
\subsection*{Part C}
|
||||
|
||||
Xenon is the more challenging poison to deal with. First, xenon has a much
|
||||
larger cross section and has a more significant effect on reactivity than
|
||||
samarium. Second, xenon-135 is not a stable element, and thus has decay effects
|
||||
when the reactor is shut down. As such, the amount of time since shutdown is an
|
||||
important factor to consider when restarting a reactor with consideration to
|
||||
xenon. A miscalculation on xenon concentration can lead to a too-large insertion
|
||||
of reactivity on start-up, which doesn't exist in the same way with samarium as
|
||||
samarium doesn't decay during shutdown. Finally, xenon is a much faster response
|
||||
dynamically compared to samarium. Xenon-135 can quickly change concentrations
|
||||
in hours, while samarium-149 concentrations can take weeks to change
|
||||
significantly. Human operators are much more likely to be able to cope with the
|
||||
longer time constant poison versus misjudging the small time constant effects of
|
||||
xenon.
|
||||
|
||||
@ -0,0 +1,22 @@
|
||||
\section*{Problem 6}
|
||||
\subsection*{Part A}
|
||||
|
||||
Withdrawing the control rods will move power production towards the upper half
|
||||
of the core. Changing boron concentration will affect all parts of the core the
|
||||
same amount, while increasing to full power does not provide enough information
|
||||
to describe what's happening in the core. That can be accomplished through rod
|
||||
position or through boron concentration.
|
||||
|
||||
Assume rods remove through the top of the core (basically all PWRs), moving the
|
||||
rods out will allow more flux higher in the core and thus move the average
|
||||
production upwards.
|
||||
|
||||
\subsection*{Part B}
|
||||
|
||||
This happens because the moderator is the coolant, and the
|
||||
coolant is colder as it enters the core at the bottom of the core. A low
|
||||
enrichment core using boron does not have as pronounced of an effect because at
|
||||
the same time because the boron is a poison. As boron enters the core, it is
|
||||
more effective compared to as it travels through the core and is 'used up'. This
|
||||
is the opposite effect of the moderator temperature, and as such they balance
|
||||
each other out to a degree.
|
||||
@ -0,0 +1,99 @@
|
||||
\section*{Problem 7}
|
||||
\subsection*{Part A}
|
||||
|
||||
Starting with the point kinetics equations with one delayed precursor group, using
|
||||
the prompt jump approximation where prompt neutrons reach equilibrium instantly:
|
||||
|
||||
\[\frac{dN(t)}{dt} = \lambda C(t)\]
|
||||
|
||||
\[\frac{dC(t)}{dt} = \frac{\beta}{\Lambda}N(t) - \lambda C(t)\]
|
||||
|
||||
Using appropriate numerical values for a PWR with low enrichment U-235 fuel:
|
||||
|
||||
\begin{itemize}
|
||||
\item \(\beta = 0.0065\) (650 pcm)
|
||||
\item \(\Lambda = 5 \times 10^{-5}\) s (50 \(\mu\)s)
|
||||
\item \(\lambda = 0.08\) s\(^{-1}\)
|
||||
\item \(\rho = +0.0005\) (+50 pcm inserted)
|
||||
\end{itemize}
|
||||
|
||||
The complete system of equations describing reactor power as a function of time is:
|
||||
|
||||
\[N(t) = \Lambda \frac{\lambda C(t)}{\beta -\rho} \]
|
||||
|
||||
\[\frac{dC(t)}{dt} = \frac{\beta}{\Lambda}N(t) - \lambda C(t)\]
|
||||
|
||||
Substituting N(t) into the precursor equation:
|
||||
|
||||
\[\frac{dC(t)}{dt} = \frac{\beta \lambda}{\beta -\rho} C(t) - \lambda C(t) = \lambda C(t) \left(\frac{\beta}{\beta - \rho} - 1\right)\]
|
||||
|
||||
Simplifying:
|
||||
|
||||
\[\frac{dC(t)}{dt} = \lambda C(t) \frac{\rho}{\beta - \rho}\]
|
||||
|
||||
With the given numerical values (\(\beta = 0.0065\), \(\rho = 0.0005\), \(\lambda = 0.08\) s\(^{-1}\), \(\Lambda = 5 \times 10^{-5}\) s):
|
||||
|
||||
\[\frac{dC(t)}{dt} = 0.08 \times C(t) \times \frac{0.0005}{0.006} = 6.67 \times 10^{-3} C(t)\]
|
||||
|
||||
\[N(t) = 5 \times 10^{-5} \times \frac{0.08 \times C(t)}{0.006} = 6.67 \times 10^{-4} C(t)\]
|
||||
|
||||
\subsection*{Part B}
|
||||
|
||||
The kinetics parameters chosen are justified as follows:
|
||||
|
||||
\begin{itemize}
|
||||
\item \(\beta = 0.0065\): This is the standard delayed neutron fraction for
|
||||
thermal fission of U-235.
|
||||
|
||||
\item \(\Lambda = 50 \mu\)s: This is the typical prompt neutron generation
|
||||
time for a pressurized water reactor.
|
||||
|
||||
\item \(\lambda = 0.08\) s\(^{-1}\): This effective decay constant represents
|
||||
the lumping of six delayed neutron precursor groups into one equivalent group.
|
||||
This gives an effective half-life of \(t_{1/2} = \ln(2)/\lambda \approx 8.7\)
|
||||
s, which is reasonable as a weighted average of the six precursor groups that
|
||||
range from \(\sim\)0.2 s to \(\sim\)80 s half-lives.
|
||||
|
||||
\end{itemize}
|
||||
|
||||
\subsection*{Part C}
|
||||
|
||||
Power will stop increasing when \(\frac{dN}{dt} = 0\), which occurs when
|
||||
temperature feedback effects add sufficient negative reactivity to cancel the
|
||||
+50 pcm reactivity insertion.
|
||||
|
||||
The reactor reaches a new equilibrium when:
|
||||
|
||||
\[\rho_{net} = \rho_{inserted} + \alpha_f \Delta T_f + \alpha_m \Delta T_m = 0\]
|
||||
|
||||
For \(\rho_{inserted} = +50\) pcm, temperature feedback must provide
|
||||
\(-50\) pcm.
|
||||
|
||||
\subsection*{Part D}
|
||||
|
||||
At end of life (EOL) with +200 pcm reactivity insertion, several key parameters
|
||||
change:
|
||||
|
||||
\begin{itemize}
|
||||
\item \(\beta_{EOL} \approx 0.005\) (500 pcm): At EOL, approximately 40-50\%
|
||||
of fissions come from Pu-239 and Pu-241 built up from neutron capture in
|
||||
U-238. Since Pu-239 has \(\beta \approx 0.0021\) (much lower than U-235's
|
||||
0.0065), the effective \(\beta\) is a weighted average that decreases to
|
||||
around 0.005.
|
||||
\end{itemize}
|
||||
|
||||
The time constant for the transient becomes:
|
||||
|
||||
\[\tau_{EOL} = \frac{\beta - \rho}{\lambda \rho} = \frac{0.0045 - 0.002}{0.08 \times 0.002} = 15.6 \text{ s}\]
|
||||
|
||||
compared to BOL:
|
||||
|
||||
\[\tau_{BOL} = \frac{0.0065 - 0.0005}{0.08 \times 0.0005} = 150 \text{ s}\]
|
||||
|
||||
The EOL transient is much faster (10x shorter period) because \(\beta - \rho\)
|
||||
is smaller relative to \(\rho\).
|
||||
|
||||
The spent core actually experiences a more dramatic transient per unit
|
||||
reactivity insertion, but the stronger feedback from increased fuel temperature
|
||||
effects provides self-limiting behavior that prevents excessive power excursion.
|
||||
Plutonium-240 absorbs a lot of neutrons as 240Pu concentration builds up.
|
||||
@ -0,0 +1,70 @@
|
||||
\section*{Problem 8}
|
||||
|
||||
\subsection*{Part A}
|
||||
|
||||
The one delayed-group model assumes all six precursor groups can be lumped into
|
||||
a single effective group with average parameters. This introduces errors:
|
||||
|
||||
\begin{itemize}
|
||||
\item Each group has different decay constants (\(\lambda_i\)) ranging from
|
||||
0.012 to 3.01 s\(^{-1}\), corresponding to half-lives from 0.2 to 80 seconds
|
||||
|
||||
\item The one-group model cannot capture the multi-timescale behavior - early
|
||||
time dynamics are dominated by fast-decaying groups, late time by slow groups
|
||||
|
||||
\item Effective parameters (\(\beta_{eff}\), \(\lambda_{eff}\)) are only
|
||||
approximate averages that work reasonably for long-term behavior but miss
|
||||
short-term details
|
||||
\end{itemize}
|
||||
|
||||
This weakness matters most for short transients where individual group dynamics
|
||||
are important, less so for long-term steady-state calculations.
|
||||
|
||||
\subsection*{Part B}
|
||||
|
||||
For six delayed groups with prompt jump approximation:
|
||||
|
||||
Power is given by:
|
||||
|
||||
\[N(t) = \frac{\Lambda}{\beta - \rho} \sum_{i=1}^{6} \lambda_i C_i(t)\]
|
||||
|
||||
Precursor concentrations evolve as:
|
||||
|
||||
\[\frac{dC_i(t)}{dt} = \frac{\beta_i}{\Lambda}N(t) - \lambda_i C_i(t), \quad i = 1,\ldots,6\]
|
||||
|
||||
In matrix form, define the state vector:
|
||||
|
||||
\[\mathbf{C}(t) = \begin{bmatrix} C_1(t) \\ C_2(t) \\ C_3(t) \\ C_4(t) \\ C_5(t) \\ C_6(t) \end{bmatrix}\]
|
||||
|
||||
The precursor equation becomes:
|
||||
|
||||
\[\frac{d\mathbf{C}}{dt} = \mathbf{A} \mathbf{C}(t)\]
|
||||
|
||||
where the matrix \(\mathbf{A}\) is:
|
||||
|
||||
{\tiny
|
||||
\[\mathbf{A} = \frac{1}{\beta - \rho} \begin{bmatrix}
|
||||
\beta_1 \lambda_1 - \lambda_1(\beta - \rho) & \beta_1 \lambda_2 & \beta_1 \lambda_3 & \beta_1 \lambda_4 & \beta_1 \lambda_5 & \beta_1 \lambda_6 \\
|
||||
\beta_2 \lambda_1 & \beta_2 \lambda_2 - \lambda_2(\beta - \rho) & \beta_2 \lambda_3 & \beta_2 \lambda_4 & \beta_2 \lambda_5 & \beta_2 \lambda_6 \\
|
||||
\beta_3 \lambda_1 & \beta_3 \lambda_2 & \beta_3 \lambda_3 - \lambda_3(\beta - \rho) & \beta_3 \lambda_4 & \beta_3 \lambda_5 & \beta_3 \lambda_6 \\
|
||||
\beta_4 \lambda_1 & \beta_4 \lambda_2 & \beta_4 \lambda_3 & \beta_4 \lambda_4 - \lambda_4(\beta - \rho) & \beta_4 \lambda_5 & \beta_4 \lambda_6 \\
|
||||
\beta_5 \lambda_1 & \beta_5 \lambda_2 & \beta_5 \lambda_3 & \beta_5 \lambda_4 & \beta_5 \lambda_5 - \lambda_5(\beta - \rho) & \beta_5 \lambda_6 \\
|
||||
\beta_6 \lambda_1 & \beta_6 \lambda_2 & \beta_6 \lambda_3 & \beta_6 \lambda_4 & \beta_6 \lambda_5 & \beta_6 \lambda_6 - \lambda_6(\beta - \rho)
|
||||
\end{bmatrix}\]
|
||||
}
|
||||
|
||||
\subsection*{Part C}
|
||||
|
||||
The prompt jump approximation error is likely smaller than the one-group error.
|
||||
|
||||
The prompt jump assumes prompt neutrons equilibrate instantly (valid when
|
||||
\(\Lambda \ll\) timescales of interest). For a 50 \(\mu\)s generation time and
|
||||
transients on the scale of seconds, this is excellent.
|
||||
|
||||
The one-group approximation loses the multi-timescale structure of the six
|
||||
groups, which significantly affects transient shape, especially in the first
|
||||
10-20 seconds where fast groups dominate.
|
||||
|
||||
For this problem (low reactivity, second-scale transient), prompt jump
|
||||
introduces \(<1\%\) error while one-group can introduce 10-20\% errors in peak
|
||||
timing and shape.
|
||||
@ -0,0 +1,23 @@
|
||||
\section*{Problem 9}
|
||||
\subsection*{Part A}
|
||||
|
||||
The boron concentration must be less than 1200ppm.
|
||||
|
||||
The plant startup within 10 hours of full power shutdown must have a higher
|
||||
xenon-135 concentration than a plant that has been shut down for 25 days. As a
|
||||
result, boron must be diluted to compensate for the xenon poisoning effect.
|
||||
|
||||
\subsection*{Part B}
|
||||
|
||||
At first boron must be concentrated to reduce power. At this point, xenon will
|
||||
increase in the core as xenon burnup reduces. To compensate with this additional
|
||||
poison as power falls, boron must be diluted to maintain 50\% power. Then, the
|
||||
xenon concentration will fall off as it decays (over days), so boron will need to be
|
||||
concentrated as the xenon negative reactivity diminishes and settles at 50\%
|
||||
power. The boron concentration will be HIGHER at the end of the transient than
|
||||
at the start. An example of the xenon concentration can be found on page 11 of
|
||||
'Xenon Transient Information Rev 2024-2'.
|
||||
|
||||
Samarium on the other hand will not have a significant effect as the amount of
|
||||
time spent on maintenance will be too short for significant samarium effects to
|
||||
build up.
|
||||
54
Class_Work/nuce2101/final/python/part_e_solver.py
Normal file
54
Class_Work/nuce2101/final/python/part_e_solver.py
Normal file
@ -0,0 +1,54 @@
|
||||
import sympy as sp
|
||||
|
||||
# Define symbolic variable
|
||||
alpha_f = sp.Symbol('alpha_f', real=True)
|
||||
|
||||
# Given values from graphs at t = 17.2s
|
||||
Delta_T_ave = 11.0 # degrees F
|
||||
d_Delta_T_dt = 1.0 # degrees F/s
|
||||
lambda_eff = 0.13 # s^-1
|
||||
dot_lambda_eff = 0.081 # s^-2
|
||||
|
||||
# Known parameters
|
||||
rho_0 = 0.008451 # initial reactivity (845.1 pcm)
|
||||
alpha_m = -10.6e-5 # in absolute units (-10.6 pcm/F = -10.6e-5 per F)
|
||||
beta = 0.0065 # delayed neutron fraction
|
||||
|
||||
print(f"alpha_m = {alpha_m} per degF = {alpha_m*1e5} pcm/degF")
|
||||
print()
|
||||
|
||||
# Reactivity at t = 17.2s (including BOTH moderator and fuel feedback)
|
||||
rho = rho_0 + alpha_m * Delta_T_ave + alpha_f * Delta_T_ave
|
||||
|
||||
# Rate of change of reactivity
|
||||
dot_rho = (alpha_m + alpha_f) * d_Delta_T_dt
|
||||
|
||||
# Power turning condition (S ≈ 0 at high power):
|
||||
# dot_rho + (dot_lambda_eff / lambda_eff) * (beta - rho) + lambda_eff * rho = 0
|
||||
|
||||
power_turning_eq = dot_rho + (dot_lambda_eff / lambda_eff) * (beta - rho) + lambda_eff * rho
|
||||
|
||||
# Solve for alpha_f
|
||||
solution = sp.solve(power_turning_eq, alpha_f)
|
||||
|
||||
print("Power turning equation:")
|
||||
print(f"dot_rho + (dot_lambda_eff/lambda_eff)*(beta - rho) + lambda_eff*rho = 0")
|
||||
print()
|
||||
print("Solving for alpha_f...")
|
||||
print()
|
||||
|
||||
for sol in solution:
|
||||
alpha_f_value = float(sol)
|
||||
alpha_f_pcm = alpha_f_value * 1e5 # Convert to pcm/degF
|
||||
print(f"alpha_f = {alpha_f_value:.6e} per degF")
|
||||
print(f"alpha_f = {alpha_f_pcm:.2f} pcm/degF")
|
||||
print()
|
||||
|
||||
# Verify the solution
|
||||
rho_check = rho_0 + alpha_m * Delta_T_ave + alpha_f_value * Delta_T_ave
|
||||
dot_rho_check = (alpha_m + alpha_f_value) * d_Delta_T_dt
|
||||
lhs = dot_rho_check + (dot_lambda_eff/lambda_eff)*(beta - rho_check) + lambda_eff*rho_check
|
||||
|
||||
print(f"Verification:")
|
||||
print(f"rho at t=17.2s = {rho_check:.6f} ({rho_check*1e5:.1f} pcm)")
|
||||
print(f"Power turning equation LHS = {lhs:.6e} (should be ~0)")
|
||||
BIN
Writing/CV/SABO_CV_OCT25.pdf
Normal file
BIN
Writing/CV/SABO_CV_OCT25.pdf
Normal file
Binary file not shown.
Loading…
x
Reference in New Issue
Block a user