diff --git a/Class_Work/nuce2101/final/latex/main.aux b/Class_Work/nuce2101/final/latex/main.aux index c31370d0f..d29c1e2a6 100644 --- a/Class_Work/nuce2101/final/latex/main.aux +++ b/Class_Work/nuce2101/final/latex/main.aux @@ -1,4 +1,4 @@ \relax \bibstyle{unsrt} \providecommand \oddpage@label [2]{} -\gdef \@abspage@last{3} +\gdef \@abspage@last{5} diff --git a/Class_Work/nuce2101/final/latex/main.fdb_latexmk b/Class_Work/nuce2101/final/latex/main.fdb_latexmk index ac60923c9..2e94b83b9 100644 --- a/Class_Work/nuce2101/final/latex/main.fdb_latexmk +++ b/Class_Work/nuce2101/final/latex/main.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 4 -["pdflatex"] 1765211608.02982 "main.tex" "main.pdf" "main" 1765211608.80257 2 +["pdflatex"] 1765217744.32161 "main.tex" "main.pdf" "main" 1765217745.56007 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 @@ "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr12.tfm" 1136768653 1288 655e228510b4c2a1abe905c368440826 "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy10.tfm" 1136768653 1124 6c73e740cf17375f03eec0ee63599741 "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs10.tfm" 1229303445 688 37338d6ab346c2f1466b29e195316aa4 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d "" "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 "" "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d "" "/usr/share/texlive/texmf-dist/fonts/type1/urw/symbol/usyr.pfb" 1136849748 33709 b09d2e140b7e807d3a97058263ab6693 "" @@ -222,14 +223,14 @@ "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1722610820.43889 128028 f533b797fba58d231669ea19e894e23e "" "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1726005817 6800784 2b63e5a224c5ad740802d8f9921962c1 "" "dane_proposal_format.cls" 1762791843.06705 2292 c3dc1b9d4d9635e69847824a1055af84 "" - 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(/usr/share/texlive/texmf-dist/tex/latex/psnfss/ts1ptm.fd File: ts1ptm.fd 2001/06/04 font definitions for TS1/ptm. -) [1] -Runaway argument? - \dot N(t) = \frac {(\rho - \beta ) N(t)}{\Lambda } + \sum _{i=1}^6 \lambda \ETC. -./problem1.tex:58: Paragraph ended before \@newline was complete. - - \par -l.58 - -I suspect you've forgotten a `}', causing me to apply this -control sequence to too much text. How can we recover? -My plan is to forget the whole thing and hope for the best. - -) (./problem2.tex) (./problem3.tex) (./problem4.tex) (./problem5.tex) (./problem6.tex) (./problem7.tex) (./problem8.tex) (./problem9.tex) (./problem10.tex) [2] (./main.aux) +) [1]) (./problem2.tex) (./problem3.tex [2]) (./problem4.tex) (./problem5.tex) (./problem6.tex) (./problem7.tex) (./problem8.tex) (./problem9.tex) (./problem10.tex) [3] [4] (./main.aux) *********** LaTeX2e <2023-11-01> patch level 1 L3 programming layer <2024-01-22> *********** ) Here is how much of TeX's memory you used: - 25417 strings out of 476182 - 527426 string characters out of 5795595 - 1932975 words of memory out of 5000000 - 46838 multiletter control sequences out of 15000+600000 - 595073 words of font info for 115 fonts, out of 8000000 for 9000 + 25425 strings out of 476182 + 527486 string characters out of 5795595 + 1933975 words of memory out of 5000000 + 46840 multiletter control sequences out of 15000+600000 + 598339 words of font info for 123 fonts, out of 8000000 for 9000 14 hyphenation exceptions out of 8191 110i,6n,107p,1008b,285s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on main.pdf (3 pages, 67404 bytes). + +Output written on main.pdf (5 pages, 84426 bytes). PDF statistics: - 77 PDF objects out of 1000 (max. 8388607) - 41 compressed objects within 1 object stream + 88 PDF objects out of 1000 (max. 8388607) + 48 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 109 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/Class_Work/nuce2101/final/latex/main.pdf b/Class_Work/nuce2101/final/latex/main.pdf index 2bd15bd27..dcb47a4bd 100644 Binary files a/Class_Work/nuce2101/final/latex/main.pdf and b/Class_Work/nuce2101/final/latex/main.pdf differ diff --git a/Class_Work/nuce2101/final/latex/main.synctex.gz b/Class_Work/nuce2101/final/latex/main.synctex.gz index e43687cb5..280d0757c 100644 Binary files a/Class_Work/nuce2101/final/latex/main.synctex.gz and b/Class_Work/nuce2101/final/latex/main.synctex.gz differ diff --git a/Class_Work/nuce2101/final/latex/problem1.tex b/Class_Work/nuce2101/final/latex/problem1.tex index d6cbd5d7a..0c82605fc 100644 --- a/Class_Work/nuce2101/final/latex/problem1.tex +++ b/Class_Work/nuce2101/final/latex/problem1.tex @@ -32,8 +32,7 @@ Now, how are these related? Well, the total number of neutrons is split between the neutrons that are prompt neutrons and those that are destined to become delayed neutrons. We can represent the fraction between the two as: -\[ N_t = (1-\beta)K_{eff} N_{in} + \sum_{i=1}^6 \lambda_i C_i \Delta t + S \Delta -t \] +\[ N_t = (1-\beta)K_{eff} N_{in} + \sum_{i=1}^6 \lambda_i C_i \Delta t + S \Delta t \] where @@ -43,18 +42,15 @@ and Then we find the change in neutron population: -\[ N_f - N_i = (1-\beta) K_{eff} N_i + \sum_{i=1}^6 \lambda_i C_i \Delta t + S -\Delta t - N_i \] +\[ N_f - N_i = (1-\beta) K_{eff} N_i + \sum_{i=1}^6 \lambda_i C_i \Delta t + S \Delta t - N_i \] and take the 'derivative': -\[ \frac{N_f - N_i}{\Delta t} = \frac{(1- \frac{1}{K_eff} - \beta) K_{eff} N_i}{\Delta t} + \sum_{i=1}^6 \lambda_i C_i + S -\] +\[ \frac{N_f - N_i}{\Delta t} = \frac{(1- \frac{1}{K_eff} - \beta) K_{eff} N_i}{\Delta t} + \sum_{i=1}^6 \lambda_i C_i + S \] then after a little more substitution found in Fundamental Kinetics Ideas: -\\[ \dot N(t) = \frac{(\rho - \beta) N(t)}{\Lambda} + \sum_{i=1}^6 \lambda_i C_i + S -\] +\[ \dot N(t) = \frac{(\rho - \beta) N(t)}{\Lambda} + \sum_{i=1}^6 \lambda_i C_i + S \] and not forgetting our precursors: @@ -80,5 +76,4 @@ precursors catch up to the reactivity increase. \subsection*{Part E} Power turning has to do with start-up rate and the rate at which the total -neutron population is changing. For this diagram, turning means that \(\dot -N_f\) changes sign. +neutron population is changing. For this diagram, turning means that \(\dot N_f\) changes sign. diff --git a/Class_Work/nuce2101/final/latex/problem2.tex b/Class_Work/nuce2101/final/latex/problem2.tex index e69de29bb..b35f36216 100644 --- a/Class_Work/nuce2101/final/latex/problem2.tex +++ b/Class_Work/nuce2101/final/latex/problem2.tex @@ -0,0 +1,20 @@ +\section*{Problem 2} + +We can use the startup rate equation assuming \(\dot \lambda_{eff}, S = 0\) to +solve this problem: + +\[SUR = 26.06 [dpm-sec] \frac{\dot \rho + \lambda_{eff} \rho}{\beta - \rho}\] + +To get from \(10^{-6}\%\) to \(10^{1}\%\) power in 50 minutes, we can find that: + +\[SUR = \frac{1-(-6)}{50} \frac{\text{decades}}{\text{minutes}} = 0.14 +\text{DPM}\] + +We then plug in our values (assume \(\dot \rho = 0\) with a step change): + +\[0.14 = 26.06 [dpm-sec] \frac{0 + 0.1 \rho}{\beta - \rho}\] +\[0.14 \beta - 0.14 \rho= 2.606 \rho\] + +\[\boxed{\rho = 0.0509\beta}\] + +Thus the correct answer is C. diff --git a/Class_Work/nuce2101/final/latex/problem3.tex b/Class_Work/nuce2101/final/latex/problem3.tex index e69de29bb..c9de8b7c4 100644 --- a/Class_Work/nuce2101/final/latex/problem3.tex +++ b/Class_Work/nuce2101/final/latex/problem3.tex @@ -0,0 +1,42 @@ +\section*{Problem 3} +\subsection*{Part A} + +The moderator temperature coefficient must be controlling power. With all other +factors constant, the moderator temperature coefficient is the only thing adding +negative reactivity to the system. + +\subsection*{Part B} + +\[\rho_{net} = \frac{\partial \rho_{net}}{\partial T}dT ++ \frac{\partial \rho_{net}}{\partial H} dH ++ \frac{\partial \rho_{net}}{\partial Poison} dPoison ++ \frac{\partial \rho_{net}}{\partial Power} dPower +\] + +But with ingoring fuel temperature feedback and no boron effects, + +\[\rho_{net} = -10 [\frac{\text{pcm}}{^\circ F}]dT ++ \frac{\partial \rho_{net}}{\partial H} dH +\] + + +Given that there is no poison or fuel temperature feedback, and steam demand +does not change, reactor power will stay the same after the control rod drops +into the core. Only moderator temperature can change reactivity in this problem. + +\subsection*{Part C} + +\[0 = -10 [\frac{\text{pcm}}{^\circ F}]dT + - 100[pcm] +\] + +\[dT = \frac{100[pcm]}{-10[\frac{\text{pcm}}{^\circ F}]}\] + +\[dT = -10^\circ F\] +\[\boxed{T_{final} = 577^\circ F}\] + +\subsection*{Part D} + +Power will remain the same, and therefore steam pressure should remain the same +as well. + diff --git a/Class_Work/nuce2101/final/latex/problem4.tex b/Class_Work/nuce2101/final/latex/problem4.tex index e69de29bb..edfc769ae 100644 --- a/Class_Work/nuce2101/final/latex/problem4.tex +++ b/Class_Work/nuce2101/final/latex/problem4.tex @@ -0,0 +1,18 @@ +\section*{Problem 4} + +The power trajectory would be exponentially positive as the reactor would become +prompt critical. One would analyze the transient by using a robot to examine the +reactor soup after the steam bomb goes off in the containment. + +But being serious, one may examine the power transient by evaluating \(\rho\) +over time using the partial addition formula we used in the last problem. +Because the reactor is prompt critical, we can essentially ignore the delayed +neutrons. The point kinetic equations can also be used, but honestly a decent +approximation will be a first order exponential growth with time constant +derived from the prompt neutron lifetime. + +For a high enrichment fuel, the growth of the curve will be impeded by basically +nothing. Fuel and moderator temperature effects will be minimal. For a low +enrichment fuel, moderator temperature and fuel temperature effects will slow +the exponential growth as temperature increases, but depending on reactor +design, will not prevent catastrophic failure. diff --git a/Class_Work/nuce2101/final/latex/problem5.tex b/Class_Work/nuce2101/final/latex/problem5.tex index e69de29bb..3fe384276 100644 --- a/Class_Work/nuce2101/final/latex/problem5.tex +++ b/Class_Work/nuce2101/final/latex/problem5.tex @@ -0,0 +1,5 @@ +\section*{Problem 5} +\subsection*{Part A} + +\subsection*{Part B} +\subsection*{Part C}