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Nuclear Reactor Theory Intermezzo. William D’haeseleer. Nuclear Reactor Theory. Intermezzo: One-Speed Diffusion Theory of a Nuclear Reactor See Duderstadt & Hamilton: § 5.III A D § 5. IV [to a large extent to be studied independently].

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## Nuclear Reactor Theory Intermezzo

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**Nuclear Reactor TheoryIntermezzo**William D’haeseleer William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor Theory**Intermezzo: One-Speed Diffusion Theory of a Nuclear Reactor See Duderstadt & Hamilton: § 5.III AD § 5. IV [to a large extent to be studied independently] William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor Theory1. One speed (mono-energetic) reactor**equation Consider mono-energetic neutrons only → time-dependent neutron diffusion equation Consider s as the neutron source due to fission William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheorySimple Geometry:Slab Reactor**Consider steady state: critical system slab geometry; thickness a a William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheorySimple Geometry:Slab Reactor**Solution: Because of symmetry: C = 0 a Second b.c.: Non-trivial solution: n: odd integer William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheorySimple Geometry:Slab Reactor**For a critical reactor, only 1-st “harmonic” remains: a Since: “buckling” For “a” very large, B1 very small, almost no buckle William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheorySimple Geometry:Slab Reactor**Customary to designate At any time in this case: “Geometric Bucling” William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheorySimple Geometry:Slab Reactor**Boundary conditions specify eigenvalues; constant A remains undetermined. Must be found from overall power considerations a William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheoryOther simple reactor shapes**Sphere and Φ finite everywhere Solution: William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheoryOther simple reactor shapes**Infinite cylinder and Φ finite everywhere Zero’s of J0; J0(xn)=0 Solution: William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheoryOther simple reactor shapes**• Bessel functions J0, Y0 • Intermezzo on Bessel functions / following slides –see also text “Bessel functions and their relatives” (GVdB & WDH) William D’haeseleer BNEN – NRT 2011-2012**Bessel Functions— Very elementary considerations —**William D’haeseleer 2007-2008 William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**Solution of in rectangular coordinates Take r as 1-D variable: William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**• Series expansion of Cosine and Sine functions William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**Solution of in cylindrical coordinates Take r as 1-D variable: (Note: error in pdf document) J0 and Y0 are Bessel functions of 0-th order William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**• Series expansions of zero-th order Bessel Functions William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**J0 behaves like a damped cosine William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**Asymptotic approximations (x sufficiently large) For x large, each behaves as “damped” cos and sin by sqrt(x) William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**Solution of in rectangular coordinates William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**Solution in cylindrical coordinates Take r as 1-D variable: (Note: error in pdf document) I0 and K0 are Modified Bessel functions of 0-th order William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**Modified Bessel Functions of 0-th order William D’haeseleer BNEN – NRT 2011-2012**Intermezzo Bessel Functions**Asymptotic approximations (x sufficiently large) For x large, each behaves as “reduced” exponentials or hyperbolic functions by sqrt (x) William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheoryOther simple reactor shapes**Sphere / Infinite cylinder / Parallelepiped / Finite cylinder William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheoryOther simple reactor shapes**William D’haeseleer BNEN – NRT 2011-2012 Ref: Lamarsh NRT**Nuclear Reactor TheoryOther simple reactor shapes**William D’haeseleer BNEN – NRT 2011-2012 Ref: Lamarsh NRT**Nuclear Reactor Theory1. One speed (mono-energetic) reactor**equation Evolution of flux dependent upon • geometry (diffusion; leakage) • material composition (absorption; fission) → For arbitrary geometry & composition, difficult to find Bg ; also in general: Steady state only arises when reactor is critical; i.e., when k = 1 William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor Theory1. One speed (mono-energetic) reactor**equation In general, k ≠ 1; therefore apply a “trick” to find k = f (dimensions, material composition) then set k = 1 relationship between dimensions & material composition for criticality ! William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor Theory1. One speed (mono-energetic) reactor**equation ≡ B² called“Material Buckling” Bm2 -- for k = 1 -- At this stage, k still unknown, since B² is not known William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor Theory1. One speed (mono-energetic) reactor**equation Note (1): • suppose one writes → one could introduce f = fuel utilization factor William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor Theory1. One speed (mono-energetic) reactor**equation Note (2):Consider an infinite reactor in such a reactor, Φ independent of position (uniform throughout) reduces to William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor Theory1. One speed (mono-energetic) reactor**equation Note (3):Consider now general case (finitereactor) • Take now as a definition: Then, source term → time-(in)dependent diffusion equation can be written as: Time independent Time dependent William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor Theory1. One speed (mono-energetic) reactor**equation Note (4):still finite reactor • Time independent case: Or for a critical reactor, k = 1 William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheoryMono-energetic critical equation**Reactor is critical if Material Buckling Geometric Buckling; B12=Bg2 for simple slab “critical equation” William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheoryMono-energetic critical equation**“critical equation” determines requirement to have a critical reactor: - material composition - dimensions etc. determines requirement to have a critical reactor: - material composition - dimensions etc. Alternative expression of critical equation: William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheoryMono-energetic critical equation**Physical meaning of critical equation Consider a bare reactor of arbitrary geometry. # of neutrons leaking out of the system # of neutrons absorbed William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheoryMono-energetic critical equation**Physical meaning of critical equation Non-leakage probability*: From: William D’haeseleer BNEN – NRT 2011-2012 *Note: symbols PL and PNL are used interchangeably and mean the same!**Nuclear Reactor TheoryMono-energetic critical equation**Physical meaning of critical equation Interpretation of Physical meaning of critical equation Interpretation of # of neutrons absorbed Gives rise to a release of fission neutrons: William D’haeseleer BNEN – NRT 2011-2012**Nuclear Reactor TheoryMono-energetic critical equation**Physical meaning of critical equation Physical meaning of critical equation Of these neutrons, only a fraction PL does not leak out, and gives rise to absorption in the next generation: Hence: For a critical reactor: William D’haeseleer BNEN – NRT 2011-2012

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