1. Nuclear Reactor TheoryIntermezzo William D’haeseleer William D’haeseleer BNEN – NRT 2011-2012

2. 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] William D’haeseleer BNEN – NRT 2011-2012

3. 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

4. Nuclear Reactor TheorySimple Geometry:Slab Reactor Consider steady state: critical system slab geometry; thickness a a William D’haeseleer BNEN – NRT 2011-2012

5. 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

6. 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

7. Nuclear Reactor TheorySimple Geometry:Slab Reactor Customary to designate At any time in this case: “Geometric Bucling” William D’haeseleer BNEN – NRT 2011-2012

8. 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

9. Nuclear Reactor TheoryOther simple reactor shapes Sphere and Φ finite everywhere Solution: William D’haeseleer BNEN – NRT 2011-2012

10. 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

11. 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

12. Bessel Functions— Very elementary considerations — William D’haeseleer 2007-2008 William D’haeseleer BNEN – NRT 2011-2012

13. Intermezzo Bessel Functions William D’haeseleer BNEN – NRT 2011-2012

14. Intermezzo Bessel Functions Solution of in rectangular coordinates Take r as 1-D variable: William D’haeseleer BNEN – NRT 2011-2012

15. Intermezzo Bessel Functions • Series expansion of Cosine and Sine functions William D’haeseleer BNEN – NRT 2011-2012

16. 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

17. Intermezzo Bessel Functions • Series expansions of zero-th order Bessel Functions William D’haeseleer BNEN – NRT 2011-2012

18. Intermezzo Bessel Functions J0 behaves like a damped cosine William D’haeseleer BNEN – NRT 2011-2012

19. Intermezzo Bessel Functions William D’haeseleer BNEN – NRT 2011-2012

20. 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

21. Intermezzo Bessel Functions Solution of in rectangular coordinates William D’haeseleer BNEN – NRT 2011-2012

22. 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

23. Intermezzo Bessel Functions Modified Bessel Functions of 0-th order William D’haeseleer BNEN – NRT 2011-2012

24. 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

25. Nuclear Reactor TheoryOther simple reactor shapes Sphere / Infinite cylinder / Parallelepiped / Finite cylinder William D’haeseleer BNEN – NRT 2011-2012

26. Nuclear Reactor TheoryOther simple reactor shapes William D’haeseleer BNEN – NRT 2011-2012 Ref: Lamarsh NRT

27. Nuclear Reactor TheoryOther simple reactor shapes William D’haeseleer BNEN – NRT 2011-2012 Ref: Lamarsh NRT

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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!

39. 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

40. 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