Equation of Continuity

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Presentation transcript:

Equation of Continuity Net flow of neutrons per second per unit area normal to the x direction: In general: Equation of Continuity Rate of change in number of neutrons Production rate Absorption rate “Leakage in/out” rate Normal to A (outwards) Source distribution function Surface area bounding  Volume Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Fundamental equation in Reactor Theory Equation of Continuity Using Gauss’ Divergence Theorem Recall: Both flux and current!! Convert current to flux?  Fundamental equation in Reactor Theory Equation of Continuity Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Equation of Continuity Steady state Non-spacial dependence Delayed sources? Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law The exact interpretation of neutron transport in heterogeneous domains is so complex. Assumptions and approximations. Simplified approaches. Simplified but accurate enough to give an estimate of the average characteristics of neutron population. Numerical solutions. Monte Carlo techniques. MCNP Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law Assumptions: The medium is infinite. The medium is uniform There are no neutron sources in the medium. Scattering is isotropic in the lab coordinate system. The neutron flux is a slowly varying function of position. The neutron flux is not a function of time. Restrictive! Applicability?? http://www.iop.org/EJ/article/0143-0807/26/5/023/ejp5_5_023.pdf Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law Lamarsh puts it more bluntly: “Fick’s Law is invalid: a) in a medium that strongly absorbs neutrons; b) within three mean free paths of either a neutron source or the surface of a material; and c) when neutron scattering is strongly anisotropic.” Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law Diffusion: random walk of an ensemble of particles from region of high “concentration” to region of small “concentration”. Flow is proportional to the negative gradient of the “concentration”. Recall: From larger flux to smaller flux! Neutrons are not pushed! More scattering in one direction than in the other. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law Number of neutrons scattered per second from d at r and going through dAz z d  r dAz Removed en route (assuming no buildup) y  Slowly varying x Isotropic Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Diffusion coefficient Fick’s Law HW 14 and show that and generalize ? Fick’s law Diffusion coefficient Total removal The current density is proportional to the negative of the gradient of the neutron flux. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law Validity: 1. The medium is infinite. Integration over all space.  after few mean free paths  0  corrections at the surface are still required. 2. The medium is uniform.   and  are functions of space  re-derivation of Fick’s law?  locally larger s  extra J cancelled by iff ??? Note: assumption 5 is also violated! 3. There are no neutron sources in the medium. Again, sources are few mean free paths away and corrections otherwise. HW 15 Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law 4. Scattering is isotropic in the lab. coordinate system. If  reevaluate D. For “practical” moderators: 5. The flux is a slowly varying function of position. a   variation in  . HW 16 Isotropic tr = t. Weekly absorbing t = s. Recall: ? Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law HW 17 Estimate the diffusion coefficient of graphite at 1 eV. The scattering cross section of carbon at 1 eV is 4.8 b. Scattering Other materials? Absorption Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). Fick’s Law 6. The neutron flux is not a function of time. Time needed for a thermal neutron to traverse 3 mean free paths  1 x 10-5 s (How?). If flux changes by 10% per second! Very small fractional change during the time needed for the neutron to travel this “significant” distance. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Back to the Continuity Equation  Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

The Diffusion Equation The Steady State Diffusion Equation If D is independent of r (uniform medium) Laplacian The Diffusion Equation The Steady State Diffusion Equation or scalar Helmholtz equation. Non-multiplying medium Buckling equation. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Non-multiplying medium Steady State Diffusion Equation Define L  Diffusion Length L2  Diffusion Area Moderation Length  Non-multiplying medium Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

The Diffusion Equation Repeated!! The exact interpretation of neutron transport in heterogeneous domains is so complex. Assumptions and approximations. Simplified approaches. Simplified but accurate enough to give an estimate of the average characteristics of neutron population. Numerical solutions. Monte Carlo techniques. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Steady State Diffusion Equation Boundary Conditions Solve DE  get . Solution must satisfy BC’s. Solution should be real and non-negative. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Steady State Diffusion Equation One-speed neutron diffusion in infinite medium Point source  HW 18 General solution A, C determined from BC’s. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Steady State Diffusion Equation HW 18 (continued) BC r      0  C = 0. Show that  neutrons per second absorbed in the ring. Show that dr r Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Scalar flux, vector current. Steady State Diffusion Equation Scalar flux, vector current. HW 19 Study example 5.3 and solve problem 5.8 in Lamarsh. Multiple Point Sources? Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).