HTR Reactor Physics. Slowing down and thermalization of neutrons. Jan Leen Kloosterman Delft University of Technology

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HTR Reactor Physics Slowing down and thermalization of neutrons Jan Leen Kloosterman Delft University of Technology J.L.Kloosterman@tudelft.

1 HTR Reactor Physics Slowing down and thermalization of neutrons Jan Leen Kloosterman Delft University of Technology Reactor Institute Delft / TU-Delft Research on nergy and Health with Radiation 1

2 Neutrons Spin-cho Small Angle Neutron Scattering (SSANS)

3 Positrons POSH-Strongest positron beam in the world Research Themes (1) nergy solar cells } batteries Materials hydrogen storage nuclear energy research 3

4 Research Themes () Health radiation and radioactive nuclides for therapy and diagnostics radiation detection systems for imaging new production routes for radionuclides new radionuclides for new applications OYSTR Power upgrade from to 3 MW Higher Density Fuel Installation of Cold Neutron Source Installation of New Instruments and Facilities 4

5 HTR Reactor Physics Pebble-bed fuel 5

Helium Temperature ( o C) 3 9 nrichment (%) 5 1 Burnup

6 Prismatic fuel Differences between LWR and HTR LWR HTR Fuel UO pin UO sphere Moderator Water Graphite Coolant Water Helium Temperature ( o C) 3 9 nrichment (%) 5 1 Burnup (MWd/kgU) 6 1 Specific power (kw/kgu) 4 8 Power density (kw/l) 1 6 6

7 Contents of this lecture Slowing down of neutrons in graphite moderated reactors Resonance shielding and cell weighting procedures in double heterogeneous geometries Implications of high temperatures on the thermal spectrum Moderation of neutrons Fission cross section (barn) U-35 Pu-39 U Fission spectrum nergy (ev) 7

Moderation of neutrons 4 1-6 f a 3.5 3 Reproduction factor.5 U-35 1-7 Fission spectrum 1.5 1 1-1 1 1 4 1 6.

collisions lastic scattering most important Conservation of energy and momentum Large energy transfer in collisions at light

8 Moderation of neutrons f a Reproduction factor.5 U Fission spectrum nergy (ev) U-35 Ferziger&Zweifel, Theory of neutron slowing down in nuclear reactors, 1966 nergy transfer in collisions lastic scattering most important Conservation of energy and momentum Large energy transfer in collisions at light nuclei Hydrogen same mass as a neutron* largest -transfer M mass nucleus A1 A m mass neutron A 1 *a neutron is.1% heavier. Think over the consequences. 8

9 nergy transfer in collisions p' ' 1 p s ' p ' s 1 for ' elsewhere Area=1 nergy transfer in collisions Average energy: 1 ' ' p ' d' 1 Average energy loss: 1 ' 1 9

10 nergy transfer in collisions A 1 xample: Hydrogen A Average energy: ' 1 Number of collisions to slow down a neutron from H L H =1 Mev to =1 ev in a hydrogeneous medium: L log 1 log n n1 collisions? Two collisions with energy loss of 5% One collisions with energy loss of 8% and one with % nergy vs Lethargy Transform nergy to a new variable that changes linearly in each collision Lethargy H u log Average lethargy gain per collision: H H 1 u log d 1 H 1 log large A 1 A 3 H 1

11 Number of collisions Now, the number of collisions to increase the average lethargy to un corresponding with energy becomes: H log n Numerically almost the same to the number of collisions needed to slow down a neutron from to H Lamarsh, Introduction to Nuclear Reactor Theory, 1965 Number of collisions Number of collisions to slow down a neutron from H =1 Mev to =1 ev in various media: L log H / n lement A n H D Be C U

12 lastic scatter cross section lastic scatter cross section (cm -1 ) water Mean free path.5 cm.5 graphite nergy (ev) Moderator power and ratio s large A good moderator has large a small Moderating power Moderator ratio s a s measure of energy transfer Moderator Power Ratio H O D O Be C

13 Space-dependent slowing down Moderating power nergy transfer to the moderator per unit path length lethargy gain by the neutron: du dx s s Monte Carlo game: Start particles at isotropic plane source Follow the particles from interval to interval In each interval, certain probability to scatter When scattering, particles can reverse direction When scattering, particles gain lethargy Space-dependent slowing down.3.5 u=4 u=6 u=8 Lethargy distribution Interval 13

14 Space-dependent slowing down Fermi-age model Age-diffusion equation Continuous slowing down model Takes the form of time-dependent diffusion eq. without absorption q r, q r, r r where q, u s u, u (slowing down density) and u is the Fermi age: 1 r 6 is mean squared distance a source neutron travels until it reaches Space-dependent slowing down Plane source Non-absorbing slab Duderstadt&Hamilton, Nuclear Reactor Analysis,

15 Space-dependent slowing down Recall from diffusion theory assumed to be known L is diffusion length 1 L r 6 L is mean squared distance a thermal neutron travels until absorption death birth L Space-dependent slowing down 1 1 Fermi age r Diffusion length L r 6 6 Migration area M L Migration length M is 1/ 6 of the rms distance a neutron travels between birth as a fission neutron and absorption in thermal range Duderstadt&Hamilton, Nuclear Reactor Analysis,

16 Resonance absorption Neutron flux depression in resonance Neutron flux spectrum in the fuel lump often calculated by collision probability method Resonance integral: I d F Flux depression in a resonance. 16

17 Neutron balance in two regions F VFt F / F F s ' F ' VF 1 PFM d' Fuel region 1 F ' / M M s ' M ' VMPMF d' 1 M ' M VMt M / F F s Moderator region VP F FM d' / M VM 1 PMF d' ' F ' 1 F ' M s ' M ' 1 ' M P P FM MF First-flight escape probabilities is probability that a neutron originating in the fuel will make its next collision in the moderator is probability that a neutron originating in the moderator will make its next collision in the fuel Both P and P are usually approximated by the first-flight FM MF escape probabilities assuming a flat source distribution. In particular: P P FM esc 17

18 Two-region slowing-down equations Three approximations in the slowing down equations: Flat source approx for P and P reciprocity theorem FM Narrow resonance approximation in the moderator No absorption in the moderator (1/ flux) Stacey, Nuclear Reactor Physics, MF Only one equation; only P ' ' 1 ' / F F F F s F PFM t t F 1 PFM d' Several approximations possible for F FM F needed ' like NR, NRIM, etc First-flight escape probability 1.9 Small lump: P 1 esc P esc Wigner rational approx: P Large lump: P. Cylinder Sphere.1 Plate F R T esc F SF /4VFt 1 S /4V 1 l F F F t SF 4V esc F F t F t R R 4R S C P F F F t t t Case et al, Introduction to the theory of neutron diffusion 18

19 lastic scatter cross section lastic scatter cross section (cm -1 ) water Mean free path.5 cm.5 graphite nergy (ev) Rod shadowing and Dancoff factors If a neutron can easily interact with neighbouring fuel lumps, the first flight escape probability is not a good estimate for P Better: P FM P esc PM 1PM 1PF PM 1PM 1PF 1PM 1 PF PM... FM P P esc M 1 1 P P P P 1 esc M F M 19

Rod shadowing and Dancoff factors i i Assuming: PM PM and PF PF this series converges to: PM 1 C PFM Pesc Pesc 1 1 1 1 1 PM PF C PF The Dancoff factor C 1 PM is the probability that a neutron emitted

20 Rod shadowing and Dancoff factors i i Assuming: PM PM and PF PF this series converges to: PM 1 C PFM Pesc Pesc PM PF C PF The Dancoff factor C 1 PM is the probability that a neutron emitted isotropically from a fuel lump, will enter a neighbouring fuel lump without interaction in between. (Bell and Glasstone, Nuclear Reactor Theory, 197) Dancoff factors for a double heterogeneous fuel design C Intra is the probability that a neutron leaving a fuel kernel will enter another kernel in the same pebble without collision with graphite C Inter is the probability that the neutron will enter a fuel kernel in another pebble without collision in between

21 * Total Dancoff factor * 11 II P * esc R T fk fk fk FZ C C 1 Pesc R1 C C 1T T T Intra Inter is probability per unit path that a neutron will collide with a moderator nuclide or will enter a fuel kernel. IO II OI Note that if the fuel pebble contains no moderator zone then: T T 1, C =1 and C C IO OI FZ fk fk Bende et al, Nucl Sci ng, 113: (1999)!=======================================! Program dancoff - calculate Dancoff factor for pebble bed HTR!! Reactor Institute Delft, Mekelweg 15, Delft! mail: J.L.Kloosterman@tudelft.nl! Website: Phone: !======================================= program dancoff parameter (pi= ) real intra,inter,infdan!-----read the radius of the fuel grain (cm) (usually.5 cm) read(5,*) rg1!-----read the number of fuel grains per pebble read(5,*) grn!-----read the radius of fuel zone of pebble (usually.5 cm) read(5,*) rp1!-----read the radius of graphite moderator shell (usually 3 cm) read(5,*) rp!-----sigma of graphite is.497 1/cm sigma =.497 rg = rp1 / (grn**(1./3.))!-----calculate dancoff factor dan1 = intra(rg1,rg,rp1,rp,sigma) dan = inter(rg1,rg,rp1,rp,sigma) cinf = grain(rg1,rg,sigma) write(6,11) dan1,dan,cinf,dan1+dan!========================================= 11 format(' Intra Dancoff factor ',f8.4,/, & ' Inter Dancoff factor ',f8.4,/, & ' Infinite med Dancoff ',f8.4,/, & ' Total Dancoff factor ', f8.4)!========================================= end real function grain(r1,r,sigma)!-----calculates the inf medium fuel grain Dancoff factor!-----q. A.1 in thesis vert Bende call trans(r1,r,sigma,tio,toi,too) grain = tio*toi/(1.-too) return end real function intra(rg1,rg,rp1,rp,sigma)!-----calculates the intra-pebble Dancoff factor!-----q. A.4 in thesis vert Bende call trans(rg1,rg,sigma,tio,toi,too) star = -alog(too)/chord(rg) intra = grain(rg1,rg,sigma)*(1.-pf(rp1*star)) VHTR urocourse, return Prague, May 1-4, 13 end real function inter(rg1,rg,rp1,rp,sigma)!-----calculates the inter-pebble Dancoff factor!-----q. A.6 and A.8 in thesis vert Bende call trans(rg1,rg,sigma,tio,toi,too) star = -alog(too)/chord(rg) prob = pf(rp1*star)!-----q. A.6 in thesis vert Bende tii = 1.-(4./3.)*rp1*star*prob fac = (1.-tii)/(1.-tii*tio*toi) inter = grain(rg1,rg,sigma)*grain(rp1,rp,sigma)*prob*fac return end subroutine trans(r1,r,sigma,tio,toi,too)!-----calculates transmission probabilities in a sphere!-----qs. A.3, A.5, A.6, and A.7 in thesis Bende nint = 1 r1 = r1** r = r** dy = r1/real(nint) tio =. do i=1,nint y = (i-.5)*dy y = y** u = sqrt(r-y)-sqrt(r1-y) tio = tio + y*exp(-sigma*u)*dy enddo tio =.*tio toi = tio/r tio = tio/r1 a = sigma*sqrt(r-r1) too = (1.-(1.+.*a)*exp(-.*a)) / (.*r*(sigma**)) return end Download from (click on reports) real function chord(r)!-----calculates mean chord length of a sphere!-----under q. A11 thesis Bende chord = 4.*r/3. return end real function pf(rlam)!-----calculates the first flight escape probability for a sphere!-----q. A.9 in thesis vert Bende twor =.*rlam pf =.75*(1.-(1.-(1.+twor)*exp(-twor))/(twor*rlam))/rlam return end 1

Dancoff factor results pebble-bed.8.7.6 5 m.5 C total.4.3..1 1 m.5 1 1.5.5 3 3.5 4 4.

22 Dancoff factor results pebble-bed m.5 C total m Number of grains x 1 4 Dancoff factor results prismatic R Sphere 3 R Cyl Average chord length for a convex body always halfwidth a radius R radius R Plate Cylinder Sphere 4 a Rc Rs 3 Kruijf&Kloosterman, Annals Nucl ner, 3: , 3 c s 4V S

23 Dancoff factor results prismatic Talamo, Annals Nucl ner, 34:68-8, 7 Doppler temperature effect 3

barn 1 5 5 Doppler temperature effect Resonance 1 barn Capture probability in broadened

24 Doppler temperature effect Due to the vibration of the nucleus, the effective resonance broadens. The area remains virtually constant. barn Doppler temperature effect Resonance 1 barn Capture probability in broadened resonance is Broadened resonance 5 barn Scatter 5 barn Capture probability in resonance is cap tot cap tot 4

25 ( T ) Doppler temperature effect ( ) ref T ref d Doppler coefficient: dt TT ref T Thermal scattering kernel 1 1*kT Hydrogen Thermal scattering kernel P( ') *kT kt '/ Lamarsh, Introduction to Nuclear Reactor Theory,

26 Thermal scattering kernel 1 4*kT Carbon Thermal scattering kernel P( ') *kT kt '/ Massimo, Physics of High Temperature Reactors, 1976 Thermal neutron spectrum Thermal region characterized by upscattering of neutrons Under the assumptions of no absorption and no sources: m ' ' d' s Result is a Maxwellian neutron number density: n s o exp 3/ and corresponding neutron flux density: M M kt kt 1/ n exp 3/ kt m kt 6

27 M Thermal neutron spectrum n 3/ Average neutron energy: 3 M d kt M 1/ n exp 3/ Most probable energy: M kt kt kt exp kt Corresponding velocity: v T m kt kt m Thermal Maxwell spectrum M () kt=.5 ev K nergy (ev) m/s for T=93 K 5 kt Thermal neutron spectrum K Thermal Maxwell spectrum M () K 9 K 1 K nergy (ev) 7

28 Duderstadt&Hamilton, Nuclear Reactor Analysis, 1976 Moderator temperature effect Pu-41 U-35 Thermal Maxwell spectrum M () K Pu K Reproduction factor a f nergy (ev) 8

29 Moderator temperature effect Moderator temperature effect T T T k Fuel Shell Reflec pcm/k Fuel Shell Pebble Reflector

30 Double cell-weighting procedure Single cell-weighting procedure 3

31 Cell weighting procedure k as a function of C/U for 1% and % enriched fuel k uniform fuel % % 1% k double heterogeneous 1% k C over U relation Core design Pressure vessel Top reflector Outer reflector Pebble bed Core barrel Inner reflector Bottom reflector Defuel shute Outlet pipe Barrel support Unloading syst. 31

32 Neutron flux density 15.4 Neutron spectrum reflector. 1 5 Neutron spectrum fuel zone nergy (ev) Concluding remarks Neutrons slowing down in HTRs need much more collisions than in LWRs; distance traveled is longer. Due to large epithermal neutron flux and homogeneously distributed fuel, resonance absorption is important => use less uranium with high enrichment and high specific power. Resonance shielding calculations need special doubleheterogeneous Dancoff factors. Physically, HTR cores are very large, but neutronically they are not. Moderator temperature reactivity effect mainly due to shift of Maxwell spectrum and non-1/v absorbers. 3

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