Chemical Engineering 693R

Transcription

1 Chemical Engineering 693 eactor Design and Analysis Lecture 8 Neutron ransport

2 Spiritual hought Moroni 7:48 Wherefore, my beloved brethren, pray unto the Father with all the energy of heart, that ye may be filled with this love, which he hath bestowed upon all who are true followers of his Son, Jesus Christ; that ye may become the sons of God; that when he shall appear we shall be like him, for we shall see him as he is; that we may have this hope; that we may be purified even as he is pure.

3 Neutron Balance 3 In reality, σ i σ i (E), ϕ ϕ(e) But for simplicity, assume one group

4 One-group eactor Equation Mono-energetic neutrons (Neutron Balance) DD φφ Σ aa φφ + ss 1 vv vv is neutron speed For reactor, ss ννσ ff φφ νν is neutrons/fission In eigenfunction form and at steady state DD φφ Σ aa φφ + νν kk Σ ffφφ 0 φφ Σ aa νν kk Σ f DD φφ φφ + BB φφ 0

5 Material Buckling φφ BB φφ φφ + BB φφ 0 νν BB kk Σ ff Σ aa DD kk ννσ ff φφ DDBB φφ + Σ aa φφ eigenfunction form of reactor equation one-group, steady-state, reactor equation material buckling (neutron generation absorption)/diffusion ννσ ff DDBB + Σ aa reactor multiplication factor multiplication factor neutron generation rate/(leakage + absorption)

6 Fuel utilization and k ss ηησ aaaa φφ ηη Σ aaaa Σ aa Σ aa φφ ηηηησ aa φφ ff Σ aaaa Σ aa kk ηηηησ aaφφ Σ aa φφ ff fuel utilization factor neutrons absorbed by fuel / (those absorbed by fuel + by other means coolant, moderator, etc.) ηηηη kk k-value for infinite (no overall leakage) reactor a material property η fission neutrons generated per absorbed neutron

7 Operating Critical eactor Equation kk 1 DDBB φφ Σ aa φφ + kk kk Σ aaφφ 1 vv DDBB φφ + kk 1 Σ aa φφ 0 eactor operating at steady state BB φφ + kk 1 Σ aa DD φφ BB φφ + kk 1 LL φφ 0 LL DD Σ aa One-group diffusion area BB kk 1 LL One-group buckling

8 Perspective Previous equations show how to solve for neutron flux profile φφ as a function of space How to determine critical reactor dimensions BB kk 11 LL kk 11 ΣΣ aa, for a bare reactor (B DD gb mat) First find solutions to the reactor equations 1D, D, or 3D hen find dimensions for a critical reactor Assumptions: Bare, homogeneous reactors Constant (special and temporal) properties None are valid but, but help to develop insight into reactor operations Because source terms are proportional to the flux, the generally inhomogeneous differential equations are now homogeneous equations.

9 Bare Slab eactor Solution dd φφ ddxx BB φφ reactor equation φφ aa φφ aa φφ 0 0 boundary conditions φφ xx AA cos BBBB + CC sin BBBB CC 0 φφ xx AA cos BBBB φφ aa/ 0 BB nn nnnn aa dd φφ general solution from symmetry or by substitution Eigenvalues from boundary conditions all n important in transient solution, only n1 important for steady solution BB 1 is buckling (prop. to flux curvature) BB 1 1 φφ ddxx he constant A is as yet undetermined and relates to the power. here are different solutions to this problem for every power level. aa aa Infinite plane indicates no net flux from sides

10 PP EE Σ ff Bare Slab eactor Power aa/ aa/ PP AA aaee Σ ff ππ φφ xx φφ xx dddd sin ππππ aa ππππ aaee Σ ff sin ππππ aa cos ππππ aa EE is the recoverable energy per fission Power Scales with flux! aa aa Infinite plane indicates no net flux from sides

11 Absorber/emitter vs eactor 1 LL φφ dd φφ ddxx ss DD dd φφ ddxx BB φφ transport in an absorber/emitter transport in a reactor source proportional to flux φφ xx ss 1 cosh xx LL aa + dd Σ aa cosh LL ππππ φφ xx aaee Σ ff sin ππππ cos ππππ aa aa flux in an absorber/emitter flux in a reactor

12 1 dd rr dddd rr dddd dddd Spherical eactor BB φφ φφ φφ 0 0 φφ rr AA sin(bbbb) + C cos(bbbb) rr rr CC 0 sin BBBB φφ rr AA rr BB nn nnnn reactor transport equation boundary conditions from symmetry or by substitution Eigen values specific solution general solution BB 1 ππ buckling φφ rr AA sin ππrr rr

13 Spherical eactor Power Integrate over symmetric dimensions transform volume integral to radial integral PP EE Σ ff φφ rr dddd 4ππEE Σ ff PP 4ππEE Σ ff AA ππ ππ sin ππππ 0 cos ππππ rr φφ rr ddrr again, power is proportional to flux and highest at center φφ rr PP sin ππππ 4 EE Σ ff rr

14 Infinite Cylindrical eactor 1 dd rr dddd dddd rr dddd BB φφ dd φφ ddrr + 1 rr dddd dddd reactor transport equation φφ φφ 0 0; φφ rr < boundary conditions dd φφ ddrr + 1 rr dddd dddd + BB mm rr φφ 0 φφ rr AAJJ 0 BBBB + CCYY 0 BBBB φφ rr AAJJ 0 BBBB BB nn xx nn BB 1 xx 1 φφ rr AAJJ rr zero-order (m0) Bessel equation general solution involves Bessel functions of first and second kind flux is finite roots of Bessel functions - φφ is zero at boundary first root solution (power production determines A)

15 Bessel Functions J0 J1 J Y0 Y1 Y Bessel Function x

16 Infinite Cylindrical eactor Power PP EE Σ ff φφ rr dddd ππee Σ ff.405rr PP ππee Σ ff rrjj 0 0 xxxjj 0 xx ddxxx xxjj 1 xx 0 PP ππee Σ ff AAJJ φφ rr 0.738PP EE Σ ff JJ.405rr 0 dddd 0 rrrr rr dddd transform volume integral to radial integral becomes power per unit length again, power is proportional to power and highest at center

17 1 rr Finite Cylindrical eactor + φφ zz BB φφ reactor transport equation φφ, zz φφ 0, zz φφ rr, HH φφ rr, HH 0 φφ rr, zz rr ZZ(zz) 1 rr 1 + φφ zz ZZ rr + 1 ZZ ZZ BB zz separation of variables boundary conditions + ZZ zz BB rr ZZ(zz) since and ZZ vary independently, both portions of the equation must equal (generally different) constants, designated as BB and BB ZZ, respectively HH HH

18 1 rr BB rr.405 rr AA JJ 0 ZZ zz BB ZZ ZZ Finite Cylinder Solution rr a problem we already solved, w/ same bcs again a problem we already solved, w/ same bcs ZZ zz AA cos ππππ HH φφ rr, zz AA JJ BB BB + BB HH rr cos ππππ HH solution is the product of the infinite cylinder and infinite slab solutions Buckling is higher than for either the infinite plane or the infinite cylinder. Buckling generally increases with increasing leakage, and there are more surfaces to leak here than either of the infinite cases. HH HH

19 Neutron Flux Contours Neutron flux in finite cylindrical reactor 3D contours of neutron flux at high power 3D contours w color scaled to magnitude intermediate power 3D contours of neutron flux at low power

20 Neutron Flux Contours Neutron flux in finite parallelepiped (cubical) reactor 3D contours of neutron flux at high power 3D contours w color scaled to magnitude intermediate power 3D contours of neutron flux at low power

21 Critical Buckling kk ννσ ff Σ aa + DDBB value of k for critical reactor BB ννσ ff Σ aa DD BB cc ννσ ff Σ aa DD BB ll kk 1 LL value of B when k 1 critical material buckling geometric buckling ννσ ff Σ aa DD kk 1 LL geometric and material buckling must be equal for a critical reactor

22 Critical Equation (One group) kk 1 + BB LL 1

23 Physical Interpretation k 1+ B k Σ φdv a a L Σa + DB Σa φdv + DB φdv k Σ k P L he probability of non-leakage is inversely proportional to k k P k L a k Σ a Σ φdv φdv P L ηfp L geometric and material buckling must be equal for a critical reactor

24 Σ a f hermal eactors Four Factor Formula Σ Σ Σ af a af + Σ Σ am af ΣaF + Σ total cross section sum of fuel and moderator am thermal fuel utilization factor η η ( E) σ ( E) φ( E) σ af af ( E) φ( E) de de neutrons emitted/thermal neutron absorbed in the fuel k pεη fσ Σ φ a a φ pεη f infinite multiplication factor proportional to probability of escaping resonance absorption, fast fission contribution,

25 Criticality Calculations ( 1 ) ( + B L 1+ B τ ) 1/ prob thermal neutron1/ prob fast neutron doesn' t leak doesn' t leak while slowing L τ k D Σ D Σ P P 1 1 F a k 1 ( )( 1 1 ) 1 ( + B L + B τ + B L + τ ) k k k wo-group equation for a bare (thermal) reactor Modified one-group critical equation k 1+ B M 1

26 Bare eactor Summary / cos 3.64 / 3.63 cos.405 J / J / 3.85 cos cos cos / 1.57 cos f f f f f E P r r A sphere VE P H z A H z D cylinder E P A D cylinder VE P c z b y a x A c b a D plate ae P a x A a D plate Σ Σ + Σ Σ + + Σ π π π π π π π π π π π geometry Buckling (B ) Flux A φ av Ω φ max

27 eflected eactors r L r A r L r C r L r A r Br A r Br C r Br A L L k B B r r r r r c c r r r c c c exp ' exp ' exp ' sin cos sin φ φ φ φ φ φ φ φ core transport equation core materials properties reflector transport equation general solution for core flux must be finite at the center general solution for the reflector flux must be finite as r increases

28 φc J c D A D ( ) φr ( ) ( ) n Jr ( ) n φ c ( ) Drφ r ( ) sinb exp( / L) c AD c c A B cosb B cot B 1 Dr B cot B 1 D 1 B cot B L r sinb D c eflected eactors r L r A' D 1 Lr + 1 r 1 + fluxes equal at core-reflector interface current densities also equal equate fluxes and current densities 1 L r 1 + exp L r divide current density by flux equation critical equation for reflected reactor (transcendental equation) critical equation when D r D c (not transcendental in )

29 eflected eactors < Bare eactors B cot B D r Dc Lr

30 Determine emaining Unknown ( ) ( ) B B B E PB A B B B B A E dr Br r A E P dv E P B L A A f f f c f cos sin 4 cos sin 4 sin 4 sin exp ' 0 0 Σ Σ Σ Σ π π π φ

31 Some details eflected reactors lend themselves less easily to analytical solution commonly reactors are considered as sphere equivalents rather than trying to solve the equations. easonable representation for fast neutrons not for thermal reactors eflector savings in size is typically about the thickness of the extrapolated distance.

32 Flux Comparisons

33 hermal Flux Variations

34 wo-energy, detailed model

35 Position 1 (center of rod) fast thermal

36 Position (outside rod at :30/7:30) fast thermal

37 In a Vacuum fast thermal

38 Azimuthal Dependence fast thermal

39 General ransient Problem Mono-energetic neutrons DD φφ Σ aa φφ + SS 1 vv For a reactor, SS ννσ ff φφ DD φφ Σ aa φφ + ννσ ff φφ 1 vv φφ BB φφ 1 DDvv

40 Power Plant Operation 40 ry to achieve steady state operation at all times Set up core and fuel to provide minimal changes Despite this, still have three scales of change 1. Short erm (Grid load transients). Intermediate erm (Fuel composition changes) 3. Long erm (Burnup & Depletion)

41 hree ime Scales (short) Short ime Constant (load change) An abrupt change in steam demand/load. A load change is seen as: eactor pressure change in the BW eactor temperature change in a PW. Higher loads lead to higher pressures/temperatures. Assumptions Shape of the flux profile is assumed constant Power changes Magnitude of the neutron flux scales everywhere in the reactor. Assumes a uniform multiplicative change everywhere Spatial variations in time in the reactor are not considered his method is called point kinetics.

42 hree ime Scales (intermediate) Intermediate ime Constant (core composition change) Changing fission product concentrations Generation rates and destruction/decay rates. Many fission products have measurable thermal neutron cross sections Change the value of k (and k ). ff ΣΣ FF aa + ΣΣ NNFF aa VV NNNN /VV FF Point kinetics can be used if spatial variations in concentrations are negligible. Otherwise, detailed spatial and temporal equations must be used! ΣΣ aa FF φφ NNNN /φφ FF

43 hree ime Scales (long) Long ime Constant (Fuel Depletion) reated as a series of steady-state problems wo things are adjusted to maintain λ 1 (i.e. λ 1/k) Material buckling eactor dimensions If λ 1, the equation is not valid Why? DD φφ Σ aa φφ λλλλσ ff φφ In operating reactor, cannot change dimensions (much) k is adjusted slowly in time by changing chemical shim conc. Chemical shim is an isotope that absorbs neutrons in the reactor k is also adjusted via the control rods. Managing fuel consumption is a classical example of this type of transient

44 Prompt Neutrons Lifetime, ll pp, is time between emission and absorption. Neutrons in thermal reactors: Spend more time (most of ll pp ) in the thermal regime ravel further as fast neutrons Average lifetime of a thermal neutron in a an infinite reactor is the mean diffusion time, tt dd, and is approximately the same as ll pp in an infinite reactor. Assuming 1/vv behavior (cross section) tt EE λλ aa EE vv(ee) 1 Σ AA EE vv EE 1 Σ AA EE 0 vv 0 ππ tt dd tt EE Σ aa vv ll pp tt dd

45 In fast reactors, prompt neutron lifetimes are much shorter, on the order of 10-7 seconds Prompt Neutrons For mixtures of fuel and moderator in thermal reactors tt dd tt(ee) ππ Σ aaaa + Σ aaaa vv ππ Σ aaaa vv ππ Σ aaaa νν Σ aaaa Σ aaaa + Σ aaaa 1 ff Σ aaaa Σ aaaa + Σ aaaa moderator diffusion time tt dd tt dddd 1 ff fffuel utilization factor

46 Simple Kinetics Model dddd tt Δnn tt l dddd dddd tt dddd kk eeeeee 1 l kk eeeeee 1 nn tt nn(tt) nn tt nn 0 exp kk eeeeee 1 l For 35 U l.1x10-4 s kk eeeeee and tt 1ss, n/n (,07 if l 10-4 as in text) Far too rapid to control!!! tt

47 Delayed Neutrons For 1-group model, 1 for 35 U is about 8.87 s and ττ is about 1.8 s.

48 Delayed Neutron Fractions

49 eactivity and Worth ρρ kk eeeeee 1 kk eeeeee δδδδ kk eeeeee reactivity ρρ and δδδδ kk($) ρρ ββ ββ is delayed neutron fraction worth can be measured in units of kk($) or kk cents? Percent Mil?

50 Power Changes 50 ββββ δδδδ ββββ kkkkkkkk ρρ ττ kkkkkkkk kk($) ~ ττ kk($) eactor Period (units of time) ime required to increase reactor power (or neutron flux) by.7 tt llll PP(tt) PP(0)

51 Example 1 51 Following a reactor scram in which all the control rods are inserted into a power reactor, how long is it before the reactor power decreases to of the steady-state power prior to shutdown? (Assume a reactor period of -80 s)

52 eactors with delayed neutrons l pp 1 ββ l pp + ββ l pp + ττ l pp + ββββ ττ is lifetime of delayed neutrons 1/ ln For δδδδ ββ nn tt nn 0 exp kk eeeeee 1 l pp exp tt For 35 U, 83 s, k eff , n/n his can be controlled! 1.8 ss l pp kk dddddd 1 ββββ δδδδ

53 ransient thermal neutron equation ( ) ( ) concentration precursor C prob escape resonance p precursor of const decay C p k s C p s k s dt d l dt d t s nv s dt d v dt dn nv dt dn s a delayed a prompt p d a a a + Σ Σ Σ Σ Σ. 1 1,, λ λ φ β λ φ β φ φ φ π φ φ π π φ φ

54 ransient thermal neutron equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ω λ ω λβ β ω ω φ ω λ ω β λ ω β ω ω φ ω λ φ β φ φ φ λ φ β p p a a decay precursor production precursor a p d a l k k t A l t C p t A k p A k C t C C t A t unknown yet as with Assume C p k dt dc dt d l dt d t C p k + + Σ + + Σ Σ Σ exp exp exp 1 exp exp ) ( λ ω β ω ω ω ω ρ p p p l l l k k k k 1 ρ eactivity definition (does not assume infinite reactor)

55 eactivity and Δk ρ k ( $ ) keff k eff ρ β 1 k( $ ) worth δk k eff ρ reactivity δk delta k β delayed neutron kk oo fraction ll ββ + ii1 GG aa ii λ ii + ωω Inhour equation

56 φ φ ρ A 1 1+ ωl eactivity Equation Solutions 1 1 dominant term as t 1 ω exp exp ωl p t p ( ω t ) + A exp( ω t ) ω + 1+ ωl approaches 0 rapdily p 6 i 1 βi ω + λ i General solution for single group of delayed neutrons Definition of reactor or stable period General solution for single group of delayed neutrons eactivity equation for six group model graphical solution on next page

57 Six-group solution Period decreases with increasing reactivity. ate of decrease increases with increasing reactivity and with decreasing prompt neutron lifetime, eventually decreasing at nearly infinite rate for short prompt neutron lifetimes. As reactivity approaches delayed neutron fraction (worth approaches $1), period becomes very short. Fast reactor behavior corresponds to prompt neutron lifetime of near zero (10-7 s) eactor worth represents fractional approach to this line, i.e., $1 at the line.

58 1-level Model Parameters ββ ττ dd (ss) 1,dd(ss) N/A h U U U Pu Pu Am Am Cm Source: Laboratoire de Physique Subatomique et de Cosmologie