Chemical Engineering 693R

Transcription

1 Chemical Engineering 693R Reactor Design and Analysis Lecture 9 Neutron Kinetics

2 Siritual hought 2

3 General ransient Problem Mono-energetic neutrons DD 2 φφ Σ aa φφ + SS vv For a reactor, SS ννσ ff φφ DD 2 φφ Σ aa φφ + ννσ ff φφ vv 2 φφ BB 2 φφ DDvv

4 hree ime Scales (short) Short ime Constant (load change) An abrut change in steam demand/load. A load change is seen as: Reactor ressure change in the BWR Reactor temerature change in a PWR. Higher loads lead to higher ressures/temeratures. Assumtions Shae of the flux rofile is assumed constant Power changes Magnitude of the neutron flux scales everywhere in the reactor. Assumes a uniform multilicative change everywhere Satial variations in time in the reactor are not considered his method is called oint kinetics.

5 hree ime Scales (intermediate) Intermediate ime Constant (core comosition change) Changing fission roduct concentrations Generation rates and destruction/decay rates. Many fission roducts 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 satial variations in concentrations are negligible. Otherwise, detailed satial and temoral equations must be used! ΣΣ aa FF φφ NNNN /φφ FF

6 hree ime Scales (long) Long ime Constant (Fuel Deletion) reated as a series of steady-state roblems wo things are adjusted to maintain λ (i.e. λ /k) Material buckling Reactor dimensions If λ, the equation is not valid Why? DD 2 φφ Σ aa φφ λλλλσ ff φφ In oerating reactor, cannot change dimensions (much) k is adjusted slowly in time by changing chemical shim conc. Chemical shim is an isotoe that absorbs neutrons in the reactor k is also adjusted via the control rods. Managing fuel consumtion is a classical examle of this tye of transient

7 Promt Neutrons Lifetime, ll, is time between emission and absortion. Neutrons in thermal reactors: Send more time (most of ll ) 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 aroximately the same as ll in an infinite reactor. Assuming /vv behavior (cross section) tt EE λλ aa EE vv(ee) Σ AA EE vv EE Σ AA EE 0 vv 0 ππ tt dd tt EE 2 Σ aa vv ll tt dd

8 In fast reactors, romt neutron lifetimes are much shorter, on the order of 0-7 seconds Promt Neutrons For mixtures of fuel and moderator in thermal reactors tt dd tt(ee) ππ 2 Σ aaaa + Σ aaaa vv ππ 2 Σ aaaa vv ππ 2 Σ aaaa νν Σ aaaa Σ aaaa + Σ aaaa ff Σ aaaa Σ aaaa + Σ aaaa moderator diffusion time tt dd tt dddd ff fffuel utilization factor

9 Simle Kinetics Model dddd tt Δnn tt l dddd dddd tt dddd kk eeeeee l kk eeeeee nn tt nn(tt) nn tt nn 0 ex kk eeeeee l For 235 U l 2.x0-4 s kk eeeeee 0.00 and tt ss, n/n 0 7 (22,027 if l 0-4 as in text) Far too raid to control!!! tt

10 Delayed Neutrons For -grou model, 2 for 235 U is about 8.87 s and ττ is about 2.8 s.

11 Delayed Neutron Fractions

12 Reactivity and Worth ρρ kk eeeeee 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?

13 Power Changes 3 ββββ δδδδ ββββ kkkkkkkk ρρ ττ kkkkkkkk kk($) ~ ττ kk($) Reactor Period (units of time) ime required to increase reactor ower (or neutron flux) by 2.72 tt llll PP(tt) PP(0)

14 Examle 4 Following a reactor scram in which all the control rods are inserted into a ower reactor, how long is it before the reactor ower decreases to of the steady-state ower rior to shutdown? (Assume a reactor eriod of -80 s)

15 Reactors with delayed neutrons l ββ l + ββ l + ττ l + ββββ ττ is lifetime of delayed neutrons /2 ln 2 For δδδδ ββ nn tt ex kk eeeeee ex tt nn 0 l For 235 U, 83 s, k eff , n/n 0.02 his can be controlled! 2.8 ss l kk dddddd ββββ δδδδ

16 ransient thermal neutron equation ( ) ( ) concentration recursor C rob escae resonance recursor of const decay C k s C s k s dt d l dt d t s nv s dt d v dt dn nv dt dn s a delayed a romt d a a a + Σ Σ Σ Σ Σ ,, λ λ φ β λ φ β φ φ φ π φ φ π π φ φ

17 ransient thermal neutron equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ λβ β φ λ β λ β φ λ φ β φ φ φ λ φ β a a decay recursor roduction recursor a d a l k k t A l t C t A k A k C t C C t A t unknown yet as with Assume C k dt dc dt d l dt d t C k + + Σ + + Σ Σ Σ + ex ex ex ex ex ) ( λ β ρ l l l k k k k ρ Reactivity definition (does not assume infinite reactor)

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

19 φ φ ρ A + l Reactivity Equation Solutions dominant term as t ex ex l t ( t ) + A ex( t ) + + l 2 2 aroaches 0 radily 6 i βi + λ i General solution for single grou of delayed neutrons Definition of reactor or stable eriod General solution for single grou of delayed neutrons Reactivity equation for six grou model grahical solution on next age

20 Six-grou solution Period decreases with increasing reactivity. Rate of decrease increases with increasing reactivity and with decreasing romt neutron lifetime, eventually decreasing at nearly infinite rate for short romt neutron lifetimes. As reactivity aroaches delayed neutron fraction (worth aroaches $), eriod becomes very short. Fast reactor behavior corresonds to romt neutron lifetime of near zero (0-7 s) Reactor worth reresents fractional aroach to this line, i.e., $ at the line.

21 -level Model Parameters ββ ττ dd (ss) 2,dd(ss) N/A h U U U Pu Pu Am Am Cm Source: Laboratoire de Physique Subatomique et de Cosmologie

22 Reactivity Equation Solutions ll kk eeeeee ll δδδδ ββββ δδδδ Reactor eriod - he time required for a neutron oulation to change by a factor of e kk eeeeee + δδδδ + ββββ ττ Lifetime of delayed neutrons ~2.8s (U235) ββββ δδδδ ββββ kk eeeeee ββββ kk eeeeee ρρ ττ kk eeeeee ρρ($) ττ ρρ $ ϕ(tt) eeeeee tt Remember, Flux is roortional to ower. CC PP(tt) eeeeee tt eeeeee PP(tt) CC PP(0) CC eeeeee PP(tt) PP(0)

23 Exloration 23 What if we add -$0. to AP000 core? 3. 4 P.i ( i s) GW 3 P( j s) GW 2 P.f ( k s) GW i, j, k t.e

24 Exloration 2 24 What if we add $0. to AP000 core? P.i ( i s) GW P( j s) GW 0 P.f ( k s) GW i, j, k t.e

25 Exloration 3 25 What if we add $0. to AP000 core, then after 0 seconds we add -$0.? P.i ( i s) GW 3 P( j s) GW 2 P.f ( k s) GW i, j, k t.e

26 Reality 26

27 Kinetics 27 his is how reactor ower is controlled Control rods add/subtract worth he circumstances we ve seen so far are not a ideal, however. Why? herefore a moderating influence is desired Feedback Mechanisms!

28 Isotoic Feedbacks (slow) Fuel Burnu (slow) Decrease in reactivity Fuel breeding (slow) Increase in reactivity Fission roduct oisons (moderate hours) 35 Xe and 49 Sm Decrease reactivity until decay away Burnable Poisons (slow) Decrease reactivity until transmuted away

29 emerature Feedbacks (fast) Atomic concentration changes Moderator coolant density Void coefficient fuel exansion Neutron energy distribution changes harden sectrum with increased RIGA reactor is extreme examle Resonance interaction changes Doler dominant feedback Burnable Poisons Geometry changes

30 Feedback Effects 30 What if we add $0. to AP000 core with void feedbacks included? P.i ( i) GW P( j) GW 3 P.f ( k) GW i, j, k t.f

31 Exotic Reactors Promt critical (suercritical) behavior refers to reactors that are critical based on romt neutrons only and hence have very short eriods. Reactors can be designed with inherent shutdown characteristics when they become suercritical. General Atomics RIGA reactor is an examle. Such reactors can roduce short but intense ulses of neutrons (see chart at left).

32 Ramifications For ositive reactivity (increases in ower), which necessarily must be small, romt neutron jum is negligible, (flux essentially unchanged in the short term) For negative reactivity (decreases in ower) can be arbitrarily large romt neutron jum can be very large U to 96% in the case of a scram over about 80 seconds. Fission roduct decay accounts for u to 6% of total ower (for an equilibrium reactor) not affected by the reactivity change cannot reduce by more than about 93% the ower outut

33 Small Reactivities i i i i i i i i i i i i i i i t t l t l l l l l β ρ β ρ β λ β ρ λ β ρ 6 Reactivity equation For small reactivities, first root is small, ignorable in denominators. Period of reactor simle exression, as tabulated below

34 Control Rods Follow load Rod worth magnitude of reactivity change required to give a secified eriod Comensate for fuel decay Rod worth magnitude in multilication factor change for which the rod can comensate hough these definitions sound different, they are very similar

35 Cluster Control Rods

36 Cruciform Control Rods

37 Cruciform Rods

38 α d ρ d d d emerature Deendence k k k dk d 2 k dk d α temerature reactivity feedback coefficient If α > 0, Unstable increases and decreases in temerature run away to meltdown or shutdown without oerator resonse. If α < 0, Stable Increases and decreases in temerature self regulate and the reactor stabilizes. reit-wigner describes absortion rofile at 0 but Doler effect broadens eaks, with Different α s for fuel/moderator ittle change in area, at higher temeratures. Different timescales 2 λ Γ Γ Fuel is most raid r g n γ σ γ ( E) 2 α 4π romt 2 Γ ( E Er ) + NRC requires negative α 4 romt values for licenses

39 Xenon (Iodine, ellurium) Xenon-35 has a high absortion cross section (2.65x0 6 b in thermal region) and is the most significant absorbing oison. 35 e β sec Fission d X dt 35 I 6.7 hr Fission λ I I λ I I β d I dt Iodine decay + γ + γ X ( ) / 2 35 Σ f Xe 9.2 sec Fission γ Σ φ I f fission yield X f fission yield Cs λ I I λ X X C ( t ) φ ( t ) ( λ + σ φ ) X, eff Σ φ β λ X 35 natural natural decay X σ φ ax 2.3x0 decay β ax ( t ) 6 yr 35 σ axφ X X Ba ( stable) absortion decay

40 Reactor Dead time Load change shutdown

41 Fuel Loading Patterns

42 Burnable (absorbing) oisons Burnable oison forms roducts with lower adsortion cross sections, comensating for accumulation of other oisons. Boron and gadolinium oxides (gadolina) are examles.

43 yical Control Worths