Nuclear Reactor Kinetics Lecture 12 (2024) PDF

Introduction to Nuclear Engineering ENGG9741 Reactor Dynamics Time dependent behaviour of nuclear reactors Edward Obbard 4th November 2024  Reactor Kinetics Part 1 Chain reaction Neutron life cycle Keff and reactivity Delayed ne...

Introduction to Nuclear Engineering ENGG9741 Reactor Dynamics Time dependent behaviour of nuclear reactors Edward Obbard 4th November 2024  Reactor Kinetics Part 1 Chain reaction Neutron life cycle Keff and reactivity Delayed neutrons Cross sections and power revision  Neutron Generation Not all the neutrons released will cause another fission event. number of neutrons current cycle k = number of neutrons previous cycle k assumes that there is no leakage of neutrons (infinite reactor) so a more common value of k is keff. The average number that are effective is keff, where, keff > 1 fission rate increases exponentially (super- critical) keff < 1 fission rate decays (sub-critical) keff = 1 criticality – the operating state of a power reactor not as you might think an accident state. Where k is known as the criticality or effective neutron multiplication factor.  Neutron Life Cycle n0 = 1040 Leaked from Resonance captures core = 140 in 238U = 180 = 1.04 Fast fission Fast non-leakage Resonance escape neutrons n0Lf = 900 n0Lfp = 720 Lf = 0.865 p = 0.8 n0 = 1000 thermal non-leakage Leaked from n0LfpLt = 620 core = 100 Lt = 0.861 Net increase of 505 reproduction neutrons from thermal n0LfpLtf = 1000 fission thermal utilisation Captured in non-fuel = 125  = 2.02 for n0LfpLtf = 495 k=1 f = 0.799  Five (4 or 6) Factor Formula keff can be expressed as keff = fpL f Lt a product of factors.  = Reproduction factor. Number This is known as the five of neutrons produced by thermal or sometimes four factor fission per neutron absorbed. formula. Or even six f = the fraction of thermal neutrons captured in the fuel (thermal factor if leakage is split utilisation factor). into thermal and fast. p = the fraction of neutrons which If L = 1 i.e. infinite reactor avoid resonance capture and are no leakage then it thermalised. reduces to the four factor  = neutron contribution from fast formula. kinf = fp fissions. Lf or Lt = the prob a neutron will leak out of the core either thermal or fast.  Reactivity It is not always useful k eff − 1 to talk about keff and this is often turned into = reactivity, symbol  k eff Therefore a sub-critical Reactivity has no units as it is a reactor  = -ve ratio. Despite this units for reactivity have been invented Critical reactor  = 0 because the units are very small. One of the most common is the Supercritical reactor pcm or percent millirho.  = +ve. To convert  to pcm you need to multiply by 100,000 i.e.  = 0.00001 = 1 pcm  Prompt and Delayed Neutrons Most neutrons over 99% are produced within 10-13 s of the fission event. These neutrons are called prompt neutrons. However, some neutrons are produced due to the decay of unstable fission products. These fission products are called precursor atoms. Usually these precursors are divided into 6 groups for reactor 6 group neutron data for kinetics calculations. thermal fission of 235U. The average time for the production of a delayed neutron is 12.7 s and the averaged decay constant, , is 0.077 s-1. Delayed neutron precursors on the chart of nuclides Delayed Neutron Fraction (DNF) As mentioned the delayed neutrons only make up a tiny fraction of the 235U = 0.0065 – 0.0075 neutrons in a given cycle. The number of precursor nuclides 233U = 0.0026 produced is a fixed fraction of the 238U number of prompt neutrons = 0.0164 (fast) produced and called  and is 239Pu = 0.0022 dependent on the fissile isotope 240Pu used. = 0.0029(fast) DNF is dependent on the reactivity 241Pu = 0.0054 of the reactor and hence is not a fixed value. In fact DNF is equal to no. of precursor atoms − = no. prompt neutrons + no. precursor atoms It is also noted that on average delayed neutrons have a lower DNF = no. of delayed neutrons energy (~0.5 MeV) than prompt no. prompt neutrons + no. delayed neutrons neutrons (average~1 MeV). Result is they are less likely to leak or be absorbed than prompt neutrons.  Prompt Neutron Lifetime, lp A prompt neutron is Moderator td (s) produced within 10-13 s of fission. It then spends some time slowing down H2O 2.1 x 10-4 from a fast to a thermal neutron. This time is relatively D2O 4.3 x 10-2 short in comparison to the time it then spends as a thermal neutron, td, Be 3.9 x 10-3 before it is absorbed. Therefore lp~ td. Note this time is very Graphite 0.017 short  Macroscopic cross-section and neutron lifetime The likelihood of interaction is defined by the macroscopic cross- section   = N n N is the atom density of the material in atoms/cm3. n  is the probability of interaction n per unit travel (cm-1) For a given particle, a reaction is expected to have occurred once in a length 1/Σ. This is the mean free path of the n particle in this material. A neutron with velocity v will traverse its mean free path before being absorbed in a mean time given by l = 1/vΣ  Mean Neutron Lifetime, l* DNF DNF The mean neutron Time Av = l p (1 − (  −  ) ) + Timedelayed ( −  ) lifetime is defined as the Note that when critical this reduces to average length of time just beta below. between the birth of a neutron and its death TimeAv = 5 10 −5 (1 − 0.0065) + (12.7  0.0065) (absorption or leakage). = 0.0826 s i.e. l* is the time between the neutron generations. t = 0, neutrons = N0, at time t = l For a reactor without number of neutrons = kN0. delayed neutrons After m =10,000 generations how l*~lp~10-5 - 10-4 s. has the neutron level changed if k Average generation time =1.001? depends also on the a) Prompt neutrons only delayed neutrons. b) Prompt and delayed neutrons  Reactor Control a) Prompt neutrons only lp = 5 x 10-5 therefore 10,000 cycles = 0.5 s. At t =0.5s N = k10,000n0 = (1.001)10,000N0 n(0.5) = 21,917n0 b) With delayed neutrons at t = 0.5, k =1.001 We use the solution to the differential eq. for rate of neutron change with t. Solution is n = n0exp(t/T) where n0 = neutron level at time t = 0, t = time and T = l*/ρ. T = 0.0826/0.001 = 82.6 n = n0exp(0.5/82.6) = 1.006n0  Neutron Flux and Power So far we have considered neutrons hitting a target. A reactor of course produces vast numbers of neutrons per second. The number of neutrons passing through a cm2 region in a second is known as neutron flux. Symbol P = ϕ Σ Ef V  units of n cm-2 s-1. Fluence is flux x time i.e. the (Power, flux, cross section, measure of the total number of energy, fuel volume) neutrons passing through the material and is useful for radiation damage models. In a reactor at power the power is also directly proportional to the neutron flux.  Burn-up Burnup is measured in E.g. 200 kg of metal U fuel, in GWd/tHM [Gigawatt-days per a 50 MWth microreactor, ton of heavy metal, i.e. usually operated continuously for 1 uranium] year… It can also be measured in 0.05 [GW] x 365 [days] / 0.2 “FIMA” [Fissions per initial [tons] = 91.25 [GWd/tHM] metal atom] Always use thermal power Note that sometimes, to ratings for burnup calculate burnup, one does not calculations (the amount of even need to know how many electricity produced is not atoms have undergone fission. relevant to changes inside the fuel)  Reactor Kinetics Part 2 Control rods and shim Reactor start-up Neutron sources Approach to criticality Prompt jump, delayed effect. Prompt criticality  Control Rods and Chemical Shim Having seen that addition of reactivity must be carefully controlled we will consider how this is achieved. The two main sources of control in a water reactor are chemical shim and control rods. Chemical shim is the addition of a neutron absorbing chemical to the coolant. Typically this is a boron containing chemical often boric acid (H 3BO3). Control by shim is a slow process as it involves varying the level of neutron absorbing chemical to the coolant. Shim is therefore often used to correct for long term reactivity changes such as fuel burn-up, and control rods for shorter time scale events such as xenon and samarium poisoning.  Control Rod Design/material Typical control rod materials Metallic – hafnium, steel containing boron, cadmium, silver-indium. Boron-carbide (B4C). Often in powder form in a metallic structure. Control rod designs are typically either a cruciform shape or cluster control rods. Toshiba ABWR cruciform control rods Cluster control rods  Alternative Absorber Designs Where space is an issue. Control rods may not be possible. An alternative is a rotating drum with absorber on one side and reflector on the other. What are the issues with such a design?  Advantages/disadvantages of Chemical Shim and Control Designs Chemical shim Cluster rod Cruciform rod Start-up (Approach to criticality) keff is always less than 1 during start up so how come the neutron level increases? This is all down to a fixed source term (S on previous slide). Natural neutron sources are boron-11(80.1% of natural boron) and deuterium. Boron-11 needs to be in close proximity to the fuel as it needs alpha particles emitted by the fuel to produce neutrons. Deuterium makes up 0.015% of natural water and up to 90% in heavy water reactors. This can be a substantial contribution to neutrons but as it requires a gamma ray it is dependent on operating history. 11 5 B + 24 →147N + 01n 2 1 ( ) H + → H → H+ n 2 1 1 1 1 0  Installed Neutron Sources The neutron sources described can be weak or dependent on reactor history. Therefore reactors usually have an installed source as well. Common sources are Be-Ra and Calfornium- 252. 252Cf is man made and very expensive, plus it has a half life of 2.65 years and therefore needs replacing. It does produce a constant initial 2.3 x 10 12 n/s/gram. The 9Be needs an alpha source and is often paired with an alpha emitter such as Po, Pu or Ra. The major reason for an installed neutron source is so the neutron detectors can be calibrated. 9 4 Be +  → 4 2 ( C )→ 13 6 12 6 C+ n 1 0 Sub-critical Multiplication Factor (M) The effect of the source Generation 1 2 3 4 5 6 7 8 9 10 1 100 60 36 21.6 13.0 7.8 4.7 2.8 1.7 1.0 on the overall steady 2 100 60 36 21.6 13.0 7.8 4.7 2.8 1.7 state neutron level in a 3 4 100 60 100 36 60 21.6 36 13.0 21.6 7.8 13.0 4.7 7.8 2.8 4.7 sub-critical reactor can 5 100 60 36 21.6 13.0 7.8 be estimated by M. 6 7 100 60 100 36 60 21.6 36 13.0 21.6 8 100 60 36 9 100 60 M = 1/(1-keff) 10 100 160 196 217.6 230.6 238.3 243 100 245.8 247.5 248.5 E.g. start with 100 neutrons and a source of strength 100 neutrons per cycle at a k value of 0.6. M = 2.5 therefore 100 neutrons become 250.  Effect of the Source Term The fixed addition of the source term means the neutron level increases when Keff is less than 1. It also shows the neutron level, for a fixed K eff, reaches an equilibrium value. 1025 Source neutrons = 51, Keff = 0.95 1020 Number of Neutrons 1015 1010 1005 1000 0 20 40 60 80 100 120 140 Cycle Number  Estimated Critical Position (ECP) On start-up the operator will always estimate S (1 − k ) ( ) = S + kS + k 2 S + k 3 S +........ where criticality will occur. To make sure this is Due to the source providing a steady state flux of neutrons a correct several source range detector will have a parameters must be count rate (CR) response known e.g. last known proportional to 1/(1-k). critical position of control i.e. the reciprocal of the count rate rods, withdrawal (CR)-1 at the detector is sequence, previous proportional to (1-k)/1. operating history to Therefore as S/CR approaches estimate effect of xenon zero k approaches 1. and other poisons. Plotting control rod % withdrawal against CR0/CR allows the critical position to be estimated.  Rate of Change of Neutron Level The rate of change of the neutron level can be represented by 2 coupled differential equations. lp is the prompt neutron lifetime,  is the precursor yield fraction that gives rise to delayed neutrons,  is the one group decay constant, C is the neutron precursor atoms, S is the source neutrons.  Approach to Criticality Using control theory it can be shown that the solution to these equations takes the form n(t ) = A + Be − xt + Ce− yt Where you can solve for the constants A, B, C and x and y for a given set of initial conditions. lp= 60 x 10-6 s,  = 0.077 s-1,  = 0.0065 And we start at a sub-critical value of  = -0.004 What does this look like if we add  = +0.001 of positive reactivity as if we were withdrawing the control rods on the approach to criticality?  Approach to Criticality For the exact case we just looked at the solution is n(t ) = 0.333 − 0.228e−0.0243t − 0.105e−158.34t 0.35 delta rho = +0.001 0.3 0.25 Delayed neutron Scaled neutron increase contribution. Time 0.2 constant = 1/0.0243 0.15 0.1 Prompt jump time constant = 1/158.34 0.05 0 0 20 40 60 80 100 120 140 160 Time (s)  Approach to Criticality What happens as we add the same amount of reactivity (+0.001) as we approach criticality? (i.e. as  tends to zero.) We can see three things. Size of prompt jump increases, time to reach equilibrium increases and final increase in neutron population is larger. 0.9 0.8 rho zero = -0.004 rho zero = -0.003 0.7 rho zero = -0.002 Scaled neutron increase 0.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 120 140 Time (s)  What happens then as we withdraw 6 control rods? -0.004 5 -0.003 -0.002 -0.0015 4 Critical Neutron Level 3 2 1 0 0 500 1000 1500 2000 Time (s)  Prompt Criticality!! What happens if you add enough reactivity such that  > 0.0065? Note the difference in timescales from previous graphs!!! Prompt Criticality 4.50E+05 4.00E+05 rho zero = 0, delta rho = +0.0067 3.50E+05 Scaled Neutron Increase 3.00E+05 2.50E+05 2.00E+05 1.50E+05 1.00E+05 5.00E+04 0.00E+00 0 0.5 1 1.5 2 2.5 Time (s)  Delayed neutron kinetics solutions (constant conditions) n/n₀ = exp( t ρ / l* ) Kinetic behaviour is governed by delayed neutrons n/n₀ = exp( t(ρ - β) / lp ) Kinetic behaviour is governed by prompt neutrons  Reactor Dynamics Closed Loop versus Open Loop Systems​ Xenon Poisoning ​Fuel Coefficient of Reactivity​ Moderator Temperature Coefficient ​Void Coefficient of Reactivity ​Saturation curve of water ​Self Regulating Behaviour ​Reactor Shutdown Decay Heat and Grace Time​  Closed Loop versus Open Loop Systems A closed loop system is one where there is a measure of the output and an error signal is then fed back into the input. An open loop system has no measure of the output. So the output is independent of the input. A closed loop system can show either negative or positive feedback. Systems are often represented using block diagrams as below. Closed loop system with negative feedback Input Output Input Output function function - error Open loop system  Reactor Dynamics Based on control theory. The primary circuit as a sub-system is a closed loop system with negative feedback. The steam generator as a sub-system is also a closed loop system with negative feedback. These sorts of systems can often be easily analysed by hand calculations using Laplace transforms. Overall system behaviour is best suited to being solved by computer.  Xenon Poisoning Xenon-135 is a strong thermal neutron absorber with a thermal capture cross-section of 2.65 x 106 barns. Xenon-135 is produced both from the decay of Iodine-135 and directly through fission. 135 Te ⎯11s − ⎯→ I ⎯⎯→ 6.7h 135 Xe ⎯ ⎯→ 9.2h − Cs ⎯ ⎯→ 135 2.3 x 10 yrs 135 Ba (stable ) − 135 − 6 fission fission fission The short half life of 135Te effectively means that all iodine can be thought of as produced directly by fission. The fission yield of 135I from 235U is ~6.4%. Xenon Poisoning  Xenon Poisoning The half lives of 135Xe and 135I are of the order of hours i.e. same order as operating reactors. This has some interesting consequences. The rate of change of xenon can be calculated by taking into account all the competing factors. dXe = 1 I +  Xe  f T −  Xe Xe −  aXeT Xe dt Decay of 135I Fission Decay of Burnup of 135Xe 135Xe by capture of a Rate of production of change 135Xe neutron of 135Xe with time How does this equation change for (a) clean core (b) at power (c) shutdown? And (d) is there an equilibrium Xenon concentration? Xenon simulations (open loop)  Xenon Poisoning Equilibrium 135Xe is usually reached fairly quickly in power reactors operating at all but low power i.e. rate of production of Xe from decay of 135I and fission is balanced by natural decay and burnout from neutron absorption. Once the reactor is switched off. The production of I and Xe from fission stops. The burnup of Xe from neutron capture also stops. Natural decay continues therefore Xe is being produced by decay of I faster than it is removed by decay or neutron burnup. This leads to a build up in the first few hours of Xe in the reactor. As xenon is a neutron absorber (poison) it redues the reactivity of the reactor and therefore excess reactivity is needed to overcome this. At the end of core life (low excess reactivity) it may not be possible to restart a reactor that has been operating at high power and then shut down.  Temperature Effects Temperature has an effect on reactivity. For safety reasons this should be a decrease in reactivity for an increase in temperature in order to give a stabilising effect. The overall coefficient of reactivity for a reactor system can be viewed as the sum of all the different coefficient terms. The temperature coefficient of reactivity is called T and is defined below with units of K-1. FUEL + MOD + POISONS + PRESS + VOIDS  T Change in Reactivity  T = = Change in Temperature T Note – temperature coefficients need not all be of same sign...  Fuel Coefficient of Reactivity Thermal The fuel coeff of reactivity is also vibration known as the prompt coeff of reactivity as the fuel is the first part to react during an increase in reactor power. For safety reasons the prompt coeff of reactivity should be negative. In fact the US NRC won’t licence a reactor where this is not the case. Fortunately for most fuel and distance thermal reactors in particular it is negative. We have seen resonances in the absorption of neutrons but this assumes the nuclei is a fixed object. In reality it moves with the atom under thermal vibration.  Doppler Broadening (simple) Even a mono-energetic beam of neutrons impinging on a moving nuclei appears to have a wide range of energies due to Doppler broadening. The result is the resonance peaks for absorption become wider and flatter as the temperature increases. Hence the resonance absorption overall increases. As resonance peaks for capture exceed those for fission this drives reactivity down.  Doppler broadening and self- shielding (more complicated) Area under the resonance absorption peak is actually very similar - even with Doppler broadening. However the flux increases in the vicinity of the weaker absorption peak - resulting in overall higher absorption Figure is from https://www.nuclear-power.net/ as temperature goes up.  Moderator Temperature Coeff The moderator As T increases temperature coeff is often density of the most important. moderator Especially where the decreases. moderator is a liquid e.g. PWR. The biggest effect here is due to density change. If boron shim is added Higher there is a positive effect Temperature on reactivity due to less less boron per cm3. moderation more leakage  Void Coefficient of ReactivityGas Water bubbles moderator The void coeff is important in liquid moderated reactors. In general as T increases the liquid turns to gas. Again as the density of the gas is much less than the liquid the moderation is reduced and more leakage occurs. Less moderation In this case the densities are so as T increases different we consider the gas as a void. In the RBMK design the void coefficient was positive at lower powers!! Which reactivity coefficients do you think are most important to safety ? How do the three coefficients in the previous slides relate to the 5-factor equation?  Zero Energy Reactor On the approach to start-up it is perfectly possible to add excess reactivity through a variety of sources. “A Zero energy reactor is a reactor which is generating insufficient power to alter its own thermal conditions”. i.e. no thermal feedback. Once criticality is achieved the neutron level is gradually increased and power increases with neutron flux. Eventually the fission energy starts to heat the water to a significant level and this controls the chain reaction. A zero energy reactor is usually subject to many operating restrictions to prevent a prompt criticality.  Start-up rules (PWR) Estimated Critical Position (ECP) Limit the start-up rate meter to ~1 DPM. Stop when the count rate doubles. No primary make-up water No steam take off. (Basically don’t add water of an unknown temperature!!) Steam Generator of a PWR The steam generator is used to exchange the heat from the primary side of the reactor system and produce steam to drive the turbines. It effectively has an inlet for primary coolant which is split into thousands of small bore tubes. These pass through the secondary side water which is at lower pressure and can boil. The steam is often cleaned so as to produce dry steam which then passes through a valve to the turbines. Steam Generator Heat Transfer TsFS Heat transfer can be crudely modelled by considering an energy qS balance of energy transferred. qPS dTS TCFPCP M S CP = qPS − qs THFPCP dt qPS = UAT = UA(TAVPRIM − TAVSEC )  (TH + TC )  Energy transferred to the secondary qPS = UA − TS  side is proportional to temperature  2  gradient through tube wall  Saturation curve of water As main steam stop valves are opened the pressure in the Steam Generator drops. Temperature of the steam will also drop due to water’s saturation curve. Water in the steam generator will boil more vigorously to try Pressure water to maintain the pressure. steam This results in a higher rate of heat transfer from primary to secondary, qPS as T has increased. Temperature  Self Regulating Behaviour This thermal feedback leads to the useful property of Self Regulating behaviour. Self regulation means the PWR reactor can compensate for any reactivity change without the need for operator action. As an example of an internal reactivity change, let us look at the effect of control rod withdrawal in a power reactor.  Self Regulating Behaviour 1. Control rod withdrawn. Positive 3. Increased reactivity. 4. No change in coolant steam demand. temperature. 2. Reactor TH Main steam stop valve power increases. To Turbine 7. Reactor power decreases. Pump Feed Water TC 6. Hotter 8. After some fluctuation, water = 5. Increased the reactor returns to negative Steam critical. Reactor power is Reactor coolant reactivity. temperature. Generator unchanged, but average temperature is increased.  Self Regulating Response for a Control Rod Withdrawal T remains constant TH Tav TC T Tav increased TS Reactor Power Steam Power Rod position Reactor power unchanged Reactor returns + to critical Reactivity _ Time (s) Self Regulating Response for a Control Rod Withdrawal 1. Control rod withdrawn. Positive reactivity. 2. Reactor power increases. 3. Increased coolant temperature (TH). 4. No change in steam demand therefore extra heat passes through the SG. 5. Results in an increased coolant temperature (T C). 6. After a lag time hotter water reaches reactor = negative reactivity. 7. Reactor power decreases. 8. After some fluctuation, the reactor returns to critical ( = 0). Reactor power is unchanged, but average temperature is increased. Q: Which of the following statements about a self regulating transient involving an addition of reactivity is false? a) The steam temperature ends up being hotter. b) The reactor becomes supercritical before returning to a stable state. c) The primary average temperature (Tav) remains the same after the transient as it was before it started. d) Steam power does not change throughout the transient.  Load Following Behaviour Thermal feedback also leads to Load Following behaviour. Load Following means the PWR reactor is able react to any change in power demand without the need for operator action. As an example, let us look at the effect of an increase in power demand.  Load Following Behaviour 6. Increased 1. Increased coolant steam demand. temperature. TH Main steam stop To Turbine 5. Reactor 2. Increased power boiling in steam increases. generator. Pump Feed Water TC 7. After some fluctuation, the 4. Colder water = 3. Decreased reactor returns to critical. positive coolant Steam Reactor power has increased Reactor to meet steam demand. reactivity. temperature. Generator Average temperature remains the same.  Load Following Response for an Increased Steam Demand TH Increased Tav difference, T TC Increased TS difference, T Reactor Power Reactor power Steam Power matches steam demand Reactivity + _ Reactor returns to critical Time (s)  Load Following Response for an Increased Steam Demand 1. Increased steam demand. 2. Increased boiling in steam generator, TS drops. 3 T between Tav prim and Tsec has increased therefore there is a greater heat transfer from the primary to secondary sides. 4. results in a decreased coolant temperature (TC) leaving the SG. 5. Colder water passes down the cold leg and after a time delay = positive reactivity at the reactor. 6. Reactor power increases and reverses fall in Tav. 7. Increased coolant outlet temperature (T H) 8. After some fluctuation, the reactor returns to critical ( = 0) and (Tav same). Reactor power has increased to meet steam demand.  Q: What would not be an end result of opening the main steam stop valves? a) Reactor power increases. b) Larger T between TH and TC. c) Tav decreases. d) Ts decreases. Q: What would happen if there was a steam leak on the secondary plant? a) The reactor would automatically scram. b) The reactor would start to load follow and power would increase. c) The reactor would self regulate in order to maintain its average temperature. d) There would be no effect on the reactor. Reactor Shutdown and Decay Heat Control rods To shut down a reactor it is inserted necessary to have enough negative reactivity to make the Prompt reactor sub-critical and have decrease enough negative reactivity Neutron level margin it is unlikely to become Delayed critical again. neutron effect Normally achieved via dropping Time all control rods into the reactor (SCRAM). The fission rate shows a sharp sudden fall followed by a more gradual decrease in flux. The thermal power however differs at this point from the fission power due to the production of decay heat.  Decay Heat and Grace Time 7% of the energy due to fission 250 is from the decay of unstable Thermal Power (MW) isotopes produced by fission. 200 These decay via mainly - and 150  rays. As a result even after a reactor 100 is fully shutdown it continues to 50 3600MWt plant produce heat for many hours. 0 If this heat is not removed the 0 10 20 30 40 50 core temperature will increase Time (h) and eventually lead to fuel   failure. − ( shut +  elap ) −0.2 The length of time this takes is P = 0.066P0  −0.2 el often known as the grace time. For a 3600 MWt plant elap is the time elapsed since immediately after shutdown the shutdown and shut is the time of decay heat is ~250 MW !! shutdown measured from the start of operation. Here it is 1 year.