Analysis of prompt supercritical process with heat transfer and temperature feedback

Nuclear Science and Techniques 20 (2009) 317-320Analysis of prompt supercritical process with heat transfer andtemperature feedbackZHU Bo1.2." ZHU Qian2 CHEN Zhiyun2'School of Nuclear Science and Engineering, Shanghai Jiao Tong Universiy, Shanghai 200240, China'Department of Nuclear Science and Engineering, Naval University of Engineering, Wuhan 430033, ChinaAbstract The prompt supercritical process of a nuclear reactor with temperature feedback and initial power as wellas heat transfer with a big step reactivity (00>B) is analyzed in this paper. Considering the effect of heat transfer ontemperature of the reactor, a new model is set up. For any initial power, the variations of output power and reactivitywith time are obtained by numerical method. The effects of the big inserted step reactivity and initial power on theprompt supercritical process are analyzed and discussed. It was found that the effect of heat transfer on the outputpower and reactivity can be neglected under any initial power, and the output power obtained by the adiabatic model isbasically in accordance with that by the model of this paper, and the analytical solution can be adopted. The resultsprovide a theoretical base for safety analysis and operation management of a power reactor.Key words Neutron kinetics, Prompt supercritical process, Point reactor, Temperature feedback, Heat transferreactor neutron kinetics equation isIntroductiondn(t)dt = [()-β] n()/I + AC()(1)The output power will increase sharply, and the coredC()Vdt = β n()/I -AC(1)(2)will be damaged, if a reactivity accident occurs, andwhere n(t) is the neutron density at time t, p(t) is theanalysis of power response to especially big reactivityreactivity, β is total fraction of the delayed neutrons, lis of great importance. Studies on prompt and delayedis the mean generation time, h is decay constant of thesupercritical process with temperature feedback weredelayed neutron precursor fission products, and C(t) iscarried out in recent years , but all the studies areaverage density of the delayed neutron precursors.mainly based on N-F modelus, in which the neutronWhen both sides of Eqs. (1) and (2) are multipliedtransient occurs in very short time under the reactivity,with the density/power ratio, n() is the reactor power.hence the adoption of the adiabatic model to relate theAssuming that the reactor has a negativepower and temperature. However, can the adiabatictemperature cofficient of reactivity a (>0) onmodel, with heat transfer all along, be suitable forintroduction of the big step reactivity Po (>P), theordinary operation state? In this paper, we propose areactivity of reactor with temperature feedback isnew physical model and mathematical description forρ=Po-a[T()-T](3)he question. Some results of significance have beenwhere T(I) is the reactor temperature at time t, and Toobtained.is initial temperature of the reactor.2 Physical model and analysisWith the big reactivity Po entering into the reactor,heat can be transferred out of the reactor by the heatThe effect of extraneous neutron source can beexchanger between first and second loops and by theneglected for a reactor operated in the critical stateheat leak, then the relation between power andwith steady output power, and one group of point中国煤化工Supported by the National Natural Science Foundation of China (No. 10575131)●Corresponding author. E-mail adress: zhubowy@ 163.comMHCNMHGReceived date: 2009-05-06318ZHU Bo et al.1 Nuclear Science and Techniques 20 (2009) 317-320temperature should beFrom Eqs.(9) and (12), the maximum power can be :dTldt = K<[n()-Qol(4)rmx= (O-B}"(2aKd)(13)where Kc is the reciprocal of thermal capacity ofreactor, and Qo is the heat being transferred out per3.2 Case 2unit time.Taking the initial power into consideration butIt is assumed that Qo is determined and Qo Srno.neglecting the heat transfer, namely no>0 and Qo~0,From the derivation of Eq.(3) with respect to t andthe intial conditions for Eq.(10) are P_Po and dpldt =using Eq.(4) one gets:-aK.no at t=0, and we have:dpldt = -aK.(n -Qo)(5) .dpldt = [(ρ -β)3-(o- β)门(2D) -aK,no(14)or: dpld2= -aK。dn/dtSubstituting Eq.(14) into Eq.(5), we have:At t=0, when a big step reactivity Po(>B) goes inton= [(Po-B)* -σ -B^](2aKD) + no(15)the reactor, a prompt supercritical state will occur, andWith Eq.(14) and the initial conditions, we can knowthe power increases to high level so sharply that at t≥0that the reactivity varies with time as:the contribution of delayed neutron precursors isp= [A+B- (A- -P)(A Po6D2e"(A+po-A)Vnegligiblel-s, and Eq. (1) can be simplifed as: ,[1+(A-po+ B) eul 1(A+po-B)](16)dn()Vdt = [p() β] n()/l(7)where A = [(0o B)'+2laK.no]!^.Combining Eqs.(5-7), one has the second orderCombining Eqs.(16) and (15), we can know that thedifferential equation about reactivity:power varies with time as:dpldr2- [(-B)IN(dJpldr -aK2Qo)=0(8)n= {(o-β)2 - A'[A+po-B-(A- PO+Be^]From Eq.(7), the corresponding Pmax to maximum[A+po B+ (A -Po+P)et"}/2aKJl+no (17)power Imax is:From Eq.(17), Imax, the time to reach the maximumPmax=β(9)power, is obtained:Although it contains only one variable, Eq.(8)lmex: = (UA) In[(A+po -B)(A -Po+B)](18)cannot be solved by an analytical solution but aand the maximum power is:numerical solution, from which transient characteristicnmax= (Po -B)"(2aK.D)+ no(19)of the prompt supercritical process can be obtained bysubstituting ρ into Eq.(7). Using Eq.9), the time toThe corresponding reactivity Pmax at the maximumreach maximum power can be obtained, too.power is obtained by Eq.(9).Eqs.11-13) and Eqs.16- 19) are results of Ref.[4]3 Analysis and discussionand Ref. [9], respectively. The two cases indicate thatthe model and the equations in this paper are correct.3.1Case 13.3Case 3When the initial power is very small, namely no=0, theheat being tansferred out of the reactor per unit timeWhen the initial power is not zero and there is the heatwill be small and negligible, namely Qo~0, Eq.(8) cantransfer, the numerical method of implicit difference isbe simplified asintroduced. For a certain kind of PWR fueled withdpId2 - [(ρ -B)V]dp/dA=0(10)U23, the parameters adopted are: β=0.0065, I=0.0001s,Solving Eq.(10) with the initial condition: ρ=p0, n=noλ=0.0774Is", K=0.05 K:MW-'.s', at =5x10~"K-. Asthe heat ransferred from the core per unit time equalsand dpldt=0 at t=0, one has (4I:to the output power under steady operation condition,dpldt = [(ρ B阳-(Po -P了](21)(11)中国煤化工r=n0 by neglectingCombining Eqs.(11) and (5), one hasotherCNMH G_n= [(0o -B3-(-B}]/(2aKd)(12)ruul u puwli asumed to be fasterthan that of heat transfer, so QoB)Sathiyasheela T. Ann Nucl Energy, 2009, 36: 246-250.introduced into the reactor, the temperature feedbackCHEN Wenzhen, KUANG Bo, GUO Lifeng, et al. Nucland heat transfer is analyzed with a new model. TheEng Design, 2006, 236: 1326-1329analysis indicates that the relation of power and10 CHEN Wenzhen, GUO Lifeng, ZHU Bo, er al. Prog Nucltemperature obtained with adiabatic model is irEnergy, 2007, 49(4): 290-302.accordance with the fact in most cases. Therefore11 U Haofeng, CHEN Wenzhen, LUO Lei, et al. Ann NuclEqs.(16 -19) can be adopted to analyze the power andEnergy, 2009, 36: 427-432.reactivity response of prompt supercritical process中国煤化工MYHCNMHG