Analysis of Reactivity Induced Accident for Control Rods Ejection with Loss of Cooling

Transcription

1 Analyss of Reactvty Induced Accdent for Control Rods Ejecton wth Loss of Coolng Hend Mohammed El Sayed Saad 1, Hesham Mohammed Mohammed Mansour 2 Wahab 1 1. Nuclear and Radologcal Regulatory Authorty, Nasr Cty, Caro 11762, Egypt and Moustafa Azz Abd El 2. Physcs Department, Faculty of Scence, Caro Unversty, Gza 12613, Egypt Abstract: Understandng of the tme-dependent behavor of the neutron populaton n nuclear reactor n response to ether a planned or unplanned change n the reactor condtons, s a great mportance to the safe and relable operaton of the reactor. In the present work, the pont knetcs equatons are solved numercally usng stffness confnement method (SCM). The soluton s appled to the knetcs equatons n the presence of dfferent types of reactvtes and s compared wth dfferent analytcal solutons. Ths method s also used to analyze reactvty nduced accdents n two reactors. The frst reactor s fueled by uranum and the second s fueled by plutonum. Ths analyss presents the effect of negatve temperature feedback wth the addton postve reactvty of control rods to overcome the occurrence of control rod ejecton accdent and damagng of the reactor. Both power and temperature pulse followng the reactvty- ntated accdents are calculated. The results are compared wth prevous works and satsfactory agreement s found. Key words: Reactvty nduced accdent, stffness confnement method, pont knetc equatons, control rods ejecton, reactvty coeffcent, and safety analyss. 1. Introducton Reactvty - ntated accdent s nuclear reactor accdent that nvolves nadvertent removal of control element from an operatng reactor, thereby causng a rapd power excurson n the nearby fuel elements and temperature. The postulated scenaros for reactvty - ntated accdents are therefore focused on few events, whch result n exceptonally large reactvty excursons, and therefore are crtcal to fuel ntegrty. In compared reference model [1], reactvty - ntated accdent was consdered to be due to negatve temperature feedback. In the present work, we consder reactvty accdent to be due to negatve temperature feedback, and the addton postve reactvty of control rods to prevent such accdents of control rods ejecton. We analyzed accdents n dfferent types of reactors, e.g. [1] modular hgh temperature gas cooled reactor desgn lke HTR-M and modular fast reactor desgn lke PRISM, [1] usng the stffness confnement method for solvng the knetc equatons. The stffness confnement method (SCM) s used to solve the knetcs equatons and overcome the stffness problem n reactor knetcs [2]. The dea s based on the observaton of stffness characterstc, whch s present only n the tme response of the prompt neutron densty, but not n the delayed neutron precursors. The method s therefore devsed to have the stffness decoupled from the dfferental equaton for precursors and s confned to the one for prompt neutrons, whch can be solved [2]. Numercal examples of applyng the method to varety problems are gven. The method s also used to analyze the reactvty nduced accdents n two reactors data, modular hgh temperature gas cooled reactor (HTR-M) whch s fueled by uranum and modular fast reactor desgn (PRISM) whch s fueled by plutonum. [1] In the next, we dscuss the mathematcal method; present the results and dscusson, and gve the concluson.

2 2. Experments The stffness confnement method s used to overcome the stffness problem n reactor knetcs for solvng the pont knetcs equatons. The pont knetcs equatons are system of coupled ordnary dfferental equatons, whose soluton gve the neutron densty and delayed neutron precursor concentratons n tghtly coupled reactor as a functon of tme. Typcally these equatons are solved usng reactor model wth at least sx delayed precursor groups, resultng n system consstng of seven coupled dfferental equatons. Obtanng accurate results s often problematc because the equatons are stff wth many technques, where very small tme steps are used. These equatons take the followng form wth an arbtrary reactvty functon [3, 4]: dn( ρ( β = n( + C ( dt Λ λ = 1 (1) dc ( β = n( λc ( dt Λ (2) where: n( s the tme-dependent neutron densty, or (power or neutron flux) all unts are (MW) as power unt; C ( s the th group delayed neutron precursor concentraton or delayed neutron emtter populaton or precursor densty ( latent-neutron densty or latent power; same unts as n the power); s the number of precursor group; ρ( s the tme-dependent reactvty; β s th group delayed neutron fracton, and β = Σ β, s the total delayed neutron fracton. In addton, Λ s the neutron generaton tme (s) and λ s decay constant of the th -group delayed neutron emtters (s -1 ). Introducng a set of Reduced precursor densty functons Ĉ ( and neutron densty, through the followng equaton [2]: t C ( t ) = Cˆ ( t ) exp[ u ( t ) dt ] (3) and defnng two auxlary functons w ( and u (, as n Eqs. (4) and (5): d w ( = ln n( (4) dt The functon w( s defned n the same way as Eq. (9) below and provdes the mechansm key o f the SCM. The functon u(, however, has nothng to do wth stffness decouplng and s not really requred theoretcally. Snce an exponental behavor s often characterstc for the frst, order dfferen tal equatons, however, a proper choce of u( may make Cˆ ( vary more slowly n tme and thus e xpedte the numercal calculaton. Choose the followng u( [2]: d u ( t ) = ln S ( t ) (5) dt Where, S ( s defned by Eq. (7) as the sum over all λ C (. We can rewrte Eqs. (1) and (2) as follows [2]: and ˆ 6 = dt = 1 dc ( β Λ w( + β ρ( dn ( t ) = dt Suppose that t s always possble to express: 6 λc ( [ u( + λ ] Cˆ ( 6 t S ( = λ C ˆ ( exp u( t ) dt (7) = 1 t ρ ( t ) β n ( t ) + S ( t ) Λ n ( t ) = exp[ w ( t ) d t ] (8) (6)

3 and rewrte Eq. (1) as: are: and 6 λc ( = 1 n( = (9) β ρ( w( + Λ Eqs. (6)-(9) form the complete set of knetc equatons for the SCM. The ntal condtons to be satsfed u ( ) = (1a) ρ() w () = Λ (1b) n ( ) = n (1c) n β ˆ () = Λλ (1d) C By usng the ntal condtons, we can obtan the numercal soluton of the equatons. We frst start by settng w and u n Eq. (7) at ther ntal values and solves Eq. (7) for Ĉ by dscretzng the equaton n t. Havng obtaned Ĉ, we calculate S( wth Eq. (1). Then, we use Eq. (5) to re-evaluate w(, plug t back nto Eq. (7), and repeat the process untl w converges (requrng 5 teratons). Calculaton for the current tme step s then fnshed wth an evaluaton of the output value of w and u va Eqs. (5) and (1). Afterward, we predct the nput values of w and u for the next tme step by lnear extrapolaton from ther output values n the prevous and current tme steps, and repeat the whole process of calculaton for the next tme step. It should be emphaszed that wthn each tme step, there s teraton to convergence on w but no teraton for the functon u, because u s not requred by the theory of (SCM) and s, n prncple, wth an arbtrary ndependent functon chosen only to expedte the computaton. Computer program s desgned wth programmng languages (FORTRAN and MATLAB) codes to solve the above equatons numercally usng Runge-Kutta method, and the output power and temperature are determned under dfferent nput reactvtes. 3. Model Problems The SCM s tested wth three types of problems whch are: (1) Step reactvty nserton, (2) Ramp nput, (3) Snusodal nput. The results are compared aganst those obtaned wth other methods, e.g., Henry s θ, weghted method [5], Exact data obtaned wth Ref. [2], and Taylor Seres Methods [4, 6], CORE [7], Mathematca s bult-n dfferental equaton solver (mplct Runge-Kutta). Each of these methods s hghly accurate, but they vary wdely n ther complexty of mplementaton. 3.1 Step Reactvty Inserton Consderng a knetc problem wth step reactvty nserton wth β =.7.In ths case, ρ( = ρ for t. The followng nput parameters were used: λ (s -1 )= (.127,.317,.155,.311, 1.4, 3.87), β = (.266,.1491,.1316,.2849,.896,.182) and Λ =.2 s. Four step reactvty nsertons are consdered: two prompt subcrtcal ρ =.3 and.55, one prompt crtcal ρ =.7, one prompt supercrtcal ρ =.8 [2, 7]. The values of n( obtaned wth the present work are compared (Table 1) wth those obtaned

4 wth a code based on the so-called Henry s θ, weghted method, whch modfes fnte dfference equatons by ntroducng tactcally chosen weghtng functons. The step sze taken was h 1 =.1. For comparson, we chose Henry ' s θ, weghtng method, and the exact values that obtaned from Ref. [2] wth the present results. The numbers presented n Table 1 are computed wth tme steps (1 s, 1 s and 2 s).the results ndcate that the present model solutons are n good agreement wth all results. The teraton n computng was used for repeatng the process untl w and u converge (requrng approxmately 1 teratons) to get step reactvty nserton wth accurate results whch are compared wth several methods. 3.2 Ramp Input of Reactvty Consder now the two cases of ramp nput. Ramp reactvty usually takes the form: ρ ( ) = ρ t Where, t Is a gven reactvty expressed n dollars [1, 11].We wll use the same parameters, whch are used n the step reactvty example, and compare our results wth those of Ref. [2]. The frst case s extremely fast and the second s moderately fast. In the frst one, t can be seen that, the response of reactor core at.1 s after a ramp nput of reactvty at the rate of $1/s s calculated (wth sx groups of delayed neutron). The computatonal results for ths case are presented n Table 2 n comparson wth the SCM soluton by Ref. [2]. ρ = ρ β Table 1 ρ Comparson of present work and dfferent methods for step reactvty nserton. Method θ-weghtng SCM Exact θ-weghtng SCM Exact θ-weghtng SCM Exact θ-weghtng SCM Exact n( t = 1 s t = 1 s t = 2 s t =.1 s t = 2 s t = 1 s t =.1 s t =.5 s t = 2 s t =.1 s t =.1 s t = 1 s The second case s a (moderately fas ramp of $.1/s to reactor core. The values of the physcal parameters are the same as those of step reactvty nserton examples. The computatonal results for ths case are presented n Table 3 along wth other methods. The teraton n computng was used for repeatng the process untl w and u

5 converge (requrng 1 teratons) to take ramp reactvty nserton whch s a tme dependent functon wth small tme step n order to get accurate results n comparson wth several methods. 3.3 Snusodal Input of Reactvty Consder the case of snusodal reactvty. In ths case the knetc parameters are used: λ (s -1 ) = (.124,.35,.111,.31, 1.14 and 3.1), β = (.215,.1424,.1274,.2568,.748, and.273), Λ =.5 s, T = 5. s and β =.652. The reactvty s a tme dependent functon of the form [2, 4, and 1]: πt ρ ( t ) = ρ sn( ) T Where, T s a half-perod and ρ = β.the results of the present method are compared wth other methods n Table 4 and showed a good agreement. The teraton n the computaton, s used for repeatng the process untl w and u converge (requrng 1 teratons) to get step reactvty nserton wth accurate results. The teraton n computng s used for repeatng the process untl w and u converge (requrng 1 teratons) to take snusodal reactvty nserton whch s a trangular functon nsde t half perod and small tme step to get accurate results. The results are compared wth several methods. Table 2 fas. Comparson of present work and SCM method n Ref. [2] for ramp nput of reactvty the frst case: ( extremely ρ Methods n ( ρ =.7 SCM Table 3 Comparson of present work and SCM method for ramp nput of reactvty the second case :( moderately fas. Methods t = 2 s t = 4 s t = 6 s t = 8 s t = 9 s θ-weghtng SCM Exact Table 4 Comparson of present work and other methods for snusodal reactvty. Methods t = 2 s t = 4 s t = 6 s t = 8 s t = 1 s Taylor Core Mathematca Analyss of Reactvty Intated Accdent 4.1 Reactvty Intated Accdent Reactvty - ntated accdent nvolves an unwanted ncrease n fsson rate and reactor power. Power ncrease may damage the reactor core, and n very severe cases, even lead to the dsrupton of the reactor. The mmedate consequence of reactvty - ntated accdent s fast rse n fuel power and temperature. The power excurson may lead to falure of the nuclear fuel rods and release radoactve materal nto prmary reactor coolant. In ths study, a new computer program has been developed for smulatng the reactor dynamc behavor durng reactvty nduced transents, and t has been used for the analyss of specfed reactvty - ntated accdents consdered n several cases. We ntroduce the two models reactors wth system parameters, whch are characterstc for modular hgh

6 temperature gas-cooled reactor desgn lke HTR-M [8] and modular fast reactor desgn lke PRISM [9]. For smplcty, we refer to these models of two reactors as HTR-M and PRISM (Tables 5 and 6). For delayed neutron parameters, t s assumed that, HTR-M s fuelled by 235 U and PRISM by 239 Pu as fssle nucldes. The dynamc equatons for the two models are the conventonal of the pont reactor knetcs equatons n combnaton wth lnear temperature feedback n reactvty, an adabatc heatng of the core after loss of coolng [1], where Eq. (13a) may be modfed to add postve reactvty of control rods. The data of the two reactor models are gven n Tables 5-7: Table U (thermal neutrons). λ (sec -1 ) β β tot =.67 Λ = 1.E-4 (s) Table PU (fast neutrons). λ (sec -1 ) β 7.6E-5 5.6E E E-4 2.6E-4 7.E-5 β tot =.2 Λ = 1.E-7 (s) Table 7 Adabatc nherent shutdown data for two model reactors. Types of reactors n (MW) c (MJ/K) α (K -1 ) HTR-M E-5 PRISM E-6 dn( ρ ( β = n( + λc ( dt Λ ext net = 1 dc ( β = n( λc ( (12) dt Λ ρ ( = ρ ( + ρ ( (13a) ρ net feed CR feed = α ( T ( T ) ext cr1 cr2 cr3 cr4 6 (11) ρ = ρ = ρ or ρ or ρ or ρ (13b) ρ feed = feedback reactvty ρ ext = external reactvty = control rods reactvty dt ( 1 = n( (14) dt c where n( = reactor power (MW); ρ net ( = the tme-dependent reactvty functon; ρ CR = addton postve reactvty of control rods; β = total delayed neutron fracton; β = Σ β β = delayed neutron facton of th group; Λ = neutron generaton tme (s); λ = decay constant of th group delayed neutron emtters (s) -1 ; C ( =delayed neutron emtter populaton (n power unts); α = negatve temperature coeffcent of reactvty (K -1 ); T = reactor temperature (K); T = crtcal reactor temperature (K) and c=heat capacty of reactor (MJ/K). In the equaton of total reactvty ρ(, the addtonal postve reactvty of control rods ρ cr has four cases to prevent the control rods ejecton accdent: ρ cr1 = ρ 1 =, ρ cr2 = ρ 2 = (β/2),

7 ρ cr3 = ρ 3 = (.8β), ρ cr4 = ρ 4 = (β). The nput parameters of the knetc equatons for two types of reactors wth dfferent fssle materals are shown n Tables 5 and Reactvty Evaluaton The reactvty of one, two and three control rods worth are calculated based on the assumptons of relatng control rods worth by the delayed neutron fracton β. Assumng that, the ejecton of one, two and three rods could nduce postve reactvty as ndcated n Table 8 for each type of reactors, n the two models. Table 8 Addtonal postve reactvty of control rods nserton. No. of control rods ρ (n $) for U 235 ρ (n $) for PU Secton (II) 5. Results and Dscusson In ths secton, we explan the result of the total energy producton whch s expressed n full-power-seconds (FPS) by dvdng the energy through nomnal power, asymptotc temperature ncrease, and equlbrum reactvty after shutdown. The relevant quantty n relaton to reactor safety s the asymptotc temperature ncrease as determned by the total energy producton durng autonomous shutdown and by the heat capacty. There s smple relaton between energy produced and temperature ncrease due to the absence of heat loss [1]. As proved n Ref. [1]: E 2ΛC = * n α 1 β 1 + Λ λ n C 2 2 Where, E = total fsson energy produced durng autonomous shutdown; n = the ntal reactor power condton dependng on the type of reactor. In case of autonomous shutdown, whch use Eq. (13a) and Eq. (14) as n Ref. [1], that the asymptotc temperature T and equlbrum reactvty after shutdown ρ are found n Eq. (16) as: 2n β 1/ 2 2α n β 2 T T = [ ], ρ = [ ] 1/ α c λ c (16) λ The total energy producton quantty s physcally more appealng; t can be seen that, both reactors, the fsson energy producton durng adabatc nherent shutdown s equvalent to one mnute full-power operaton. As a consequence of the hgh heat capacty of the cores, ths energy s easly accommodated wth temperature ncrease about 13 K. So, the results of the two reactors models are gven n Table Reactvty Addton at Full Power Condton 6. 1 Frst Reactor (PRISM Reactor) PRISM reactor s assumed to be operatng at equlbrum power condton equal to 47 (MW) and the lmted value of tme (s) on x axs equals to 3 (s) at full power condton. Where, control rod nserton ncreases the α β λ (15)

8 thermalzaton of neutrons, and thus, results n a postve reactvty addton. Control rod nserton requres a certan drvng force. The drvng forces on control rods n the reactors are the buoyancy from the fuel materal and the supportng force from the control system of reactor. If the control system should lose the support of control rods or control rods should break, control rods would be flown out of the reactor. Thus, n PRISM reactor, accdental nsertons can result from the ejecton of control rod drve, and/or control rod control system or operator error. Reactvty s also added step by step. The full power transents for one, two, and three control rods ejecton are shown n Fg. 1. When control rods are ejected, power pulse ndcates n the four cases: Frst, wth negatve temperature feedback and wthout postve reactvty of control rods at ntal condton t = (s), n = 47 (MW), power decreases and after t = 3 (s), power approaches to saturaton wth n = 1 (MW). Second, wth negatve temperature feedback and addton control rod reactvty equal (β/2), the maxmum power rato ncrease by 2. tmes from the ntal value of power at t =.768 (s). Thrd, wth negatve temperature feedback and addton control rod reactvty equal (.8β), the maxmum power rato ncrease by tmes from the ntal value of power at t =.582 (s). Fourth, wth negatve temperature feedback and addton control rod reactvty equal (β), the maxmum power rato ncrease by tmes from the ntal value of power at t =.616 (s). When control rods are ejected, power pulse s ncreased many tmes than; the rated power s generated n a very short tme. Ths s because the accdent s reactvty accdent. The temperature transents are shown n Fg. 2 for four cases. Because of temperature s proportonal wth power at Eq. (14) and wth Eq. (13a) wth the net reactvty, so that, power n Eq. (11) ncreases due to the postve reactvty addton n reactvty equaton of control rod worth. The maxmum temperature exceeded 1,521 K for about 25 s.the results ndcate that, n the frst case at ntal condton t = (s), T = 95 (K), after that temperature ncreases untl t = 25 s become at frst case: T = 1,195 (K), second case: T = 1,33 (K), thrd case: T = 1,43 (K), fourth case: T=1,53 (K). After t = 2 (s), temperature approaches to saturaton. Table 9 Adabatc nherent shutdown results for two models reactors. Types of reactors E /n (fps) T T (K) ρ (%) H. Van Dam Present Results HTR-M H. Van Dam PRISM

9 Fg. 1 The power (MW) transent as a functon of tme at full power condton wth dfferent values of postve of control rods ejecton for PRISM reactor. Fg. 2 The temperature (K) as a functon of tme durng the transents at full power for PRISM reactor. 6.2 Second reactor (HTR-M Reactor) HTR-M reactor s assumed to be operatng at equlbrum power condton equal to 2 (MW) and the lmted value of tme (s) on x axs equals to 3 (s) at full power condton. Reactvty s also added step by step as, explaned above n PRISM reactor. The full power transents for one, two, and three control rods ejecton are shown n Fg. 3. Control rods are ejected, power pulse ndcates n four cases: Frst, wth negatve temperature feedback and wthout postve reactvty of control rods at ntal condton t = (s), n = 2 (MW), power decreases due to negatve temperature. Second, wth negatve temperature feedback and addton control rod reactvty equal (β/2), the maxmum power rato ncrease by tmes from the ntal value of power at t = (s). Thrd, wth negatve temperature feedback and addton control rod reactvty equal (.8β), the maxmum power rato ncrease by tmes from the ntal value of power at t = (s). Fourth, wth negatve temperature feedback and addton control rod reactvty equal (β), power rato ncreases by 53.5 tmes

10 from the ntal value of power at t =.76 (s). When control rods are ejected, power pulse s ncreased many tmes of the rated power s generated n a very short tme. Ths s because the accdent s a reactvty accdent. Fg. 3 The power (MW) transent as a functon of tme at full power condton wth dfferent values of postve reactvty of control rods ejecton for HTR-M reactor. Fg. 4 The temperature (K) as a functon of tme durng the transents at full power for HTR-M reactor. The temperature transents are shown n Fg. (4) for four cases. Because of temperature s proportonal wth power at Eq. (14) and wth Eq. (13a), wth the net reactvty, so that power n Eq. (11) ncreases due to the postve reactvty addton n reactvty equaton of control rod worth. The maxmum temperature exceeded 998.4K for about 25 s. The results ndcate that, n the frst the ntal condton t = (s), T = 35 (K), after that temperature ncrease untl t = 25 (s) become at frst case: T = (K), second case: T = (K), thrd case: T = (K), fourth case: T = (K). After t = 2 (s), temperature approaches to saturaton. 7. Conclusons Computer program s desgned to solve the pont reactor dynamcs equatons usng the stffness confnement method (SCM) and dfferent nput reactvty s appled (step, ramp and snusodal), the resultant powers are determned and llustrated. Good accuracy n comparson wth reference values s obtaned. The model s appled to the two types of reactors. There are modular of fast reactor desgn lke PRISM reactor [9] and modular hgh temperature gas-cooled reactor desgn lke HTR-M reactor [8]. PRISM reactor s fuelled by 239 Pu, the HTR-M reactor s fuelled by 235 U as fssle nucldes.

11 In the work of Van Dam [1] (we used t for comparson purpose), the author obtaned reactvty accdent due to negatve temperature feedback after loss of coolng to dfferent reactors wth dfferent fssle materal. Reactvty - ntated accdent s consdered to be due to lnear temperature feedback and an adabatc heatng of the core after loss of coolng. In the present work, we consder reactvty accdent due to lnear temperature feedback, an adabatc heatng of the core after loss of coolng and wth addton of postve reactvty due to control rods ejecton. We analyzed accdents n dfferent types of reactors (HTR-M and PRISM), usng the stffness confnement method for solvng the knetcs equatons. In the present work, one obtans reactvty nduced accdent due to control rods ejecton wth negatve temperature feedback and addton of postve reactvty of the control rods to overcome the occurrence of control rods ejecton accdent and prevent reactors from damage. The addton of postve reactvty s used for four cases: (, β/2,.8β, β), where at the zero case only negatve temperature feedback as the case of the Ref. [1] and the other cases of negatve temperature feedback and the addton value of control rods reactvty. Ths s called reactvty nduced accdent. The power for 239 Pu fueled reactor, when reactvty of reactor s ncreased by β, the reactor peak power ncreased by 83. 8,85 tmes the ntal value wth the saturated temperature of 1,53 (K). For HTR-M reactor ncrease by factor of 53.5 tmes the ntal value at equlbrum temperature of 1, (K), when reactvty s ncreased by β. References [1] H. Van Dam, Dynamcs of passve reactor shutdown, Prog. Nucl. Energy 3 (1996) 255. [2] Y. Chao, Al. Attard, A resoluton to the stffness problem of reactor knetcs, Nuclear Scence and Engneerng 9 (1985) [3] J.J. Duderstadt, L.J. Hamlton, Nuclear Reactor Analyss. John Wley & Sons, 1976, pp [4] D. McMahon, A. Person, A Taylor seres soluton of the reactor pont knetcs equatons, arxv: (21) [5] T.A. Porschng, The numercal soluton of the reactor knetcs equatons by dfference analogs: A comparson of methods, WAPD-TM-564, U.S. Natonal Bureau of Standards, U.S. Department of Commerce (1966) [6] B. Mtchell, Taylor seres methods for the soluton of the pont reactor knetc equatons, Annals of Nuclear Energy 4 (1977) [7] B. Quntero-Leyva, CORE: A numercal algorthm to solve the pont knetcs equatons, Annals of Nuclear Energy 35 (28) [8] K. Kugeler, R. Schulten, Hgh Temperature Reactor Technology, Sprnger, Berln, 1989, pp [9] G.J. Van Tuyle, G.C. Slovk, R.J. Kennett, B.C. Chan, A.L. Aronson, Analyses of unscrammed events postulated for the PRISM desgn, Nuclear Technology, 91 (199) [1] M. Knard, E.J. Allen, Effcent numercal soluton of the pont knetcs equatons n nuclear reactor dynamcs, Annals of Nuclear Energy 31 (24) [11] D.L. Hetrck, Dynamcs of Nuclear Reactors, Unversty of Chcago Press, Chcago, 1971.