2. Neutronic calculations at uranium powered cylindrical reactor by using Bessel differential equation

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1 Trasworld Research Network 37/661 (), Fort P.O. Trivadrum Kerala, Idia Nuclear Sciece ad Techology, 01: 15-4 ISBN: Editor: Turgay Korkut. Neutroic calculatios at uraium powered cylidrical reactor by usig Bessel differetial equatio Aybaba Haçerlioğulları Kastamou Uiversity, Kastamou Arts & Scieces Faculty, Physics Dept Kastamou, Turkey Abstract. Nuclear reactors are the complex machie-equipmet systems costructed through the use of advaced egieerig techologies. Fissio-type reactors are devices developed to geerate eergy at a stable power by takig the chai reactio uder cotrol. Therefore, K, eutro multiplicatio coefficiet is a importat uclear parameter for a uclear reactor to sustai geeratig eergy by itself at a stable power. K, is the ratio of the umber of eutros geerated i a geeratio to the umber of eutros absorbed i the previous geeratio. I this study, It has bee used differet cylidrical fuel cofiguratios. These cofiguratios are Uraium isotopes (%98,13 33 U, %1,4 34 U, %0,03 35 U, %0,60 38 U), respectively. For these cofiguratios it has bee calculated criticality (K EFF ) usig Bessel differetial equatio, Neutro flux (Ø) averaged over cylidrical surface ad total fissio eergy depositio over cylidrical ca be applied. The Bessel equatio is formulated as follows ad this equatio is a special case of Bessel s equatio [1, ]. Correspodece/Reprit request: Dr. Aybaba Haçerlioğulları, Kastamou Uiversity, Kastamou Arts & Scieces Faculty, Physics Dept., Kastamou, Turkey. aybaba@kastamou.edu.tr

2 16 Aybaba Haçerlioğulları Itroductio I a commercially available fissio reactor, oly a small percetage of Uraium is utilized for eergy geeratio. More tha 97% of Uraium fuel is removed from the reactor as spet fuel. I the study Bessel differetial equatios are used for the calculatios of eutro flux (Ø) ad criticality coefficiet (K) ad cylidrical geometric structure are take ito accout as the reactor geometry. Neutro flux (Ø) i the reactor chages accordig to the geometry of the reactor cylidrical the type of the fuel used ad physical properties of the reactor. Oe -group method ca be applied fairly effectively to the determiatio of the critical size of a fast reactor, provided that properly averaged cross-sectio values for the eutro spectrum are used [5]. The priciple task of a reactor cotrol system is to maitai cotrol over the chai reactio, that is, to cotrol the umber of eutros i oe geeratio relative to the umber of eutros i the previous geeratio. With all the above coditios ad simplificatios, the Neutro Diffusio Equatio for oe eergy group bares a ifiite critical reactor model. I our day, the productio of uclear eergy is mostly met by light ad heavy water reactors [3]. Fossil fuel eergy source, which ca be divided with ormal eutros, is used i these reactors. Reactio effect sectios geerated with euros i fusio ad fusio eergy reactors has a importat place i reactor desig. Durig the geeratio of these importat reactios, ot oly the structural edurace of the materials but also their geometrical desig is of importace. As part of uclear eergy raw materials, U 33, U 35, Th 3 ad Pu-39 cores take their place as fuel. I this study, by takig certai percetages from the cores metioed, we have calculated the total flux i the core of the reactor ad its reflector, the gai of the reflector, its critical volume ad its total power at the heart of the reactor. While makig eutroic calculatios, we have made use of R radius reflective spherical reactor geometrical structure. We have take advatage of spherical Bessel Differetial Equatio, which was modified uder oe group method approach becomes [1,, 8]. This cylidrical reactor has radius R. I this reactor flux depeds oly o the distace r from the axis. Bessel differetial approach to solutio The Bessel equatio is formulated as follows ad this equatio is a special case of Bessel s equatio, d ø/dr + 1/r.dø/dr + (B - /r ). Ø =0 (1)

3 Neutroic calculatios for reactors 17 I which is iteger(=0,1,,3 ), if we let r x, φ y, ad α m = B, after multiplicatio by r. Usig these approaches, we ca reach the balace equatio. From our recet discussios, we recogize this as Bessel s differetial equatio. x y '' ' + xy + ( x ) y = 0 () The solutios of this equatio are called Bessel Fuctios of order. Sice Bessel's differetial equatio is a secod order ordiary differetial equatio, two sets of fuctios, the Bessel fuctio of the first kid Y 1 =A J (x) ad Y =CY (x) are the solutios to the above formulated equatio. Y1 ad Y are respectively called as the fuctios of the Bessel fuctio of the first kid ad the Bessel fuctio of the secod kid. The solutio to (*) y( x) = AJ ( x) + CY ( x) (3) Equatio Bessel fuctio of the first kid of order ca be expressed as a series of gamma fuctios. The Bessel fuctio of the secod kid of order ca be expressed i terms of the Bessel fuctio of the first kid. As illustrated i Fig.1 ad the Bessel fuctio of the first kid ad secod kid. J ( x) = 4 x x x = Γ( + 1) ( + ) x4( + )( + 4) k = 0 k ( 1) ( x / ) k! Γ( + k + 1) + k (4) m ( 1) m x 1 ( m 1)! x m m + x Y( x) = J( x) l y + π + π m= 0 m! π m= 0 m!( m+ )! J p( x)cos pπ J p( x) = lim. p si pπ m+ The Bessel fuctio of the secod kid of order ca be expressed i terms of the Bessel fuctio of the secod kid also kow as the Weber Fuctio. Bessel Equatio ca be expadig ito series. J ( Br) = k= 0 Br 1+ k ( ) k!( + k)! Thus, J (z) which satisfies Bessel s equatio is a cylider fuctio.

4 18 Aybaba Haçerlioğulları That for the real physical system, the eutro flux must be real ad oegative, ad the oly Eige fuctio that is positive over the full domai, 0<r<R is related to the fudametal mode sice the first zero of the J 0 (x) Bessel fuctio occurs at X 1 =.4048, the real eutro flux is the distributio i the physical system [6, 7, 1]..405 φ ( r ) = φ MAX J 0 r R (5) Determiatio of the maximum eutro flux The simplest form of the eutro balace equatio is called Bessel Differetial Equatio. Oe of the steady state critical ideal reactor geometries that ca be treated via aalytical meas is a log cylidrical core model, as illustrated i Fig.1. All the adjectives used to describe the system are eeded to reduce the geeral, very complicated, particle balace equatio ito a form that ca be treated aalytically. The ifiite homogeeous descriptio implies that the axial height is large relative to the radius ad that the eutro desity i the axial ad azimuthally directios is egligible, leavig a fuctioal depedece material properties are costat throughout the system. Figure 1. Basic geometry for the cylidrical critical reactor model A ad B. These coditios suggest that the variatio of the oly oe variable or φ( r, θ, z) φ( r), where φ is the symbol used to represet the eutro flux. Also, symmetry i the system suggests that the eutro populatio will be the largest i the ceter of the reactor, which implies that the flux gradiet is zero at r =0 critical reactor boudary coditios as show i Fig.1.

5 Neutroic calculatios for reactors 19 dφ r = 0, dr r = 0 = 0 ad at r = R, φ ( R) = 0, i mathematical terms, this system is a d order homogeeous boudary. I other words, K multiplicatio coefficiet is the ratio of the previous eutro geeratio to the ext eutro geeratio. Neutro flux (φ ) i the reactor chages accordig to the geometry of the reactor. The reaso why Bessel differetial equatios are used i the study is that the Bessel differetial equatios are relatively easier to solve ad depedet o boudary value data tha other mathematical equatios ad that other equatio modelig. Offer advaced mathematical solutios (itegral + differetial). For critical reactor, P is the possibility of ot leakig ad expressed as followig formula [4, 5]. P K eff = = K Rate of eutro absorptio Rate of eutro absorptio + Rate of eutro loss (6) As the umber of eutros will be steady-state i fiite medium whe the reactor is totally critical which is studied, followig solutio ca be show accordig to oe-group diffusio method [3] D 0 (7) ϕ ϕ+ S = a I the equatio, the first term refers to rate of the productio of eutros i Volume, ad the secod term refers to rate of absorptio of eutros i volume, ad the third term refers to the rate of leakage of eutros from Volume. As the rate at which eutros are lost by absorptio per cm 3 / sec is equal to S = K φ (8) a Thus, by usig diffusio coefficiet, we ca fid L, diffusio legth. I K 1 this situatio, the formula B = α is called as the bucklig of the reactor or K L. B L α the formula B = = 1 ca be used. This equatio is geerally called as Critical Equatio. For the ifiite reactor, the formula must be the followig B =.405 ( ) R

6 0 Aybaba Haçerlioğulları If the reactor is the compositio of the core ad the reflector, the 9 th equatio ca be defied i two differet ways. These equatios are kow as oe-group modelig Critical specificatios of some moderators which are used i the reactor are show o the Table 1. Table 1. Critical specificatios of some moderators [5]. A separate calculatio usig the total power of the reactor P, which is a desig criterio, should be carried out. I practice, the total power for a oboilig reactor ca be easily determied by measurig the flow rate of the coolat as well as its ilet ad outlet temperatures to the reactor core. Let us determie ow the maximum thermal eutro flux for a homogeeous reactor equipped with axial ad radial reflectors. Usig Eq-7, we ca write the total power of the reactor as P = E Σ φ( r dv R f ) where dv is the differetial volume elemet. I the view of the geometry of the problem, dv is the give by dv=πrdr for the ifiite cylider. This itegral ca be evaluated usig the formulas below P = π E Σ R A J (.405 ) /.405 = 1,35 E Σ R f 1 1 R f R A Fial expressio for the flux (Ø(r)) is the. 0,738P φ ( r) = J 0 (.405 r / R) E R R f For the fiite cylider, the bucklig of the reactor ad fiite flux is show usig the formulas below,.405 π B = ( ) + ( ) R H

7 Neutroic calculatios for reactors 1 π. z ( z) φmaxj 0( Br) cos( ) H φ = (9) Numerical calculatios I this study, by beefittig from Numerical Bessel Equatio, critical calculatios are calculated usig the four differet variatios of uraium material. I Table, microscopic cross sectio values are show as bar type. Table. Microscopic cross sectio values of fuel material [5]. For K critical calculatio, K EFF values are give i Table 4. The umber of the total eutro umber crossig over the surface of cylider. Durig the study, we used Matemice-7 Wolfram ad Numeric Bessel s programmers. The values are i accordace with those from Mote Carlo Mcp\code system of ENDF-V-VI (RSIC computer code collectio Mcp-4b). I Fig., K EFF values are reflected i accordace with crash*legth of the way/ absorptio of active track umber. The results are compared usig the K EFF results which are gaied usig aalytic Bessel fuctio with those from MCNP4b/ ENDF-V-VI [13, 14]. Table 3. The Flux Distributio (H=4.81, R=.405).

8 Aybaba Haçerlioğulları Table 4. K EFF, active track umber. 1,05 1,00 0,95 K effect 0,90 0,85 0,80 Bessel Calc. Mote Carlo Calc. 0, Track umber Figure. K EFF values compared track umber. Figure 3. Flux rate compariso for ifiite cylidrical reactor.

9 Neutroic calculatios for reactors 3 I Table 3, H=1 ad R=1are cosidered to be a referece, whe we compare the flux value i fiite ad ifiite cylider, as see o Fig.3, =0 Bessel fuctio is equal to at the C critic =.405. This shows that the first critical legth does t chage. However, as see i Fig.3, aother harmoics of Bessel, flux rates ca chage i these legths (=1,, 3..) That is, flux value depeds o geometrical structure ad used fusio fuel variatio. I Table 4, active track umber of the reactor compared with those of calculated (K EFF ) values. As see i Fig., betwee the track umbers 0-0 ad 60-80, Bessel calculatios ad Mote Carlo Calculatios are almost the same ad K EFF of the reactor is Fig.1. Coclusios ad discussio I the study, the reaso why Bessel differetial equatios are used i the study is that the Bessel differetial equatios are relatively easier to solve ad depedet o boudary value data tha other mathematical equatios ad that other equatio modelig offer advaced mathematical solutios (itegral + differetial). Bessel differetial equatios are secod order ordiary differetial equatios ad they offer solutios i the cylidrical, spherical ad polar coordiates easily ad also required physical parameters i the reactor ca easily be obtaied through the use of Bessel differetial equatios. Bessel differetial equatios are used for the calculatios of eutro flux (φ) ad criticality coefficiet (K) ad cylidrical geometric structure is take ito accout as the reactor geometry. Bessel differetial equatio of higher order ca be expressed by Bessel fuctio of lower orders. Keepig the first terms i the series expasios the behavior of a Bessel fuctio at small or large ca be captured ad expressed as elemetary fuctios which are much easier to be uderstood ad calculated tha the more abstract symbols [8]. K, multiplicatio coefficiet is the ratio of the previous eutro geeratio to the ext eutro geeratio. Neutro flux (φ) i the reactor chages accordig to the geometry of the reactor (the type of the fuel used ad physical properties of the reactor. Several mathematical equatios are required to obtai criticality coefficiet (K) regardig the chages metioed. The importat equatios ad theories used are as follows; Bessel differetial equatios, the Mote Carlo method, geeral diffusio equatios, Fourier ad Taylor Series, the perturbatio theory. Refereces 1. Gray, A., et al. 199, Bessel fuctios Lodo.. Watso, G.N. 19, Theory of Bessel fuctios, Cambridge.

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