My first LaTeX equation!

Baru aja nyobain nulis persamaan pake LaTeX, n ini dia persamaan LaTeX pertamaku!!

Sungguh saat2 yg bersejarah.. hihihi…

Persamaan apa hayo..??

Helium density in pellet-clad gap of nuclear fuel rod

Kemarin2 lg butuh data densitas Helium di dalem fuel rod nuklir, heran, udah kesana-kemari keliaran pake google ko ga ktemu2 yah.. ada yg punya datanya kah..? perlu neh..

Hmm.. apa itu termasuk data “rahasia” yah.. hmm aneh..

Yah daripada kerjaan mampet gara2 ga ada data, kepaksa bikin perkiraan sendiri deh.. sy pake persamaan gas ideal yg paling sederhana aja^[1]:

Harusnya sih persamaan gas ideal lumayan bagus untuk dipake ngitung densitas Helium, kan Helium gas monoatomik, jadi harusnya kelakuannya ya mirip sama gas ideal kan.. biar yakin benchmark dulu lah, sama data Helium pada keadaan STP^[2]:

• T = 273.15 K
• p = 101325 Pa

Nilai parameter lainnya:

• M Helium = 4.002602E-3 kg/mol ^[3]
• R = 8.314472 J/(K.mol) ^[1]

Trus itung deh..

Klo liat di wikipedia^[3], densitas Helium (STP) tu 0.1786 g/L, wah ternyata cocok ma perhitungan gas ideal!! siip!!!

Berarti sekarang saya bisa buat perkiraan yg lumayan akurat tentang densitas Helium di dalam fuel rod dong ya! data2 yang dipake:

• p = 3 MPa (dapet dr sebuah jurnal)
• T = 650 K (ada itungan nya, kpn2 di post deh..)

Itung..

Hmm.. berarti densitas Helium di dalam pellet-clad gap tu sekitar 0.00222185 g/cc dong ya..

Bener ga yah..? hmm…

Referensi:

1. http://en.wikipedia.org/wiki/Ideal_gas_law
2. http://en.wikipedia.org/wiki/Standard_conditions_for_temperature_and_pressure
3. http://en.wikipedia.org/wiki/Helium

Neutronic Study of the IRIS Reactor Core

Disclaimer:
Information presented in this article are based on publicly available data of the IRIS reactor project, as properly cited from the original source. This article is NOT part of the official IRIS project led by Westinghouse. For more reliable information, the reader should refer to any official websites and information sources of the IRIS project and/or the IRIS consortium. All trademarks and registered trademarks shown in this article are the property of their respective owners.

Syeilendra Pramuditya (シエイレンドラ  -  プラムディティア)
Energy Engineering Division
Research Laboratory for Nuclear Reactors
Tokyo Institute of Technology
JAPAN

Abstract

The neutronic analysis of the integral primary system PWR has been performed. The reactor analyzed is a modular, integral, light water cooled, low-to-medium power (~1000 MWth) reactor, which emphasizes proliferation resistance and enhanced safety. The comprehensive neutronics code system SRAC was used to develop a full-core model of the reactor core, and cross section data generated from JENDL-3.2 nuclear data library were used. The calculation results show that the core design has a relatively high power peaking factor, which is a disadvantage in terms of safety and thermal hydraulic performance. The reactivity coefficients are found to be negative, which indicates that the reactor core shows inherent safety features.

1. Introduction

Over the past decades, there have been several projects involving the integral reactor concept. Advantages of integral reactors include increased safety, more compact layout and reduced construction costs. Increased safety for integral reactors comes from the following design features: low power density, passive safety features of the containment, and of course the very key feature of the integral core configuration – no large pipe penetration into the reactor vessel. The elimination of all reactor coolant piping removes that piping from any loss of coolant accident (LOCA) possibility. The compact plant layout is derived primarily from the elimination of the reactor coolant piping and by placing equipment normally external to the RPV such as the steam generator (S/G), reactor coolant pump (RCP), and pressurizer (PZR) within the vessel. The elimination of the requirements for large on-site welds on reactor coolant piping, as well as the modular configuration of the reactor vessel assembly, is expected to lead to a shorter construction time. This, in conjunction with the overall smaller physical footprint, is expected to lead to lower construction costs. This work describes the neutronic calculation of the integral primary system PWR core, without thermal hydraulic feedback.

2. Reactor description

The reactor analyzed is the reference design of a modular, integral, light water cooled, low-to-medium power (~350 MWe) reactor, which emphasizes proliferation resistance and enhanced safety, currently known as the International Reactor Innovative and Secure or the IRIS reactor (Carelli et al., 2004; Carelli, 2009). A distinguishing characteristic of the IRIS reactor is the integral design: The steam generators (S/Gs), reactor coolant pumps (RCPs) and pressurizer (PZR) are all contained within the reactor pressure vessel (RPV) (Carelli, U.S. DOE Final Technical Progress Report-STD-ES-03-40, 2003). This configuration is different from a conventional PWR where the S/Gs, PZR, and RCPs are all mounted outside of the RPV, connected by reactor coolant piping of varying diameter, all located within a containment. Summary of the IRIS reference design is shown in Table 1.

Table 1. IRIS reference design

Nominal reload strategy	Two-batch
Number of fresh FAs	40–45
Actual number of batches	1.98–2.22
FAs with 4.95% 235U enrichment	40–45
FAs with reduced 235U enrichment	-
Cycle length (Years)	3.0–3.5
Average discharge burnup (MWd/tU)	48–53,000
Lead rod average burnup (MWd/tU)	< 62,000

More detailed description and technical specification of the IRIS reactor could be found in the listed references.

3. Methodology

3.1. Reactor simulation codes

The methodology comprises two major parts, i.e. generation of group constants for various core regions, and whole core calculations. The Japanese Standard Reactor Analysis Code, the SRAC code system (Tsuchihashi et al., 1986), was used to perform the cell and whole core calculation. The SRAC code system was designed and developed at the Japan Atomic Energy Research Institute (JAERI, now JAEA) to permit overall neutronics calculation for various types of thermal reactors. The system covers generation of group constants, cell and core calculations including burnup. The SRAC code system is composed of the collision probability method (CPM) cell calculation code, named PIJ, and several whole core calculation codes. For the current study, we use the CITATION code for whole core calculation. The CITATION code evaluates the neutron multiplication factor, k-eff, by solving the neutron flux eigen-value problem by using finite-difference multigroup neutron diffusion theory approximation of the neutron transport equation, by direct iteration method. The code computes the effective multiplication factor, flux and power profiles in the core by using group constants generated by the PIJ code. In addition to this, the code can also be used to calculate reactivity feedback coefficients, effective delayed neutron fraction, and prompt neutron generation time (Fowler et al., 1971). Detailed description of these codes could be found in the listed references.

3.2. Neutron energy group

The JENDL-3.2 evaluated nuclear data library (Shibata et al., 1990) was used for CPM cell calculation and to generate the few group constants. Four energy groups were used in this work (Table 2).

Table 2. Energy group structure

No.	EU (eV)	EL (eV)	Group type
1	1E+7	6.74E+4	Fast
2	6.74E+4	130	ResolvedResonance
3	130	2.38	Unresolved Resonance
4	2.38	1E-5	Thermal

3.3. Geometrical modeling

3.3.1. Modeling of the fuel cell

The reference core design of the IRIS reactor use the Westinghouse standard fuel assembly for PWR (Carelli, 2009), in which the fuel rods are arranged in 17×17 rectangular array (Carelli et al., 2004). Hence, the most appropriate geometrical model for cell calculation is the square cell, with several concentric circles representing regions for fuel, cladding, and moderator (Fig. 1).

Figure 1. Fuel cell modeling

3.3.2. Modeling of the reactor core

The IRIS reactor core consists of 89 fuel assemblies (FAs). Each fuel assembly contains 264 fuel rods and 25 control elements, arranged in 17×17 matrix (Carelli et al., 2004). The geometrical model for whole core calculations which was used in this work is mainly based on the work of Jecmenica et al., 2003, in which the core is modeled in 3D-XYZ geometry (Fig. 2). Active core height is 426.7 cm with uniform enrichment of 4.95 w/o 235U. The total core height, including top and bottom axial reflector regions, is 506.7 cm. Radial reflector was modeled using reflector cells of the same dimensions as FA.

Figure 2. Reactor core modeling

3.4. Core depletion analysis

The core depletion calculation can be divided into two main parts: (a) solution of the isotopic depletion equation, which requires information of the neutron flux; and (b) solution of the static multigroup diffusion equation for the neutron flux. Hence, we decoupled those calculations such that the depletion equations are solved over a specified time interval in which the power is assumed to be constant. Then, at the end of each time interval, the depleted densities and local average power level are used to calculate new group constants, and again, the multigroup diffusion equation is solved to determine a new neutron flux distribution and power distribution for the next time interval (Duderstadt and Hamilton, 1976; Zaki Su’ud, 2008).

3.5. Calculation of reactivity coefficients

Reactivity coefficients were determined by performing a sequence of static criticality calculations, using the CITATION code, to calculate the core effective multiplication factor, k-eff, for different parameters under consideration, i.e. fuel temperature, coolant temperature, and void fraction, as explained by Muhammad and Majid, 2008; Muhammad and Majid, 2009; and Duderstadt and Hamilton, 1976. The change in reactivity was calculated as follows (IAEA TECDOC-643, 1992):

(eq. 1)

Where k₀ is k_eff at the reference condition (888.586 K), and k₁ is k_eff at a specified condition. Reactivity coefficient is defined as change in reactivity for given change in parameter (Ott and Neuhold, 1985), and generally expressed as:

(eq. 2)

Here is any parameter that affects reactivity, and is corresponding change in reactivity.

4. Results and discussions

4.1. Criticality calculation

The group constants and infinite multiplication factor, k-inf, were calculated as a function of P/D (or H₂O/U) at a single calculational cell. In this work, P/D was increased from 1.05285 (corresponding to H₂O/U=0.59671) to 3.5797 (corresponding to H₂O/U=22.21507), while keeping all other parameters unchanged. The results are given in Table 3.

Table 3. k-inf as a function of P/D

Pitch (mm)	P/D	k-inf
10	1.05285	1.137346
11	1.15814	1.269726
12	1.26342	1.358791
12.54	1.32028	1.395168
14	1.47399	1.462823
16	1.68457	1.505699
18	1.89514	1.515622
20	2.10571	1.504098
25	2.63213	1.423757
30	3.15856	1.30966
32	3.36913	1.260744
34	3.5797	1.211802

The value of k-inf as a function of fuel pitch is plotted in Fig. 3.

Figure 3. k-inf as a function of P/D

The underlined values in Table 3 and the red dot in Figure 3 are calculation results for the current reference core design at its operating condition. Figure 3 shows that for current reference core, reactivity decreases as P/D decreases, this is corresponding to the decrease in reactivity as coolant density decreases, or as coolant temperature increases, which is a good point for safety performance.

4.2. Core power distribution

Power distribution and peaking factor are important parameters in terms of safety and thermal hydraulic performance. The maximum power density is found from the calculation at location (35, 1, 55), which is physically at the center of the core. The maximum power density is 175.225 Watt/cc, therefore, the calculated power peaking factor is 3.418.

4.3. Reactivity coefficients

4.3.1. Fuel temperature coefficient of reactivity

To calculate the coefficients for change of fuel temperature, only the fuel temperature was varied from 848.586 K to 948.586 K. The results of reactivity calculation for various fuel temperatures are given in Table 4 and plotted in Figure 4.

Table 4. Fuel temperature coefficient of reactivity

Tfuel (K)	keff
848.586	1.362786	0.266209	0.00087
868.586	1.361958	0.265763	0.000423
888.586	1.361173	0.26534	0
908.586	1.360401	0.264923	-0.00042
928.586	1.359604	0.264492	-0.00085
948.586	1.358848	0.264083	-0.00126

Figure 4. Fuel temperature coefficient of reactivity

The underlined values in Table 4 and the red dot in Figure 4 are calculation results for the current reference core design. Table 4 and Figure 4 show that the core reactivity decreases as the fuel temperature increases, this is due to Doppler broadening effect on the absorption cross section (Duderstadt and Hamilton, 1976), in which the energy range of neutrons to be absorbed in resonance is increased. Therefore, more neutrons are absorbed by the resonance, this will eventually lead to the decrease of core reactivity.

The reactivity coefficient for fuel temperature change from 848 K to 948 K, denoted as , is then determined as the slope of the curve in Figure 4:

(eq. 3)

Therefore, .

4.3.2. Moderator temperature coefficient of reactivity

To calculate the coefficients for change of moderator temperature, only the moderator temperature was varied from 544 K to 644 K. The results of reactivity calculation for various fuel temperatures are given in Table 5 and plotted in Figure 5.

Table 5. Moderator temperature coefficient of reactivity

Tmod (K)	keff
544	1.361577	0.265558	0.000218
564	1.361378	0.26545	0.000111
584	1.361173	0.26534	0
604	1.360987	0.265239	-0.0001
624	1.360788	0.265132	-0.00021
644	1.360583	0.265021	-0.00032

Figure 5. Moderator temperature coefficient of reactivity

Figure 5 shows that the core reactivity decreases as the moderator temperature increases, this is because an increase in moderator temperature, keeping the density constant, will lead to a hardened neutron spectrum, resulting in increased resonance absorption cross section. The hardened spectrum will cause an increase in the capture-to-fission ratio of 235U, which means a decrease in eta value, and hence a decrease in core reactivity.

The reactivity coefficient for moderator temperature change from 544 K to 644 K, denoted as , is then determined as the slope of the curve in Figure 5:

(eq. 4)

Therefore, .

4.3.3. Void coefficient of reactivity

To calculate the coefficients for change of void fraction in the coolant, the void fraction was varied from 0% to 10%. The results of reactivity calculation for various coolant void fraction are given in Table 6 and plotted in Figure 6.

Table 6. Void coefficient of reactivity

Void (%)	keff
0	1.361173	0.265340	0
2	1.356340	0.262722	-0.00262
4	1.351344	0.259996	-0.00534
6	1.345924	0.257016	-0.00832
8	1.340822	0.254189	-0.01115
10	1.335250	0.251077	-0.01426

Figure 6. Void coefficient of reactivity

Figure 6 shows that the core reactivity decreases as the coolant void fraction increases, this is because void formation in the coolant will decrease the average density of the coolant, and because coolant also acts as moderator in thermal reactor, this will lead to a spectrum hardening, and further causes an increase in resonance cross section, and hence reduces the core reactivity.

The reactivity coefficient for coolant void fraction from 0% to 10%, denoted as , is then determined as the slope of the curve in Figure 6:

(eq. 5)

Therefore, .

5. Conclusions

The calculation results show that the core has power peaking factor of 3.418, which is relatively high and could be considered as a disadvantage in terms of safety and thermal hydraulic performances. The fuel temperature coefficient of reactivity, coolant temperature coefficient of reactivity, and void coefficient of reactivity were all found to be negative. The Doppler coefficient was found to be more negative than the moderator temperature coefficient, which means that the fuel temperature change plays more roles on the inherent safety feature of the reactor core.

References

1. Carelli, M.D., 2009. The exciting journey of designing an advanced reactor. Nuclear Engineering and Design 239, 880-887.
2. Carelli, M.D., Conway, L.E., Oriani, L., Petrovic, B., Lombardi, C.V., Ricotti, M.E., Barroso, A.C.O., Collado, J.M., Cinotti, L., Todreas, N.E., Grgic, D., Moraes, M.M., Boroughs, R.D., Ninokata, H., Ingersoll, D.T., Oriolo, F., 2004. The design and safety features of the IRIS reactor. Nuclear Engineering and Design 230, 151–167.
3. Duderstadt, J.J., Hamilton, L.J., 1976. Nuclear Reactor Analysis. Wiley, New York.
4. Fowler, T.B., Vondy, D.R., Cunningham, G.W., 1971. Nuclear Reactor Core Analysis Code-CITATION, USAEC Report ORNL-TM-2496, Revision 2. Oak Ridge National Laboratory.
5. IAEA, 1992 IAEA, 1992. Research Reactor Core Conversion Guide Book. IAEA-TECDOC-643. International Atomic Energy Agency, Vienna.
6. Ječmenica, R., Trontl, K., Pevec, D., Grgić, D., 2003. IRIS Core Criticality Calculations. In: Int. Conf. Nuclear Energy for New Europe, Portorož, Slovenia. September 8-11. pp 105.1-105.5.
7. Muhammad, F., Majid, A., 2008. Reactivity feedback coefficients of a material test research reactor fueled with high-density U₃Si₂ dispersion fuels. Nuclear Engineering and Design 238, 2583-2589.
8. Muhammad, F., Majid, A., 2009. Reactivity feedbacks of a material test research reactor fueled with various low enriched uranium dispersion fuels. Annals of Nuclear Energy, In Press, Corrected Proof, DOI: 10.1016/j.anucene.2009.03.006.
9. Ott, K.O., Neuhold, R.J., 1985. Introductory Nuclear Dynamics. American Nuclear Society, La Grange Park, Illinois, USA.
10. Shibata et al., 1990 Shibata, K. et al., 1990. Japanese Evaluated Nuclear Data Library, Version-2, JENDL-3, JAERI 1319, JAERI, Tokai-mura, Naka-gun, Ibaraki-ken, pp. 319-11, Japan.
11. Tsuchihashi et al., 1986 Tsuchihashi, K. et al., 1986. Revised SRAC Code System: JAERI Thermal Reactor Standard Code System for Reactor Design and Analysis, JAERI 1302, Japan.
12. Zaki Su’ud, 2008. Neutronic performance comparation of MOX, nitride and metallic fuel based 25–100 MWe Pb–Bi cooled long life fast reactors without on-site refuelling. Progress in Nuclear Energy 50, 276-278.

Calculation code for nuclear cross section

Code package:

• The code [ncs.doc], written in Delphi (download, rename to “ncs.zip”, then extract)

Calculation code for spherical nuclear reactor

The governing equation being used is the steady state neutron diffusion equation:

Numerical schemes being used are:

• Central finite difference for flux calculation
• Gauss-Siedel and S.O.R for flux calculation
• Power method for criticality calculation

Code package:

• Manual (English)
• Code [snr.doc], written in Delphi (download, rename to “snr.zip”, then extract)

Flowchart of the code:

Some previews: