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Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2013, Article ID 487604, 6 pageshttp://dx.doi.org/10.1155/2013/487604

Research ArticleModification of Neutron Kinetic Code forPlate Type Fuel Nuclear Reactor

Salah Ud-Din Khan,1 Shahab Ud-Din Khan,2 and Yang Zhifei3

1 King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia2 Institute of Plasma Physics, Chinese Academy of Sciences, P.O. Box 1126, Hefei, Anhui 230031, China3 College of Nuclear Science and Technology, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Salah Ud-Din Khan; [email protected]

Received 16 July 2013; Revised 2 September 2013; Accepted 17 September 2013

Academic Editor: Wael H. Ahmed

Copyright © 2013 Salah Ud-Din Khan et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The research is conducted on themodification of neutron kinetic code for the plate type fuel nuclear reactor. REMARK is a neutronkinetic code that works only for the cylindrical type fuel nuclear reactor. In this research, our main emphasis is on the modificationof this code in order to be applicable for the plate type fuel nuclear reactor. For this purpose, detailed mathematical studies havebeen performed and are subjected to write the program in Fortran language. Since REMARK code is written in Fortran language,so we have developed the program in Fortran and then inserted it into the source library of the code. The main emphasis is on themodification of subroutine in the source library of the code for hexagonal fuel assemblies with plate type fuel elements in it. Thenumber of steps involved in themodification of the code has been included in the paper.The verification studies were performed byconsidering the small modular reactor with hexagonal assemblies and plate type fuel in it to find out the power distribution of thereactor core. The purpose of the research is to make the code work for the hexagonal fuel assemblies with plate type fuel element.

1. Introduction

Real-time multigroup advanced reactor kinetics (REMARK)code is developed by the GSE power system used for the real-time simulation of nuclear reactor core. This code utilizestwo-group, 3D time-dependent diffusion theory in the formof finite difference equations to simulate nuclear reactorcore. This code has two neutron energy groups to accuratelysimulate the characteristics of fast and thermal neutron undernormal and abnormal operating conditions of the nuclearreactor. The core geometry can be modeled as 3D mesh cellstructure under the limited capacity of computer resourcesfor real-time applications.Mesh cell sizes are chosen to ensurethat the material properties within cell are homogenous orheterogeneous depending on the fuel, reflector, shielding, andso forth. In the REMARK code, there is improved quasistaticsolution for obtaining the flux distribution and six delayedneutrons to be calculated in each mesh cell. Reactivity feed-back is based on the core thermal hydraulic conditions likefuel temperature, coolant temperature, moderator density,

and void fraction. This code can calculate the reactivityeffect of control rod movement and soluble boron. It canalso give the exact neutron flux readings at in-core andex-core detector points and utilizes neutron cross-sectionfor neutronic calculations [1, 2]. This is 3D neutron kineticcode, but it cannot examine the special kind of geometrysuch as hexagonal fuel assemblies. Therefore, the currentresearch is focused on the modification of the code to makeit applicable for the hexagonal fuel assemblies. The flowdiagram of REMARK code is given in Figure 2.

Themain concern of our research is tomodify the nuclearreactor core for hexagonal fuel assemblies. The researchon the new developed hexagonal fuel assembly design hasalready been done on the supercooled water reactor (SCWR)to show the moderation capability for the hexagonal fuelassemblies [3, 4].

The benchmark studies of different calculation schemesrelated to reactor design, collision probability, power recon-struction, and so forth were investigated by ten differentinstitutes. The focus was to find out the difficulties caused by

2 Science and Technology of Nuclear Installations

AB

C

D E

FR

a a

Δx

O

a

Figure 1: Hexagonal plan.

different neutron data reductionmethods and for the homog-enization of the fuel assemblies [5, 6].

2. Reactor Core Simulation Model

For the hexagonal fuel assemblies we consider the hexagon𝐴𝐵𝐶𝐷𝐸𝐹 with sides “𝑎” and center “O” as shown in Figure 1.The distance between opposite sides is denoted by “Δ𝑥” andthe height is denoted by “Δ𝑧” [3]. The area and volume of thehexagon can be written as

𝑆𝐴𝐵𝐶𝐷𝐸𝐹

=√3

2(Δ𝑥)2,

𝑉𝐴𝐵𝐶𝐷𝐸𝐹

= 𝑆 × Δ𝑧 =√3

2(Δ𝑥)2Δ𝑧.

(1)

The one-sided area of hexagon can be justified as

𝑆1= 𝑎Δ𝑧 =

Δ𝑥

√3× Δ𝑧 =

√3

3Δ𝑥Δ𝑧. (2)

We have considered eight sections of adjacent block of hexa-gon and calculated the neutron diffusion equations.Thereforethe leakage of the neutron diffusion equation integral withvolume of the node is given as

∫(𝑖,𝑗,𝑘)

∇ ⋅ 𝐷∇𝜙𝑑𝑉 = ∫𝑠

(𝐷𝜕𝜙

𝜕𝑛)𝑑𝑆 = ∑

𝑘

∫𝑆𝑘

𝐷𝜕𝜙

𝜕𝑛𝑑𝑆

= (𝑇11 + 𝑇12 + 𝑇13 + 𝑇14

+𝑇15 + 𝑇16 + 𝑇17 + 𝑇18) ,

(3)

where T11 can be calculated as

𝑇11 = 𝐷𝑖+1,𝑗,𝑘

𝜙𝑖+1,𝑗,𝑘

− 𝜙𝑖+1/2,𝑗,𝑘

Δ𝑥/2

√3Δ𝑦Δ𝑧

3, (4)

where 𝜙𝑖+1/2,𝑗,𝑘

is neutrons flux density at the interface.

Since the net flux of neutrons at the adjacent interface iscontinuous, so

𝐷𝑖+1/2,𝑗,𝑘

𝜙𝑖+1/2,𝑗,𝑘

− 𝜙𝑖,𝑗,𝑘

Δ𝑥/2= 𝐷𝑖+1,𝑗,𝑘

𝜙𝑖+1,𝑗,𝑘

− 𝜙𝑖+1/2,𝑗,𝑘

Δ𝑥/2,

𝜙𝑖+1/2,𝑗,𝑘

=

𝐷𝑖,𝑗,𝑘

𝜙𝑖,𝑗,𝑘

+ 𝐷𝑖,𝑗,𝑘

𝜙𝑖,𝑗,𝑘

𝐷𝑖,𝑗,𝑘

+ 𝐷𝑖+1,𝑗,𝑘

.

(5)

Substituting (5) in (4) we get

𝑇11 =

2𝐷𝑖,𝑗,𝑘

𝐷𝑖+1,𝑗,𝑘

(𝐷𝑖,𝑗,𝑘


) Δ𝑥

(𝜙𝑖+1,𝑗,𝑘


)√3Δ𝑦Δ𝑧

3,

𝑇11

𝑉=

2𝐷𝑖,𝑗,𝑘


(𝐷𝑖,𝑗,𝑘


) Δ𝑥

(𝜙𝑖+1,𝑗,𝑘


)

×

(√3Δ𝑦Δ𝑧) /3

((√3/2) Δ𝑥Δ𝑦Δ𝑧)

=

4𝐷𝑖,𝑗,𝑘


3 (𝐷𝑖,𝑗,𝑘


) Δ𝑥2(𝜙𝑖+1,𝑗,𝑘


) .

(6)

Similarly T12, T13, T14, T15, and T16 can be calculated as T11and have the same value as T11. T17 is an integral on the top,whereas T18 is an integral on the bottom and can be writtenas

𝑇17 = 𝐷𝑖,𝑗,𝑘

𝜙𝑖,𝑗,𝑘+1/2


Δ𝑧/2

√3

2Δ𝑥Δ𝑦, (7)

where 𝜙𝑖,𝑗,𝑘+1/2

is neutrons flux density at the interface.Here, the net neutrons flux at the adjacent interface is

continuous; therefore

𝐷𝑖,𝑗,𝑘



Δ𝑧/2= 𝐷𝑖,𝑗,𝑘

𝜙𝑖,𝑗,𝑘+1

− 𝜙𝑖,𝑗,𝑘+1/2

Δ𝑧/2,


=

𝐷𝑖,𝑗,𝑘

𝜙𝑖,𝑗,𝑘

+ 𝐷𝑖,𝑗,𝑘+1

𝜙𝑖,𝑗,𝑘+1

𝐷𝑖,𝑗,𝑘


.

(8)

Substituting (8) in (7) we get

𝑇17 =

2𝐷𝑖,𝑗,𝑘

𝐷𝑖,𝑗,𝑘+1

(𝐷𝑖,𝑗,𝑘


) Δ𝑧

(𝜙𝑖,𝑗,𝑘+1


)

×√3

2Δ𝑥Δ𝑦,

𝑇17

𝑉=

2𝐷𝑖,𝑗,𝑘


(𝐷𝑖,𝑗,𝑘


) Δ𝑧

(𝜙𝑖,𝑗,𝑘+1


)

×

(√3/2) Δ𝑥Δ𝑦

((√3/2) Δ𝑥Δ𝑦Δ𝑧)

,

𝑇17

𝑉=

2𝐷𝑖,𝑗,𝑘


(𝐷𝑖,𝑗,𝑘


) Δ𝑧2(𝜙𝑖,𝑗,𝑘+1


) ,

(9)

where T18 will have the same value as T17.

Science and Technology of Nuclear Installations 3

REMARK code

CRDCY1 CRDCY4 CRDCYS

CNTLRD THTOCR XSECT

Cross-section for each mesh

KEFF

AMPL

CRTOTH

CRTONM

CRTONI

TH propertiesControl rod position

Modified

Modified

Modified

Control module

Shape1 Shape2

Power

Poison

DELAYN

Figure 2: Flow diagram of REMARK code with modifications.

Now we will calculate the extrapolated length, that is, 𝑑 =

Δ𝑥extd, to be considered as core boundary conditions:

𝐽 =

−𝐷𝑖,𝑗,𝑘

𝜙𝑖,𝑗,𝑘

𝑑 + Δ𝑥/2=

−2𝐷𝑖,𝑗,𝑘

𝜙𝑖,𝑗,𝑘

2𝑑 + Δ𝑥,

∫

(𝑖,𝑗,𝑘)

∇ ⋅ 𝐷∇𝜙𝑑𝑉 = ∫𝑆

(𝐷𝜕𝜙

𝜕𝑛)𝑑𝑆 = ∑

𝑘

∫𝑆𝑘

𝐷𝜕𝜙

𝜕𝑛𝑑𝑆,

𝑇1 = −𝐽 ∗ 𝑆

𝑉

=

2𝐷𝑖,𝑗,𝑘

𝜙𝑖,𝑗,𝑘

2𝑑 + Δ𝑥×

(√3Δ𝑦Δ𝑧/3)

(√3/2Δ𝑥Δ𝑦Δ𝑧)

=

4𝐷𝑖,𝑗,𝑘

3 (Δ𝑥2 + 2Δ𝑥Δ𝑥extd)𝜙𝑖,𝑗,𝑘

.

(10)

In three-dimensional hexagonal grid, 𝐷 is called diffusioncoefficient and ∇⋅𝐷∇Ψ is the leakage term in four directions.The leakage term is calculated in four directions as

∇ ⋅ 𝐷∇𝜙 =1

𝑉∫

(𝑖,𝑗,𝑘)

∇ ⋅ 𝐷∇𝜙𝑑𝑉

= (𝑇11 + 𝑇12 + 𝑇13 + 𝑇14 + 𝑇15

+𝑇16 + 𝑇17 + 𝑇18) × 𝑉−1

,

∇ ⋅ 𝐷∇𝜙 = 𝑎−

𝑖𝜙𝑖−1,𝑗,𝑘

+ 𝑎+

𝑖𝜙𝑖+1,𝑗,𝑘

+ 𝑎−

𝑗𝜙𝑖,𝑗−1,𝑘

+ 𝑎+

𝑗𝜙𝑖,𝑗+1,𝑘

+ 𝑎−

𝑗2𝜙𝑖−1,𝑗−1,𝑘

+ 𝑎+

𝑗2𝜙𝑖+1,𝑗+1,𝑘

+ 𝑎−

𝑘𝜙𝑖−1,𝑗,𝑘

+ 𝑎+

𝑘𝜙𝑖+1,𝑗,𝑘

− (𝑎𝑖+ 𝑎𝑗+ 𝑎𝑗2

+ 𝑎𝑧) 𝜙𝑖,𝑗,𝑘

.

(11)

For the middle section block grid,

𝑎+

𝑖=

4𝐷𝑖,𝑗,𝑘




) Δ𝑥2,

𝑎−

𝑖=

4𝐷𝑖,𝑗,𝑘

𝐷𝑖−1,𝑗,𝑘


+ 𝐷𝑖−1,𝑗,𝑘

) Δ𝑥2,

𝑎+

𝑘=

2𝐷𝑖,𝑗,𝑘


(𝐷𝑖,𝑗,𝑘


) Δ𝑧2,

𝑎−

𝑘=

2𝐷𝑖,𝑗,𝑘

𝐷𝑖,𝑗,𝑘−1

(𝐷𝑖,𝑗,𝑘

+ 𝐷𝑖,𝑗,𝑘−1

) Δ𝑧2.

(12)

Similarly, 𝑎+𝑗, 𝑎−𝑗, 𝑎+𝑗2, and 𝑎

−

𝑗2will have the same values.


For the grid block boundary section:

X direction left border

𝑎+

𝑖=

4𝐷𝑖,𝑗,𝑘




) Δ𝑥2,

𝑎−

𝑖=

4𝐷𝑖,𝑗,𝑘

3 (Δ𝑥2 + 2Δ𝑥Δ𝑥extd);

(13)

X direction right border

𝑎+

𝑖=

4𝐷𝑖,𝑗,𝑘

3 (Δ𝑥2 + 2Δ𝑥Δ𝑥extd),

𝑎−

𝑖=

4𝐷𝑖,𝑗,𝑘

𝐷𝑖−1,𝑗,𝑘


+ 𝐷𝑖−1,𝑗,𝑘

) Δ𝑥2;

(14)

Z direction border at the bottom

𝑎+

𝑘=

2𝐷𝑖,𝑗,𝑘




) Δ𝑧2,

𝑎−

𝑘=

2𝐷𝑖,𝑗,𝑘

(Δ𝑧2 + 2Δ𝑧Δ𝑧extd);

(15)

Z direction border at the top

𝑎+

𝑘=

2𝐷𝑖,𝑗,𝑘

(Δ𝑧2 + 2Δ𝑧Δ𝑧extd),

𝑎−

𝑘=

2𝐷𝑖,𝑗,𝑘

𝐷𝑖,𝑗,𝑘−1

(𝐷𝑖,𝑗,𝑘

+ 𝐷𝑖,𝑗,𝑘−1

) Δ𝑧2.

(16)

Similarly, 𝑎+𝑗, 𝑎−𝑗, 𝑎+𝑗2, and 𝑎

−

𝑗2will have the same values:

𝑎𝑖= 𝑎+

𝑖+ 𝑎−

𝑖,

𝑎𝑗= 𝑎+

𝑗+ 𝑎−

𝑗,

𝑎𝑗/2

= 𝑎+

𝑗/2+ 𝑎−

𝑗/2,

𝑎𝑧= 𝑎+

𝑧+ 𝑎−

𝑧.

(17)

In order to solve the diffusion and convection diffusionproblems,we used alternating direction implicitmethod.Thismethod was first discovered by Peaceman and Rechford [4]especially to solve parabolic and 2D elliptical problems.

The reactor core is a typical case of nonuniform problemand the parameters involved are not continuous. Therefore,ADI method was considered for calculating the neutron fluxof each section in the block of reactor core. Neutron flux inwhole reactor core is not continuous and there is a need todivide the coarse grid of core assembly. Hence, ADI methodwas employed to calculate the data and to give the accuracyof results, and, consistently, an algorithm should be adjustedhereafter [5].

3. Modification of REMARK Code

In real-time simulation, the computational time of each stepis less than the simulation time, so it is difficult that thenodal expansion method, the nodal green function method,and other methods meet the requirements for the real-timesimulation of the reactor core.

In REMARK code, neutron diffusion time-space dynam-ics equations with two-group and six-group delayed neutronsare utilized for the simulation [5]. These equations are givenas below.

Fast Group with 𝐸 ≥ 1. We have

𝜕𝜙1 (𝑟, 𝑡)

V1𝜕𝑡

= ∇ ⋅ 𝐷1 (𝑟, 𝑡) ∇𝜙 (𝑟, 𝑡) − ∑

𝑎1

(𝑟, 𝑡) 𝜙1 (𝑟, 𝑡)

− ∑

12

(𝑟, 𝑡) 𝜙1 (𝑟, 𝑡) + (1 − 𝛽) V∑

𝑓1

(𝑟, 𝑡) 𝜙1 (𝑟, 𝑡)

+ (1 − 𝛽) V

× ∑

𝑓2

(𝑟, 𝑡) 𝜙2 (𝑟, 𝑡) + 𝑆𝑑 (𝑟, 𝑡) + 𝑆 (𝑟, 𝑡) .

(18)

Thermal Group with E < 1. We have

𝜕𝜙2 (𝑟, 𝑡)

V2𝜕𝑡

= ∇ ⋅ 𝐷2 (𝑟, 𝑡) ∇𝜙

2 (𝑟, 𝑡) − ∑

12

(𝑟, 𝑡) 𝜙1 (𝑟, 𝑡) .

(19)

The six-group delayed neutron precursor concentration canbe calculated as

𝜕𝐶𝑘(𝑟,𝑡)

𝜕𝑡= − 𝜆𝑘𝐶𝑘 (𝑟, 𝑡) + 𝛽

𝑘V∑

𝑓1

(𝑟, 𝑡) 𝜙1 (𝑟, 𝑡)

+ 𝛽𝑘V∑

𝑓2

(𝑟, 𝑡) 𝜙2 (𝑟, 𝑡) .

(20)

In our research, we have experienced modelling of the real-time simulation technique for hexagonal fuel assembliesespecially account to solve neutron diffusion equation. Inhexagonal assembly, there are eight leakage directions whichchange the diffusion terms in diffusion equations. For thecalculation of the reactor core, there are several steps thatshould be taken.

(i) Grid homogenization (fuel/control grid in assembliesso as to make 1D model which consists of fuel, clad,and moderator) should be performed.

(ii) Results from grid homogenization are used to calcu-late few-group or two-group diffusion calculations.

(iii) The finally originating group calculations are used indiffusion equation to solve the diffusion calculation ofthe whole reactor core.

(iv) Effective multiplication factor and power or neutronflux distribution will be calculated.

(v) Neutron transport equations will be employed after aseries of approximations to get the diffusion calcula-tions.

Science and Technology of Nuclear Installations 5

3.1. Assumptions. There are some assumptions that need to beconsidered formodification in the REMARK code in terms ofhexagonal assemblies.

(i) Ignore transfer of neutrons from thermal to fastgroup; that is, transfer cross-section is zero.

(ii) Delayed or prompt neutron should be considered asfast neutron.

(iii) Poison concentrations like Xe and Sm from I-135 andPm-149 should be considered in calculations.

(iv) Taking into consideration other phenomena, fissionpower and decay power can be calculated.

(v) Now the effective multiplication factor will be calcu-lated by neutrons originating from fission divided byneutrons from absorption and leakage.

In this paper, we have compiled all the data in separateprogram of hexagonal fuel assemblies into many subroutineslike:

(i) neutron flux distribution,(ii) group 1 and group 2 leakage,(iii) delayed neutrons,(iv) discontinuity factor,(v) method to solve diffusion equation.

Since REMARK code was written in Fortran language, sowe developed the program in Fortran language against eachfactor stated above (Figure 3).These programs are then addedin the source code of REMARK code and they compile thewhole subroutines. The code was then tested by consideringsimulation of hexagonal fuel assemblies and it gave goodresults, thus pointing to the accuracy of modification ofREMARK code for hexagonal fuel assemblies.

4. Verification Studies of REMARK Code

For verification of REMARK code, we have considered smallmodular nuclear reactor of power 220MW designed by theCollege of Nuclear Science and Technology, Harbin Engi-neering University, China [6, 7].Themain design parametersof the reactor core are illustrated in Table 1.

For the verification studies, we have considered someof the parameters of the reactor core design and utilizedin the lattice physic code, that is, HELIOS [8]. This codegives neutron cross-sectional data which is then used inthe REMARK code for obtaining the power distribution ofthe reactor core. In the current research, the reactor corehas hexagonal fuel assemblies with plate type fuel in it. TheREMARK code in its original form cannot work for theplate type fuel nuclear reactor; therefore, the modificationstudies have been done in the REMARK code in order tobe applicable for plate type fuel reactor. The cross-sectionaldata from HELIOS code exchanges with modified versionof REMARK code which gives the power distribution of thereactor core and this verifies the modification of the code,although REMARK code needs some thermal hydraulic data

Figure 3: Modified REMARK code in Fortran language.

Table 1: Design parameter of reactor core.

Core parameters Designing valuesCore power 220MWFuel type Plate typeFuel used Zr2 + UO2(Nb)25%Number of fuel assemblies 55Number of fuel plates in one assembly 3 × 20 = 60Total number of fuel plates 3300Number of control rod groups 6Height of each fuel assembly 1.5mFuel meat width 1.2mmCladding thickness 0.4mmTotal heat transfer area of core 937.431m2

Single channel flow width 2mmCirculation area of core 0.6562017m2

Primary coolant pressure 15.5MPaCore inlet/outlet temperature 558/597KPrimary coolant flow rate 1004.3 Kg/STotal number of pumps 6

which come from the plant design data and are used to getpower distribution of the reactor core. The flow diagram ofthe process is shown in Figure 4.

In order to generate cross-sectional data, HELIOScode has been used which is two-dimensional (2D) latticephysics high-order transport code and is able to solve anykind of geometry. This code utilizes neutron and gammagroups for the burnup calculations, flux distribution, andmicro/macro cross-sectional data. It works via two separatecodes AURORA and ZENITH which are input and outputprocessing codes.The data exchanged between the two codesare accessed by the subroutine package HERMIS [8].

The obtained power distribution of the reactor corefrom combination of HELIOS and modified version of theREMARK code is shown in Figure 5.


Neutron kineticcode

REMARK

Lattice physics code

HELIOS

Power distribution

Generatecross-sections

Generate powerdistribution

Thermal hydraulicparameters

Figure 4: Flow diagram for verification studies.

1 2 3 4 50.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Axi

al p

ower

dist

ribut

ion

Axial nodes

Axial power distribution

Figure 5: Axial power distribution of the reactor core.

5. Conclusion

In this paper, the research is focused on the modification andverification of the neutron kinetics code, that is, REMARKcode. Originally this code only works for the cylindrical typefuel elements, so we have modified this code to be applicableto plate type fuel nuclear reactor. The modification has beendone by detailed mathematics and the modified files in thesource library of the code have been computed.Themodifiedcode has been verified by coupling with HELIOS code to getthe power distribution of the plate type fuel nuclear reactor.The result obtained from coupling analysis gives the accuracyof the modified version of the code.

Acknowledgment

The authors would like to extend their sincere appreciation tothe Deanship of Scientific Research at King Saud Universityfor its funding of this research through the Research GroupProject no. RGP-VPP-255.

References

[1] REMARKModeling Techniques Handbook, GSE Power systems,Baltimore, Md, USA.

[2] REMARK Code Structure Manual Users Guide, GSE Powersystems, Baltimore, Md, USA.

[3] L. Wang, D. Qin, Q. Li, and B. Xia, “A new SCWR fuel assemblywith two-row fuel rods between the hexagonal moderatorchannels,” Annals of Nuclear Energy, vol. 58, pp. 60–64, 2013.

[4] S. Kaichao, P. Seltborg, andA. Libal,MCNPmodeling of hexagonVVER fuel [M.S. thesis], Reactor Physics Department, RoyalInstitute of Technology, Stockholm, Sweden, 2008.

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