Multigroup diffusion equation in spherical geometry

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Nuclear Reactor Theory

Multigroup Diffusion Equation in Spherical geometry

Syeilendra Pramuditya


2009

Problem 8A

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According to the instruction, this problem should be solved by using the one-group diffusion

equation. To condense the group constants (5 groups to 1 group), we need to calculate the

fluxes for each group again, because the geometry in this problem (spherical) is different from

geometry of previous problem (cylindrical), which means that the geometrical buckling for each

group must be different from previous problem.

The general multi-group neutron diffusion equation is expressed as follow:

()

And, by assuming that we are dealing with a steady state homogenous system, the equations

simplify as follow:

()

First, we need to find the expression of geometrical buckling for spherical geometry, we

introduce the Helmhotz equation as follow:

()

The above equations is actually about the Eigen value problem.

For spherical geometry, the Laplacian takes the form:

()

And by assuming that the system under consideration only depends on radial coordinate, and

does not depend on any angular coordinates, the Laplacian simplifies as follow:

()

And now substitute eq. () to eq. ():

()

Eq. () has the boundary condition of:

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()

The general solution of eq. () is as follow:

()

Where C1 and C2 are constants, the second term of the above equation becomes infinity as r

approaches zero, therefore, C2 should be zero, then:

()

Applying the boundary condition, eq. ():

()

Therefore, B should be:

()

Therefore, the solutions of this Eigen value problem are as follow:

()

()

The geometrical buckling is then defined as the fundamental Eigen value of the Laplace

operator, which is when n = 1:

()

C1 in eq. () is the normalization factor for flux, and can be calculated from reactor power (1000

MWt), as follow:

()

()

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Substitute eq. () and () to eq. ():

()

()

By using the method of Integral by Parts:

()

()

()

()

As r approaching zero:

()

Therefore:

()

Therefore, the flux distribution can be calculated analytically by using eq. (), (), and (), and the

results are as follow:

r (cm) Flux

0 1.32823E+14

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25 1.29576E+1450 1.20121E+1475 1.05283E+14

100 8.63406E+13125 6.48949E+13150 4.27057E+13175 2.15143E+13200 2.87349E+12

0

Problem 8B

First, the removal cross sections for each group are calculated as follow:

()

And the extrapolation distance for each group are calculated as follow:

()

d[1] = 5.3280000000E+00 cm

d[2] = 2.1312000000E+00 cm

d[3] = 1.9180800000E+00 cm

d[4] = 1.7049600000E+00 cm

d[5] = 5.1148800000E-01 cm

And geometrical buckling for each group is calculated as follow:

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()

B[1] = 2.3410106613E-04

B[2] = 2.4156445856E-04

B[3] = 2.4207465940E-04

B[4] = 2.4258647831E-04

B[5] = 2.4548288893E-04

Then the fluxes for each group are calculated as follow:

()

()

()

()

()

The results are as follow:

flux[1] = 1.4855100217E+01 #/cm2.s

flux[2] = 6.0866413556E+00 #/cm2.s

flux[3] = 5.9419121579E-01 #/cm2.s

flux[4] = 3.4902315167E-03 #/cm2.s

flux[5] = 5.8169382911E-08 #/cm2.s

And, by assuming that ,

()

One group constants are calculated as follow:

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()

()

()

()

The result of k-eff (1g) is slightly different from k-eff (5g), this is caused by error from group

constants condensation process.

Problem 8C

The numerical method to solve this problem consists of two coupled iteration schemes:

1. Inner iteration to calculate the neutron flux

2. Outer iteration to calculate the keff, which is performed after the inner iteration is

numerically converged

This coupled equation system to calculate keff and the corresponding flux is known as the power

iteration method.

1. The Inner Iteration: Flux Calculation

The one-group neutron diffusion equation in a homogenous system is expressed as follow:

()

In an angular-independent spherical coordinate, eq. () takes the form as follow:

()

()

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For r = 0, the second term in the left hand side of eq. () will become infinity, to handle this

problem, we will use the L’Hospital Theorem:

()

Therefore, we will divide eq. () into two parts:

()

And,

()

With boundary conditions as follow:

()

Next, we will use the central finite difference approximation:

()

()

Substitute eq. () to eq. ():

()

By using the principle of symmetry, eq. () becomes as follow:

()

By rearranging the terms, we will get:

()

Now, substitute eq. () and eq. () to eq. ():

()

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By rearranging the terms, we will get:

()

Therefore, eq. () and () will be used in the inner iteration to evaluate the flux distribution

numerically. The inner iteration is continued until the error in flux decreases below some

specified amount:

()

Note that eq. () and () are known as the Jacobian Iteration Scheme, we can improve the rate of

convergence by employing the Gauss-Siedel Iteration Scheme to eq. ():

()

We can improve the rate of convergence even further by employing the Successive over

Relaxation (SOR) scheme:

()

Where is the SOR constant, and unique for each equation system .

2. The Outer Iteration: k-eff Calculation

First, we will re-write eq. () as follow:

()

()

()

We do not really know the source term, , since it involves the flux itself, hence, we must

make an initial estimate of source term, keff, and flux to start the calculation:

()

Next, we evaluate the flux by using these initial estimations, and employing the inner iteration

scheme, which are eq. () and (), that is to solve the following equation:

()

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Once the inner iteration converged, we will get the new flux distribution, . By using this

newly calculated flux, then we can generate improved estimates of source:

()

As the iteration number becomes large, it is expected that we eventually will get the true Eigen

function of flux, which satisfies eq. (). The keff is defined as the ratio of the number of neutrons

in two consecutive fission generations, hence, by using eq. (), keff can be evaluated as follow:

()

By performing the volume integration to eq. (), we will get:

()

Recognizing that , and make use of eq. (), we can re-write eq. () as follow:

()

Notice that by scaling the source term in eq. () and () by a factor of (1/keff) in each iteration, we

prevent the numerical overflow or underflow and assure that this iteration scheme will stable

and converge. The outer iteration is continued until the error in keff and/or S decreases below

some specified amount:

()

()

Therefore, eq. () and () are the algorithmic basis of the outer iteration to determine keff.

The flowchart of this power iteration method is shown below:

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The Power Iteration Method

After the iteration converged, we must normalize the flux to the reactor power (1000 MWt), by

using eq. () and ():

()

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Integral in the above equation is solved by using the Trapezoidal Method for numerical

integration:

()

Therefore the normalization factor for the flux is as follow:

()

To perform this numerical calculation, I developed a code based on the Pascal language, please

refer to the appendix section at the end of this report.

From analytical calculation, we have these data:

The outer iterations were then performed by employing two kinds of convergence criteria:

And here are some parameters used in the calculation:

maximum error for flux calculation: 1E-6

maximum error for source calculation: 1E-6

maximum error for k-eff calculation: 1E-6

maximum inner iteration: 10000

maximum outer iteration: 10000

The results are as follow:

Table 1. Calculation Results

Mesh Width Error TypeTotal

IterationOuter

Iteration

Average Inner

Iterationcalculated keff Difference* (%)

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residual 4830 332 14.55 1.16456841772010 0.026427372

residual 12877 337 38.21 1.16491056028794 0.002944209

residual 48124 335 143.65 1.16485925288219 0.001460328

relative 14498 10000 1.45 1.16456721274325 0.026530814

relative 22540 10000 2.25 1.16490671885187 0.002614438

relative 57789 10000 5.78 1.16483730750576 0.003148314

*Relative difference from analytical keff

As shown in Table 1, by using relative error, it is very difficult to reach numerical convergence

with the accuracy for keff calculation of 10-6, even up to outer iteration of 10000, it has not

converged yet. It is found that the best result was achieved when we use mesh size of

, and maximum residual error of 10-6, which only produce relative difference from

analytical keff of 0.001460328%, as shown in the third row of Table 1.

Neutron flux distribution calculated with numerical method is shown below:

This result is almost identical to that from analytical solution.

Appendix: The Code

program spherical_diffusion;

uses crt;

var

file1:text;

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i,nr,iter_in,itermax_in,iter_total:integer;

konvergen,conv_source:boolean;

radius,dr,sigma_a,D,error,alpha,errmax_flux:real;

FluxOld,FluxNew:array[0..1000] of real;

Sold,Snew:array[0..1000] of real;

nusigma_f,sum1,sum2,kold,knew,err_source,errmax_source,err_k:real;

errmax_k,sigma_f,fluxnorm,volume,power,dummy1:real;

iter_out,itermax_out:integer;

function calculate_center(a,b,c,d,e:real):real;

begin

calculate_center:=(a*sqr(b)/(6*a+c*sqr(b)))*(6*d/sqr(b)+e/a);

end;

function calculate_region(a,b,c,d,e,f,g:real):real;

begin

calculate_region:=(a*sqr(b)/(2*a+c*sqr(b)))*((d+e)/(sqr(b))+(d-e)/(g*sqr(b))+f/a);

end;

begin

clrscr;

assign(file1,'out.txt');

rewrite(file1);

{preparing all required data}

alpha:=0;

radius:=204.3299897447;

sigma_a:=4.5948564225E-02;

D:=2.0317144072;

errmax_flux:=1E-6;

errmax_source:=1E-6;

errmax_k:=1E-6;

itermax_in:=10000;

itermax_out:=10000;

nusigma_f:=5.4083863335E-02;

sigma_f:=nusigma_f/2.5;

volume:=(4/3)*PI*radius*radius*radius;

power:=1E+9;

dr:=2.8812956642;

nr:=round(radius/dr);

{boundary condition}

FluxNew[nr]:=0;

FluxOld[nr]:=0;

Sold[nr]:=0;

Snew[nr]:=0;

{initial guess}

for i:=0 to nr-1 do

begin

FluxOld[i]:=1;

FluxNew[i]:=1;

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Sold[i]:=1;

Snew[i]:=Sold[i];

end;

kold:=1;

writeln('Calculation started...');

writeln;

iter_total:=0;

{outer iteration}

iter_out:=0;

repeat

begin

inc(iter_out);

conv_source:=true;

{inner iteration}

iter_in:=0;

repeat

begin

inc(iter_in);

konvergen:=true;

inc(iter_total);

{evaluete flux}

FluxNew[0]:=calculate_center(D,dr,sigma_a,FluxOld[1],Sold[0]/kold);

for i:=1 to nr-1 do

begin

{Jacobian}

FluxNew[i]:=calculate_region(D,dr,sigma_a,FluxOld[i+1],FluxOld[i-1],Sold[i]/kold,i);

{Gauss-Siedel}

{FluxNew[i]:=calculate_region(D,dr,sigma_a,FluxOld[i+1],FluxNew[i-1],Sold[i]/kold,i);}

end;

{SOR subroutine}

for i:=0 to nr-1 do

begin

FluxNew[i]:=FluxNew[i]+alpha*(FluxNew[i]-FluxOld[i]);

end;

{convergence test for flux}

for i:=0 to nr-1 do

begin

error:=abs(FluxNew[i]-FluxOld[i])/FluxNew[i];

if (error > errmax_flux) then konvergen:=false;

end;

{update flux}

for i:=0 to nr do FluxOld[i]:=FluxNew[i];

end;

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until (konvergen=true) OR (iter_in > itermax_in);

{convergence test for source}

for i:=0 to nr-1 do

begin

Snew[i]:=nusigma_f*FluxNew[i];

err_source:=abs((Snew[i]-Sold[i])/Snew[i]);

if (err_source>errmax_source) then conv_source:=false;

end;

sum1:=0;

sum2:=0;

for i:=0 to nr do

begin

sum1:=sum1+Sold[i];

sum2:=sum2+Snew[i];

end;

{convergence test for k-eff}

knew:=kold*(sum2/sum1);

err_k:=abs((knew-kold)/knew);

{err_k:=abs((knew-1.1648762639)/1.1648762639);}

{update source and k-eff}

for i:=0 to nr do Sold[i]:=Snew[i];

kold:=knew;

end;

until (iter_out=itermax_out) or ((conv_source=true) and (err_k<=errmax_k));

{flux normalization to reactor power}

sum1:=0;

for i:=0 to nr-1 do

begin

{trapezoid method for numerical integration}

sum1:=sum1+(dr/2)*(FluxNew[i]*sqr(i*dr)+FluxNew[i+1]*sqr((i+1)*dr));

end;

fluxnorm:=(3.204E-11)*sigma_f*4*PI*sum1;

fluxnorm:=(1E9)/fluxnorm;

{saving results to file}

append(file1);

writeln(file1,'k-eff = ',knew);

writeln(file1);

writeln(file1,' r (cm) Flux');

for i:=0 to nr do

begin

FluxNew[i]:=FluxNew[i]*fluxnorm;

append(file1);

writeln(file1,i*dr:10:3,' ',FluxNew[i]:10:0);

end;

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close(file1);

writeln('Calculation finished! result saved to out.txt');

readln;

end.

Sample output:

k-eff = 1.1645684177

r (cm) Flux

0.000 1.32989E+14

5.763 1.32811E+14

11.525 1.32276E+14

17.288 1.31388E+14

23.050 1.30151E+14

28.813 1.28570E+14

34.576 1.26654E+14

40.338 1.24412E+14

46.101 1.21854E+14

51.863 1.18992E+14

57.626 1.15841E+14

63.389 1.12414E+14

69.151 1.08730E+14

74.914 1.04804E+14

80.676 1.00655E+14

86.439 9.63036E+13

92.201 9.17690E+13

97.964 8.70725E+13

103.727 8.22357E+13

109.489 7.72808E+13

115.252 7.22303E+13

121.014 6.71068E+13

126.777 6.19334E+13

132.540 5.67328E+13

138.302 5.15278E+13

144.065 4.63408E+13

149.827 4.11941E+13

155.590 3.61091E+13

161.353 3.11070E+13

167.115 2.62079E+13

172.878 2.14315E+13

178.640 1.67962E+13

184.403 1.23195E+13

190.166 8.01799E+12

195.928 3.90682E+12

201.691 0.00000E+00

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References

1. Duderstadt, James J. and Louis J. Hamilton. (1976), Nuclear Reactor Analysis, John Wiley

& Sons, Inc, New York.

2. DeVries, Paul L., (1993), A First Course in Computational Physics, John Wiley & Sons, Inc,

New York.

3. Press, William H., Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. (1989),

Numerical recipes in Pascal, Cambridge University Press, New York.

4. http://en.wikipedia.org/wiki/Laplacian

5. http://en.wikipedia.org/wiki/Integrate_by_parts

6. http://en.wikipedia.org/wiki/Finite_difference

7. http://en.wikipedia.org/wiki/Numerical_integration

8. http://en.wikipedia.org/wiki/L'Hôpital's_rule

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Multigroup diffusion equation in spherical geometry