General group collapsing expression

[madhisoka]

How can I collapse macroscopic absorption cross-section of 4 groups into two ?
Assuming that the first two groups are fast groups and the other twos are thermal .

I am suffering with the following :
1- Do I have to assume that the groups are directly coupled ?
2-Does what apply on the two groups model apply on the 4 ?

 

[rpp]

To preserve reactions rates, you want to collapse the cross sections by "flux weighting" them.

You didn't mention what groups collapse to other groups, but assume you want to collapse the first two cross sections in the 4-group structure down to the first group in the 2-group structure.
$$\overline{\Sigma_{x1}} = \frac{ \Sigma_{x1} \phi_1 + \Sigma_{x2} \phi_2}{\phi_1 + \phi_2} $$
Likewise, collapsing the second two cross sections in the 4-group structure to the second group in the 2-group structure
$$\overline{\Sigma_{x2}} = \frac{ \Sigma_{x3} \phi_3 + \Sigma_{x4} \phi_4}{\phi_3 + \phi_4} $$
where:
* The LHS is in the two-group structure,
* The RHS is in the four-group structure, and
* ##x## is the type of reaction (absorption, fission, etc.)

The scattering cross sections are a little more complicated.
$$\overline{\Sigma_{1\rightarrow1}} = \frac{ \Sigma_{1\rightarrow1} \phi_1 + \Sigma_{1\rightarrow2} \phi_1 + \Sigma_{2\rightarrow1} \phi_2 + \Sigma_{2\rightarrow2} \phi_2}{\phi_1 + \phi_2} $$
etc. for ##\overline{\Sigma_{1\rightarrow2}}##, ##\overline{\Sigma_{2\rightarrow1}}##, and ##\overline{\Sigma_{2\rightarrow2}}##

These equations only accounts for energy, it doesn't include any spatial dependence to the cross sections.

There is a more general formula that includes energy and space, but it is difficult to write in this forum..

 

[madhisoka]

To preserve reactions rates, you want to collapse the cross sections by "flux weighting" them.

You didn't mention what groups collapse to other groups, but assume you want to collapse the first two cross sections in the 4-group structure down to the first group in the 2-group structure.
$$\overline{\Sigma_{x1}} = \frac{ \Sigma_{x1} \phi_1 + \Sigma_{x2} \phi_2}{\phi_1 + \phi_2} $$
Likewise, collapsing the second two cross sections in the 4-group structure to the second group in the 2-group structure
$$\overline{\Sigma_{x2}} = \frac{ \Sigma_{x3} \phi_3 + \Sigma_{x4} \phi_4}{\phi_3 + \phi_4} $$
where:
* The LHS is in the two-group structure,
* The RHS is in the four-group structure, and
* ##x## is the type of reaction (absorption, fission, etc.)

The scattering cross sections are a little more complicated.
$$\overline{\Sigma_{1\rightarrow1}} = \frac{ \Sigma_{1\rightarrow1} \phi_1 + \Sigma_{1\rightarrow2} \phi_1 + \Sigma_{2\rightarrow1} \phi_2 + \Sigma_{2\rightarrow2} \phi_2}{\phi_1 + \phi_2} $$
etc. for ##\overline{\Sigma_{1\rightarrow2}}##, ##\overline{\Sigma_{2\rightarrow1}}##, and ##\overline{\Sigma_{2\rightarrow2}}##

These equations only accounts for energy, it doesn't include any spatial dependence to the cross sections.

There is a more general formula that includes energy and space, but it is difficult to write in this forum..

Thank you man, but I am already struggling with the scattering cross sections, like idk if there is a direct coupling or they aren't directly coupled also I don't think that the term where it scatters from 1 to 1 is a right term since there is no up scattering.

And what about the flux ? I don't have the flux of any of the groups

 

[Astronuc]

And what about the flux ? I don't have the flux of any of the groups

Is there a source term?

Neutrons scatter down in energy, and they do not interact. Only at thermal energies is upscatter possible.

In a reactor or critical system. Each energy group would contribute to some fission, and for fissile materials like U-235 and Pu-239/-241, that's mostly in the thermal energies, but some fast fissions do occur. One solves a system of equations for the fluxes.

 

[madhisoka]

Is there a source term?

Neutrons scatter down in energy, and they do not interact. Only at thermal energies is upscatter possible.

In a reactor or critical system. Each energy group would contribute to some fission, and for fissile materials like U-235 and Pu-239/-241, that's mostly in the thermal energies, but some fast fissions do occur. One solves a system of equations for the fluxes.

Up scattering isn't an issue , I am more concerned with the direct coupling thing also This is the question I have 2 thermal groups and 2 fast, I have to collapse the 2 fast groups with one of the thermal into one group ? so 3 groups into one ? I am trying to derive an expression for segma absorption but the fluxes of 1 2 and 3 are extremely hard to play around with .

 

[rpp]

You have to have the group fluxes in order to collapse. You need the group fluxes to preserve the reaction rates.

The equation I gave is correct, but the upscatter cross section may be zero. I gave you the general form. Just set the upscatter cross section to zero if it is zero.

 

[rpp]

Up scattering isn't an issue , I am more concerned with the direct coupling thing also This is the question I have 2 thermal groups and 2 fast, I have to collapse the 2 fast groups with one of the thermal into one group ? so 3 groups into one ? I am trying to derive an expression for segma absorption but the fluxes of 1 2 and 3 are extremely hard to play around with .

Looking at the question from the book, you are not given enough information to collapse the cross sections yourself. It isn't even clear if the thermal group includes two or three of the groups from the 4-group table. The cross sections are just given to you in this problem.

Why are you trying to collapse them? If you do want to collapse them, you will have to solve a problem with 4 groups and then use the flux spectrum to collapse down to 2 groups. It would help if you had the group boundaries so you know which groups to collapse to.

 

[madhisoka]

Looking at the question from the book, you are not given enough information to collapse the cross sections yourself. It isn't even clear if the thermal group includes two or three of the groups from the 4-group table. The cross sections are just given to you in this problem.

Why are you trying to collapse them? If you do want to collapse them, you will have to solve a problem with 4 groups and then use the flux spectrum to collapse down to 2 groups. It would help if you had the group boundaries so you know which groups to collapse to.

Thank you for the replying, Question is about collapsing the first 3 groups noticing that the third one is a thermal one and the first two are fast.

I have to derive a general expression for collapsing N groups like this one :

 

[rpp]

We are slowly getting there...

In order to collapse cross sections, you need a neutron flux. In the example you gave in the last comment, the flux came from the analytic solution for the following situation:
* 2 energy groups
* Homogeneous bare core
* no upscatter
* all fission neutrons appear in the fast group (##\chi_1=1, \chi_2=0##)

For this situation, the 2-group diffusion equations are
$$
D_1 B^2 \phi_1 + \Sigma_{R1} \phi_1 = \nu \Sigma_{f1} \phi_1 + \nu \Sigma_{f2} \phi_2
$$
$$
D_2 B^2 \phi_2 + \Sigma_{R2} \phi_2 = \Sigma_{21} \phi_1
$$
Now, to collapse cross sections, you need to flux weight them. To collapse 2-groups to one-group
$$
\overline{ \Sigma_a }= \frac{\Sigma_{a1} \phi_1 + \Sigma_{a2}\phi_2}{\phi_1 + \phi_2}
$$

If you use the 2nd diffusion equation to solve for ##\phi_1## and substitute it into the collapse equation, you will get
$$
\overline{ \Sigma_a }= \frac{\Sigma_{a1} (D_2 B^2 + \Sigma_{a2}) + \Sigma_{a2} \Sigma_{21} }{ D_2 B^2 + \Sigma_{a2} + \Sigma_{21} }
$$
Which is equivalent to what you showed. All of the flux terms cancel, and the average cross section is just an expression with cross sections.

So what you need to do:
1. Come up with an analytic solution to the 4-group equations, which will include approximations for ##\chi## and the downscatter cross sections
2. Insert the resulting flux values into the collapse equations.

Is this a homework problem? You will probably not get the answers you showed in comment 5.

I still don't think you have enough information. You need to know the group boundaries so you know what 4-groups collapse to what 2-groups,
and you will also need to know the correct 4-group fission spectrum (##\chi##).

Let me know how it goes.

And out of curiosity, what book are you quoting from?

 

[madhisoka]

We are slowly getting there...

In order to collapse cross sections, you need a neutron flux. In the example you gave in the last comment, the flux came from the analytic solution for the following situation:
* 2 energy groups
* Homogeneous bare core
* no upscatter
* all fission neutrons appear in the fast group (##\chi_1=1, \chi_2=0##)

For this situation, the 2-group diffusion equations are
$$
D_1 B^2 \phi_1 + \Sigma_{R1} \phi_1 = \nu \Sigma_{f1} \phi_1 + \nu \Sigma_{f2} \phi_2
$$
$$
D_2 B^2 \phi_2 + \Sigma_{R2} \phi_2 = \Sigma_{21} \phi_1
$$
Now, to collapse cross sections, you need to flux weight them. To collapse 2-groups to one-group
$$
\overline{ \Sigma_a }= \frac{\Sigma_{a1} \phi_1 + \Sigma_{a2}\phi_2}{\phi_1 + \phi_2}
$$

If you use the 2nd diffusion equation to solve for ##\phi_1## and substitute it into the collapse equation, you will get
$$
\overline{ \Sigma_a }= \frac{\Sigma_{a1} (D_2 B^2 + \Sigma_{a2}) + \Sigma_{a2} \Sigma_{21} }{ D_2 B^2 + \Sigma_{a2} + \Sigma_{21} }
$$
Which is equivalent to what you showed. All of the flux terms cancel, and the average cross section is just an expression with cross sections.

So what you need to do:
1. Come up with an analytic solution to the 4-group equations, which will include approximations for ##\chi## and the downscatter cross sections
2. Insert the resulting flux values into the collapse equations.

Is this a homework problem? You will probably not get the answers you showed in comment 5.

I still don't think you have enough information. You need to know the group boundaries so you know what 4-groups collapse to what 2-groups,
and you will also need to know the correct 4-group fission spectrum (##\chi##).

Let me know how it goes.

And out of curiosity, what book are you quoting from?

The question is all about deriving an equation similar to the one you mentioned but for N-groups. the other equation is collapsing the first 3 groups into one group noticing that the first two groups are fast groups . about the spectrum the fact that X1+X2=1 may help the thing is about K I have to end up with an equation that doesn't have K . Book I am using is Nuclear reactor Analysis "James Duderstadt"

 

[madhisoka]

Looking at the question from the book, you are not given enough information to collapse the cross sections yourself. It isn't even clear if the thermal group includes two or three of the groups from the 4-group table. The cross sections are just given to you in this problem.

Why are you trying to collapse them? If you do want to collapse them, you will have to solve a problem with 4 groups and then use the flux spectrum to collapse down to 2 groups. It would help if you had the group boundaries so you know which groups to collapse to.

The first 3 groups are surely fast , so now I have to collapse 3 fast groups into one group

 

[madhisoka]

Is there a source term?

Neutrons scatter down in energy, and they do not interact. Only at thermal energies is upscatter possible.

In a reactor or critical system. Each energy group would contribute to some fission, and for fissile materials like U-235 and Pu-239/-241, that's mostly in the thermal energies, but some fast fissions do occur. One solves a system of equations for the fluxes.

Hi sir, I am still stuck with the same problem , didn't find anything online . In a brief I have to collapse 3 fast groups constants into one without having the fluxes .

 

[rpp]

I'm not sure what else to add. You first need to solve for the neutron fluxes, then collapse the cross sections with the neutron fluxes.
You will also need values of the fission spectrum (##\chi##) for each energy group. If there are three fast groups, each group may have a non-zero ##\chi##.