Geometry of Fast Reactor: Calculating Critical Dimensions & Masses

[thanphongvt]

a fast reactor composed of U-235 in form of a cube(point source,strength So) .
calculate the :
critical dimensions,
critical volumes,
critical masses.
discuss more in cases of spherical , cylindrical and cubical cores.can i solve this practice by separating in 3 dimensions?
and what is the condition at the center of the cubic.?

 

[Astronuc]

a fast reactor composed of U-235 in form of a cube(point source,strength So) .
calculate the :
critical dimensions,
critical volumes,
critical masses.
discuss more in cases of spherical , cylindrical and cubical cores.


can i solve this practice by separating in 3 dimensions?
and what is the condition at the center of the cubic.?

The problem is a bit vague. Is one assuming that the fast reactor is simply pure U-235 (100% enrichment)? Or are there structural materials and coolant?

The different geometries will give different leakage rates depending on the fast flux at the boundaries and the total surface area.

What should the net current be at the center?

 

[edgepflow]

You could get basic results with 1D diffusion theory with "fast" cross sections. There is a simple formula for geometric buckling in a cube.

 

[Uranium]

Classic problem.

You just need to set the material buckling to the geometrical buckling for an homogeneous reactor. I would assume it's all 235. By setting these equal, just solve for the length of the cube's side...then you can get volume and mass.

Though this doesn't handle the point source.

 

[QuantumPion]

Though this doesn't handle the point source.

It's given in the problem statement as an initial condition, phi(r=0)=So

 

[Uranium]

I was indicating that my solution doesn't involve the given So...so I'm not sure how correct it would be.

I'm pretty sure my previous solution is correct. That's how it's usually done in practice problems like that.

I guess you could solve a 1-D diffusion equation to get a flux profile using So, but that wouldn't really answer any of the questions.

I'm not completely sure what is meant by "what is the condition at the center...".

 

[statphys]

Classic problem.

You just need to set the material buckling to the geometrical buckling for an homogeneous reactor. I would assume it's all 235. By setting these equal, just solve for the length of the cube's side...then you can get volume and mass.

Though this doesn't handle the point source.

That sounds about right to me. Cube and sphere geometry are the easiest to solve (the flux shapes in the cube reactor are simple cosines in each direction). I'm not sure how important the point source is in the determination of the parameters of a critical system.

 

[Uranium]

Kind of what I was thinking.

I think the point source only comes into play in solving the condition considering a uniformly distribution source (from the 235) and the point source at the center.

 

[QuantumPion]

If I remember my reactor physics correctly (I'll have to break out my D&H tonight), you can only use the geometric/material buckling equivalence if there is no source. Otherwise, you have a time-dependent problem for which you must solve the diffusion equation directly.

edit- forget that, he's not trying to solve for flux, he just wants the critical dimensions, which only depend on geometry and materials. The source is irrelevant unless you want to solve for the flux itself.

 

[Uranium]

That's quite likely.

 

[QuantumPion]

That's quite possible, and would definitely make sense since the buckling would be thrown off...

Well the geometric buckling would not change, it is not dependent on a source. It is the fundamental mode, the shape the flux takes after infinite time purely based on the geometry of the core. What would change would be the flux distribution as a function of power. But this is irrelevant when you just want the critical parameters.

 

[Uranium]

The buckling should change at least a little since you can accept greater leakage for criticality. The shape would still be a cosine, just one with greater curvature, right?

Or perhaps it would be more like a hump...not cosine.

 

[QuantumPion]

The buckling should change at least a little since you can accept greater leakage for criticality. The shape would still be a cosine, just one with greater curvature, right?

Or perhaps it would be more like a hump...not cosine.

It doesn't work that way. A fixed source has no impact on the multiplication factor or the probability of non-leakage. The flux distribution would change but the definition of buckling is a solution of the diffusion equation that assumes S=0.

 

[Astronuc]

I was struggling with a fast reactor made of pure U-235! As the question is posed, there is no other structural material or coolant. That's not so much a reactor as a recipe for a nuclear explosive. And by critical, is one referring to prompt critical? Normally we don't take reactors there. I was hoping the OP would elaborate.

For a given volume, the cube has the highest leakage, followed by cylinder, then the sphere which has the lowest leakage.