# How to calculate critical mass?

#### richphys

1. The problem statement, all variables and given/known data

How (or if) I can calculate critical mass with the information I'm given?

I have to write a criticality safety evaluation for an assignment. The assignment is about decomissioning a dissolver which was used to recover plutonium. On the contingencies section I need to apply 'magic merv'. These are the parameters which can effect criticality - mass, absorption, geometry, interaction, concentration, moderation, enrichment, reflection and volume.

For the mass section I'm not sure if I need to calculate the critical mass to determine how much solution (which contains some plutonium nitrate) should go in a storage drum. I don't know how to do this.

2. Relevant equations

I have calculated that there is 36.5 gallons of liquid in the dissolver which includes plutonium nitrate. I intend to put the liquid into a 55 gallon (lead-lined 2 inch thick) drum measuring 43 inches in height and 31 inches in diameter. I need to know if this is safe.

I am told that plutonium nitrate concentrations are less than 1 kg/L.

3. The attempt at a solution

I have attached text from the assignment.

Thank you!

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#### anorlunda

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Gold Member
Many people here wisely refuse to open a .docx file from an unknown person on the Internet. Can you convert it to .PDF and post that? Or better still, can you make it text/latex in the body of your post?

#### richphys

I've attached the pdf below.

In the mid-1960s a secret nuclear research facility was built around a remote inlet on the Russian Arctic coast. Shortly before the break-up of the Soviet Union decommissioning of the facility commenced and. Due to its remote location, and a lack of road access, it was decided to transport these drums to a reprocessing site by sea through the use of. The drums were loaded onto the barge where, due to the political and economic turmoil of the time, they were left and forgotten until 2013. After successful removal of the drums it was decided by the authorities that remediation of the rest of the plant should be undertaken and, after initial site surveys, two steel skips containing fissile materials were removed from the site yard, two waste dumps were cleared, and a fuel pond was emptied.

Adjacent to the main facility building, in a separate building, is a metal machining and recovery lab. This lab was used to recover and machine plutonium that is (originally) from natural uranium metal fuel rods irradiated in a graphite moderated reactor. This lab was in use from 1964 until abandonment of the site. The lab had low throughput and there is no criticality alarm system in place. Previously, during operations, neutron monitors were used when handling fissile materials. The nine metre square, six metre high lab was on the ground floor and had 30 cm thick floors, walls and ceilings. The site was susceptible to flooding of no more than 6” at a frequency of once per decade.

In the middle of the lab, surrounded by a steel mezzanine, and accessed (to fill) by a ladder, there is a batch dissolver. Made of 1” thick 316 stainless steel, this 60 cm diameter, Teflon-lined, dissolver is mounted on four, 2” diameter steel legs (located on a 90 degree pitch), as shown in Figure 2. The dissolver has a lid which can be removed and repositioned using a block and tackle (not shown), mounted directly above. When used previously, decanting of fissile liquors from the dissolver was accomplished by opening a valve (tap) at the bottom of the dissolver (flow under gravity). Debris was removed from the liquor by use of a filter placed between it and the receiving container. This dissolver currently remains half-filled with liquid (assumed to be nitric acid, potentially bearing fissile material) and the lid remains in place. Inspection indicates that, over time, debris has settled at the bottom of the dissolver. Directly beneath the dissolver is a spherical, Teflon-lined 180 cm diameter, 1” thick walled, stainless steel sump. The sump is filled to the top with borosilicate glass Raschig rings with diameter and height both 4 cm. The glass wall thickness is 0.5 cm. Spillages have accumulated in the sump over the many years of use although the waste volume or evaporation rates are not known. Access to the sump is via a 15 cm diameter drain from a sloped floor. The sump is buried in the ground beneath the floor. Previous experience showed that soluble boron was frequently used on site and that solutions with concentrations <1 kg/L plutonium nitrate were often present. The total fissile mass in the lab is not known.

The dissolver is scheduled for demolition and hence it, and the sump, needs to be drained and decontaminated. A chemical waste removal company has tendered for the work. The proposed process is to drain the sump into the dissolver using a binda-super-lario-b-piston-hand-pump and a 2” diameter nylon hose. This will be diluted with light water as required. The dissolver will then be drained in the normal way into any number of 15 cm diameter, 30 cm tall flat-topped polypropylene bottles (each of which has a 1” diameter, 1” tall cap protruding from its top). The waste engineers propose transporting the filled bottles from site in 45 x 30 x 35 cm cardboard boxes. They propose stacking these boxes three high, starting in the corner, in the north east corner of the room. As the bottles are filled, the process states that they are immediately placed into a cardboard box. Due to difficulties in shipping, the timetable for removal of the fissile waste from the room is not known and hence you should assume that the boxes will remain in position throughout.

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#### anorlunda

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Gold Member
Thanks, but the assignment is not your attempt at solution. You must show your attempt.

#### richphys

You haven't read my post. I'm asking for help on how (or if) I can calculate the critical mass with the information I'm given. I don't want someone to do my assignment for me I just need direction regarding critical mass.

#### anorlunda

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Gold Member
It sounds like you think there is just a formula to plug into. How much nuclear engineering have you studied so far? It is hard to imagine a professor giving that assignment without you having the proper background first.

There are some very simple examples here: https://en.wikipedia.org/wiki/Critical_mass#Critical_mass_of_a_bare_sphere

But the real answer is here:
https://en.wikipedia.org/wiki/Nuclear_criticality_safety#Calculations_and_analyses said:
To determine if any given system containing fissile material is safe, its neutron balance must be calculated. In all but very simple cases, this usually requires the use of computer programs to model the system geometry and its material properties.

The analyst describes the geometry of the system and the materials, usually with conservative or pessimistic assumptions. The density and size of any neutron absorbers is minimised while the amount of fissile material is maximised. As some moderators are also absorbers, the analyst must be careful when modelling these to be pessimistic. Computer codes allow analysts to describe a three-dimensional system with boundary conditions. These boundary conditions can represent real boundaries such as concrete walls or the surface of a pond, or can be used to represent an artificial infinite system using a periodic boundary condition. These are useful when representing a large system consisting of many repeated units.

Computer codes used for criticality safety analyses include COG (US),[2] MONK (UK),[3] KENO (US),[4] MCNP (US),[5] and CRISTAL (France).[6]

#### richphys

I have studied very little of it so far.

Without access to a programme like MCMP what should I say regarding critical mass? I have formula for geometric buckling, dimensions of the storage drums and the density of plutonium, but there isn't much else to go on...

I have a formula in the notes given as critical mass/density = volume... can this be applied to a storage cylinders?

#### anorlunda

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Gold Member
I have a formula in the notes given as critical mass/density = volume... can this be applied to a storage cylinders?
Normally, I would say no that is not good enough. But you must assume that your assignment requires only methods that you have been taught so far, so it should be in your notes.

#### richphys

Normally, I would say no that is not good enough. But you must assume that your assignment requires only methods that you have been taught so far, so it should be in your notes.
Are there any tips you could offer regarding a discussion on critical mass, given the data I have? The data I have is as follows:

Plutonium nitrate exists in concentrations < 1 Kg/L
The total mass of plutonium nitrate is unknown.
There will be 36.5 gallons of plutonium nitrate waste from the dissolver.
The plutonium nitrate waste will be moved from the dissolver to 30 gallon drums measuring 39.25 inches in height by 27 inches in diameter.
Raschig rings will be added in concentrations from 4-85 g/L.

#### anorlunda

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Gold Member
There are others here on PF who have professional experience with radiation safety. I'll leave it to them to answer. It may take some time before they read this thread.

#### anorlunda

Mentor
Gold Member
By the way, what kind of course are you taking where they gave you that assignment? It seems illogical to ask you to do that if you have not studied nuclear engineering.

"How to calculate critical mass?"

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