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In “Conditions for criticality by uranium deposition in water-saturated geological formations“, Xudong Liu; Joonhong Ahn of the Dept of Nuclear Engineering, U. Cal. Berkeley; & Fumio Hirano of Japan Atomic Energy Agency, Geological Isolation Research and Development Directorate, Tokai-mura, (2014), raise the question of risk of a criticality event, an uncontrolled nuclear reaction, in a deep geologic repository for nuclear waste. Despite some seemingly faulty assumptions, which appear to understate risk, they rather bravely, considering their affiliations and funding, conclude that there could be a problem. At the end is the statement: “This study was carried out under a contract with METI (Ministry of Economy, Trade and Industry) of Japanese Government in the fiscal year of 2012 as part of its R&D supporting program for developing geological disposal technology.”

Little wonder that Japan was allegedly already trying to get shot of its high level nuclear waste by sending it to Scotland in the 1980s. Scots didn’t like this idea of burying Japanese nuclear waste in the Ayrshire Hills, and with the help of Willie McRae, appear to have blocked it (unless it was later dumped illegally). Liu et. al. do not tell us why an uncontrolled nuclear reaction in a deep geological nuclear waste facility is a problem, but clearly they, and others, appear to believe it is. A similar study was done for the proposed US nuclear waste facility at Yucca Mountain. Liu et. al. (2014) let themselves off the hook by pointing out that a waste site must be selected first because “detailed information about geological formations, geohydrology, and geochemistry is required, which can only be obtained after determining a disposal site…,” which is clearly true, but has been largely ignored by most countries.

Willie McRae, the lawyer who successfully advocated against the burial of Japanese, and other, nuclear waste in Ayrshire, near Mullwarcher, died mysteriously, reportedly shortly after alleging that nuclear waste would be (illegally?) buried further north in the sands at Applecross (or offshore). https://miningawareness.wordpress.com/2016/01/26/willie-mcrae-death-nuclear-waste-dumping-in-scotland-group-continues-to-call-for-inquiry/
Willie McCrae car OGL N. Constab.
Willie McRae’s car found on the side of the road with him in it, shot in the head. https://web.archive.org/web/20120415074535/http://www.northern.police.uk/Default.aspx.LocID-0axnew00o.RefLocID-0ax00b00d004.Lang-EN.htm (OGL)

Illustration of critical mass public domain via wikimedia
Illustration of the concept of “critical mass” in respects to nuclear weapons design. The circles represent spheres of critical material (enriched uranium or plutonium), arrows represent neutron paths, and stars represent fission reactions. In the top frame, there is too little active material and so the fission reaction quickly ends. In the middle frame the addition of more material allows for more fission reactions and more neutrons, thus opening up the possibility of a self-perpetuating reaction (criticality and super-criticality). In the bottom frame, the addition of a neutron reflector to the original material has increased the efficiency of the reaction by preventing as many neutrons from escaping, also opening up the possibility of criticality. The diagram is meant to illustrate that there is not necessarily one set “critical mass” — the amount of mass needed to go critical can depend on the exact arrangement used in assembling the material.” Released to the public domain via Wikimedia: https://commons.wikimedia.org/wiki/File:Critical_mass.svg
nuclear fizzle public domain via wikimedia
Diagram illustrating nuclear predetonation (fizzle) in a very-schematic gun-type setup. At the top, a piece of fissile material (i.e. enriched uranium) is accelerated towards another piece to create a critical mass. However, because the acceleration is too slow, stray neutrons caused by spontaneous fissioning cause the surfaces of the two pieces to begin to react before a full critical mass is formed (middle). Because of this, a small explosion results, destroying the containment apparatus and sending the bulk of the material, which did not fission, away from itself, preventing further nuclear reaction. This explosion would be much smaller than a full nuclear explosion and result in the dispersal of expensive nuclear material.” Released to public domain via Wikimedia: https://commons.wikimedia.org/wiki/File:Nuclear_predetonation.svg

Xudong Liu, Joonhong Ahn & Fumio Hirano (2014), “Conditions for criticality by uranium deposition in water-saturated geological formations“, Journal of Nuclear Science and Technology, Abstract states that “This study focuses on neutronic analysis to examine the criticality conditions for uranium depositions in geological formations resulting from geological disposal of damaged fuels from Fukushima Daiichi reactors. MCNP models are used to evaluate the neutron multiplication factor (keff) and critical mass for various combinations of host rock and geometries. It has been observed that the keff for the deposition become greater with (1) smaller concentrations of neutron-absorbing materials in the host rock, (2) larger porosity of the host rock, (3) heterogeneous geometry of the deposition, and (4) greater mass of uranium in the deposition. This study has revealed that the planar fracture geometry applied in the previous criticality safety assessment for geological disposal would not necessarily yield conservative results against the homogeneous uranium deposition“. This study is now available in full for free here: http://escholarship.org/uc/item/8zf4q0jm (Heterogeneous geometry is irregular shape. Clearer explanation of the issues found further below).

While they conclude that there could be a problem, as well as that a more detailed study is warranted, multiple seemingly faulty assumptions spring quickly to the eye, upon only a cursory reading of their research: Apparently Faulty assumption 1 – transuranics will have been degraded BEFORE corrosion! Apparently Faulty assumption 2 – admittedly low temperature assumption of 20 C (roughly 68 F). Probable Faulty assumption 3 – assumption that container designs are sub-critical. Possible Faulty assumption 4 – 70 cm spacing. There are probably others which we have overlooked.

It is important to note that the US NRC is allowing the packing of damaged nuclear fuel in spent fuel casks.

Apparently faulty assumption (1) is that long-lived transuranic radioactive materials will have been degraded to uranium BEFORE waste containers are corroded through. This may or may not be true for some thick containers but is clearly false for the flimsy 1/2 inch thick Holtec spent fuel casks.

Apparently faulty assumption (2) – admittedly arbitrarily low temperature of 20 C (roughly 68 F). NASA space math hints at how low an estimation 20 C is compared to real conditions: “One of the deepest laboratories in the world is in the small town of Soudan, Minnesota. It is a physics research facility and is at a depth of 690 meters (2,263 feet) below the surface. It is operated by the University of Minnesota. If the geothermal gradient is +25 C/km, how warm are the walls of this laboratory if the average surface temperature is +55 F (+13 C)? Answer: T = +13 C + 0.69km x (+25 C/km) = +30 C (or +86 F)http://spacemath.gsfc.nasa.gov/Insight/Insight10.pdf

Probable faulty assumption (3) is that container designs are sub-critical. As routine readers of this blog know, Holtec has asked the US NRC for, and received, various exemptions which decrease safety. The only part of the container which is sealed is a flimsy 1/2 inch thick (and these spent fuel casks are huge). Furthermore, changes relating to quality of fuel basket welds could lead to basket collapse, resulting in a bad configuration. Additionally, Holtec has asked for, and received, exemptions from quality control for the neutron absorber Metamic. The basket itself is aluminum, which may constitute a corrosion hazard (read about Lasagna cells) – lack of adequate boron neutron absorber would increase this hazard. Do a search for Holtec in the search window of Mining Awareness blog to read more details-documentation. See for instance: https://miningawareness.wordpress.com/2015/03/10/holtec-nuclear-waste-snf-casks-friction-stir-welds-kissing-bonds-other-safety-concerns-comment-now/

4: Liu et. al. (2014) also assume that “if two uranium depositions are more than 70 cm (27.5 in) apart, they are neutronically independent of each other“. In an intentionally or unintentially collapsing facility, such as WIPP, will they remain separated?

A few important statements found in Xudong Liu, Joonhong Ahn & Fumio Hirano (2014), “Conditions for criticality by uranium deposition in water-saturated geological formations“: “Fission neutrons have greater chance to be thermalized with greater amount of water… heterogeneity of the uranium deposition has sensitive effects on neutron transport… In summary, the systems keff will increase, when (1) the rock contains fewer neutrons absorbing materials, (2) the rock has larger porosity, (3) the deposition has heterogeneous geometry, and (4) the deposition contains larger amount of uranium… detailed information about geological formations, geohydrology, and geochemistry is required, which can only be obtained after determining a disposal site… low host rock porosity is preferred for criticality safety“.

What is a Criticality Accident?

From Wikipedia:
A criticality accident is an uncontrolled nuclear chain reaction. It is sometimes referred to as a critical excursion or a critical power excursion and represents the unintentional assembly of a critical mass of a given fissile material, such as enriched uranium or plutonium, in an unprotected environment. A critical or supercritical fission reaction (one that is sustained in power or increasing in power) generally only occurs inside reactor cores and occasionally within test environments; a criticality accident occurs when the same reaction is achieved unintentionally and in an unsafe environment. Though dangerous and frequently lethal to humans within the immediate area, the critical mass formed is still incapable of producing a nuclear detonation of the type seen in fission bombs, as the reaction lacks the many engineering elements that are necessary to induce explosive supercriticality. The heat released by the nuclear reaction will typically cause the fissile material to expand, so that the nuclear reaction becomes subcritical again within a few seconds. In the history of atomic power development, 60 criticality accidents have occurred, including 22 in collections of fissile materials located in process environments outside of a nuclear reactor or critical experiments assembly. Although process accidents occurring outside of reactors are characterized by a large release of radiation, the release is localized and has caused fatal radiation exposure only to persons very near to the event (less than 1 m), resulting in 14 fatalities. No criticality accidents have resulted in nuclear explosions.” CC: https://en.wikipedia.org/wiki/Criticality_accident (Emphasis added. Recall that Wikipedia tends to be occupied by pro-nuclear people, but this is the best summary we could find.) See too: http://www.orau.org/ptp/Library/accidents/la-13638.pdf

Los Alamos Nuclear Lab explains the importance of geometry in the context of historic criticality accidents:
The geometry and material specifications provided in accident documentation fall far short of qualifying as criticality benchmarks as accepted by the international criticality safety community
Only primary parameters affecting criticality are considered in our estimates–fissile species (235U or 239Pu), fissile density, shape of fissile material, and degree of moderation. Uranium enrichment is also considered in the case of accidents 9, 15, and 22….
Vessel Shape: The vessel shape, e.g., cylindrical, vertical axis. Although this designation is accurate for most accidents, some accidents are known to have occurred when the axis of cylindrical symmetry was neither vertical nor horizontal, but rather tilted at some angle from the vertical. Vessel Volume: Vessel volume denotes the total volume of the vessel.
Fissile Volume: This heading could be more properly described as fissile material volume. It is an estimate of the volume occupied by the fissile material that dominated the neutronic reactivity of the system. In some cases (accidents 5 and 18), fissile material was present in low concentration exterior to this volume. This additional material had a secondary impact on the system reactivity and was therefore ignored. For those accidents that occurred or were modeled with a vertical axis of cylindrical symmetry and the fissile material was in solution or slurry form, an additional parameter, h/D, is provided. In those cases the fissile material was modeled as a right-circular cylinder (lower case h designates the height of the cylinder and capital D designates the diameter of the vessel).
Shape Factor: The shape factor was used to convert actual shape to equivalent spherical shape as a method to compare these 21 accidents in terms of geometri-cally equivalent spherical systems. For the 18 accidents where h/D is specified, the unreflected curve in Figure 3634 was used to determine the shape factor. The curve in Figure 36 is based directly on experimental results minimizing depen-dence on calculations. For the remaining 3 accidents (numbers 2, 6, and 20), buckling or other mathemati-cally simple approximations were used to estimate the shape factor…
” “A Review of Criticality Accidents, 2000 Revision” LA-13638 Issued: May 2000, http://www.orau.org/ptp/Library/accidents/la-13638.pdf

An Easy to Understand Summary of Criticality Issues
Changing the point of criticality
The mass where criticality occurs may be changed by modifying certain attributes such as fuel, shape, temperature, density and the installation of a neutron-reflective substance. These attributes have complex interactions and interdependencies. This section explains only the simplest ideal cases.
* Varying the amount of fuel
It is possible for a fuel assembly to be critical at near zero power. If the perfect quantity of fuel were added to a slightly subcritical mass to create an “exactly critical mass”, fission would be self-sustaining for one neutron generation (fuel consumption makes the assembly subcritical).
If the perfect quantity of fuel were added to a slightly subcritical mass, to create a barely supercritical mass, the temperature of the assembly would increase to an initial maximum (for example: 1 K above the ambient temperature) and then decrease back to room temperature after a period of time, because fuel consumed during fission brings the assembly back to subcriticality once again.
* Changing the shape
A mass may be exactly critical without being a perfect homogeneous sphere. More closely refining the shape toward a perfect sphere will make the mass supercritical. Conversely changing the shape to a less perfect sphere will decrease its reactivity and make it subcritical.
* Changing the temperature
A mass may be exactly critical at a particular temperature. Fission and absorption cross-sections increase as the relative neutron velocity decreases. As fuel temperature increases, neutrons of a given energy appear faster and thus fission/absorption is less likely. This is not unrelated to Doppler broadening of the U238 resonances but is common to all fuels/absorbers/configurations. Neglecting the very important resonances, the total neutron cross section of every material exhibits an inverse relationship with relative neutron velocity. Hot fuel is always less reactive than cold fuel (over/under moderation in LWR is a different topic). Thermal expansion associated with temperature increase also contributes a negative coefficient of reactivity since fuel atoms are moving farther apart. A mass that is exactly critical at room temperature would be sub-critical in an environment anywhere above room temperature due to thermal expansion alone.
* Varying the density of the mass
The higher the density, the lower the critical mass. The density of a material at a constant temperature can be changed by varying the pressure or tension or by changing crystal structure (see Allotropes of plutonium). An ideal mass will become subcritical if allowed to expand or conversely the same mass will become supercritical if compressed. Changing the temperature may also change the density; however, the effect on critical mass is then complicated by temperature effects (see “Changing the temperature”) and by whether the material expands or contracts with increased temperature. Assuming the material expands with temperature (enriched uranium-235 at room temperature for example), at an exactly critical state, it will become subcritical if warmed to lower density or become supercritical if cooled to higher density. Such a material is said to have a negative temperature coefficient of reactivity to indicate that its reactivity decreases when its temperature increases. Using such a material as fuel means fission decreases as the fuel temperature increases.
* Use of a neutron reflector
Surrounding a spherical critical mass with a neutron reflector further reduces the mass needed for criticality. A common material for a neutron reflector is beryllium metal. This reduces the number of neutrons which escape the fissile material, resulting in increased reactivity.
* Use of a tamper
In a bomb, a dense shell of material surrounding the fissile core will contain, via inertia, the expanding fissioning material. This increases the efficiency. A tamper also tends to act as a neutron reflector. Because a bomb relies on fast neutrons (not ones moderated by reflection with light elements, as in a reactor), because the neutrons reflected by a tamper are slowed by their collisions with the tamper nuclei, and because it takes time for the reflected neutrons to return to the fissile core, they take rather longer to be absorbed by a fissile nucleus. But they do contribute to the reaction, and can decrease the critical mass by a factor of four.[1] Also, if the tamper is (e.g. depleted) uranium, it can fission due to the high energy neutrons generated by the primary explosion. This can greatly increase yield, especially if even more neutrons are generated by fusing hydrogen isotopes, in a so-called boosted configuration.
Critical size
The critical size is the minimum size of a nuclear reactor core or nuclear weapon that can be made for a specific geometrical arrangement and material composition. The critical size must at least include enough fissionable material to reach critical mass. If the size of the reactor core is less than a certain minimum, fission neutrons escape through its surface and the chain reaction is not sustained
CC; https://en.wikipedia.org/wiki/Critical_mass The link also gives a summary of weights at which criticality occurs.
Weights at which criticality events occur wikipedia

Of related interest: http://www.bbc.co.uk/schools/gcsebitesize/science/add_ocr_gateway/radiation/fissionrev2.shtml

For another source of the Willie McRae photo see: https://www.whatdotheyknow.com/request/155337/response/382528/attach/10/macrae%2012.jpg