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[Prof. Hafemeister’s manuscript was prepared on March 22, 2011, just eleven days after the Japanese earthquake and tsunami. The situation at the Fukushima Daiichi nuclear power plant will no doubt evolve rapidly over the coming weeks. For readers wishing to keep up with the latest developments, updates on the situation in Japan prepared by the MIT department of Nuclear Science and Engineering are available at http://mitnse.com/. Information on safety and oversight at US nuclear plants in 2010 is given in a Union of Concerned Scientists (UCS) website. Another UCS website, has successive news stories and helpful graphics, as well as links to other useful sites – Ed.]
On March 11, 2011, a magnitude 9.0 earthquake from a “reverse fault” struck northeastern Japan. The Fukushima Daiichi site on the Pacific Ocean, 240 km north-east of Tokyo, houses six boiling water reactors, three of which were in operation and three in maintenance at the time of the earthquake. It appears that the three-operating reactors shut down without a loss of coolant accident (LOCA), but the accompanying tsunami, which arrived 15 minutes later, disabled back-up electrical generators. This prevented pumping of cooling water to the reactors and their spent fuel ponds. The ensuing damage and radioactive release is the worst accident since the Chernobyl accident in Ukraine on 26 April 1986. The Chernobyl accident was particularly bad since the burning of its carbon moderator propelled 3-4% of the radioactive core into the atmosphere from a reactor without a containment dome. The radioactive releases from the Fukushima site will, most likely, be far less than was the case Chernobyl, but the clean-up of the Fukushima reactors and spent fuel ponds will be very significant. The Three Mile Island Accident of 28 March 1979 released only minor amounts of radioactivity, but the cleanup cost $1 billion and took eight years.
In this article, I use some basic reactor and thermal physics to estimate the available response time before a light water reactor core begins to melt. I consider two types of accidents: A loss of coolant accident (LOCA) and a loss of power accident (LOPA). The calculated response time for a LOCA is about 1 minute, and, for a LOPA, about 0.5-1 day, but individual circumstances cause variations on these results. These calculations are based on material presented in Ref. 1.
Helicopters and water cannons failed to cool the reactors and the spent fuel ponds. As a last-gasp effort, corrosive seawater with boric acid was flooded on reactors 1, 2 and 3, but without the use of their internal pumps. It is speculated that rapid deployment of portable generators on land or ships to give power to the reactors’ pumps could have lessened the severity of the Fukushima accident. First, I will summarize the status, one week after the earthquake-tsunami of March 11, of the six Fukushima Daiichi (FD) reactors and spent fuel ponds :
FD-1: Hydrogen from oxidation of zircaloy cladding (over 95% zirconium) exploded on March 12, destroying the secondary confinement roof, but the primary containment was said to be intact.
FD-2: Hydrogen explosion on March 15 breached primary containment, causing a partial meltdown. Iodine-131 (8-day half-life) was observed in Tokyo at a distance of 240 km on March 19 and its level in spinach at a distance of 70 km was 27 times the limited level. On March 20, electrical power was reestablish at unit 2, with the other units to follow. Water was added to pond 2 on March 20.
FD-3: Hydrogen explosion on March 14 destroyed the secondary confinement roof and walls. Pond 3 was filled on March 20.
FD-4: Reactor was shut down three months ago with the transfer of all of its spent fuel to the spent fuel pond. Full-core discharges are rarely done in the U.S., where only the oldest fuel is usually removed. The young, very hot spent fuel heated pond water to boiling, with a report that pond had no water, starting a fire. This is consistent with the last-measured temperature of 84oC (183oF), as compared to the usual temperature of 25oC (77oF). Pond 4 was filled on March 20.
FD-5: Reactor was shut down, but the pool’s last-measured temperature was 63oC (145oF), as compared to the usual temperature of 25oC (77oF).
FD-6: Reactor was shut down, but the pool’s last measured temperature was 60oC (140oF), as compared to the usual temperature of 25oC (77oF).
Loss-of-Coolant Rise TimeI first calculate the thermal rise time of a light water reactor (LWR) core after a loss-of-coolant accident. Thermal rise is the time for the core to get sufficiently hot to begin an exothermal reaction between zircaloy and water. The calculation is based on the following assumptions :
These equations give 7.7% of operating thermal power at 1 second, 3.7% at 1 minute, 1.3% at 1 hour, 0.5% at 1 day, 0.3% at 1 week, and 0.2% at one month, all of which conform to the measured data.
The thermal rise time is obtained by equating the heat needed to raise the core to 1370oC to the time integral of thermal power P. The heat needed to raise the core to 1370oC is
Q = Nc(ΔT), (3)
where N is the number of moles of UO2, c is the UO2 molar specific heat, and ΔT is the temperature rise for the core to be 1370°C, that is, ΔT = 1370°C – 400°C = 970°C. The number of moles of UO2 in the core is
N = (108 g)/(238 + 32)g/mole = 3.7 x 105 moles. (4)
The high-temperature specific heat, c = 3R = 24.9 J/mole-°C, is used since the temperatures are considerably above the UO2 Debye temperature of 100 K. Thus, the heat needed to raise the core to its critical temperature is
Qrise = Nc(DT) = (3.7 x 105 moles)(24.9 J/mole-°C) (970°C) = 8.9 x 109 J. (5)
The thermal rise time is obtained by equating Qrise, to the time integral of the beta decay power,
Qbeta decay = ∫t0 P dt = ∫t0 0.0766(3 x 109)t-0.181dt = (2.8 x 108)t0.819 J = 8.9 x 109 J. (6)
Solving for t gives a thermal rise time of 68 sec, which is close to the published values of 1 minute, calculated with the heat equation . Since the time scale for a LOCA is only a minute, essentially all beta-decay heat is trapped in the core.
Loss-of-Power Rise Time
A more gradual LOCA almost happened in 1975 when a workman at the Brown’s Ferry, Alabama, boiling water reactor (BWR) used a candle to check airflow and inadvertently set fire to electrical cables, cutting off electrical power for cooling pumps. Beta-decay heat began evaporating the water coolant, which in turn initiated a process that would have uncovered the core and begun a LOCA. The beta-decay heat needed to evaporate 700 tonnes of water is
Qevap = mLevap = (7 x 105 kg)(2.27 MJ/kg) = 1.6 x 1012 J. (7)
Setting Qevap equal to the integrated beta-decay heat, over the two time regions gives t = 19 hours, similar to the stated 13 hours available to recover the situation.
LOCA in Spent Fuel Ponds
The Fukushima spent fuel ponds are 12 meters deep, with 8 meters of water over the tops of the spent fuel assemblies. Pond water can be lost through holes in the concrete and by evaporation from the radioactive heat of the spent fuel. After one year, spent fuel radioactive heating is 15 kW/tonne, and at 10 years it falls to 2 kW/tonne. The spent fuel problem was exacerbated in the United States because the density of spent fuel in the ponds was increased as a result of the 1977 decision not to reprocess spent fuel. Increasing the density of fuel rods gives additional heating density and reduces the paths to remove heat by radiation and convection. Some parameters give temperatures over 900oC in a spent fuel pond after a LOCA, a point where zircaloy cladding spontaneously ignites in air . The problem could be lessened by moving some rods to a geological repository, or by placing them in surface storage, which is happening in the U.S. at this time. Damage could be mitigated after loss of coolant in the ponds by plugging pond holes with quick-setting material, pouring water on the ponds, or using large air blowers.
A Final Comment
Further data and analysis by experts are needed before serious conclusions on the Fukushima accident can be made. However, it initially seems that rapid deployment of portable generators on trucks or on ships to the Fukushima site could have made a considerable difference. Cables would be needed to deliver the power, but this should have been possible. Crews would experience some radiation, but less than that received by the fifty workers inside the plants. These generators could have provided electricity to drive the internal pumps at the reactors to bring water into the reactors and the spent fuel ponds. It took nine days to establish a 1.5 km power line to reactor 2 on March 20, with power to follow at the other reactors.
 D. Hafemeister, Am. J. Phys. 50(8), 713-720 (1982) and Physics of Societal Issues (Springer, New York, 2007), pp. 171-3.
 New York Times: 16 March 2011 (pp. A1, A7-9) and 18 March 2011 (A1, A11-12).
 A. Nero, A Guidebook to Nuclear Reactors (University of California Press, Berkeley, 1979), p. 54.
 R. Alvarez, et al, “Reducing the hazards from stored spent powerreactor fuel in the United States,” Science and Global Security 11, 1–51, 2003.
David W. Hafemeister
California Polytechnic State University, San Luis Obispo