|Fig. 1: Simple schematic of once-through reactor, and two ways in which waste heat could be utilized
Gas turbine power plants work by igniting fuel, and expanding the hot gas through a turbine to generate work. However, since the exhaust gas from the turbine is still quite hot, it still holds a lot of energy. Not harvesting this energy significantly reduces the energy efficiency of the power plant. Hence the "combined cycle gas turbine": a gas turbine which also has a bottoming cycle that uses the hot turbine exhaust to simply boil water in a heat exchanger, and this steam is sent through a steam turbine to generate more electricity. The addition of this cycle has led to significantly higher efficiencies in gas turbine power plants, currently nearing 60%.
It is easy to draw a similar analogy to nuclear power plants. The fuel cannot be "burnt" to completion in the primary reactor, for various reasons. It exits the reactor incredibly hot, and still has a large energy content. And yet, no secondary cycle for nuclear power plants has been implemented. The main way nuclear energy is burned in the U.S. is known as the "once-through" route, and utilizes only approximately 5% of the energy of the reactor fuel.  Plutonium recycling, which extracts plutonium from the spent uranium fuel, raises this number to a mighty 6%.  There is a proposed method to utilize upwards of 99% of the fuel, however it involves a polymetallurgical reprocessing of the waste before burning it in a fast-neutron reactor.  The process is only in the prototype phase and is likely to have significant costs of research and implementation. All at a time when the public's fear over nuclear energy has made research funding increasingly hard to secure.
However, even without recovering fissionable fuel, there is still energy to be recouped which requires little to no new research. Fuel rods continue to beta decay and gamma decay long after they are considered spent and pulled from the reactor. This generates heat, which goes completely unused. It could be used to generate steam and run a separate cycle, or simply to preheat the main working fluid. A few simple schematics of how this heat could be used are shown in Fig. 1. An expression for the power generated by the beta and gamma decay of the fuel is given below: 
In this equation, P is the power generated, P0 is the nominal reactor power, t is the time elapsed since reactor startup, and ts is the time of reactor shutdown. Let's try to put in some numbers. First, we assume that shutdown of the reactor and removal of a spent rod from the reactor are the same things for any individual fuel rod. Many power plants plan to refuel approximately ever two years, so let us take a modest number of 500 days for our refueling time.  However, typically only 1/3 to 1/4 of the fuel is placed during a refueling event, so let us assume that any individual rod remains in the reactor for 2000 days, or 1.73 × 108 seconds. This will be the shutdown time, ts. A plot of P/P0 for this value of ts is shown in Fig. 2. The power is plotted for a time of 1 × 108 seconds, as that is the duration for which this equation remains accurate. It shows that for a significant duration of time, the power emitted by this fuel is between 1 and 10 percent of the reactor power, which is a significant fraction.
|Fig. 2: Decay heat as fraction of reactor power, as a function of time
Furthermore, integrating this curve for the time shown yields the total amount of energy lost, and for the parameters stated this integration yields a value of 6.7 × 104. This means that for a 1 GW plant, 6.7 × 104 Gigajoules of energy will be lost over this time, or 1.86e7 kWh. This is a very large number; however, it needs to be put into perspective. Over this same time, a 1 GW plant will produce 1 × 108 Gigajoules, meaning the power due to beta and gamma decay of spent fuel amounts to only 0.067% of the energy produced by the plant (a number for an individual fuel rod would have to be divided by the number of rods in the reactor, but this number represents the entire ancillary cycle). Another way of putting this number into perspective is that 6.7 × 104 Gigajoules over a time of 1 × 108 seconds averages out to a 0.67 MW power plant. It should be noted that values for the power of this ancillary cycle can be easily changed by integrating to different final times - if you integrate for very short times while the rods are hotter, you will obtain a significantly higher average power. However this number would be artificial, as it only makes sense to integrate over a length of time similar to the time between refuelings. This integration is carried out to a value comparable to the time between refuelings, and thus is a reasonable estimate for the average expected power from such a cycle. Not many people are clamoring to build a 0.67 MW-average power plant - especially one with such high risks involved.
And make no mistake, the risks are indeed high. Cooling pools are so ubiquitous because of their simplicity. The simpler things are, they less chance they have of breaking. And regarding nuclear waste, you don't want things to break. If the turbine of the bottoming steam cycle, or some other part of the heat exchanging process, were to fail, it would be difficult to repair without getting close to the radioactive waste. Furthermore, if active cooling malfunctions, It also may require that the plant be shut down for an extended period of time in order to delay the production of waste. Furthermore, in cooling pools, the water not only serves the purpose of cooling, but also of radioactive shielding. If the proposed heat exchanger were to be implemented, some rods would inevitably be surrounded by steam, not liquid water. Therefore, radiation shielding would be diminished, and alternative means of shielding would need to be constructed. The water could alternatively be pressurized, to prevent it from boiling, but this increases the dangers of waste storage. Implementation might be easiest for plants with molten salt, which would not boil in an ancillary heat exchanger even at atmospheric pressure. Nevertheless, there are so many other risks, and such little payoff, that a bottoming cycle from nuclear waste heat release is not likely to be a fertile endeavor, especially at a time when public faith in nuclear energy is fickle.
© Matt Tilghman. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
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