# Energy density calculations of nuclear fuel

Energy density is like the miles-per-gallon rating of a power plant. It measures how much energy is released (in megajoules) given a certain mass of fuel (in kilograms). Perhaps the most physically unique thing about nuclear power is that the energy density of nuclear fuel is millions of times higher than any chemical (like fossil fuel, biofuel, or batteries).

## Details of the computation

The easy way to compute energy density of nuclear fuels is to figure out how much fission energy can be released from 1 mole of the fuel. The equation for energy density in MJ/kg is:

$\text{ED} = \frac{\kappa_{fis} \text{[MeV/fission]} N_A \text{[fissions/mol]}}{A \text{[g/mol]}} \times \frac{1.60217\times10^{-19} \text{[Mega Joules/MeV]}}{0.001 \text{kg/g}} = \text{MJ/kg}$

where:

• $$\kappa_{fis}$$ is the energy release per fission for the nuclide of interest. These values are measured by scientists and collected in nuclear data files such as the ones available at the National Nuclear Data Center. Look for the (n,fis.ene.release) Interpreted field for each nuclide here.

• $$N_A$$ is Avogadro’s number, or 6.022e23. This is the number of atoms per mole. Since we’re assuming 100% of atoms fission, this is equal to the number of fissions per mole.

• $$A$$ is the atomic mass of the nuclide of interest. This can be found on any Chart of the Nuclides, like this one.

## Some nuclear energy density calculations

Running the equation, here’s how much energy is contained in certain amounts of nuclear fuel.

MaterialEnergy released per fission (MeV) Atomic weight (g/mol) Energy density (MJ/kg)
U-235193.4235.0479,390,000
U-238/Pu-239198.9238.0580,620,000
Th-232/U-233191.0232.0479,420,000

Table 1. Energy densities of nuclear fuels. Energy per fission does not include the energy lost to neutrinos since it is practically unrecoverable.

## Influence of burnup

Another factor worth considering is how complete the fuel is consumed. For example, a wood fire may burn out before all the energy is extracted from the wood. In traditional LWR nuclear power plants, usually only 5-7% of the energy of the fuel is extracted. Furthermore, the fuel has already gone through an enrichment process so only about 1% of the energy of the mined resource is used. Advanced nuclear power plants called breeder reactors such as the liquid metal fast breeder reactor (LMFBR) or the molten salt breeder reactor (MSBR) can extract much more of the mined energy. The fraction of the energy extracted from the fuel in a reactor is called the burnup.

So in a LWR, the effective energy density is around 5% of 80 million, or 4 million MJ/kg.

## Comparison with other fuels

A single fuel pellet may weigh about 10 grams so it contains 40,000 MJ in a typical reactor and 800,000 MJ in a breeder reactor. Thus we can compare its content to other fuels.

Material Energy Density
(MJ/kg)
Equivalent to
fuel pellet in LWR
Equivalent to fuel
pellet in breeder
Coal 30 1.4 tons 30 tons
Oil 42 285 gallons 5700 gallons
Natural Gas 53.5 38,800 cubic ft 775,000 cubic ft
Lithium 43 1 ton 20 tons

Note: The characteristics of the power generation system affect exactly how much usable energy is extracted. For instance, if a power plant makes heat to be converted to electricity, the thermal efficiency ($$\epsilon_{th}$$) determines how much of the heat gets converted to electricity. These values vary from around 35-45% for coal plants to 34% for nuclear plants, to above 60% for combined cycle natural gas plants. Thus, from a electricity point of view, the values above would be 20-50% less.

## Other Complications

In a nuclear reactor, fission isn’t the only process that releases energy. The actinides, fission products, and even structural and coolant nuclides often undergo capture reactions that release energy without fissioning. The fraction of energy released by a nuclear reactor by these reactions can be on the order of 10% of the total power of the reactor.