Energy density

Energy density is the amount of energy stored in a given system or region of space per unit volume. It may also be used for energy per unit mass, though the accurate term for this is specific energy. Often only the useful or extractable energy is measured, which is to say that inaccessible energy (such as rest mass energy) is ignored.[1] In cosmological and other general relativistic contexts, however, the energy densities considered are those that correspond to the elements of the stress–energy tensor and therefore do include mass energy as well as energy densities associated with the pressures described in the next paragraph.

Energy per unit volume has the same physical units as pressure, and in many circumstances is a synonym: for example, the energy density of a magnetic field may be expressed as (and behaves as) a physical pressure, and the energy required to compress a compressed gas a little more may be determined by multiplying the difference between the gas pressure and the external pressure by the change in volume. In short, pressure is a measure of the enthalpy per unit volume of a system. A pressure gradient has the potential to perform work on the surroundings by converting enthalpy to work until equilibrium is reached.

Introduction to energy density

There are different types of energy stored in materials, and it takes a particular type of reaction to release each type of energy. In order of the typical magnitude of the energy released, these types of reactions are: nuclear, chemical, electrochemical, and electrical.

Nuclear reactions take place in stars and nuclear power plants, both of which derive energy from the binding energy of nuclei. Chemical reactions are used by animals to derive energy from food, and by automobiles to derive energy from gasoline. Liquid hydrocarbons (fuels such as gasoline, diesel and kerosene) are today the most dense way known to economically store and transport chemical energy at a very large scale (1 kg of diesel fuel burns with the oxygen contained in ~15 kg of air). Electrochemical reactions are used by most mobile devices such as laptop computers and mobile phones to release the energy from batteries.
Types of energy content

There are several different types of energy content. One is the theoretical total amount of thermodynamic work that can be derived from a system, with a given temperature and pressure for the surroundings. This is called exergy. Another is the theoretical amount of work that can be derived from reactants that are initially at room temperature and atmospheric pressure. This is given by the change in standard Gibbs free energy. But as a source of heat or for use in a heat engine, the relevant quantity is the change in standard enthalpy or the heat of combustion.

There are two kinds of heat of combustion:

The higher value (HHV), or gross heat of combustion, includes all the heat released as the products cool to room temperature and whatever water vapor is present condenses.
The lower value (LHV), or net heat of combustion, does not include the heat which could be released by condensing water vapor, and may not include the heat released on cooling all the way down to room temperature.

A convenient table of HHV and LHV of some fuels can be found in the references.[2]
Energy density in energy storage and in fuel
Selected energy densities plot[3][4][5][6][7][8][9]

In energy storage applications the energy density relates the energy in an energy store to the volume of the storage facility, e.g. the fuel tank. The higher the energy density of the fuel, the more energy may be stored or transported for the same amount of volume. The energy density of a fuel per unit mass is called the specific energy of that fuel. In general an engine using that fuel will generate less kinetic energy due to inefficiencies and thermodynamic considerations—hence the specific fuel consumption of an engine will always be greater than its rate of production of the kinetic energy of motion.
Broad implications

Energy density differs from energy conversion efficiency (net output per input) or embodied energy (the energy output costs to provide, as harvesting, refining, distributing, and dealing with pollution all use energy). Large scale, intensive energy use impacts and is impacted by climate, waste storage, and environmental consequences.

No single energy storage method boasts the best in specific power, specific energy, and energy density. Peukert's law describes how the amount of useful energy that can be obtained (for a lead-acid cell) depends on how quickly it is pulled out. To maximize both specific energy and energy density, one can compute the specific energy density of a substance by multiplying the two values together, where the higher the number, the better the substance is at storing energy efficiently.

Alternative options are discussed for energy storage to increase energy density and decrease charging time.[10][11][12][13]

Gravimetric and volumetric energy density of some fuels and storage technologies (modified from the Gasoline article):

Note: Some values may not be precise because of isomers or other irregularities. See Heating value for a comprehensive table of specific energies of important fuels.
Note: Also it is important to realise that generally the density values for chemical fuels do not include the weight of oxygen required for combustion. This is typically two oxygen atoms per carbon atom, and one per two hydrogen atoms. The atomic weight of carbon and oxygen are similar, while hydrogen is much lighter than oxygen. Figures are presented this way for those fuels where in practice air would only be drawn in locally to the burner. This explains the apparently lower energy density of materials that already include their own oxidiser (such as gunpowder and TNT), where the mass of the oxidiser in effect adds dead weight, and absorbs some of the energy of combustion to dissociate and liberate oxygen to continue the reaction. This also explains some apparent anomalies, such as the energy density of a sandwich appearing to be higher than that of a stick of dynamite.

Tables of energy content
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Unless otherwise stated, the values in the following table are lower heating values for perfect combustion not counting oxidizer mass or volume. The following unit conversions may be helpful when considering the data in the table: 3.6 MJ = 1 kW⋅h ≈ 1.34 hp⋅h.

Energy densities of energy media
Storage type	Specific energy (MJ/kg)	Energy density (MJ/L)	Specific energy (W⋅h/kg)	Energy density (W⋅h/L)	How energy is released and Comments
Antimatter	89,875,517,874	Depends on the density of the antimatter's form	24,965,421,631,578	Depends on the density of the antimatter's form	Annihilation, counting both the consumed antimatter mass and ordinary matter mass
Hydrogen (fusion)	639,780,320^[14] but at least 2% of this is lost to neutrinos.	Depends on conditions	177,716,755,600	Depends on conditions	Reaction 4H→⁴He
Deuterium	571,182,758^[15]	Depends on conditions	158,661,876,600	Depends on conditions	Proposed fusion scheme for D+D→⁴He, by combining D+D→T+H, T+D→⁴He+n, n+H→D and D+D→³He+n, ³He+D→⁴He+H, n+H→D
Deuterium+tritium	337,387,388^[16]	Depends on conditions	93,718,718,800	Depends on conditions	D + T → ⁴He + n Being developed.
Plutonium-239	83,610,000	1,300,000,000–1,700,000,000 (Depends on crystallographic phase)	23,222,915,000	370,000,000,000–460,000,000,000 (Depends on crystallographic phase)	Heat produced in Fission reactor
Plutonium-239	31,000,000	490,000,000–620,000,000 (Depends on crystallographic phase)	8,700,000,000	140,000,000,000–170,000,000,000 (Depends on crystallographic phase)	Electricity produced in Fission reactor
Uranium	80,620,000^[17]	1,539,842,000	22,394,000,000		Heat produced in breeder reactor
Thorium	79,420,000^[17]	929,214,000	22,061,000,000		Heat produced in breeder reactor (Experimental)
Plutonium-238	2,239,000	43,277,631	621,900,000		Radioisotope thermoelectric generator. Note that the heat is only produced at a rate of 0.57 W/g.
Hydrogen, liquid^[18]	141.86 (HHV) 119.93 (LHV)	10.044 (HHV) 8.491 (LHV)	39,405.6 (HHV) 33,313.9 (LHV)	2,790.0 (HHV) 2,358.6 (LHV)	Energy figures apply after reheating to 25°C.
Hydrogen, at 690 bar and 25°C^[18]	141.86 (HHV) 119.93 (LHV)	5.323 (HHV) 4.500 (LHV)	39,405.6 (HHV) 33,313.9 (LHV)	1,478.6 (HHV) 1,250.0 (LHV)
Hydrogen, gas, 1 atm, 25°C^[18]	141.86 (HHV) 119.93 (LHV)	0.01188 (HHV) 0.01005 (LHV)	39,405.6 (HHV) 33,313.9 (LHV)	3.3 (HHV) 2.8 (LHV)
Diborane^[19]	78.2		21,722.2
Beryllium	67.6	125.1	18,777.8	34,750.0
Lithium borohydride	65.2	43.4	18,111.1	12,055.6
Boron^[20]	58.9	137.8	16,361.1	38,277.8
Methane (1.013 bar, 15 °C)	55.6	0.0378	15,444.5	10.5
LNG (NG at −160 °C)	53.6^[21]	22.2	14,888.9	6,166.7
CNG (NG compressed to 250 bar/~3,600 psi)	53.6^[21]	9	14,888.9	2,500.0
Natural gas	53.6^[21]	0.0364	14,888.9	10.1
LPG propane^[22]	49.6	25.3	13,777.8	7,027.8
LPG butane^[22]	49.1	27.7	13,638.9	7,694.5
Gasoline (petrol)^[22]	46.4	34.2	12,888.9	9,500.0	Combusted inside Internal combustion engines. 20 to 40% thermal efficiency.
Polypropylene plastic	46.4^[23]	41.7	12,888.9	11,583.3
Polyethylene plastic	46.3^[23]	42.6	12,861.1	11,833.3
Residential heating oil^[22]	46.2	37.3	12,833.3	10,361.1
Diesel fuel^[22]	45.6	38.6	12,666.7	10,722.2	Combusted inside Internal combustion engines. 25 to 40% thermal efficiency.
100LL Avgas	44.0^[24]	31.59	12,222.2	8,775.0
Jet fuel (e.g. Kerosene)	43^[25]^[26]^[27]	35			Aircraft engine
Gasohol E10 (10% ethanol 90% gasoline by volume)	43.54	33.18	12,094.5	9,216.7
Lithium	43.1	23.0	11,972.2	6,388.9
Biodiesel oil (vegetable oil)	42.20	33	11,722.2	9,166.7
DMF (2,5-dimethylfuran)	42^[28]	37.8	11,666.7	10,500.0
Crude oil (according to the definition of tonne of oil equivalent)	41.868	37^[21]	11,630	10,278
Polystyrene plastic	41.4^[23]	43.5	11,500.0	12,083.3
Body fat	38	35	10,555.6	9,722.2	Metabolism in human body (22% Efficiency^[29])
Butanol	36.6	29.2	10,166.7	8,111.1
Gasohol E85 (85% ethanol 15% gasoline by volume)	33.1	25.65	9,194.5	7,125.0
Graphite	32.7	72.9	9,083.3	20,250.0
Coal, anthracite^[6]	26–33	34–43	7,222.2–9,166.7	9,444.5–11,944.5	Figures represent perfect combustion not counting oxidizer, but efficiency of conversion to electricity is ~36%
Silicon^[30]	1.790	4.5	500	1,285	Energy stored through solid to liquid phase change of silicon
Aluminum	31.0	83.8	8,611.1	23,277.8
Ethanol	30	24	8,333.3	6,666.7
DME^[31]^[32]	31.7 (HHV) 28.4 (LHV)	21.24 (HHV) 19.03 (LHV)	8,805.6 (HHV) 7,888.9 (LHV)	5,900.0 (HHV) 5,286.1 (LHV)
Polyester plastic	26.0^[23]	35.6	7,222.2	9,888.9
Magnesium	24.7	43.0	6,861.1	11,944.5
Coal, bituminous^[6]	24–35	26–49	6,666.7–9,722.2	7,222.2–13,611.1
PET plastic (impure)	23.5^[33]		6,527.8
Methanol	19.7	15.6	5,472.2	4,333.3
Hydrazine (combusted to N₂+H₂O)	19.5	19.3	5,416.7	5,361.1
Liquid ammonia (combusted to N₂+H₂O)	18.6	11.5	5,166.7	3,194.5
PVC plastic (improper combustion toxic)	18.0^[23]	25.2	5,000.0	7,000.0
Wood^[34]	18.0		5,000.0
Peat briquette^[35]	17.7		4,916.7
Sugars, carbohydrates, and protein	17	26.2 (dextrose)	4,722.2	7,277.8	Metabolism in human body (22% Efficiency^[36])
Calcium	15.9	24.6	4,416.7	6,833.3
Glucose	15.55	23.9	4,319.5	6,638.9
Dry cow dung and camel dung	15.5^[37]		4,305.6
Coal, lignite	10–20		2,777.8–5,555.6
Sodium	13.3	12.8	3,694.5	3,555.6	burned to wet sodium hydroxide
Peat	12.8		3,555.6
Nitromethane	11.3		3,138.9
Sulfur	9.23	19.11	2,563.9	5,308.3	burned to sulfur dioxide^[38]
Sodium	9.1	8.8	2,527.8	2,444.5	burned to dry sodium oxide
Battery, lithium-air rechargeable	9.0^[39]		2,500.0		Controlled electric discharge
Household waste	8.0^[40]		2,222.2
Zinc	5.3	38.0	1,472.2	10,555.6
Iron	5.2	40.68	1,444.5	11,300.0	burned to iron(III) oxide
Teflon plastic	5.1	11.2	1,416.7	3,111.1	combustion toxic, but flame retardant
Iron	4.9	38.2	1,361.1	10,611.1	burned to iron(II) oxide
Gunpowder	4.7–11.3^[41]	5.9–12.9
TNT	4.184	6.92
ANFO	3.7		1,027.8
Battery, zinc-air^[42]	1.59	6.02	441.7	1,672.2	Controlled electric discharge
Liquid nitrogen	0.77^[43]	0.62	213.9	172.2	Maximum reversible work at 77.4K with 300K reservoir
Sodium sulfur battery	0.54–0.86		150–240
Compressed air at 300 bar	0.5	0.2	138.9	55.6	Potential energy
Latent heat of fusion of ice (thermal)	0.335	0.335	93.1	93.1
Lithium metal battery	1.8	4.32			Controlled electric discharge
Lithium-ion battery	0.36–0.875^[46]	0.9–2.63	100.00–243.06	250.00–730.56	Controlled electric discharge
Flywheel	0.36–0.5	5.3			Potential energy
Alkaline battery	0.48^[47]	1.3^[48]			Controlled electric discharge
Nickel-metal hydride battery	0.41^[49]	0.504–1.46^[49]			Controlled electric discharge
Lead-acid battery	0.17	0.56			Controlled electric discharge
Supercapacitor (EDLC)	0.01–0.030^[50]^[51]^[52]^[53]^[54]^[55]^[56]	0.006–0.06^[50]^[51]^[52]^[53]^[54]^[55]	up to 8.57^[56]		Controlled electric discharge
Water at 100 m dam height	0.000981	0.000978	0.272	0.272	Figures represent potential energy, but efficiency of conversion to electricity is 85–90%^[57]^[58]
Electrolytic capacitor	0.00001–0.0002^[59]	0.00001–0.001^[59]^[60]^[61]			Controlled electric discharge
Storage type	Energy density by mass (MJ/kg)	Energy density by volume (MJ/L)	Specific energy (W⋅h/kg)	Energy density (W⋅h/L)	How energy is released and Comments

Divide joule/m3 by 106 to get MJ/L. Divide MJ/L by 3.6 to get kW⋅h/L.

The mechanical energy storage capacity, or resilience, of a Hookean material when it is deformed to the point of failure can be computed by calculating tensile strength times the maximum elongation dividing by two. The maximum elongation of a Hookean material can be computed by dividing stiffness of that material by its ultimate tensile strength. The following table lists these values computed using the Young's modulus as measure of stiffness:

Mechanical energy capacities
Material	Energy density by mass (J/kg)	Resilience: Energy density by volume (J/L)	Density (kg/L)	Young's modulus (GPa)	Tensile yield strength (MPa)
Rubber band	1,651–6,605^[62]	2,200–8,900^[62]	1.35^[62]
Steel, ASTM A228 (yield, 1 mm diameter)	1,440–1,770	11,200–13,800	7.80^[63]	210^[63]	2,170–2,410^[63]
Acetals	908	754	0.831^[64]	2.8^[65]	65 (ultimate)^[65]
Nylon-6	233–1,870	253–2,030	1.084	2–4^[65]	45–90 (ultimate)^[65]
Copper Beryllium 25-1/2 HT (yield)	684	5,720^[66]	8.36^[67]	131^[66]	1,224^[66]
Polycarbonates	433–615	520–740	1.2^[68]	2.6^[65]	52–62 (ultimate)^[65]
ABS plastics	241–534	258–571	1.07	1.4–3.1^[65]	40 (ultimate)^[65]
Acrylic		1,530		3.2^[65]	70 (ultimate)^[65]
Aluminum 7077-T8 (yield)	399	1120^[66]	2.81^[69]	71.0^[66]	400^[66]
Steel, stainless, 301-H (yield)	301	2,410^[66]	8.0^[70]	193^[66]	965^[66]
Epoxy resins		113–1810		2–3^[65]	26–85 (ultimate)^[65]
Douglas fir Wood	158–200	96	.481–.609^[71]	13^[65]	50 (compression)^[65]
Steel, Mild AISI 1018	42.4	334	7.87^[72]	205^[72]	370 (440 Ultimate)^[72]
Aluminum (not alloyed)	32.5	87.7	2.70^[73]	69^[65]	110 (ultimate)^[65]
Pine (American Eastern White, flexural)	31.8–32.8	11.1–11.5	.350^[74]	8.30–8.56 (flexural)^[74]	41.4 (flexural)^[74]
Brass	28.6–36.5	250–306	8.4–8.73^[75]	102–125^[65]	250 (ultimate)^[65]
Copper	23.1	207	8.93^[75]	117^[65]	220 (ultimate)^[65]
Glass	5.56–10.0	13.9–25.0	2.5^[76]	50–90^[65]	50 (compression)^[65]

Table on energy content of batteries:

Battery energy capacities
Storage device	Energy content (Joule)	Energy type	Typical mass (g)	Typical dimensions (diameter × height in mm)	Typical volume (mL)	Energy density by volume (MJ/L)	Energy density by mass (MJ/kg)
Alkaline AA battery^[77]	9,360	Electrochemical	24	14.2 × 50	7.92	1.18	0.39
Alkaline C battery^[77]	34,416	Electrochemical	65	26 × 46	24.42	1.41	0.53
NiMH AA battery	9,072	Electrochemical	26	14.2 × 50	7.92	1.15	0.35
NiMH C battery	19,440	Electrochemical	82	26 × 46	24.42	0.80	0.24
Lithium-ion 18650 battery	28,800–46,800	Electrochemical	44–49^[78]	18 × 65	16.54	1.74–2.83	0.59–1.06

Nuclear energy sources

The greatest energy source by far is mass itself. This energy, E = mc2, where m = ρV, ρ is the mass per unit volume, V is the volume of the mass itself and c is the speed of light. This energy, however, can be released only by the processes of nuclear fission (0.1%), nuclear fusion (1%), or the annihilation of some or all of the matter in the volume V by matter-antimatter collisions (100%). Nuclear reactions cannot be realized by chemical reactions such as combustion. Although greater matter densities can be achieved, the density of a neutron star would approximate the most dense system capable of matter-antimatter annihilation possible. A black hole, although denser than a neutron star, does not have an equivalent anti-particle form, but would offer the same 100% conversion rate of mass to energy in the form of Hawking radiation. In the case of relatively small black holes (smaller than astronomical objects) the power output would be tremendous.

The highest density sources of energy aside from antimatter are fusion and fission. Fusion includes energy from the sun which will be available for billions of years (in the form of sunlight) but so far (2018), sustained fusion power production continues to be elusive.

Power from fission of uranium and thorium in nuclear power plants will be available for many decades or even centuries because of the plentiful supply of the elements on earth,[79] though the full potential of this source can only be realised through breeder reactors, which are, apart from the BN-600 reactor, not yet used commercially.[80] Coal, gas, and petroleum are the current primary energy sources in the U.S.[81] but have a much lower energy density. Burning local biomass fuels supplies household energy needs (cooking fires, oil lamps, etc.) worldwide.
Thermal power of nuclear fission reactors

The density of thermal energy contained in the core of a light water reactor (PWR or BWR) of typically 1 GWe (1 000 MW electrical corresponding to ~3 000 MW thermal) is in the range of 10 to 100 MW of thermal energy per cubic meter of cooling water depending on the location considered in the system (the core itself (~30 m3), the reactor pressure vessel (~50 m3), or the whole primary circuit (~300 m3)). This represents a considerable density of energy which requires under all circumstances a continuous water flow at high velocity in order to be able to remove the heat from the core, even after an emergency shutdown of the reactor. The incapacity to cool the cores of three boiling water reactors (BWR) at Fukushima in 2011 after the tsunami and the resulting loss of the external electrical power and of the cold source was the cause of the meltdown of the three cores in only a few hours, even though the three reactors were correctly shut down just after the Tōhoku earthquake. This extremely high power density distinguishes nuclear power plants (NPP's) from any thermal power plants (burning coal, fuel or gas) or any chemical plants and explains the large redundancy required to permanently control the neutron reactivity and to remove the residual heat from the core of NPP's.
Energy density of electric and magnetic fields
Main article: Radiant energy density

Electric and magnetic fields store energy. In a vacuum, the (volumetric) energy density is given by

\( {\displaystyle u={\frac {\varepsilon _{0}}{2}}\mathbf {E} ^{2}+{\frac {1}{2\mu _{0}}}\mathbf {B} ^{2}} \)

where E is the electric field and B is the magnetic field. The solution will be (in SI units) in Joules per cubic metre. In the context of magnetohydrodynamics, the physics of conductive fluids, the magnetic energy density behaves like an additional pressure that adds to the gas pressure of a plasma.

In normal (linear and nondispersive) substances, the energy density (in SI units) is

\( {\displaystyle u={\frac {1}{2}}(\mathbf {E} \cdot \mathbf {D} +\mathbf {H} \cdot \mathbf {B} )} \)

where D is the electric displacement field and H is the magnetizing field.

In the case of absence of magnetic fields, by exploiting Fröhlich's relationships it is also possible to extend these equations to anisotropic and nonlinear dielectrics, as well as to calculate the correlated Helmholtz free energy and entropy densities.[82]

When a pulsed laser impacts a surface, the radiant exposure, i.e. the energy deposited per unit of surface, may be called energy density or fluence.[83]
See also

Energy content of biofuel
Energy density Extended Reference Table
Figure of merit
Food energy
Heat of combustion
High Energy Density Matter
Power density and specifically
Power-to-weight ratio
Rechargeable battery
Solid-state battery
Specific energy
Specific impulse

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Calculated from fractional mass loss times c squared.
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