# Energy density

Physical size
Surname volumetric energy density
Formula symbol ${\ displaystyle w,}$ ${\ displaystyle \ rho}$
Derived from Energy per volume
Size and
unit system
unit dimension
SI J · m -3 M · L −1 · T −2
Physical size
Surname gravimetric energy density, specific energy
Formula symbol ${\ displaystyle w,}$ ${\ displaystyle \ rho}$
Size and
unit system
unit dimension
SI J · kg -1 L 2 · T −2

In physics, the energy density describes the distribution of energy over a certain quantity and consequently always has the shape ${\ displaystyle E}$${\ displaystyle X}$

${\ displaystyle w = {\ frac {\ mathrm {d} E} {\ mathrm {d} X}}.}$

Most often it is used as a

• volumetric energy density, a measure of the energy per room volume of a substance (SI unit: joules per cubic meter)
• Gravimetric energy density or specific energy, a measure of the energy per mass of a substance (SI unit: joules per kilogram).

But ultimately you can define a corresponding energy density for each physical quantity. According to DIN 5485 , the term is the energy density of the dimensional , in particular volumetric indication reserved for specific energy especially in mass; see " Energy " and " Related Quantity ".

The energy density of energy storage devices such as fuels and batteries used in technology is of great practical interest . In vehicle construction in particular , the energy density of the energy storage device used is decisive for the achievable range.

## Energy density in electrodynamics

### Energy density of electromagnetic waves

From Maxwell's equations one can conclude that the maximum energy output of electromagnetic waves in a substance is proportional to the square of the field amplitudes . Electric and magnetic fields contribute equally:

${\ displaystyle w = {\ frac {1} {2}} \ left ({\ vec {E}} \ cdot {\ vec {D}} + {\ vec {H}} \ cdot {\ vec {B} } \ right)}$

### Energy density in the plate capacitor

The energy of a charged plate capacitor is calculated as

${\ displaystyle W = {\ frac {1} {2}} CU ^ {2}.}$

The following applies to the capacity:

${\ displaystyle C = \ varepsilon _ {0} \ varepsilon _ {r} {\ frac {A} {d}}}$

The voltage U results from E · d. By inserting it you get for the energy:

${\ displaystyle W = {\ frac {1} {2}} \ varepsilon _ {0} \ varepsilon _ {r} {\ frac {A} {d}} E ^ {2} d ^ {2}}$

This leads to the energy density:

${\ displaystyle w_ {el} = {\ frac {W} {V}} = {\ frac {1} {2}} \ varepsilon _ {0} \ varepsilon _ {r} E ^ {2}}$

### Energy of the magnetic field of a coil

For the energy of the magnetic field of a coil with the amount of the magnetic flux density , the cross-sectional area , the length , the number of turns, the current strength , the magnetic field constants and the relative permeability results ${\ displaystyle W}$ ${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle l}$${\ displaystyle n}$${\ displaystyle I}$ ${\ displaystyle \ mu _ {0}}$ ${\ displaystyle \ mu _ {r}}$

${\ displaystyle W = {\ frac {B ^ {2}} {2 \ cdot \ mu _ {0} \ cdot \ mu _ {r}}} \ cdot A \ cdot l = {\ frac {(n \ cdot I) ^ {2}} {2}} \ cdot \ mu _ {0} \ cdot \ mu _ {r} \ cdot {\ frac {A} {l}}}$

and then continue

${\ displaystyle w_ {B} = {\ frac {B ^ {2}} {2 \ cdot \ mu _ {0} \ cdot \ mu _ {r}}} = {\ frac {(n \ cdot I) ^ {2}} {2}} \ cdot \ mu _ {0} \ cdot \ mu _ {r}}$

for the energy density of the flux density . ${\ displaystyle w_ {B}}$${\ displaystyle B}$

## Energy density of energy storage and primary energy carriers

Energy densities of selected energy stores

The energy density of fuels is called the calorific value or calorific value , that of batteries capacity per volume or capacity per mass. For example, the energy density of a lithium polymer battery is 140–180 watt hours per kg mass (140–180 Wh / kg) and that of a nickel metal hydride battery (NiMH) is 80 Wh / kg. In comparison with other types of electrical energy storage of the battery cuts right from favorable.

A high energy density is desired in order to keep transport costs for the energy carrier low, but also in order to achieve long operating times for mobile devices and long ranges for vehicles. For example, model helicopters can fly longer with a lithium polymer battery than with a NiMH battery of the same mass.

The energy density of nutrients is also known as the physiological calorific value .

In addition to accumulators, energy storage devices to support the power grid include superconducting magnetic energy storage devices (SMES), pumped storage power plants and compressed air storage power plants .

### Examples

Substance / system Energy density in MJ / kg Energy density in MJ / l comment Note * credentials
NdFeB and SmCo magnets 0.000 055 Range: 200-400 kJ / m 3 BH max , i.e. 30-55 J / kg like
Electrolytic capacitor 0.000 4 Range: 0.01-0.1 Wh / kg, i.e. 0.04-0.4 kJ / kg el
Double layer capacitor 0.01 Range: 0.1-3 Wh / kg, i.e. 0.4-10 kJ / kg el
Lead accumulator 0.11 0.25 a) Range: 3–30 Wh / kg, i.e. 10–110 kJ / kg
b) 30–40 Wh / kg

Energy density (MJ / l) calculated on the basis of the gross density of a starter battery

chem a)
b)
Adenosine triphosphate (ATP) 0.128 = 64.6 kJ / mol (with cleavage of both bonds) at 0.507 kg / mol chem see adenosine triphosphate
NiCd battery 0.14 a) 40 Wh / kg
b) Range: 4–70 Wh / kg, i.e. 15–250 kJ / kg
c) 40–70 Wh / kg
chem a)
b)
c)
Flywheel storage with CFRP 0.18 49 Wh / kg mech
Carbon-zinc battery 0.23 65 Wh / kg, i.e. 230 kJ / kg chem
NiMH battery 0.28 a) 2,300 mAh · 1.0 V / 30 g = 76.7 Wh / kg
b) 60 Wh / kg
c) Range: 15–120 Wh / kg, i.e. 50–400 kJ / kg
d) 60–80 Wh / kg
chem a)
b)
c)
d)
Li-titanate battery 0.32 90 Wh / kg, i.e. 0.32 MJ / kg chem
Melting energy ice 0.33 at 1013.2 hPa and 0 ° C Phase transition
Zebra battery 0.43 Range: 100-120 Wh / kg, i.e. 0.36-0.43 MJ / kg chem , (but with an unclear unit)
Alkaline manganese battery 0.45 125 Wh / kg, i.e. 450 kJ / kg chem
Compressed air 0.46 0.14 a) 138 · 10 6 Ws / m 3 at 300 kg / m 3
b) However, the gravimetric energy density is up to a factor of 10 lower if the pressure vessel is also taken into account
mech a)
b) without ref.
Li-polymer battery 0.54 a) 150 Wh / kg, i.e. 540 kJ / kg
b) 130–200 Wh / kg
chem a)
b)
Li-ion battery 0.65 0.7-1.8 a) 180 Wh / kg
b) 100 Wh / kg
c) Range: 40–200 Wh / kg,
i.e. 150–700 kJ / kg d)> 160 Wh / kg
chem a)
b)
c)
d)
Hydrogen (including hydride tank ) 1.19 chem, O
Zinc-air battery 1.2 a) 340 Wh / kg, i.e. 1,200 kJ / kg
b) three times as large as conventional Li batteries
chem, O a)
b)
Lithium-sulfur accumulator 1.26 350 Wh / kg chem
Solid state accumulator (mass production) 1.44 > 400 Wh / kg chem, O
Lithium-air accumulator 1.6 a)> 450 Wh / kg
b) should reach 1,000 Wh / kg
chem, O a)
b)
Heat of condensation of water 2.26 at 1013.2 hPa and 100 ° C. 40.7 kJ / mol Phase transition
Lithium thionyl chloride battery 2.34 650 Wh / kg chem
Thermite 4.0 18.4 chem (?)
Trinitrotoluene (TNT) 4.6 6.92 1,046 kJ / mol / (227 g / mol). Oxidizer is contained in the molecule. Note: for the TNT equivalent, an energy density of 4.18 MJ / kg = 1.0 Mcal / kg is used. chem see TNT equivalent
Aluminum-air battery 4.7 a) 1,300 Wh / kg, i.e. 4,700 kJ / kg
b) Future goal: 8,000 Wh / kg = 28 MJ / kg
chem, O a)
b)
strongest explosives 7th Oxidizer is contained in the molecule. see explosives
Residual waste (moist) 11 Range 8-11 MJ / kg O, Hw
Brown coal 11.3 a) Range 8.4-11.3 MJ / kg
b) 9.1 MJ / kg
O, Hw a)
b)
Wood (air dry) 16.8 a) Range 14.6-16.8 MJ / kg
b) 14.7 MJ / kg
O, Hw a)
b)
sugar 16.7 O
Sewage sludge 17th Range 11–17 MJ / kg for dry matter (digested lower value, not digested upper value) O, Hw
straw 17th O, Hw
Wood pellets and wood briquettes 18th O, Hw
Brown coal (briquette) 19.6 O, Hw
Methanol 19.7 15.6 O, Hw
Ammonia (liquid) 22.5 15.3 −33 ° C or 9 bar O, Hw
Ethanol 26.7 21.1 O, Hw
Old tires 29.5 O, Hw
Silicon 32.6 75.9 O
carbon 32.8 74.2 O
Propane (liquid) 46.3 23.4 liquid at 15 ° C O, Hw
Hard coal 34 a) Range 27-34 MJ / kg
b) 29.3 MJ / kg
c) 30 MJ / kg, coke 28.7 MJ / kg, briquettes 31.4 MJ / kg
O, Hw a)
b)
c)
diesel 43 35-36 O, Hw
petrol 40-42 29-32 O, Hw
crude oil 41.9 Heavy oil, bunker oil, residual oil has approx. 40 MJ / kg O, Hw
Heating oil , light 42.8 36 O, Hw
Methane (main component of natural gas ) 50 0.0317 a) 50 MJ / kg / 35.9 MJ / m 3
b) 55.5 MJ / kg / 39.8 MJ / m 3
c) 31.7 MJ / m 3
O, Hw a)
b)
c)
Hydrogen 1 bar (without tank) 120 0.01079 O , (?)
Hydrogen 700 bar (without tank) 120 5.6 O , (?)
Liquid hydrogen (without tank) 120 10.1 O , (?)
Hydrogen (liquid, bound to LOHC ) 13.2 10.4 The calorific value of the LOHC carrier ( methanol ) is not taken into account. Energy density calculated on the basis of the maximum loading of 0.11 kg H2 / kg methanol . O, Hw
Atomic hydrogen 216 spontaneous reaction to form molecular hydrogen chem
Radioisotope generator 5,000 electrical (60,000 MJ / kg thermal) nucl.
Nuclear fission natural uranium (0.72% 235 U) 648,000 corresponds to 7.5 GWd / t SM nucl.
Burn-up (nuclear technology) 3,801,600 Value based on the average burn-up of around 40 GWd / t today. Fissile material up to 500  GWd / t SM corresponds to 43,200,000 MJ / kg. nucl.
Decay of the free neutron 74,600,000 780 keV (1.250 10 −13 J) per neutron (1.674 10 −27  kg) nucl.
Nuclear fission 235 U 79,390,000 1,500,000,000 corresponds to 1,042 GWd / t SM nucl.
Nuclear fission 232 Th 79,420,000 929,214,000 nucl.
Nuclear fusion (nuclear weapon, nuclear fusion reactor) 300,000,000 corresponds to 3,472 GWd / t SM nucl.
Proton-proton reaction 627,000,000 Most important fusion reaction in the sun ; corresponds to 7,256 GWd / t SM nucl.
complete conversion of mass into energy 89,875,000,000 maximum possible energy density; corresponds to 1,042,000 GWd / t SM nucl. E = mc²

Remarks:

• likes: magnetic energy
• el: electrical energy
• chem: enthalpy of reaction
• mech: with mechanical conversion
• nukl: conversion of atomic nuclei or elementary particles
• Hw: calorific value
• O = oxidizer is air and is not taken into account in the reference mass.

1 J = 1 W s ; 1 MJ = 0.2778 kWh ; 1 kWh = 3.6 MJ ; 1 G W d = 24 GWh = 86.4 T J

## More energy densities

• Mechanical energy density: The elastic energy that is stored in a certain volume of a material is referred to as mechanical energy density (formula symbol mostly ) and is determined in mechanical tests (e.g. in a tensile test ) , whereby the mechanical stress and the elongation are . The mechanical energy density in the event of material failure serves as an easily measured parameter for the toughness of a material, but does not always correlate with the fracture toughness measured by fracture mechanics .${\ displaystyle U}$${\ displaystyle \ textstyle U = \ int \ sigma \ cdot \ mathrm {d} \ epsilon}$${\ displaystyle \ sigma}$${\ displaystyle \ epsilon}$
• Spectral energy density: Dependence of the energy of a radiation spectrum on the frequency.${\ displaystyle \ epsilon _ {\ nu} = {\ tfrac {\ mathrm {d} E} {\ mathrm {d} \ nu}}.}$
• Sound energy density : The energy density of the sound field.
• Calorific value , calorific value (there also the comparison of different energy densities of typical fuels)
• Specific or molar latent heat : The energy stored in the physical state.
• Gravimetric energy density of food , used in the Volumetrics Diet
• Shear energy density : The energy density during a shear .

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