
Energy = E Joules Time = T seconds Power = P = E/T Watts Mass = M kilograms Energy/Mass = e = E/M Joules/kilogram Power/Mass = p = P/M Watts/kilogram
Energy per mass Carbons (MJoule/kg) Antimatter 90000000000 Fusion bomb 250000000 Maximum for a deuterium+tritium fusion bomb Fission bomb 83000000 Maximum for a uranium bomb Nuclear battery 589000 Strontium90, beta decay, 29 year half life Hydrogen 141.8 0 Methane 55.5 1 Ethane 51.9 2 Propane 50.4 3 Butane 49.5 4 Octane 47.8 8 Gasoline 47 8±2 Kerosene 46 12±2 Diesel 46 16±3 Lubricating oil 46 36±16 Fat 37 20±10 9 Calories/gram Ethanol 29 2 7 Calories/gram Sugar 17 6 4 Calories/gram Protein 17 4 Calories/gram Coal 32 Wood 22 Pure carbon 32.8 1 Plastic explos. 8.0 Smokeless powder 5.2 Modern gunpowder TNT 4.7 Aluminumair battery 4.6 Black powder 2.6 Medieval gunpowder Lithiumion battery 1.0 Phosphocreatine .137 ATP .057 Adenosine triphosphate Supercapacitor .026 Aluminum capacitor .012 Spring .0003
The energy sources that can be used by vehicles are:
Energy/Mass Power/mass Energy/$ Rechargeable Charge Maximum charging MJoule/kg Watt/kg MJoule/$ time cycles Gasoline 45 60 Battery, aluminum 4.6 130 No Battery, lithiumion .8 1200 .010 Yes 1 hour 1000 Supercapacitor .026 14000 .0005 Yes Instant Infinite Aluminum capacitor .010 50000 .0001 Yes Instant Infinite
MJoules Rocket Shock Density Boil /kg km/s km/s g/cm^{3} Kelvin Beryllium+ O2 23.2 5.3 Aluminum + O2 15.5 Magnesium+ O2 14.8 Hydrogen + O2 13.2 4.56 .07 20 Kerosene + O3 12.9 Octanitrocubane 11.2 10.6 1.95 Methane + O2 11.1 3.80 .42 112 Octane + O2 10.4 .70 399 Kerosene + O2 10.3 3.52 .80 410 Dinitrodiazeno. 9.2 10.0 1.98 C6H6N12O12 9.1 1.96 China Lake compound Kerosene + H2O2 8.1 3.2 Kerosene + N2O4 8.0 2.62 HMX (Octogen) 8.0 3.05 9.1 1.86 RDX (Hexagen) 7.5 2.5 8.7 1.78 Al + NH4NO3 6.9 Nitroglycerine 7.2 8.1 1.59 Unstable PLX 6.5 1.14 95% CH_{3}NO_{2} + 5% C_{2}H_{4}(NH_{2})_{2} Composition 4 6.3 8.04 1.59 91% RDX. "Plastic explosive" Kerosene + N2O 6.18 Dynamite 5.9 7.2 1.48 75% Nitroglycerine + stabilizer PETN 5.8 8.35 1.77 Smokeless powder 5.2 6.4 1.4 Used after 1884. Nitrocellulose TNT 4.7 6.9 1.65 Trinitrotoluene Al + Fe2O3 4.0 Thermite H2O2 2.7 3.1 1.45 423 Hydrogen peroxide Black powder 2.6 3.08 .6 1.65 Used before 1884 Al + NH4ClO4 2.6 NH4ClO4 2.5 N2O 1.86 1.76 N2H4 1.6 2.2 1.02 387 Hydrazine NH4NO3 1.4 2.0 2.55 1.12 Ammonium nitrate Bombardier beetle .4 Hydroquinone + H_{2}O_{2} + protein catalyst N2O4 .10 1.45 294 Rocket: Rocket exhaust speed Shock: Shock speed
High explosives have a large shock velocity.
MJoules Shock Density /kg km/s g/cm^{3} Octanitrocubane 11.2 10.6 1.95 Dinitrodiazeno. 9.2 10.0 1.98 C6H6N12O12 9.1 1.96 China Lake compound HMX (Octogen) 8.0 9.1 1.86 RDX (Hexagen) 7.5 8.7 1.78 PLX 6.5 1.14 95% CH_{3}NO_{2} + 5% C_{2}H_{4}(NH_{2})_{2} Composition 4 6.3 8.04 1.59 91% RDX. "Plastic explosive" Dynamite 5.9 7.2 1.48 75% Nitroglycerine + stabilizer PETN 5.8 8.35 1.77
The best oxidizer is liquid oxygen, and the exhaust speed for various fuels when burned with oxygen is:
Exhaust Energy Density of fuel + oxidizer speed /mass km/s MJ/kg g/cm^{3} Hydrogen H2 4.46 13.2 .32 Methane CH4 3.80 11.1 .83 Ethane C2H6 3.58 10.5 .9 Kerosene C12H26 3.52 10.3 1.03 Hydrazine N2H4 3.46 1.07Liquid hydrogen is usually not used for the ground stage of rockets because of its low density.
We use kerosene as a standard fuel and show the rocket speed for various oxidizers. Some of the oxidizers can be used by themselves as monopropellants.
Energy/Mass Energy/Mass Rocket Rocket Boil Density with kerosene as monopropellant with kerosene as monopropellant Kelvin g/cm^{3} MJoule/kg MJoule/kg km/s km/s O3 12.9 2.97 161 O2 10.3 0 3.52 0 110 1.14 H2O2 8.1 2.7 3.2 1.6 423 1.45 N2O4 8.00 .10 2.62 294 1.44 N2O 6.18 1.86 1.76 185 N2H4  1.58 2.2 387 1.02
MJoules Rocket Density /kg km/s g/cm^{3} C6H6N12O12 9.1 1.96 China Lake compound HMX (Octogen) 8.0 3.05 1.86 RDX (Hexagen) 7.5 2.5 1.78 Al + NH4ClO4 2.6 NH4ClO4 2.5 NH3OHNO3 2.5 1.84 Hydrxyammonium nitrate Al + NH4NO3 6.9 NH4NO3 1.4 2.0 1.12 Ammonium nitrate
~808 Qing Xuzi publishes a formula resembling gunpower, consisting of 6 parts sulfur, 6 parts saltpeter, and 1 part birthwort herb (for carbon). ~850 Incendiary property of gunpower discovered 1132 "Fire lances" used in the siege of De'an, China 1220 alRammah of Syria publishes "Military Horsemanship and Ingenious War Devices", describes the purification of potassium nitrate by adding potassium carbonate with boiling water, to precipitate out magnesium carbonate and calcium carbonate. 1241 Mongols use firearms at the Battle of Mohi, Hungary 1338 Battle of Arnemuiden. First naval battle involving cannons. 1346 Cannons used in the Siege of Calais and the Battle of Crecy 1540 Biringuccio publishes "De la pirotechnia", giving recipes for gunpowder 1610 First flintlock rifle 1661 Boyle publishes "The Sceptical Chymist", a treatise on the distinction between chemistry and alchemy. It contains some of the earliest modern ideas of atoms, molecules, and chemical reaction, and marks the beginning of the history of modern chemistry. 1669 Phosphorus discovered 1774 Lavoisier appointed to develop the French gunpowder program. By 1788 French gunpowder was the best in the world. 1832 Braconnot synthesizes the first nitrocellulose (guncotton) 1846 Nitrocellulose published 1847 Sobrero discovers nitroglycerine 1862 LeConte publishes simple recipes for producing potassium nitrate. 1865 Abel develops a safe synthesis of nitrocellulose 1867 Nobel develops dynamite, the first explosive more powerful than black powder It uses diatomaceous earth to stabilize nitroglycerine 1884 Vieille invents smokeless gunpowder (nitrocellulose), which is 3 times more powerful than black powder and less of a nuisance on the battlefield. 1902 TNT first used in the military. TNT is much safer than dynamite 1930 RDX appears in military applications 1942 Napalm developed 1949 Discovery that HMX can be synthesized from RDX 1956 C4 explosive developed (based on RDX) 1999 Eaton and Zhang synthesize octanitrocubane and heptanitrocubane Black powder = .75 KNO_{3} + .19 Carbon + .06 Sulfur
Above 550 Celsius, potassium nitrate decomposes. 2 KNO_{3} ↔ 2 KNO_{2} + O_{2}.
Potassium nitrate KNO_{3} 75% (Saltpeter) Charcoal C_{7}H_{4}O 15% Sulfur S 10% Oversimplified equation: 2 KNO_{3} + 3 C + S → K_{2}S + N_{2} + 3 CO_{2} Realistic equation: 6 KNO_{3} + C_{7}H_{4}O + 2 S → KCO_{3} + K_{2}SO_{4} + K_{2}S + 4 CO_{2} + 2 CO + 2 H_{2}O + 3 N_{2}Nitrite (NO_{3}) is the oxidizer and sulfur lowers the ignition temperature.
MJoules /kg Hydrogen + Oxygen 13.16 Gasoline + Oxygen 10.4 Mass Energy Energy/Mass kg MJ MJ/kg MOAB 9800 46000 4.7 8500 kg of fuel
Form Ignition Density (Celsius) White 30 1.83 Red 240 1.88 Violet 300 2.36 Black 2.69Red phosphorus is formed by heating white phosphorus to 250 Celsius or by exposing it to sunlight. Violet phosphorus is formed by heating red phosphorus to 550 Celsius. Black phosphorus is formed by heating white phosphorus at a pressure of 12000 atmospheres. Black phosphorus is least reactive form and it is stable below 550 Celsius.
The safety match was invented in 1844 by Pasch. The match head cannot ignite by itself. Ignitition is achieved by striking it on a rough surface that contains red phosphorus. When the match is struck, potassium chlorate in the match head mixes with red phosphorus in the abrasive to produce a mixture that is easily ignited by friction. Antimony trisulfide is added to increase the burn rate.
Match head Fraction Striking surface Fraction Potassium chlorate KClO3 .50 Red phosphorus .5 Silicon filler Si .4 Abrasive .25 Sulfur S small Binder .16 Antimony3 trisulfide Sb2S3 small Neutralizer .05 Neutralizer small Carbon .04 Glue smallA "strike anywhere" match has phosphorus in the match head in the form of phosphorus sesquisulfide (P_{4}S_{3}) and doesn't need red phosphorus in the striking surface. P_{4}S_{3} has an ignition temperature of 100 Celsius.
Before the invention of iron, fires were started by striking flint (quartz) with pyrite to generate sparks. Flintlock rifles work by striking flint with iron. With the discovery of cerium, ferrocerium replaced iron and modern butane lighters use ferrocerium, which is still referred to as "flint".
Cerium .38 Ignition temperature of 165 Celsius Lanthanum .22 Iron .19 Neodymium2 .04 Praseodymium .04 Magnesium .04
Nitrous oxide is stored as a cryogenic liquid and injected along with gaoline into the combustion chamber. Upon heating to 300 Celsius the nitrous oxide decomposes into nitrogen and oxygen gas and releases energy. The oxygen fraction in this gas is higher than that in air (1/3 vs. .21) and the higher faction allows for more fuel to be consumed per cylinder firing.
Air density = .00122 g/cm^{3} Nitrous oxide gas density = .00198 g/cm^{3} Diesel density = .832 g/cm^{3} Gasoline density = .745 g/cm^{3} Diesel energy/mass = 43.1 MJoules/kg Gasoline energy/mass = 43.2 MJoules/kg Nitrous oxide boiling point = 88.5 Celsius Air oxygen fraction = .21 Nitrous oxide oxygen fraction= .33 Nitrous oxide decompose temp = 300 Celsius Nitrous oxide liquid pressure= 52.4 Bars Pressure required to liquefy N2O at room temperature
Hydroquinone and peroxide are stored in 2 separate compartments are pumped into the reaction chamber where they explode with the help of protein catalysts. The explosion vaporizes 1/5 of the liquid and expels the rest as a boiling drop of water, and the pquinone in the liquid damages the foe's eyes. The energy of expulsion pumps new material into the reaction chamber and the process repeats at a rate of 500 pulses per second and a total of 70 pulses. The beetle has enough ammunition for 20 barrages.
2 H_{2}O_{2} → 2 H_{2}O + O_{2} (with protein catalyst) C_{6}H_{4}(OH)_{2} → C_{6}H_{4}O_{2} + H_{2} (with protein catalyst) O_{2} + 2 H_{2} → 2 H_{2}O Firing rate = 500 pulses/second Number of pulses in one barrage = 70 Firing time = .14 seconds Number of barrages = 20
A turbojet engine compresses air before burning it to increase the flame speed and make it burn explosively. A ramjet engine moving supersonically doesn't need a turbine to achieve compression.
Airbus A350 compression ratio = 52 Air density at sea level = 1 bar Air density at 15 km altitude = .25 bar Air density in A350 engine = 13 barFrom the thermal flame theory of Mallard and Le Chatelier,
Temperature of burnt material = T_{b} Temperature of unburnt material = T_{u} Temperature of ignition = T_{i} Fuel density = D_{fuel} Oxygen density = D_{oxygen} Reaction coefficient = C Reaction rate = R = C D_{fuel} D_{oxygen} Thermal diffusivity = Q = 1.9⋅10^{5} m^{2}/s Flame speed = V V^{2 = Q C Dfuel Doxygen (Tb  Ti) / (Ti  Tu) }
If the pressure front moves supersonically then the front forms a discontinuous shock, where the pressure makes a sudden jump as the shock passes.
Metal powder is often included with explosives.
Energy/mass Energy/mass not including including oxygen oxygen (MJoule/kg) (MJoule/kg) Hydrogen 113.4 12.7 Gasoline 46.0 10.2 Beryllium 64.3 23.2 Aluminum 29.3 15.5 Magnesium 24.5 14.8 Carbon 12.0 3.3 Lithium 6.9 3.2 Iron 6.6 4.6 Copper 2.0 1.6
An oxygen candle is a mixture of sodium chlorate and iron powder, which when ignited smolders at 600 Celsius and produces oxygen at a rate of 6.5 manhours of oxygen per kilogram of mixture. Thermal decomposition releases the oxygen and the burning iron provides the heat. The products of the reaction are NaCl and iron oxide.
The properties of the best commercial lithium ion batteries are:
Energy/Mass = .8 Joule/kg Power/Mass = 1200 Watt/kg Energy/$ = .01 MJoule/kg Density = 3.5 gram/cm^{3} Recharges =1000 Shelf life = 1.0 year Voltage = 3.7 VoltEnergy/Mass and Power/Mass are an engineering tradeoff. One can be increased at the expense of the other.
Battery energy is often given in "Watt hours" or "Ampere hours".
Voltage = V Volts Charge = C Coulombs (1 Amphour = 3600 Coulombs) Electric current = I Amperes Electric power = P = VI Watts Time = T seconds Energy = E = PT Joules = CV Joules1 Watt hour = 3600 Joules = 1 Watt * 3600 seconds
1 Amp hour = 3600 Coulombs = 1 Coulombs/second * 3600 seconds
A battery with a voltage of 3.7 Volts that delivers
1 Ampere for 1 hour has an energy of
Energy = 1 Ampere * 3.7 Volts * 3600 seconds = 13320 Joules
A single battery is a "cell" and a set of cells is a "pack". Packs are used to multiply the energy and power of cells.
Battery packs are notorous for catching fire, but cell technology has reached the point where it's now possible to make safe battery packs, and the design is simple enough so that anyone can construct their own packs.
Cells can be combined in series and/or parallel. Connecting in series multiples voltage, and voltage is helpful for achieving high power in a motor.
Connecting in series is easier than in parallel. If it's possible to achieve the required power without parallelization then one should do so, and this is usually possible with modern cells.
Series packs have the advantage that the cells can easily be extracted and charged individually, and cells can be interchanged between packs. One can also construct a set of series packs and swap them in like gun clips.
High power electric bikes use a voltage of 72 Volts. If we use one series array of C cells then a pack provides 4440 Watts and 1.2 MJoules. Any electric device requiring less than this much power can be powered by a series pack.
The properties of a modern highpower cell are:
Type = "C" Voltage = 3.7 Volts Energy = 60 kJoules Power = 155 Watts Mass = 92 grams Energy/mass = 650 kJoules/kg Power/mass =1680 Watts/kg Current = 42 Amperes Manufacturer = "Basen"When the cells are connected in series the values for voltage and power are:
Cells Voltage Power Volts kWatts 1 3.7 .15 2 7.4 .30 3 11 .45 Electric kick scooter 4 15 .60 6 24 .90 Electric bike 10 36 1.5 20 72 3.0 Compact electric car 96 356 15.0 Large electric car
Electric bike motors use either 36, 48, or 72 Volts. The following table shows how to build a battery pack for each motor power.
Power Volts Cells Series Parallel Current Cell Cell Cell Cell Cell kWatt Amperes Amperes Amphours $ type ID# .5 36 10 .75 36 10 10 1 21 25 2.1 4 A LG HD4 1.5 48 13 13 1 31 30 2.0 4.5 A Sony VTC4 3 72 20 20 1 42 60 4.5 4.5 C Basen 6 72 40 20 2 83 120 4.5 4.5 C Basen 12 72 80 20 3 167 180 4.5 4.5 C Basen Cells Total number of cells, equal to the number of cells connected in series times the number of cells connected in parallel. Series Number of cells connected in series. For example, 20 batteries with 3.6 volts each connected in series produces a voltage of 72 Volts. Parallel Number of cells connected in parallel. Current Current required to provide given power Cell Maximum current of a cell
The Gigafactory in Nevada has a production target for 2020 of:
Battery production = 200 TeraJoules/year Energy of one car = 310 MJoules (Tesla Model S) Cars supplied/year = .64 millionThe Solar City Factory produces solar panels.
Panel production = 1.0 GWatts/year Panel duty factor = .25 (Average solar intensity over peak solar intensity) Effective power production = .25 GWatts/year
Voltage = V Volts Capacitance = C Farads Total energy = E = ½ C V^{2} Joules Effective = E_{e} = ¼ C V^{2} JoulesNot all of the energy in a capacitor is harnessable because the voltage diminishes as the charge diminishes, hence the effective energy is less than the total energy.
A = Plate area Z = Plate spacing Ke = Electric force constant = 8.9876e9 N m^{2} / C^{2} Q = Max charge on the plate (Coulombs) Emax= Max electric field = 4 Pi Ke Q / A V = Voltage between plates = E Z = 4 Pi Ke Q Z / A En = Energy = .5 Q V = .5 A Z E^{2} / (4 π Ke) e = Energy/Volume = E / A Z = .5 E^{2} / (4 π Ke) q = Charge/Volume = Q / A / Z C = Capacitance = Q/V = (4 Pi Ke)A capacitor can be specified by two parameters:
The maximum electric field is equal to the max field for air times a dimensionless number characterizing the dielectric
Eair = Maximum electric field for air before electical breakdown Emax = Maximum electric field in the capacitor Rbohr= Bohr radius = Characteristic size of atoms = 5.2918e11 m = hbar^{2} / (ElectronMass*ElectronCharge^{2}*Ke) Ebohr= Bohr electric field = Field generated by a proton at a distance of 1 Bohr radius = 5.142e11 Volt/m Maximum energy density = .5 * 8.854e12 Emax^{2} Emax (MVolt/m) Energy density (Joule/kg) Al electrolyte capacitor 15.0 1000 Supercapacitor 90.2 36000 Bohr limit 510000 1.2e12 Capacitor with a Bohr electric field
One has to choose a wise balance for the masses of the motor, battery, fuselage, and payload. The properties of the electrical components are:
Energy/Mass Power/mass Energy/$ Power/$ $/Mass MJoule/kg kWatt/kg MJoule/$ kWatt/$ $/kg Electric motor  10.0  .062 160 Lithiumion battery .75 1.5 .009 .0142 106 Lithium supercapacitor .008 8 .0010 .09 90 Aluminum capacitor .0011 100If the battery and motor have equal power then the battery has a larger mass than the motor.
Mass of motor = M_{mot} Mass of battery = M_{bat} Power = P (Same for both the motor and the battery) Power/mass of motor = p_{mot} = P/M_{mot} = 8.0 kWatt/kg Power/mass of battery = p_{bat} = P/M_{bat} = 1.5 kWatt/kg Battery mass / Motor mass= R =M_{bat}/M_{mot} = p_{mot}/p_{bat} = 5.3The "sports prowess" of a drone is the drone power divided by the minimum hover power. To fly, this number must be larger than 1.
Drone mass = M_{dro} Motor mass = M_{mot} Motor power/mass = p_{mot} = 8000 Watts/kg Hover minimum power/mass = p_{hov} = 60 Watts/kg Drone power = P_{dro} = p_{mot} M_{mot} Hover minimum power = P_{hov} = p_{hov} M_{dro} Sports prowess = S = P_{dro}/P_{hov} = (p_{mot}/p_{hov}) * (M_{mot}/M_{dro}) = 80 M_{mot}/M_{dro}If S=1 then M_{mot}/M_{dro} = 1/80 and the motor constitutes a negligible fraction of the drone mass. One can afford to increase the motor mass to make a sports drone with S >> 1.
If the motor and battery generate equal power then the sports prowess is
S = (p_{bat}/p_{hov}) * (M_{bat}/M_{dro}) = 25 M_{bat}/M_{dro}If M_{bat}/M_{dro} = ½ then S=12.5, well above the minimum required to hover.
Suppose a drone has a mass of 1 kg. A squash racquet can have a mass of as little as .12 kg. The fuselage mass can be much less than this because a drone doesn't need to be as tough as a squash racquet, hence the fuselage mass is negligible compared to the drone mass. An example configuration is:
kg Battery .5 Motors .1 To match the battery and motor power, set motor mass / battery mass = 1/5 Rotors <.05 Fuselage .1 Camera .3 Drone total 1.0Supercapacitors can generate a larger power/mass than batteries and are useful for extreme bursts of power, however their energy density is low compared to batteries and so the burst is short. If the supercapacitor and battery have equal power then
Battery power/mass = p_{bat} = 1.5 kWatts/kg Supercapacitor power/mass = p_{sup} = 8.0 kWatts/kg Battery power = P Battery mass = M_{bat} = P / p_{bat} Supercapacitor mass = M_{sup} = P / p_{sup} Supercapacitor/Battery mass= R =M_{sup}/ M_{bat} = p_{bat}/p_{sup} = .19The supercapacitor is substantially ligher than the battery. By adding a lightweight supercapacitor you can double the power. Since drones already have abundant power, the added mass of the supercapacitor usually makes this not worth it.
If a battery and an aluminum capacitor have equal powers,
Aluminum capacitor mass / Battery mass = .015If a battery or supercapacitor is operating at full power then the time required to expend all the energy is
Mass = M Energy = E Power = P Energy/Mass = e = E/M Power/Mass = p = P/M Discharge time= T = E/P = e/p Energy/Mass Power/Mass Discharge time Mass MJoule/kg kWatt/kg seconds kg Lithium battery .75 1.5 500 1.0 Supercapacitor .008 8.0 1.0 .19 Aluminum capacitor .0011 100 .011 .015"Mass" is the mass required to provide equal power as a lithium battery of equal mass.
Mass Energy E/M Power P/M Price Energy/$ C Voltage kg kJoule kJ/kg kWatt kW/kg $ kJoule/$ Farads Volts PM5R0V105R .000454 .0062 13.8 5.7 .0011 1 5.0 Maxwell BCAP0350 .060 .638 10.6 .459 7.65 16 .040 350 2.7 Adafruit .135 .984 7.3 20 .049 630 2.5 BMOD0006E160B02 5.2 37.1 7.1 2.08 .40 1170 .032 5.8 160 XLM62R1137R 15 125.3 8.4 124.2 8.3 1396 .090 130 62.1