
A flying car powered by lithiumion batteries can fly for 45 minutes and cover 100 km. The minimum price of the car is set by the battery. The smallest battery capable of powering a 1person car costs $8000.
Flying cars will be capable of vertical takeoff and landing and will have 2, 3, or 4 rotors. The rotor number is determined by a tradeoff between efficiency (fewer rotors is better) vs. stability and failsafe (more rotors is better). The car will also have a wing to help with horizontal flight.
The properites of flying cars are determined by the properties of lithiumion batteries and rotors. In the sections below we use these to construct a concrete design for a flying car.
The design of the car depends on the physics of rotors. For a rotor,
Power required to hover = Constant * LiftForce^{3/2} / RotorRadiusThe larger the rotor radius the better, so long as it's not so large as t dominate the mass of the car. We choose the design so that the total mass in rotors is half the mass of the pilot.
The most efficient copter has one lift rotor (a "monocopter"). Increasing the number of rotors while preserving the total rotor mass means that each rotor becomes smaller, hence it takes more power to fly.
Increasing the rotor number increases stability and redundancy. Most drones use 4, 6, or 8 rotors. 4 rotors offers good stability and failsafe and there is no point to a flying car with more than 4 rotors. Flying cars can be expected to have 2, 3, or 4 rotors.
The flight time is proportional to the battery mass, hence the battery should be as large as possible but not so large so as to dominate the car mass. We choose a design with a battery mass equal to the pilot mass. With this mass, the battery power is twice that required to hover, and so power isn't a problem.
Stateoftheart lithiumion batteries have an energy/mass of .8 MJoules/kg and can fly a car for 44 minutes. In the future, lithiumsulfur batteries will take over with an energy/mass of 1.4 MJoules/kg.
We outline a design using 2 large lift rotors plus a few small stability rotors, with the following masses:
Flying car mass = 120 kg (Excluding battery and pilot) Battery mass = 100 kg Pilot mass = 80 kg Total car mass = 300 kg Total aircraft mass = M = 300 kg (Includes passenger) # of large rotors = N = 2 Rotor radius = R = 1.5 meters Gravity constant = g = 9.8 meters/second^{2} Rotor force = F = Mg/N =1470 Newtons Rotor quality = q = 1.02 Air density = D = 1.22 kg/meter^{3} Rotor power = P_{r}=(qDR)^{1}F^{3/2}= 30.2 kWatts Hover power = P_{h}= N P_{r} = 60.4 kWatts Hover power/mass = P/M = 101 Watts/kg battery mass = m = 100 kg Battery power/mass = p =1200 Watts/kg Battery power = P_{b}= p m = 120 kWatts Battery energy/mass = e = .8 MJoules/kg Battery energy = E = e m = 80 MJoules Battery $/energy = c = 100 $/MJoule Battery cost = C = c E =8000 $ Hover time = T = E/P_{h} =2650 seconds = 44 minutesThe properties of propellers are discussed in the propeller section. The rotor tip speed is
Rotor lift/drag = Q = 5.5 Rotor tip speed = V = PQ/F = 113 m/sThe ideal horizontal cruise speed is around 1/3 of the rotor tip speed. If we assume a cruise speed of 40 meters/second and a flight time of 44 minutes then the range is 106 km.
Rotor radius = R Air density = D = 1.22 kg/meter^{3} Rotor tip speed = V Rotor width param= W Rotor lift force = F_{l} = D W R^{2} V^{2} Rotor drag force = F_{d} Rotor lift/drag = Q = F_{l} / F_{d} Rotor power = P = F_{d} V = F V / Q Rotor quality = q = Q W^{½} D^{½} = F_{l}^{3/2} P^{1} R^{1} Rotor force/power= Z = F_{l}/ P = Q / V = D^{½} W^{½} Q R F^{½} = q R F_{l}^{½}The physical parameters of a propeller are {R,Q,W,q}, with typical values of
Q = 5.5 W = .045 q = 1.29Most propellers have 2 blades and some have 3. If there are 4 or more blades then q declines.
A measurement of F_{l} and V determines W.
A measurement of P, F_{l}, and V determines Q.
A measurement of F_{l}, P, and r determines q.
Q and W are not independent. They are related to the blade aspect ratio.
Q ≈ Aspect ratio W ≈ Q^{½} q ≈ Q^{½}
A commonlyappearing quantity is the power/mass ratio, which is inversely proportional to the force/power ratio.
Mass = M Gravity = g Hover force = F = M g Hover power = P Force/Power ratio = Z = F/P Power/Mass ratio = p = P/M = g/Z
Using data for commercial drone propellers,
Propeller radius = R Propeller mass parameter = C = 5 kg/meter^{3} Propeller mass = M = C R^{3}
Energy/Mass Power/Mass Recharge Year Anode Cathode Market fraction of MJoule/kg Watt/kg Lithiumion batteries Lithium air 6.12 No Future Li O2 Aluminum air 4.68 200 No 1970 Al O2 Lithium thionyl 2.00 700 No 1973 Li SOCl2 Zinc air 1.59 No 1932 Zn O2 Lithiumion sulfur 1.44 670 Yes Future Li S 0 Lithium metal 1.01 400 No 1976 Li MnO2 Lithiumion CoNiAlO2 .79 Yes 1999 Li CoNiAlO2 .10 Lithiumion CoNiMnO2 .74 1200 Yes 2008 Li CoNiMnO2 .29 Lithiumion CoO2 .70 200 Yes 1991 Li CoO2 .29 Lithiumion Mn2O4 .54 1200 Yes 1999 Li Mn2O4 .10 Lithiumion FePO4 .47 1200 Yes 1996 Li FePO4 .22 Alkaline .40 Yes 1992 Zn MnO2 NiMH .34 1000 Yes 1990 MH NiO(OH) Lead acid .15 180 Yes 1881 Pb PbO2 NiCd .14 200 Yes 1960 Cd NiO(OH)
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
Size Charge Current Price Amphours Amps $ Basen C 4.5 60 8.0 Panasonic B 4.0 15 8.0 Sony VTC6 A 3.0 30 8.0 Panasonic A 3.5 10 5.5 Efest IMR AA .65 6.5 3.5 Efest IMR AAA .35 3 3.0Prices from www.liionwholesale.com