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Flying electric cars

A flying car powered by lithium-ion 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 1-person 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 lithium-ion batteries and rotors. In the sections below we use these to construct a concrete design for a flying car.

Number of rotors

The design of the car depends on the physics of rotors. For a rotor,

Power required to hover  =  Constant * LiftForce3/2 / RotorRadius
The 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.

State-of-the-art lithium-ion batteries have an energy/mass of .8 MJoules/kg and can fly a car for 44 minutes. In the future, lithium-sulfur 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/second2
Rotor force         =  F = Mg/N =1470    Newtons
Rotor quality       =  q        =   1.02
Air density         =  D        =   1.22 kg/meter3
Rotor power         =  Pr=(qDR)-1F3/2= 30.2  kWatts
Hover power         =  Ph= N Pr =  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       =  Pb= 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/Ph =2650    seconds  =  44 minutes
The 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/s
The 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.
Propeller lift and power

The properties of a propeller are:

Rotor radius     =  R
Air density      =  D  =  1.22 kg/meter3
Rotor tip speed  =  V
Rotor width param=  W
Rotor lift force =  Fl =  D W R2 V2
Rotor drag force =  Fd
Rotor lift/drag  =  Q  =  Fl / Fd
Rotor power      =  P  =  Fd V  =  F V / Q
Rotor quality    =  q  =  Q W½ D½
                       =  Fl3/2 P-1 R-1
Rotor force/power=  Z  =  Fl/ P
                       =  Q / V
                       =  D½ W½ Q R F
                       =  q R Fl
The physical parameters of a propeller are {R,Q,W,q}, with typical values of
Q  = 5.5
W  =  .045
q  = 1.29
Most propellers have 2 blades and some have 3. If there are 4 or more blades then q declines.

A measurement of Fl and V determines W.
A measurement of P, Fl, and V determines Q.
A measurement of Fl, P, and r determines q.

Q and W are not independent. They are related to the blade aspect ratio.

Q  ≈  Aspect ratio
W  ≈  Q
q  ≈  Q½

Power/Mass ratio

A commonly-appearing 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

Rotor mass

Using data for commercial drone propellers,

Propeller radius          =  R
Propeller mass parameter  =  C  =  5   kg/meter3
Propeller mass            =  M  =  C R3

Battery types
              Energy/Mass  Power/Mass  Recharge  Year  Anode  Cathode   Market fraction of
               MJoule/kg    Watt/kg                                     Lithium-ion 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
Lithium-ion sulfur   1.44     670       Yes    Future  Li    S               0
Lithium metal        1.01     400       No     1976    Li    MnO2
Lithium-ion CoNiAlO2  .79               Yes    1999    Li    CoNiAlO2         .10
Lithium-ion CoNiMnO2  .74    1200       Yes    2008    Li    CoNiMnO2         .29
Lithium-ion CoO2      .70     200       Yes    1991    Li    CoO2             .29
Lithium-ion Mn2O4     .54    1200       Yes    1999    Li    Mn2O4            .10
Lithium-ion 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 and power

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  Joules
1 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

Battery packs

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 high-power 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

Commercial lithium batteries

           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     3.0
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