Fuel consumption of a 1500 kg car

1
Fuel consumption of a 1500 kg car

• Coefficient of rolling friction mr is 0.02.
• Force of rolling friction Fr mr mg 0.02
  1500 10 m/s2 300
• Drag at 70 km/h is
• R (1/2) ? CDAv2.
• For a car at 70 km/h 20 m/s
• R 0.5 1.2 kg/m3 0.7m2 (20m/s)2 170 N
• A total force resisting the motion of the car at
  this speed is 470 N.
• Cars usually use at 80 km/h on an open road 6
  10 L of gasoline per 100 km depending on size
  and engine.
• Lets check if it makes sense.

2
Fuel consumption of a 1500 kg car

• In order to maintain speed, the cars engine must
  do work equal to the work done by friction and
  air drag
• W F d 470 N x 100 km 47 x 106 J 47 MJ
• Gasoline has an energy of 34.6 MJ per liter The
  consumption would be thus 1.4 L.
• But car engines have the efficiency of only 20
  
• Taking the efficiency into account leads to a
  realistic answer of 7 L/100km.

3
Questions

• Gasoline consumption is larger in the city than
  on an open highway although speeds (and air drag)
  are lower. Why?
• Lets calculate how much energy is required to
  accelerate from 0 50 km/h? We only want to know
  what is due to the acceleration itself and apart
  from the energy needed to overcome rolling
  friction and air drag.
• We can get it directly from the work- energy
  principle
• W DK ½ m v2 150 000J (m 1500 kg)

4
Gasoline consumption is larger in the city than
on an open highway although speeds (and air drag)
are lower. Why?

• Or we can calculate it from the work done to
  accelerate the car
• W Fd mad
• vf2 vi2 2ad
• vi 0
• vf2 2ad
• ad vf2/2
• W Fd amd m vf2/2
• Which is the same result!
• W 150 000 J
• How much energy is consumed if this distance is
  traveled at a constant 50 km/h?
• The rolling resistance and air drag combine to
  about 400 N
• W F d 470 N 35 m 14000 J.
• Compare to 3 x 105 J during acceleration phase
  Car needs at least 10 times more energy for
  acceleration.

5
Car needs at least 10 times more energy for
acceleration.

• In the city we accelerate and brake much more
  often than on the highway, we also stop and keep
  the car idling a lot, which is using fuel without
  propelling the car.

6
Hybrid Cars

• Why are hybrid cars more energy efficient?
• Efficiency of combustion engines depends on rpm.
• Most efficient at relative high speeds 90 km/h.
• Electric motors are much more efficient at all
  speeds.
• The combustion engine of the hybrid car is off
  when we stop for the traffic lights or drive very
  slowly
• Some kinetic energy of the car is recovered back
  to the battery when braking.

7

• An SUV can drive 23 miles on one gallon of fuel.
  What is its approximately fuel efficiency in L
  per 100 km.
• (1 gallon 3.8 4 L, 1 mile 1.6 km)
• 100 L
• 50L
• 1000L
• 10L
• 1 L

8

• An SUV can drive 23 miles on one gallon of fuel.
  What is its fuel efficiency in L per 100 km.
• (1 gallon 3.8 4 L, 1 mile 1.6 km)
• 23 miles/gallon 23 1.6 km/3.8 L
• 23 0.42 9.7 km/L
• 100 km / (9.7 km/L) 10.3 L per 100 km
• Conversion
• (Liters / 100 km) 238/(Miles per gallon)

9

• Toyota Prius have a fuel efficiency of 4.5 L per
  100 km. Approximately how many miles it can drive
  on one gallon of fuel.
• (1 gallon 3.8 L 4 L, 1 mile 1.6 km)
• 10
• 25
• 50
• 75
• 100

10

• Toyota Prius have a fuel efficiency of 4.5 L per
  100 km. How many miles it can drive on one gallon
  of fuel.
• (1 gallon 3.8 L 4 L, 1 mile 1.6 km)
• (Liters / 100 km) 238/(Miles per gallon)
• (Miles per gallon) 238 (Liters / 100 km)
• 52.9 miles per gallon

11

• A 4 000 kg car and a 2 000 kg car are rolling
  over a horizontal surface with friction (same mr)
  and are brought to rest by it. The heavier car
  has an initial speed of 2 m/s and the lighter car
  has an initial speed of 4 m/s. Which statement
  best describes their respective stopping
  distances?
• The 4 000 kg car travels twice as far as the 2
  000 kg car before stopping.
• The 2 000 kg car travels twice as far as the 4
  000 kg car before stopping.
• Both cares travel the same distance before
  stopping.
• The 2 000 kg car travels four times as far as the
  4 000 kg car before stopping.
• The 4 000 kg car travels four times as far as the
  2 000 kg car before stopping.

12
The change in kinetic energy is equal to the work
done by the friction force

• 0 - mv2/2 Frd -mg mrd
• d mv2/2 mg mr v2/2gmr
• So the mass does not matter and the two times
  faster car will roll four times farther (the
  distance depends on the square of velocity)
  Answer 4 is correct

13
What is the maximum speed of the 1000 kg car with
60 hp engine?

• Assume rolling friction mr 0.02

14
Power

• Power can be also defined as the rate at which
  work is done. Average power is the amount of work
  done in a time interval Dt or the amount of
  energy transferred in a time interval Dt
• SI unit 1 watt 1W 1J/s
• Mechanical power

15
What is the maximum speed of the 1000 kg car with
60 hp engine?

• Assume rolling friction mr 0.02
• 60 hp 60750W 45 000 W kgm2/s3
• Force of rolling friction
• Fr Nmr mgmr 200 N
• P Frv
• V P/Fr 45000/(1010000.02) 225 m/s
• 810 km/h!
• kgm2/s3 /(m/s2 kg) m/s
• What is wrong?

16
Drag force

• R (1/2) ? CDAv2.
• For a car at 225 m/s
• R 0.5 1.2 kg/m3 0.7m2 (225m/s)2
• 21000N
• 100 time more than rolling friction force
• Back to calculations
• What will be the drag force at 50m/s

17
Drag force

• What will be the drag force at 45m/s
• R (1/2) ? CDAv2.
• For a car at 50 m/s
• R 0.5 1.2 kg/m3 0.7m2 (50m/s)2
• 850N
• Total force in this case 850200 1050
• P 1050N 50 m/s 52 kW
• 50 m/s 180 km/h
• Makes sense!

18
Fuel Economy of Airplanes

• A Boeing 747 has a maximum range of 13 450 km and
  a maximum fuel capacity of 216 840 L.
• Cruising speed 530 mph ( 850 km/h)
• We can calculate the fuel economy from this 16
  L/km or 1600 L per 100 km.
• The number of passengers is 500, so each
  passenger uses 3.2 L per 100 km when the plane is
  full.
• This is similar to the fuel economy of a car!

19
Fuel Economy of Airplanes
How much fuel is needed for climbing to an
altitude of 10.5 km? The mass of the plane is
390 000 kg at take-off. Potential energy m g h
4 ? 1010 J How much energy is needed for the
acceleration to the cruising speed of 915 km/h
(254 m/s)? K ½ m v2 1.26 ?1010 So total
energy needed for accelerating and climbing to
the cruising altitude is 4 ? 1010 1.26 ?1010
5.26 ?1010 J 5.26 ?104 MJ How much fuel is
needed for this? The energy efficiency of the
jet engines is 25 (estimated) Energy in jet
fuel 37.6 MJ/L So we need 5.26 ?104 MJ/ (37.6
MJ/L ? 0.25) 1400/0.25L 5600L of jet fuel to
accelerate to 915 km/h and go up to 10 500 m.
20
Fuel Economy of Airplanes
The rest of the fuel (210,000L) is used to
maintain speed against air drag. Lets estimate
the drag force A 747 has a fuel capacity of 216
840 L and a max. range of 13 450 km. So 210
000 L are used to counter air drag. W F d, so
F W/d 210 000L 37.6 MJ/L 0.25/13 450 000
m 0.147 MN 147 kN Notice that the maximum
engines thrust of this airplane is 4 x 282
kN) From this we see that minimizing drag force
is key. The density of air is less at higher
altitude. Thats why planes fly relatively high
although their exhaust is more harmful if it is
emitted at such high altitudes.
21
Fuel Economy of Airplanes
How much power is a Boeing 747 using in level
flight? Cruising speed 850 km/h 236 m/s P F
v 0.147 MN 236 m/s 35 MW
22

• Comparison of Energy Efficiency
• For meaningful comparison Efficiency per
  passenger.
• Two numbers from using average occupancy and
  maximum occupancy.
• Number with average occupancy more meaningful for
  political decisions.
• Units Miles per gallon (mpg)
• 1.0 mpg (US) 0.425 km/L
• Airplane 67 mpg 28.5 km/L or 100 km/28.5 km/L
  3.51L per 100 km.
• From Wikepedia Fuel efficiency in transportation
  (http//en.wikipedia.org/wiki/Fuel_efficiency_in_t
  ransportation)

23
Data for the Skytrain

• cars are 40' long
• typically 4 car trains
• each car holds 80 people comfortably
• Mass of the car is around 15,000 kg
• top speed 90 km/h
• coefficient of drag 1.8
• coefficient of rolling friction 0.001
• efficiency of the linear induction motors for the
  skytrain, 85 (estimated) at 40 km/h
• use of regenerative braking could save 5 - 20
  of overall energy costs
• http//www.railway-energy.org/static/Regenerative_
  braking_in_DC_systems_103.php