Pyramid Science

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Wednesday, March 31, 2010

Economical Driving - The Physics

Car Fuel Efficiency
Cars - The Real Purchase Price
Economical Driving
Fast Driving And Increased Fuel Cost
Fuel Consumption Ratings
Head Injury
Internal Combustion Engine

A heavy large boulder requires a great deal more energy to initiate movement than a lighter stone. This demonstrates the property of inertia. Inertia relates to a rotation motion, but as a body moves horizontally while being pulled down by gravity the vectors combine. The result is similar in principle to the orbit of a satellite around a planet. So it is with a vehicle. The forward motion and gravitational downward motion that is dependent on the weight (mass) of the vehicle.

A tank has a continuous track and this spreads the load
so minimising the downward gravity influence (friction)

A small lightweight car will require less energy to start movement than a heavy one and anyone who has ever pushed a car will appreciate this. The kerb weight defines the gross standing weight of a vehicle without passengers or cargo, and can be anything between 1 and 2.5 tonnes. The M1 Abrams battle tank weighs in excess of 60 tonnes. Once a vehicle is moving it acquires momentum and is essentially a measure of resistance to stopping. Energy is consumed to initiate this movement and requires the same energy to stop the motion. This is generally braking, but will also incorporate drag and is the result of gravity pulling down on the vehicle and the resulting friction of the tyres with the road. This resistance to movement is ever-present so a heavy vehicle requires a greater amount of energy just to maintain movement. The drag forces also include wind resistance once in motion. The faster the forward motion, the greater the resistance to this motion: friction of the tyres on the road and resistance caused by static air. On a windy day, the energy requirement changes and depends on whether a head-wind, following-wind or cross-wind applies.

Kinetic energy relates the mass and speed of an object and this relationship is not linear, but a square function. If the speed is doubled, the KE is quadrupled. Three times the velocity and the KE is increased by 9 times. To provide this force, liquid (or gaseous) fuel is consumed and converted into a different form with the production of energy. Not an efficient process. On a windless day, a 2.5 tonne vehicle propelled from 0-60kmph in 5 seconds will use an enormous quantity of fuel when compared to a 1 tonne vehicle accelerated from standing to 60kmph in 10 seconds and will consequently require much less 'fuel'. The problem here is what speed is the base x1 so allowing the estimate of x2 or x3? The vehicle mpg figure that has a defined speed does not allow the mpg at half that speed to be estimated. A heavy (2.5 tonne) and large engine (+3L) moving at 60kmph in top gear may be just "ticking over". A smaller engine (1.2-1.4L) will generate much higher rpm and will consume more fuel even though the vehicle may only be 1 tonne. The fuel consumption figures to achieve this speed (60kmph) is not available and it may be easy, intermediate or hard acceleration.

The figure is meaningless when such a variety of engine
sizes and undefined driving conditions apply

The 60 tonnes M1 Abrams battle tank accelerated to 30kmph uses a tremendous quantity of fuel. These tanks do not use a clutch (the intense friction would cause it to burn out) and the connection between the engine and gears is simply on or off.

Hard acceleration and high revs will result in a very high rate of consumption. A more gentle velocity rate increase (acceleration) will produce much improved economy. At constant velocity, a 5km journey at 30kmph will take 10 minutes (approximately, since the relatively short time for 0-30, 60, 90kmph is ignored), at 60kmph takes 5 minutes and travelling at 90kmph will take 3.33 minutes. Clearly, the greater the speed, the shorter the journey time. The fuel consumption will increase from x1 to x4 to x9, respectively (30 -> 60 -> 90). A journey may be x3 quicker, but the cost in fuel increases (at least) by a factor of x9. However, the base speed will clearly make a major difference: 30kmph -> 60kmph is x2 and -> 90kmph is x3 if 30kmph is the base speed. If 60kmph is the base speed then 90kmph is x1.5 this value and not x3. As an instance the Bentley W12 (Continental Flying Spur) has a published fuel economy of 17mpg at 60mph (96kmph) and so this can be converted to 34mpg at 30mph or around 7-8mpg at 90mph (60mph -> 90mph = x1.5 velocity, but x2.25 KE = 1.5 x 1.5 square function). The picture can be massaged to provide a 'best' answer. This single example illustrates the essentially meaningless nature of such figures when comparing very different vehicles.

Simplistically, if a car capable of 50mph required 50bhp, a similar (size/weight) would need at least 100bhp (x2) to reach 100mph (x2). But to move at 200mph (x4) power in excess of 800bhp (x16) would be necessary. An additional 50mph to a top speed of 250mph would require 1250bhp (50x16). A small increase of 10mph (+5.16%) to 260mph would need 1350bhp  (+8% = 50x27). This may seem to be a modest increment of just 100bhp (+8%), but is all that is necessary to power the vehicle to initially 100mph. The engine that powers the vehicle to the highest speed adds significantly to the kerb weight. Power is a requirement not just to increase the overall velocity of the car, but also propel the increase in weight to this speed.
  • In other words, to increase the speed from 100mph to 260mph (x2.6) requires a power (and concomitant fuel) increase of 100bhp to 1350bhp (x13.5).

50mph -> 100mph (x2)
  • Power: 50 x 4 = 200bhp
  • 100% increase in speed, 400% increase in power

50mph -> 200mph (x4)
  • Power: 50 x 16 = 800bhp 
  • 400% increase in speed, 1600% increase in power
50mph -> 250mph (x5)
  • Power: 50 x 25 = 1250bhp
  • 500% increase in speed, 2500% increase in power

50mph -> 260mph (x5.2)
  • Power: 50 x 27 = 1350bhp
  • 520% increase in speed, 2700% increase in power

So, to increase the top speed by x5.2 from 50mph to 260mph requires a x27 power output of 50bhp to 1350bhp. The increase in engine size and weight conspire to create a huge increase in fuel cost. Crudely, up to x27 the volume (litres) of fuel. Reaching a high speed requires energy, but maintaining any velocity needs fuel and power to overcome the resistance caused by movement through the air (the value of the square of the speed) and the downward influence of gravity: the heavier the vehicle, the greater the downward force and the consequential increase in friction between the tyres and road surface.

The acquired momentum will require efficient braking to slow and stop the moving vehicle. The faster the moving vehicle, the heavier the braking required as the momentum (mv) is that much greater. The major consequence of a heavier vehicle arises from the combination of drag forces: wind resistance and tyre-friction on the road, which are in opposition to maintaining the achieved velocity after acceleration becomes zero (constant velocity). Wind resistance increases dramatically (square function) as speed increases as does the friction: tyres get hot at speed (the faster, the hotter). These forces will slow the vehicle to an eventual standstill unless energy (fuel consumption) is applied and all this assumes a flat road: ie no upward and no downward inclination (hills). As a vehicle moves downward, the potential energy changes into kinetic energy and does work. Momentum = mv and the greater the weight (mass) of the vehicle, the greater the momentum for a given speed. Movement uphill will require an energy input (acceleration) to maintain a constant velocity as gravity pulls the vehicle down onto the ground increasing tyre friction and raising the potential energy while decreasing the kinetic energy. The drag in opposition causes a decrease in the velocity (deceleration)  An acceleration will be required to maintain constant velocity. Fuel load goes UP considerably.

As an economy comparison for the 5km journey, the only obvious difference in the energy requirement between the 30kmph, 60kmph or 90kmph journey is the acceleration time: slightly longer to achieve a higher speed. The fuel consumption when the steady speed has been reached will appear to be the same for two comparative vehicles (1 tonne and 2.5 tonnes). It isn't. To maintain progress, energy must be consumed and the heavier the vehicle, more energy is necessary.

If the speed is raised from 30 kmph to 60kmph (+100%), for a 1 tonne vehicle the KE increases from 1*30*30 (900 joules) to 1*60*60 (3600 joules). A fourfold increase. Increasing the acquired velocity to 90kmph (+50%) the energy requirement changes considerably: 1* 90* 90 joules (8100 joules) and is a 9 times increase when compared to the slower speed. Speed increases by 3 times and fuel load goes up by 9 times. Energy increase is 900 to 3600  to 8100 joules. The change is actually in the ratio 1:4:9 or a square function of the velocity for a given mass. If the car is heavier (2.5 tonne) then the energy requirement is substantially greater: the KE increases from 2.5 * 30 * 30 (2250 joules) to 2.5 * 60 * 60 (9000 joules) to 2.5* 90*90 (20250 joules). Again an increase of 1:4:9 when compared to the slowest speed or just x2.25 the energy requirement compared to the next lowest speed (90kmph = 20250 joules -> 60kmph = 9000 joules). However, for true comparison between the 1 tonne and 2.5 tonnes vehicle the energy increase is potentially) rather different than it 'appears':

  • 1 tonne: 900, 3600, 8100 joules
  • 2.5 tonnes: 2250, 9000, 20250 joules 

Although the increment is still 1:4:9 for each weight, the difference between the two cars of different weights and low/high speed is actually 900 joules and 20250 joules. This shows that a heavier (x2.5) vehicle driven at x3 speed (90 kmph) of the lighter vehicle (30kmph) requires 22.5 times the energy from fuel: 2.5 tonnes at 90kmph requires x22.5 the fuel load of 1 tonne at 30kmph. Another way of stating this is that at constant speed over a fixed distance the fuel demand for a 2.5 tonne vehicle is potentially 22.5 times greater than the 1 tonne vehicle depending on the driving speed and conditions. This may be basic physics, but the scenario itself is complex and very easily massaged.

    Renault Clio (Dynamique: 1.2 16V 3dr hatchback)
    • 1090kg (kerb weight), 47mpg, 73bhp, 50L
    • Theoretical distance/tank = 47 * 50/4.54518 miles
    • Top speed = 102mph , 0-60mph [59% potential] = 12.6 seconds
    •  Car tax: 139 g/km CO2
    Ford Focus (Studio: 1.4 16V 3dr hatchback)
    • 1150kg (kerb weight), 42mpg, 73bhp, 55L
    • Theoretical distance/tank = 42 * 55/4.54 = 509 miles
    • Top speed = 102mph, 0-60mph [59% potential] = 13.6 seconds
    • Car tax: 158 g/km CO2
    Peugeot 207 (1.4 16V S 3dr hatchback)
    • 1139kg (kerb weight), 44mpg, 90bhp, 50L
    • Theoretical distance/tank = 44 * 50/4.54 = 485 miles
    • Top speed = 112mph, 0-60mph [54% potential] = 13.7 seconds
    • Car tax: 152 g/km CO2
    BMW X6M (4.4 xDrive X6M 5d Auto)
    • 2305kg (kerb weight), 20mpg, 547bhp, 85L
    • Theoretical distance/tank = 20 * 85/4.54 = 375 miles
    • Top speed = 155mph, 0-60mph [39% potential] = 4.5 seconds
    • Car tax: 325 g/km CO2
    Hummer H2 (6.2 V8 Adventure 5d Auto)
    •  2850kg (kerb weight), 16mpg, 392bhp, 121L
    • Theoretical distance/tank = 16 * 121/4.54 = 426 miles
    • Top speed = unknown, 0-60mph = 7.6 seconds
    • Car tax: 412 g/km CO2
     Bentley W12 (Continental Flying Spur)
    •  2475kg (kerb weight), 17mpg, 601bhp, 90L
    • Theoretical distance/tank = 17 * 90/4.54 = 337 miles
    • Top speed = 200mph, 0-60mph [30% potential]= 4.5 seconds
    • Car tax: 396 g/km CO2
      • The Renault Clio, Ford Focus and Peugeot 207 have very similar statistics. The BMW X6M, Hummer H2 and Bentley W12 are included for comparison of high performance and very poor economy and demonstrates that there has never been any attempt to reduce petrol consumption or consider speed restrictions in the UQ (aka UK) Ltd. The objective has always been to maximise revenue and this is very clearly demonstrated reviewing the car tax levied (March 2010).
      •  Considering these figures alone it appears that the Peugeot 207 (1.4) requires a 23% bhp increase (90bhp) to produce a similar performance to the Renault Clio (1.2, 73bhp) or Ford Focus (1.4, 73bhp). The Renault has the smallest engine (1.2) and is possibly a reason for it being the lightest of the three, but this is traded off against the potentially lower revving engine of the Peugeot 207 (1.4) at a comparative speed: 59%  against 54%.
      • The BMW X6M is more than twice the kerb weight of the Ford Focus or Renault Clio and for its performance statistics requires an engine that produces x7.5 the power output (547 v 73bhp). The fuel consumption figures support this: 42 - 47mpg v 20mpg. This comparison is at 60mph, but the potential (155mph) shows that this 20mpg will drop considerably (the rpm is not known).
      • The Hummer H2 is a tank, but even so has greater potential than the Bentley W12, which has truly absurd fuel economy regardless of its potential performance. At nearly its top (published) speed (potential = 200mph) at 180mph, the fuel economy (theoretically) plummets to 1/36th the 30mph figure (speed x6, KE = x36). Even at 60mph it is only 17mpg.
    This is probably the non-disclosed reason why such performance cars
    are allowed to be built since they can never in reality be driven at
    such speeds without requiring frequent refuelling

    Performance statistics for diesel engines appears to compare well with petrol powered cars, claiming 42mpg for combined driving, but at an unspecified speed and declared rpm (revs are assumed to be per minute). The 2993cc (3L) V6 24v turbodiesel engine delivers 271bhp @ 4000rpm, 443lb ft @ 2000rpm. The standing-start acceleration to 62mph = 5.9 seconds and a top speed (electronically governed) of 155mph. This car has a kerb weight of 1820kg and emissions at 179g/km CO2. At 4000rpm what is the speed? The potential may be 155mph, though what rpm are produced at this 'top speed'? Electronically governed implies there is more theoretically available and so the rpm by themselves do not impart any significant information. Note: mixing imperial and metric units.

    • Describing a car this way makes it sound like a strong performance, yet not declaring details of fuel consumption except 42mpg for combined driving, which is absolutely meaningless when presented as a single statistic.
    Another highly relevant factor is the operating temperature of the engine. Short journeys barely reach the optimal temperature with longer journeys being the most economical based only on engine temperature. The best combination is uninterrupted motion at moderate speed.

    • Why is the acceleration statistic (0-60mph) of a car so important? It is the most fuel uneconomic and simply means that the next red trafffic light is more quickly reached. And a longer wait until the following green light - DA


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