What about rolling resistance
Some things we've completely ignored so far are the energy consumed in the tyres and bearings of the car, the energy that goes into the noise of wheels against asphalt, the energy that goes into grinding rubber off the tyres, and the energy that vehicles put into shaking the ground. Collectively, these forms of energy consumption are called rolling resistance. The standard model of rolling resistance asserts that the force of rolling resistance is simply proportional to the weight of the vehicle, independent of
wheel 
Crr 
train (steel on steel) 
0.002 
bicycle tyre 
0.005 
truck rubber tyres 
0.007 
car rubber tyres 
0.010 
Table A.8. The rolling resistance is equal to the weight multiplied by the coefficient of rolling resistance, Crr. The rolling resistance includes the force due to wheel flex, friction losses in the wheel bearings, shaking and vibration of both the roadbed and the vehicle (including energy absorbed by the vehicle's shock absorbers), and sliding of the wheels on the road or rail. The coefficient varies with the quality of the road, with the material the wheel is made from, and with temperature. The numbers given here assume smooth roads. [2bhu35]
i 40
i 40
Figure A.9. Simple theory of car fuel consumption (energy per distance) when driving at steady speed. Assumptions: the car's engine uses energy with an efficiency of 0.25, whatever the speed; Cd Ac31 = 1 m2; mcar = 1000 kg; and Crr = 0.01.
Figure A.10. Simple theory of bike fuel consumption (energy per distance). Vertical axis is energy consumption in kWh per 100 km. Assumptions: the bike's engine (that's you!) uses energy with an efficiency of 0.25,; the dragarea of the cyclist is 0.75 m2; the cyclist+bike's mass is 90 kg; and Crr = 0.005.
Figure A.11. Simple theory of train energy consumption, per passenger, for an eightcarriage train carrying 584 passengers. Vertical axis is energy consumption in kWh per 100 pkm. Assumptions: the train's engine uses energy with an efficiency of 0.90; Cd Atrain = 11 m2; mtrain = 400 000 kg; and Crr = 0.002.
the speed. The constant of proportionality is called the coefficient of rolling resistance, Crr. Table A.8 gives some typical values.
The coefficient of rolling resistance for a car is about 0.01. The effect of rolling resistance is just like perpetually driving up a hill with a slope of one in a hundred. So rolling friction is about 100 newtons per ton, independent of speed. You can confirm this by pushing a typical oneton car along a flat road. Once you've got it moving, you'll find you can keep it moving with one hand. (100 newtons is the weight of 100 apples.) So at a speed of 31m/s (70mph), the power required to overcome rolling resistance, for a oneton vehicle, is force x velocity = (100 newtons) x (31m/s) = 3100 W;
which, allowing for an engine efficiency of 25%, requires 12 kW of power to go into the engine; whereas the power required to overcome drag was estimated on p256 to be 80 kW. So, at high speed, about 15% of the power is required for rolling resistance.
Figure A.9 shows the theory of fuel consumption (energy per unit distance) as a function of steady speed, when we add together the air resistance and rolling resistance.
The speed at which a car's rolling resistance is equal to air resistance is
Crrltlcg = ~pCdAv2,
given by that is,
Bicycles
For a bicycle (m = 90 kg, A = 0.75 m2), the transition from rollingresistancedominated cycling to airresistancedominated cycling takes place at a speed of about 12km/h. At a steady speed of 20km/h, cycling costs about 2.2 kWh per 100 km. By adopting an aerodynamic posture, you can reduce your drag area and cut the energy consumption down to about 1.6 kWh per 100 km.
Trains
For an eightcarriage train as depicted in figure 20.4 (m = 400 000 kg, A = 11 m2), the speed above which air resistance is greater than rolling resistance is v = 33 m/s = 74 miles per hour.
For a singlecarriage train (m = 50 000 kg, A = 11 m2), the speed above which air resistance is greater than rolling resistance is v = 12 m/s = 26 miles per hour. Dependence of power on speed
When I say that halving your driving speed should reduce fuel consumption (in miles per gallon) to one quarter of current levels, some people feel sceptical. They have a point: most cars' engines have an optimum revolution rate, and the choice of gears of the car determines a range of speeds at which the optimum engine efficiency can be delivered. If my suggested experiment of halving the car's speed takes the car out of this designed range of speeds, the consumption might not fall by as much as fourfold. My tacit assumption that the engine's efficiency is the same at all speeds and all loads led to the conclusion that it's always good (in terms of miles per gallon) to travel slower; but if the engine's efficiency drops off at low speeds, then the most fuelefficient speed might be at an intermediate speed that makes a compromise between going slow and keeping the engine efficient. For the BMW 318ti in figure A.12, for example, the optimum speed is about 60 km/h. But if society were to decide that car speeds should be reduced, there is nothing to stop engines and gears being redesigned so that the peak engine efficiency was found at the right speed. As further evidence
S 80
f 70
g 40
Figure A.12. Current cars' fuel consumptions do not vary as speed squared. Prius data from B.Z. Wilson; BMW data from Phil C. Stuart. The smooth curve shows what a speedsquared curve would look like, assuming a dragarea of 0.6 m2.
400 300
100 80
60 50 40
30 20
400 300
100 80
60 50 40
30 20
Ferrari Porsche Kadett Panda
100 200 300 400 top speed (km/h)
Ferrari Porsche Kadett Panda
100 200 300 400 top speed (km/h)
Figure A.13. Powers of cars (kW) versus their top speeds (km/h). Both scales are logarithmic. The power increases as the third power of the speed. To go twice as fast requires eight times as much engine power. From Tennekes (1997).
that the power a car requires really does increase as the cube of speed, figure A.13 shows the engine power versus the top speeds of a range of cars. The line shows the relationship "power proportional to v3."
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