Super-Mileage Vehicle Fuel Economy as a Function of Speed

Super-Mileage Vehicle Fuel Economy 
as a Function of Speed

By Nick Demma - Feb 24th 2006 

The purpose of this analysis is to figure out what kind of fuel economy a well-designed super-mileage vehicle would get as a function of its speed. By well-designed, I mean that an appropriate engine is used. In the actual super-mileage championships, a 3.5 Hp Briggs and Stratton engine is used. This is makes the vehicles grossly overpowered, so they either have to run at a very low throttle setting, which is thermodynamically inefficient, or they arrange to bump start the engine and they run the engine intermittently. The reason for all of this lunacy is that the super-mileage championship is sponsored by Briggs and Stratton, so they are going to use one of their engines, however inappropriate it may be. A proper design might require a tenth Hp engine.

There are many good sources of information about bicycle physics, like this:
http://www.legslarry.beerdrinkers.co.uk/tech/SpeedAndPower.htm 

Procedure: Use the information from the web site (above) to figure out the equations which describe the power Vs speed for the best hard shell streamliner. Use the energy that you can get out of a gallon of gasoline divided by the power to get the amount of time that the vehicle could run on a gallon, then multiply the speed by the time to get the distance that is would go. Do this for a variety of speeds to get a plot of the fuel economy Vs the speed.

Data: The web site gives the speed at 250 Watts of power output and the power at 
40 Km/Hr. From these two points we can find an equation that describes the power as a function of the speed. Let's work through this for the best hard shell streamliner. All calculations are done using the meter-kilogram-second (i.e. metric) because our system is utterly idiotic. In deference to those who want to understand the results, the answers are converted to familiar units. At 250 Watts, the streamliner bike goes 69 Km/Hr or 19.17 m/s. To go 40 Km/Hr or 11.111 m/s, the streamliner needs 75 Watts. The drag force due to rolling resistance is not a function of the speed and the drag force due to air resistance is proportional to the square of the speed. Since the power is the force times the speed, the power needed to overcome the rolling resistance is proportional to the speed and the power needed to overcome the air resistances is proportional to the cube of the speed. Let's let x = the speed squared and let's let 
y = the speed cubed. The power as a function of speed is therefore:
Power = a * x + b * y = z


We know the power for two different speeds, so we have two equations and two unknowns:

a * (11.111) + b * (11.111)^3 = 75
a * (19.167) + b * (19.167)^3 = 250

Solving this system of equations gives us the coefficients a and b. We now have an equation that gives us the power at any speed:

Streamliner Power = 3.5649 * speed + 0.0258 * speed^3

The power is in Watts and the speed is in meters per second. Gasoline has about 20,000 BTU/pound and weighs about 6.17 pounds per gallon, so 6.17 pounds of gasoline has about 123457 BTUs, or about 130,000,000 Joules. If we figure that an internal combustion engine is about 25% efficient under optimal conditions, then we get about 32.5 million Joules of usable energy per gallon. For any speed, we figure the power, then divide the energy by the power to get the time it takes to use a gallon of gas. Multiply this time by the speed to get the distance and the results are shown in Figure 1.


Figure 1: This is the fuel economy Vs the speed.

Conclusions: Although the streamliner that could get this kind of fuel economy would not be practical, it does show what it possible. Compromises like leaving the bottom open so the rider could stop by himself would reduce the fuel economy somewhat, but getting 2000 miles per gallon at 25 MPH would probably be realistic. Improving on this efficiency is somewhat pointless because a person who puts on 4000 miles per year would have to buy only two gallons of gas at a cost of less than $6, so getting infinite miles per gallon would save only $6 per year.

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