The CAFE Formula--Aircraft Flight Efficiency
For the CAFE 400 races of the 1980's CAFE wanted a simple, fair and valid mathematical formula to rate aircraft flight efficiency in a way based on the laws of aeronautical physics. The first step in creating a formula for vehicular efficiency is to decide what one wants to be efficient about. The fundamental efficiency quantities of any vehicle are its velocity, payload and energy consumption. These quantities were used by CAFE to define a generic vehicular efficiency that applies to all vehicles, cars, watercraft and aircraft. This concept formed the basis for the 1982 CAFE 400 formula of velocity x MPG x payload, a formula that was subsequently found to be in accord with Nobel Laureate Teodor von Karman’s universal formula for vehicular efficiency. The formula is auto-constraining, meaning that each term grows at the expense of both other terms.
For a comprehensive discussion of the auto-constraining relationship between speed, MPG and payload and how it relates to the laws of aerodynamics, click [Here]
Dick VanGrunsven astutely noted that the CAFE formula did not predict which aircraft would subsequently be winners in the marketplace. Attracting consumers to buy small aircraft meant they needed other features beyond flight efficiency. Because the main purpose of the NASA Personal Air Vehicle (PAV) Challenge is to popularize PAVs, NASA engineers and CAFE collaborated to set qualifying standards for features deemed essential to marketability and safety. These qualifying standards include stall speed, utility, handling qualities, passenger comfort, size, weight, runway length, and number of seats. These must be met by each PAV Challenge team vehicle before it becomes eligible to compete in the PAV Challenge for the CAFE Efficiency Prize. The details about these standards are described in the PAV Challenge Rules that are posted at www.cafefoundation.org.
The Formula for the CAFE Efficiency Prize in the PAV Challenge
After a great deal of discussion, CAFE Board member and engineer Steve Williams proposed a new CAFE Efficiency formula for use in the Personal Air Vehicle Challenge. The new formula retained the three key parameters of velocity, payload and energy consumption, but corrected for vehicle size and occupancy.
The vehicle size and occupancy correction deserves special mention here. In aircraft, the effect of scale unduly favors aircraft of larger payload for several reasons. The first of these is what aerodynamicists call Reynolds number, which basically refers to the size difference between the vehicle itself and the molecules of air through which it travels. Similar issues of molecular size apply to vehicles that travel on land or sea. What this means is that the payload factor in the universal formula necessarily needs to be slightly de-emphasized.
A second effect of scale is related to the "Square-Cube Law," which basically refers to the fact that as a vehicle's external wetted surface area increases in square feet, its interior cargo room increases even more so, being a cube function of its dimensions. This effect would argue that payload should be credited only as the 2/3 root of payload.
A third effect of scale is the matter of occupancy, i.e., the fact that 4 and 6 seat vehicles most often travel with only 1 person aboard. On the highway, vehicles average only 1.3 people per car. These three size effects on payload have led CAFE to modify its historic efficiency formula for the NASA PAV Challenge.
The CAFE Efficiency Prize in the PAV Challenge will be based on the formula:
Taking the cube root of the Payload in pounds corrects for the above-mentioned vehicle size and occupancy issue. In addition, a range of 400 miles, similar to that of most new cars, was included in the formula so that the efficiency of PAVs could be better compared to cars.
This new formula prioritizes the things that matter to a prospective buyer of a PAV. The astute reader will recognize that this formula places a value on Velocity by its use of "minutes" in the denominator. This results in a value of $60 per hour for time. The "fuel cost" term in the denominator will be the dollar cost of fuel consumed during the competition.
The new formula removed some non-linear effects of the previous CAFE formulae. A rationale for this is found in the use of a sum rather than a product of terms in the denominator. Consider the following example: On a 90 mile trip from Santa Rosa to San Jose, a PAV can travel at either 90 mph, 180 mph or 360 mph. At 90 mph, the trip takes 60 minutes. At 180 mph the trip takes 30 minutes. At 360 mph, the trip takes only 15 minutes. To save just 15 minutes, the 360 mph PAV would spend a hugely larger amount for fuel. A denominator product that credited the PAV in proportion to its Velocity would excessively and inappropriately reward that PAV for saving only 15 minutes, whereas crediting the PAV in proportion to the minutes saved properly reflects the relevant use to a consumer.
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