Fuel consumption and mathematics
Fuel consumption refers to the amount of fuel used per unit of measurement, typically time or distance, and is often confused with fuel economy, which is the distance or time achieved per unit of fuel. Understanding fuel consumption is crucial for budgeting and maintenance in various fields, as it helps estimate fuel costs and predict machinery issues before they escalate. The calculation of fuel consumption is straightforward: it involves measuring the fuel used while traveling a certain distance, using the formula C = f ÷ D, where C is fuel consumption, f is the fuel used, and D is the distance traveled. In countries like the United States, fuel economy is often expressed in miles per gallon, while other regions may use liters per 100 kilometers.
In addition to practical applications, mathematicians study fuel optimization, employing principles such as Pontryagin's Maximum Principle to determine optimal fuel usage trajectories. Interestingly, the Jevons Paradox highlights that greater efficiency can sometimes lead to increased overall fuel consumption. Furthermore, the analysis extends beyond traditional fuels, with research being conducted into alternative fuel sources like biodiesel and electricity, reflecting a growing interest in sustainable energy solutions. Overall, both mathematical calculation and optimization play significant roles in understanding and improving fuel consumption.
Fuel consumption and mathematics
Summary: Vehicle fuel consumption and efficiency are often mathematically investigated.
Fuel consumption can be defined as the amount of fuel used for each unit of measurement (usually time or distance). An often mistaken meaning is fuel economy, which is the reciprocal of fuel consumption: the amount of distance or time for each unit of fuel used. In addition to people in all walks of life using mathematics to measure fuel consumption, mathematicians research fuel optimization.
In one case, Pontryagin’s Maximum Principle, named after mathematician Lev Pontryagin, characterizes optimum values that determine a trajectory, such as fuel consumption or flight time. On the other hand, the counterintuitive assertion that greater fuel efficiency often results in increased fuel consumption is sometimes known as the Jevons Paradox, after economist and logician William Jevons. Mathematicians are also involved in research for alternative fuel sources for vehicles, such as biodiesel and electrical power.
Fuel consumption is calculated for various reasons, including budgeting and maintenance. If a business that uses fuel knows the average amount of time or average distance traveled by its machines and the fuel consumption for each unit, it can calculate the approximate amount of money needed for purchasing fuel over the next fiscal period. Tracking fuel consumption on a regular basis can indicate a potential breakdown of internal engine parts before the issue becomes a major repair or hazardous situation.
Calculating Fuel Consumption
Calculating fuel consumption is a fairly simple process if you have a way to measure both the time or distance the machine was used and the amount of fuel used to refuel the machine. For example, machines that are designed for travel, like cars, trucks, vans, or tractor-trailers contain an odometer to record the number of miles or kilometers traveled. Many even have a trip odometer that can be reset after refueling. To calculate fuel consumption, start by having the vehicle completely filled with fuel and the odometer reading recorded or reset.
After using the vehicle, fill its tank with fuel and measure the amount of fuel that has been added. The assumption here is that the amount of fuel added to bring the tank back to its full position would approximate the amount of fuel used since the last time the vehicle was fueled. At the same time that the vehicle is refueled, also record the odometer. A trip odometer indicates the exact distance traveled since the last fill-up (the distance traveled since it was last reset). If not using a trip odometer, take the current total distance traveled and subtract the previous reading taken at the last fill-up.
Now that the distance and amount of fuel has been measured, calculating fuel consumption is the simple division problem C = f ÷ D, where C is the fuel consumption, f is the amount of fuel used, and D is the distance traveled.
For example, a vehicle that traveled 400 miles on 20 gallons of fuel has a fuel consumption of 20 ÷ 400 = 0.05 gallons per mile, meaning that five-hundredths of a gallon (6.4 fluid ounces) of fuel was used to travel each mile. In Europe, Australia, and other countries (like Canada and China) fuel consumption is calculated in liters per 100 kilometers traveled.
A vehicle that traveled 600 kilometers on 75 liters of fuel would have a fuel consumption of 75 ÷ 600 = 0.125 liters per kilometer. To get liters per 100 kilometers, multiply the result by 100 to get 12.5 liters per 100 kilometers. When looking at fuel consumption, a lower number is better than a higher number, meaning you use less fuel to achieve the same distance.
Some countries use fuel economy; for example, the United States uses miles per gallon and Japan uses kilometers per liter. The formula for fuel economy (E) is E = D ÷ F.
In the above examples, 400 ÷ 20 = 20 miles per gallon (mpg), and 600 ÷ 75 = 8 kilometers per liter. For fuel economy, a larger number is desired, meaning a greater distance can be traveled using the same amount of fuel.
Not all machines were designed to travel, such as forklifts and construction equipment. Generally, these machines do not measure the distance they have traveled but rather the number of hours the machine has been in use. Many of these machines have an “hour meter” that measures the time the machine operates. For example, if a forklift uses 5 gallons of fuel over an 8-hour shift, fuel consumption is found by the formula C = F ÷ T where T is the time the machine is in use. In the above example, 5 ÷ 8 = 0.625 gallons per hour.
Bibliography
O’Hayre, Ryan, Suk-Won Cha, Whitney Colella, and Fritz B. Prinz. Fuel Cell Fundamentals. Hoboken, NJ: Wiley, 2009.
Ross, Michael. A Primer on Pontryagin’s Principle in Optimal Control. Carmel, CA: Collegiate Publishers, 2009.
U.S. Department of Energy. “Save Money.” http://www.fueleconomy.gov/feg/savemoney.shtml.
Woodsie, Christine. Energy Independence: Your Everyday Guide to Reducing Fuel Consumption. Guilford, CT: Lyons Press, 2009.