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Best Speed for Fuel Economy

Road Rail Air Water

Do emission factors always provide a good measure of environmental performance? This page provides a closer look at issues involved in comparing the greenhouse gas emissions associated with different transportation modes.

Specifically, the page looks at the relation between speed and fuel economy. Better fuel economy always means lower greenhouse gas emissions. But slower speeds do not always mean better fuel economy. The page considers speed and mileage for large freight transport vehicles in each transportation mode. In addition, TERC has developed a spreadsheet, the Best Speed for Fuel Economy Calculator, that allows users to estimate best speed and mileage for a wide range of vehicle types, and provides more detail on how the calculator generates the estimates.

A fair comparison

Emissions calculators, such as the TERC Intermodal Emissions Calculator, provide one way of comparing the greenhouse gas impacts of alternative transportation modes, based on an assumed average rate of emissions per mile traveled for each mode (see the TERC page on Intermodal Choices and Greenhouse Gases). But comparisons of impact per mile only tell part of the story. There may be environmentally sound reasons to prefer a transportation mode with a higher rate of emissions per mile to one with a lower rate, depending on the circumstances. For example, spoilage of perishable goods can easily outweigh the transportation impact, when the greenhouse emissions associated with replacing the goods is factored in. In cases where shipping by a faster mode can reduce spoilage, it may be preferable. It can be argued that different transportation modes provide essentially different services. If each transportation mode is considered to serve a different purpose, then simply comparing emissions per mile does not provide a meaningful comparison.

So on what basis can we compare the impacts of transportation modes? Here is a way to compare the typical performance of each mode to an "ideal" standard suitable for that mode: Suppose we can estimate what kind of emission factor we would expect if trucks, trains, ships, and aircraft were specifically designed to move goods with the sole objective of delivering the best fuel economy possible. Of course, fuel economy is already an important design factor. But from an economic standpoint, other factors can be even more critical. Speed is a prime example. As long as shippers are willing to pay a premium for faster delivery, the time it takes to move a load from A to B will outcompete fuel economy as a design consideration. That economic advantage comes at an environmental cost.

Speed is generally not treated as a variable in emissions calculators -- the effect of speed is simply averaged into the emission factor. But if we want to estimate the best attainable fuel economy for a transportation mode, we can't just consider the average speed that vehicles travel in practice. Instead, we have to look at a broader range of possible speeds. It is important to note that both very low and very high speeds tend to waste fuel, but for different reasons. At very low speeds, the engine will burn fuel for a long time getting from A to B, and fuel economy will suffer. And at high speeds, as every driver knows, extra speed means fewer miles per gallon. There is a speed in between where fuel economy is best. If we can estimate what that speed is for each mode, we can compare what the fuel consumption would be at that ideal speed to what it is when vehicles run at speeds more typical for that mode. This provides a measure for each mode of how much the environmental impact of greenhouse gas emissions has been traded off against the economic advantages of speed.

What is the best speed for fuel economy?

To understand the relation between speed and fuel economy, consider what forces are trying to slow the vehicle down. There are three main components. There is friction with whatever is supporting the vehicle, there is resistance from the air (or water) that the vehicle is moving through, and, if the vehicle is climbing, there is gravity. For wheeled vehicles, friction and (of course) gravity stay approximately constant over a wide range of speeds, but air and water resistance increase more and more sharply as speed increases. So to get from A to B, about the same amount of energy, and therefore the same amount of fuel, is spent overcoming friction and gravity, whatever the speed. But the force needed to overcome air or water resistance increases with the square of the speed. That force is responsible for most of the extra fuel consumed at higher speeds.

The best speed for fuel economy, as noted above, is determined by a balance between the extra fuel needed to overcome air or water resistance as speed increases, set against the fuel saved by getting there faster, and not having to run the engine as long. To appreciate how much fuel is saved by decreasing travel time, consider that even very efficient engines will use only about 25-40% of the energy that it generates by burning the fuel for moving the vehicle. The other 60-75% is mostly wasted as heat, or is used to run air conditioners and other accessories -- wasted or not, it's fuel burned that doesn't contribute to the motion. It's that non-motive part of the fuel demand that makes running at low speeds so expensive.

So as speed increases, travel time decreases, and more fuel is saved. But, because of air resistance, each additional mile per hour of speed needs more of an increase in power (and thus in fuel consumption rate) than the last one did. There comes a point at which the fuel saved by decreasing travel time is no longer enough to compensate for the rising amount of fuel needed to overcome air resistance. Speeding up any further will begin to decrease fuel economy rather than improve it. So that is the speed which provides the best fuel economy.

Assuming that the drag force is the dominant speed-dependent force in the range of speeds that are relevant here, the calculations take on a relatively simple form. It turns out that the numerical value of the optimal speed can be found by taking the non-motive power produced by the engine, and dividing by the effective mass of the air displaced by the vehicle per distance traveled. The resulting quantity has units of distance per time cubed; taking the cube root gives the value of the best speed for fuel economy. The details of the calculation are provided on the tab labeled "explanation" in the spreadsheet "Best Speed for Fuel Economy Calculator".

How do the modes compare?

Following the method described in the spreadsheet, examples of best speed calculations for each mode are summarized below. Estimates of power requirements and drag forces are provided for typical large freight vehicles, and typical speeds for those vehicles are compared with the calculated best fuel economy speed. The examples can provide a broad overview of how the modes compare.

Additional information is available from the Best Speed Calculator spreadsheet, including sample data for a variety of vehicle types in each mode. For users that have power and drag data for a specific vehicle, the spreadsheet is designed so that users can enter those data and get the best speed result for that vehicle.

Road

Assuming that the friction ("rolling resistance") is approximately independent of speed, we can use the simple cube root form of the solution quoted above. The instructions, as outlined in the Best Speed Calculator, are to take the power term (the non-motive power generated by the engine, say 60% of its average total power), divide by the drag term (the effective mass of air displaced per distance traveled, as defined below) and take the cube root.

Power term: A truck that consumes about 10 gallons of diesel fuel per hour during normal operation is delivering total power (based on the combustion energy of the fuel) at about 419 horsepower, of which about 40%, or about 160 hp, is actually moving the load. The rest, about 250 hp, is the non-motive fraction, which is the value to be used in the best speed calculation.

Drag term: A typical class 8 tractor-trailer combination has a cross section that measures 8.5 feet wide and 13.5 feet high, or about 115 square feet in cross section. The total mass of air displaced by the truck per distance traveled is given by the density of the air (mass per volume) times the cross-sectional area. That total is then multiplied by a dimensionless coefficient (the drag coefficient) to account for streamlining, which can lower the resulting drag force substantially. ("Effective mass per distance" as used here means the total mass of air in the volume swept out by the truck as it travels one unit of distance, multiplied by the drag coefficient.) The density of air is 0.075 pounds per cubic foot, so the total mass of air that the truck must displace per foot of travel is 115 x .075 = 8.6 pounds of air per foot of travel (that's about 23 tons per mile!). Streamlining typically results in drag coefficients of around 0.65, cutting down the effective mass of air displaced to 5.6 pounds per foot of travel (15 tons per mile).

Speed up and save gas? – Passenger cars, with much lower drag than large trucks, have best fuel economy speeds well in excess of typical highway speeds. But don’t expect to see those savings if you travel at that speed in your present vehicle – even driving in top overdrive gear, at those speeds your engine would be racing and its efficiency would suffer accordingly. The calculator indicates the mileage that your vehicle, with its existing engine and body, could approach if it had been designed specifically to cruise at its best economy speed. But since there are few places where driving that speed is (legally) possible, there wouldn’t be much demand for a vehicle built to take advantage of it.

Of course, rather than using an existing engine, one of today’s typical passenger cars would do even better on mileage with a smaller engine – one just large enough to maintain its best economy speed uphill. Moreover, a smaller engine means a lower best economy speed. So why can’t we find cars on the market whose best economy speed falls within the legal limit? Unfortunately, the market for sporty-looking cars that accelerate like trucks at highway speeds is limited, so manufacturers will probably not be offering them in the near future. On the other hand, when self-driving cars are available that can handle the higher speeds safely and that don’t have impatience programmed into their software, manufacturers might revisit the issue.

That is all we need for the calculation. At this point, it is convenient to continue in metric units (see box) and convert back to miles per hour at the end. Instead of dividing 250 horsepower by 5.6 pounds per foot, it will be much easier to follow the calculation if we divide the equivalent 219,000 watts of power by 8.3 kilograms per meter. The result is 219,000/8.3 or 26,386 in units of meters per second cubed. The cube root is 29.8 meters per second, which is equivalent to 67 miles per hour.

How accurate should we expect this estimate to be? The Best Speed Calculator allows users to substitute other values for the input variables (engine power and efficiency, vehicle cross section and drag coefficient), and to compare the results with the values quoted here. Because the optimal speed depends on the cube root of the input variables, relatively large changes to the inputs will produce smaller changes to the result. Suppose, for example, that the value used above for the drag coefficient of a typical truck underestimated the performance of an aerodynamically advanced design by, say, 20%. Lowering the drag coefficient by 20% would raise the optimal speed, but only by about 7%, from 67 to 71 miles per hour.

The estimate corresponds reasonably well to data collected on class 8 trucks from a fleet engaged in normal freight operations, according to a report issued by Oak Ridge National Laboratories (ORNL), which found that the best fuel efficiency for trucks carrying heavy loads was attained in the range of 57 - 66 mph. So large trucks traveling at typical highway speeds are operating close to the best speed possible for fuel efficiency.

Highway speed limits depend more on drivers' reaction times than on any economic or technological constraints. The close match between this estimate of the optimal speed for fuel economy and actual highway speeds is probably a reflection of how well engine power has been matched to vehicle size and shape for large trucks. Given an operating speed set by non-economic factors, class 8 trucks appear to have evolved toward the most fuel efficient way to move large amounts of freight by road.

Rail

Rolling resistance for rail is even lower (in fact much lower) than for road vehicles, so we can continue to use the simple cube root estimate. Again, we need to find the non-motive power generated by the engine, divide by the effective mass of the air displaced by the train per distance traveled, and take the cube root.

Power term: A typical large locomotive (SD-40) delivering 3,100 hp of motive power at full throttle burns 168 gallons of diesel fuel per hour (see for example figure 5 in this reference). The energy released by burning 168 gallons of diesel fuel amounts to 23.8 megajoules. Producing that much energy in one hour equates to an average total (motive plus non-motive) power of 6,615 kilowatts or 8,867 hp. The efficiency is therefore 3,100/8,867 or 35%, and the non-motive power, the remaining 65% of the total, is 4,302 kilowatts (5,767 hp).

Drag term: Finding the air resistance of a train is somewhat more complicated than for a tractor-trailer, since it depends not only on the cross-sectional areas of the locomotive and the railcars, but also on the spacing between them. Gaps between intermodal containers on flatcars, for example, can have a significant effect on the drag force. The drag force due to a line of rail cars can be significantly less than the sum of the forces that would be generated by each car moving alone (because of the effect of slipstreaming), but frequent or irregular gaps can reduce much of the potential advantage.

All of the components of a train's resistance force are summarized in the Davis Equation, originally based on measurements made in the 1920s, but periodically revised to reflect improvements in equipment and operations (see for example the section "Intermodal Rail Model" starting on page 4 of this reference). The model recommended in the reference for planning purposes provides a representative modern set of coefficients for use in the Davis equation. Numerical values are provided for locomotives and for cars of various types (container, trailer on flatcar, etc.). (The calculation does not take car spacing into account -- the effect on drag of adding cars is approximated by simply adding the individual contributions from each car.) According to the model, each railcar produces an amount of drag equivalent to a cross sectional area that displaces the equivalent of about 12.6 kg of air per meter of travel. (The locomotive, plowing through the air in the lead, with no slipstreaming from cars in front of it, produces about twice the drag of a single railcar.)

Using these values, we find that the best speed for mileage is the cube root of 4,302 kilowatts divided by about 12.6 kg/meter per railcar, which works out to 57 miles per hour for the typical locomotive pulling twenty cars, and 50 mph for thirty cars (see the Best Speed Calculator for more details). Like road transportation, these estimates are close to typical speeds attained in practice.

Water

The same calculation applies to water transportation, except that the major source of drag is the water rather than the air. For moderate speeds, the drag on a ship is still proportional to the square of the speed, as it is for land vehicles. At higher speeds, ships pay an extra penalty that does not apply to truck or trains (push on land and it pushes back; push on water and some of it rolls away in waves, carrying energy with it that would otherwise be moving the ship forward). The calculation does not take this extra wave resistance into account. Since it is an additional penalty for excess speed, including it in the calculation would tend to decrease the speed that would produce the best mileage. So the best speed estimate given below for a large container ship is, if anything, an overestimate.

Power term: Specifications provided by the shipbuilder for what was, at one time, the largest container ship in service, indicate that the vessel has a total engine power of 110,628 kilowatts. If 40% of that is converted to motion, the non-motive power generated by the engines is the remaining 60%, or 66,377 kW.

Drag term: Using the same data source, the beam (width) of the large container ship is 56 meters. Estimating the effective drag area as 560 square meters, the ship must displace 574,000 kilograms of seawater for every meter it travels.

According to the formula, the best cruising speed for fuel economy is found by dividing 66,377 kilowatts by 574,000 kilograms per meter and taking the cube root. The result is 4.87 meter per second, or 10.9 miles per hour. In this case, the actual design speed for this ship is closer to 30 miles per hour (25 knots).

In contrast to large land transport vehicles, large oceangoing vessels appear to have evolved toward improving speed at the expense of fuel economy. However, it may be of interest to note that more recent giant container ships from the same shipbuilder have been designed to cruise at somewhat slower speeds, and that the consequent decreases in both fuel consumption and emissions are being promoted as significant features of their design.

Air

Three factors complicate the estimate for air transportation. First, aircraft spend a significant proportion of their travel time climbing to cruising altitude, whereas surface vehicles get up to speed relatively quickly. Fuel consumption while cruising thus provides a fairly good estimate of overall speed for surface vehicles, but climbing and descending will be more of a factor, and will more significantly affect the overall fuel consumption for aircraft, especially for short flights. Second, aircraft burn a greater proportion of their total weight as fuel than do surface vehicles. As the aircraft becomes lighter during the flight, less power is required to maintain lift, and the fuel consumption rate decreases steadily during the flight. Up to this point, we have been assuming a constant fuel consumption rate while the vehicle is cruising. Taking a changing power profile into account would complicate the calculation. But we can still get a reasonable estimate by using the average fuel consumption rate over the entire flight and treating it as if it were constant. Third, as an aircraft's speed approaches the speed of sound, the drag force begins to increase even more rapidly than the square of the velocity, because of the compressibility of air. Although the mechanism is different, compressibility does for aircraft what wave resistance does for ships -- it imposes an extra penalty on speed. Thus our best speed estimate formula, which ignores this extra penalty, is again likely to be an overestimate. (Large jets typically fly at about 85% of the speed of sound.)

Power term: According to Boeing, a 747-8 freighter carries 400,000 pounds of jet fuel, and has a maximum range of 8,130 kilometers. (The maximum range of the passenger jet version of the 747-8 is nearly twice that of the freighter, presumably because airlines have not yet succeeded in packing passengers as densely as cargo.) Assuming that the maximum range allows 15% fuel reserve, and that the overall efficiency of the engines in converting fuel energy to forward motion is 25%, the aircraft is generating power at an average rate of 213 megawatts, of which 75%, or 160 MW, is non-motive power.

Drag term: Just as ships lose some of their propulsive energy to their wakes, aircraft lose energy to downwashed air. Air pushed backward contributes to forward motion, while air pushed downward does not. But while ships' wakes are nuisances that designers try to minimize, some downwash from an aircraft is essential, being the air's reaction to the same lifting force that keeps the plane aloft. The net result of sending air down rather than back has the same effect as drag (less forward motion for a given propulsive force), and is in fact called "lift-induced drag". However, in contrast to the atmospheric drag we have been considering so far, which increases with the square of the velocity, the relative importance of the lift-induced drag decreases (at higher speeds, less of the total propulsive force is needed to maintain the required lift). With one term increasing while the other decreases, the total drag actually passes through a minimum point as speed increases. Pilots who want to extend their time of flight as long as possible will fly at a speed close to that minimum, since that is the speed that requires the least power, and thus burns fuel at the slowest rate that will keep the plane in the air. But if the goal is using the least fuel to cover a fixed distance, it may pay to travel somewhat faster. To estimate when speeding up might save fuel, we can use the same calculation for aircraft as we used above for surface vehicles. The key quantity is the mass of air that the aircraft acts on as it travels a unit distance.

To calculate the amount of air displaced by an aircraft in flight, the frontal area of the aircraft does not provide a suitable measure, since streamlining is much more significant for planes than for trucks, railcars, and ships. Drag for an aircraft is typically specified in terms of a drag coefficient referred to an area equal to the footprint spanned by the wings. For example, the Boeing website provides a figure of 5,650 square feet for the wing area of a 747-8 freighter. The total drag coefficient can be calculated by adding the drag at the minimum power point, given as 0.0310, to a term involving the lift-induced drag, which raises the coefficient to 0.0499. The product of the reference area and the drag coefficient, 282 square feet, or 26.2 square meters, is called the "drag area", and represents the area of a flat frontal surface that would provide the same amount of drag as the plane does. A tube of air with that cross section contains the amount of mass that the flat surface sweeps out when it travels a distance equal to the length of the tube, and is the appropriate mass to use for the best speed calculation. For the air at a typical 747 cruising altitude of 40,000 feet, the plane acts (as far as drag is concerned) as if it were displacing the mass in a tube containing 26.2 cubic meters of air for every meter of travel. Air in that neighborhood of stratosphere contains 0.303 kilograms per cubic meter, so the mass of air displaced by the plane equals 7.9 kilograms per meter. (Note that, between the effects of the thinner air and the streamlining, a 200 foot wingspan 747 actually puts up less resistance to being dragged through the atmosphere than a ten by twelve foot cross-section railcar.)

The best speed for fuel economy estimate for the 747-8 is the cube root of the non-motive power, 160 megawatts, divided by the drag term, 7.9 kilograms per meter, which comes to 272 meters per second, or 608 miles per hour. The actual design cruising speed is 560 miles per hour. Since, as noted above, when we factor in the extra drag penalty for speeds close to the speed of sound, the best speed estimate will go down somewhat, we can conclude that the 747-8 freighter typically flies at something very close to the best speed for fuel economy, given its size and shape.

Conclusions

For road, rail, and air transportation, the largest vehicles currently in service for moving freight travel at speeds close to their respective best speeds for fuel economy. Looked at another way, each vehicle's engine power and aerodynamics has been well matched to its cruising speed. In contrast, large ships, as currently designed and operated, pay a significant penalty in terms of fuel consumption and carbon emissions in order to travel as fast as they do. If we are measuring fuel consumption and carbon emissions per ton of goods transported, ocean transportation is still a good option compared to the other modes. But if the goal were to transport a ton of goods with the least possible emissions, ocean transportation could do much better with slower ships.

More resources

A report by Oak Ridge National Laboratories (ORNL) contains the results of detailed measurements of fuel consumption by class 8 trucks in normal operations at highway speeds. The results show how additional factors like the inertia acquired on downslopes (not considered in the calculations above, which reflect average mileage over long distances) can affect fuel economy over short stretches.

An overview of factors affecting fuel economy for rail, including the effects of gaps on air resistance, can be found in a set of presentation slides by Chris Barkan of the University of Illinois at Urbana-Champaign.

A useful summary of the Davis Equation and associated parameters (with additional references) is included in a 2010 report, Measuring the Impact of Intermodal Rail Movements in State Transportation Planning available from the Transportation Research Forum.

A good summary of drag forces on ships can be found in the document Basic Principles of Ship Propulsion provided by MAN Diesel and Turbo, a manufacturer of large marine diesel engines.

Web pages from the website Aerospaceweb.org provide values for the lift and drag coefficients for the 747.

Additional references can be found in the Best Speed Calculator spreadsheet.

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